Implementability of correlated and communication equilibrium outcomes in incomplete information games

Abstract

In a correlated equilibrium, the players’ choice of actions is directed by correlated random messages received from an outside source, or mechanism. These messages allow for more equilibrium outcomes than without any messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also reflect the types of the players, either because they are affected by extraneous factors that also affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Mechanisms may be further differentiated by the connections between the messages that the players receive and their own and the other players’ types, by whether the messages are statistically dependent or independent, and by whether they are random or deterministic. Consequently, whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the corresponding number is 14 or 15 for correlated equilibria and even larger—15, 16 or 17—for communication equilibria. For both solution concepts, the implication relations between the different kinds of equilibria form a two-dimensional lattice, which is considerably more intricate than the single-dimensional one of the complete information case.

Appendices

Appendix A: Correlated strategy distributions

For a subset $\mathcal P$ of the six fundamental properties of mechanisms (see Sect. 4.2), a correlated strategy distribution is $\mathcal P$-implementable if there is some mechanism with all the properties in $\mathcal P$ that implements it. If $\mathcal P$ and $\mathcal Q$ are two subsets of properties, $\mathcal P$-implementability implies $\mathcal Q$-implementability if in every Bayesian game every $\mathcal P$-implementable CSD is also $\mathcal Q$-implementable. Shorthand for this relation is

$$\begin{aligned} \mathcal P \Rightarrow \mathcal Q . \end{aligned}$$

A trivial sufficient condition for implication is reverse inclusion, $\mathcal P \supseteq \mathcal Q$. $\mathcal P$-implementability and $\mathcal Q$-implementability are comparable if the former implies the latter or vice versa, and equivalent if both implications hold. Shorthand for equivalence is

$$\begin{aligned} \mathcal P \Leftrightarrow \mathcal Q \end{aligned}$$

The connection between properties of mechanisms and attributes of CSDs is extended by considering pairs of subsets of the fundamental properties. Each such pair, $\mathcal P$ and $\mathcal P ^{{\prime }}$, defines an attribute of CSDs , namely, the conjunction of $\mathcal P$-implementability and $\mathcal P ^{{\prime }}$-implementability, which is denoted by

$$\begin{aligned} \mathcal P \wedge \mathcal P ^{\prime }. \end{aligned}$$

A CSD has this attribute if it is implementable both by a mechanism with the properties in $\mathcal P$ and by a (generally different) mechanism with the properties in $\mathcal P ^{{\prime }}$.²⁴ Lemma 5 at the end of this section shows that such a CSD is always implementable by a mechanism that has both the properties in $\mathcal P$ and those in $\mathcal P ^{{\prime }}$. It follows that conjunctions actually do not give rise to new attributes of CSDs.

If a particular implementing mechanism for a CSD does not have a certain fundamental property, it does not generally follow that the CSD lacks the corresponding attribute, for it still may be that some other implementing mechanism does have the property. An exception to this rule is presented by the following lemma. The lemma identifies several attributes of CSDs that can be determined by looking at a particular implementing mechanism, namely, the default one (see Sect. 4.2.1).

Lemma 4

A CSD is $\tilde{O}$-, $D$- or $I$- implementable if and only if its default mechanism has property $\tilde{O}$, $D$ or $I$, respectively.

Proof

Consider a CSD $\eta$ and its default mechanism ${\varvec{m}}$. By definition, $\eta$ is equal to the joint distribution of the random type profile ${\varvec{t}}$ and the default random action profile ${\varvec{m}}\left( {\varvec{t}} \right)$. Let $\left( {{\varvec{m}}^{{\prime }},\sigma ^{{\prime }}} \right)$ be any correlated strategy other than the default one whose CSD is $\eta$, which means that $\eta$ is equal to the joint distribution of ${\varvec{t}}$ and the random action profile ${\varvec{a}}^{{\prime }}$ defined by

$$\begin{aligned} {\varvec{a}}_i^{\prime } =\sigma _i^{\prime } \left( {{\varvec{t}}_i ,{\varvec{m}}_i^{\prime } \left( {\varvec{t}} \right) } \right) ,\qquad i\in N. \end{aligned}$$

The two equalities together imply that

$$\begin{aligned} {\varvec{m}}\left( t \right) \mathop {=}\limits ^{d} \left( {\sigma _j^{\prime } \left( {t_j ,{\varvec{m}}_j^{\prime } \left( t \right) } \right) } \right) _{j\in N} ,\qquad t\in \text{ supp }(\eta _T ). \end{aligned}$$

(15)

By property (5) of the default mechanism, for every $t\in T$ and $i$ there is some $t^{{\prime }}$ with $\left( {t_i ,t_{-i}^{\prime } } \right) \in \text{ supp }(\eta _T )$ such that (2) holds, which by (15) implies that

$$\begin{aligned} {\varvec{m}}_i \left( t \right) \mathop {=}\limits ^{d} \sigma _i^{\prime } \left( {t_i ,{\varvec{m}}_i^{\prime } (t_i ,t_{-i}^{\prime } )} \right) . \end{aligned}$$

If ${\varvec{m}}^{{\prime }}$ satisfies $D$, then the expression on the right-hand side has a degenerate distribution, which proves that ${\varvec{m}}$ satisfies $D$. If ${\varvec{m}}^{{\prime }}$ satisfies $\tilde{O}$, then the distribution of the expression on the right-hand side is unaffected by replacing $t^{{\prime }}$ with any other type profile. The equality that results from this replacement proves that ${\varvec{m}}$ also satisfies $\tilde{O}$. If ${\varvec{m}}^{{\prime }}$ satisfies $I$, then for every $t\in \text{ supp }(\eta _T )$ the entries on the right-hand side of the equality in (15) are independent, and therefore (3) holds. Since by property (6) of the default mechanism (3) holds also for $t\notin \text{ supp }(\eta _T )$, this means that ${\varvec{m}}$ satisfies $I$. $\square$

Unfortunately, Lemma 4 cannot be extended to all attributes of CSDs. In particular, as indicated at the end of Sect. 4.2.1, the default mechanism of an $O$-implementable CSD does not necessarily have property $O$.

1.1 Intrinsic characterizations

An intrinsic characterization of an attribute of CSDs presents an alternative to the attribute’s original definition in terms of properties of implementing mechanisms. This subsection presents the proofs of the three propositions in Sect. 5.1 and adds to them intrinsic characterization for the remaining two attribute in Fig. 1.

Proof of Proposition 1

1  To prove the sufficiency of the two conditions, consider a CSD $\eta$, with default mechanism ${\varvec{m}}$ and default random action profile ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}} \right)$. Suppose that (i) or (the stronger condition) (ii) holds. For every type profile $t\in \text{ supp }\left( {\eta _T } \right)$ and action profile $a$,

$$\begin{aligned}&\Pr \left( {{\varvec{a}}=a|{\varvec{t}}=t} \right) =\Pr \left( {{\varvec{a}}_1 =a_1 |{\varvec{a}}_{-1} =a_{-1} ,{\varvec{t}}=t} \right) \Pr \left( {{\varvec{a}}_{-1} =a_{-1} |{\varvec{t}}=t} \right) \nonumber \\&\qquad =\Pr \left( {{\varvec{a}}_1 =a_1 |{\varvec{t}}_1 =t_1 } \right) \Pr \left( {{\varvec{a}}_{-1} =a_{-1} |{\varvec{t}}=t} \right) =\cdots \nonumber \\&\qquad =\Pr \left( {{\varvec{a}}_1 =a_1| {\varvec{t}}_1 =t_1 } \right) \Pr \left( {{\varvec{a}}_2 =a_2 |{\varvec{t}}_2 =t_2 } \right) \cdots \Pr \left( {{\varvec{a}}_n =a_n |{\varvec{t}}_n =t_n } \right) , \end{aligned}$$

(16)

where the second equality follows from (i) (with $i=1$), and the subsequent equalities follow from using an identical trick for the other entries of $a$. Again by (i), for every player $i$, type $t_i$, action $a_i$, and type profile $t^{{\prime }}$ for which $\left( {t_i ,t_{-i}^{\prime } } \right) \in \text{ supp }\left( {\eta _T } \right)$,

$$\begin{aligned} \Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}_i =t_i } \right) =\Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}=\left( {t_i ,t_{-i}^{\prime } } \right) } \right) =\Pr \left( {{\varvec{m}}_i \left( {t_i ,t_{-i}^{\prime } } \right) =a_i } \right) . \end{aligned}$$

It follows from property (5) of the default mechanism that the equality between the left- and right-hand sides continues to hold even if $\left( {t_i ,t_{-i}^{\prime } } \right) \notin \text{ supp }\left( {\eta _T } \right)$. This proves that, if (ii) holds, then ${\varvec{m}}$ satisfies $D$. In addition, by (16), for $t\in \text{ supp }\left( {\eta _T } \right)$, $a\in A$ and any $t^{{\prime }}\in T$,

$$\begin{aligned} \text{ Pr }({\varvec{a}}=a\mid {\varvec{t}}=t)=\Pr \left( {{\varvec{m}}\left( t \right) =a} \right) =\prod \limits _{i=1}^n \Pr \left( {{\varvec{m}}_i \left( {t_i ,t_{-i}^{\prime } } \right) =a_i } \right) . \end{aligned}$$

(17)

Assume, without loss of generality, that the indexing of the (finite) type space $T_i =\left\{ {t_i^1 ,t_i^2 ,\ldots } \right\}$ of each player $i$ (see Sect. 4.1) is such that $t^{1}=\left( {t_1^1 ,t_2^1 ,\ldots ,t_n^1 } \right) \in \text{ supp }\left( {\eta _T } \right)$. Define a mechanism $\bar{{\varvec{m}}}$ by

$$\begin{aligned} \bar{{\varvec{m}}}_i \left( t \right) =\left( {{\varvec{m}}_i \left( {t_i^1 ,t_{-i}^1 } \right) ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i}^1 } \right) ,\ldots } \right) ,\qquad i\in N. \end{aligned}$$

(18)

Thus, the message that each player $i$ receives from $\bar{{\varvec{m}}}$ is a pure strategy of the form $\left( {a_i^1 ,a_i^2 ,\ldots } \right) \in A_i^{T_i }$, where the $j$th action $a_i^j$ ($j=1,2,\ldots )$ coincides with the message player $i$ would receive from the default mechanism ${\varvec{m}}$ if his type were $t_i^j$ and the other players’ types were given by $t_{-i}^1$. Since this description does not involve in any way the actual, realized type profile, the mechanism $\bar{{\varvec{m}}}$ satisfies $S$ and $O$. Clearly, it also satisfies $D$ if ${\varvec{m}}$ does the same, which, as shown, is the case if (ii) holds. Even if only (i) holds, $\bar{{\varvec{m}}}$ satisfies $I$. To prove this, it has to be shown that for any $a_1^1 ,a_1^2 ,\ldots \in A_1$, $a_2^1 ,a_2^2 ,\ldots \in A_2 , {\ldots } , a_n^1 ,a_n^2 ,\ldots \in A_n$,

$$\begin{aligned}&\Pr \left( {\varvec{m}}_1 \left( t_{1}^{1} ,t_{-1}^{1} \right) =a_1^1 ,{\varvec{m}}_1 \left( {t_1^2 ,t_{-1}^1 } \right) =a_1^2 ,\ldots ;\right. \nonumber \\&\qquad \left. {\varvec{m}}_2 \left( {t_2^1 ,t_{-2}^1 } \right) =a_2^1 ,{\varvec{m}}_2 \left( {t_2^2 ,t_{-2}^1 } \right) =a_2^2 ,\ldots ;\ldots \right) \nonumber \\&\qquad =\prod \limits _{i=1}^n \Pr \Big ({\varvec{m}}_i \left( {t_i^1 ,t_{-i}^1 } \right) =a_i^1 ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i}^1 } \right) =a_i^2 ,\ldots \Big ). \end{aligned}$$

(19)

By (4), the right-hand side is equal to

$$\begin{aligned} \prod \limits _{i=1}^{n} \prod \limits _j \Pr \left( {\varvec{m}}_i \left( {t_i^j ,t_{-i}^1 } \right) =a_i^j \right) \end{aligned}$$

(20)

and the left-hand side is equal to

$$\begin{aligned} \Pr \left( {{\varvec{m}}\left( {t^{1}} \right) =\left( {a_1^1 ,a_2^1 ,\ldots ,a_n^1 } \right) } \right) \cdot \prod \limits _{i=1}^n \prod \limits _{j\ne 1} \Pr \left( {{\varvec{m}}_i \left( {t_i^j ,t_{-i}^1 } \right) =a_i^j } \right) . \end{aligned}$$

By the second equality in (17), used with $t=t^{{\prime }}=t^{1}$, the last expression is equal to (20). Thus, the mechanism $\bar{{\varvec{m}}}$ satisfies $I$. To prove that it implements the CSD $\eta$, define a corresponding correlated strategy $\left( {\bar{{\varvec{m}}},\bar{\sigma }} \right)$ by

$$\begin{aligned} \bar{\sigma }_i \left( {t_i^j ,\left( {a_i^1 ,a_i^2 ,\ldots } \right) } \right) =a_i^j ,\qquad i\in N,j=1,2,\ldots . \end{aligned}$$

(21)

According to $\bar{\sigma }_i$, of all the entries in the message, player $i$ takes the one corresponding to his actual type. It has to be shown that for every $t\in \text{ supp }\left( {\eta _T } \right)$ and $a\in A$,

$$\begin{aligned}&\Pr \left( {{\varvec{a}}=a|{\varvec{t}}=t} \right) \\&\qquad =\Pr \left( {\bar{\sigma }_1 \left( {{\varvec{t}}_1 ,\bar{{\varvec{m}}}_1 \left( {\varvec{t}} \right) } \right) =a_1 ,\bar{\sigma }_2 \left( {{\varvec{t}}_2 ,\bar{{\varvec{m}}}_2 \left( {\varvec{t}} \right) } \right) =a_2 ,\ldots ,\bar{\sigma }_n \left( {{\varvec{t}}_n ,\bar{{\varvec{m}}}_n \left( {\varvec{t}} \right) } \right) =a_n |{\varvec{t}}=t} \right) . \end{aligned}$$

By (17), used with $t^{{\prime }}=t^{1}$, this equality is equivalent to

$$\begin{aligned}&\prod \limits _{i=1}^n \Pr \left( {{\varvec{m}}_i \left( {t_i ,t_{-i}^1 } \right) =a_i } \right) \\&\qquad =\Pr (\bar{\sigma }_1 \left( {t_1 ,\bar{{\varvec{m}}}_1 \left( t \right) } \right) =a_1 ,\bar{\sigma }_2 \left( {t_2 ,\bar{{\varvec{m}}}_2 \left( t \right) } \right) =a_2 ,\ldots ,\bar{\sigma }_n \left( {t_n ,\bar{{\varvec{m}}}_n \left( t \right) } \right) =a_n ). \end{aligned}$$

By property $I$ of the mechanism $\bar{{\varvec{m}}}$, the right-hand side is equal to

$$\begin{aligned} \prod \limits _{i=1}^n \Pr \left( {\bar{\sigma }_i \left( {t_i ,\bar{{\varvec{m}}}_i \left( t \right) } \right) =a_i } \right) , \end{aligned}$$

and by (18) and (21), the left-hand side is also equal to this product. Therefore, the above equality holds, which proves that $\bar{{\varvec{m}}}$ implements $\eta$.

It remains to prove the necessity of the two conditions. For every CSD $\eta$ there is a correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ such that $\eta$ is the joint distribution of ${\varvec{t}}$ and the random action profile ${\varvec{a}}$ defined by (7). By that definition,

$$\begin{aligned} \Pr ({\varvec{a}}_i =a_i \mid {\varvec{t}}=t)=\Pr \left( {\sigma _i \left( {t_i ,{\varvec{m}}_i \left( t \right) } \right) =a_i } \right) ,\qquad i\in N,t\in \text{ supp }\left( {\eta _T } \right) ,a_i \in A_i . \end{aligned}$$

Therefore, if the mechanism ${\varvec{m}}$ satisfies $D$, then the probability on the left-had side is either 0 or 1, and if ${\varvec{m}}$ satisfies $\tilde{O}$, then

$$\begin{aligned} \Pr ({\varvec{a}}_i =a_i \mid {\varvec{t}}=t)=\Pr ({\varvec{a}}_i =a_i \mid {\varvec{t}}_i =t_i ),\quad i\in N,t\in \text{ supp }\left( {\eta _T } \right) ,a_i \in A_i . \end{aligned}$$

(22)

If ${\varvec{m}}$ satisfies $I$, then for every $i\in N$, $t\in \text{ supp }\left( {\eta _T } \right)$ and $a\in A$

$$\begin{aligned} \Pr \left( {{\varvec{a}}=a|{\varvec{t}}=t} \right)&= \prod \limits _{j=1}^{n} \Pr \left( {\sigma _j \left( {t_j ,{\varvec{m}}_j \left( t \right) } \right) =a_j } \right) \\&= \Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}=t} \right) \Pr \left( {{\varvec{a}}_{-i} =a_{-i} |{\varvec{t}}=t} \right) . \end{aligned}$$

These equalities and the one in (22) together give

$$\begin{aligned}&\Pr \left( {{\varvec{t}}_{-i} =t_{-i} ,{\varvec{a}}=a|{\varvec{t}}_i =t_i } \right) =\Pr \left( {{\varvec{a}}=a|{\varvec{t}}=t} \right) \Pr \left( {{\varvec{t}}_{-i} =t_{-i} |{\varvec{t}}_i =t_i } \right) \\&\qquad =\Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}=t} \right) \Pr \left( {{\varvec{a}}_{-i} =a_{-i} |{\varvec{t}}=t} \right) \text{ Pr }\left( {{\varvec{t}}_{-i} =t_{-i} |{\varvec{t}}_i =t_i } \right) \\&\qquad =\Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}_i =t_i } \right) \Pr \left( {{\varvec{t}}_{-i} =t_{-i} ,{\varvec{a}}_{-i} =a_{-i} |{\varvec{t}}_i =t_i } \right) . \end{aligned}$$

This proves that if $\eta$ is both $D$- and $\tilde{O}$-implementable (and, a fortiori, if it is $S,O,D$-implementable), then (ii) holds, and if $\eta$ is $\tilde{O}$- and $I$-implementable (and, a fortiori, if it is $S,O,I$-implementable), then (i) holds. $\square$

Proof of Proposition 2

To prove the sufficiency of the condition, suppose that a measure $\mu$ satisfying (14) exists for a CSD $\eta$. Restrict $\mu$ to its support $R$, and let the random variable ${\varvec{r}}$ be the identity map on $R$. By construction, ${\varvec{r}}$ is independent of the random type profile ${\varvec{t}}$. Define a mechanism ${\varvec{m}}$ by

$$\begin{aligned} {\varvec{m}}_i \left( t \right) ={\varvec{r}},\qquad i\in N,t\in T. \end{aligned}$$

(23)

This mechanism clearly has properties $S$ and $O$. The message space of each player is $R$, each element $r$ of which is a pure-strategy profile $(a_1^1 ,a_1^2 ,\ldots ;a_2^1 ,a_2^2 ,\ldots ;\ldots ;a_n^1 ,$ $a_n^2 ,\ldots )$ (where, for each $i$ and $j$, $a_i^j$ is the action prescribed to player $i$’s $j$th type). Define a correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ with this mechanism by

$$\begin{aligned} \sigma _i \left( {t_i^j ,r} \right) =a_i^j ,\quad i\in N,j=1,2,\ldots . \end{aligned}$$

(24)

Strategy $\sigma _i$ simply instructs player $i$ to take the action that his strategy prolife prescribes to his actual type. It has to be shown that the joint distribution of ${\varvec{t}}$ and the random action profile ${\varvec{a}}$ corresponding to this correlated strategy is equal to $\eta$. By (14), this equality is equivalent to

$$\begin{aligned} \text{ Pr }({\varvec{a}}=a\mid {\varvec{t}}=t)=\mu ^{t}\left( {\{a\}} \right) ,\qquad t\in \text{ supp }(\eta _T ),a\in A. \end{aligned}$$

(25)

By (7) and (23), for any type profile $t=(t_1^{j_1 } ,t_2^{j_2 } ,\ldots ,t_n^{j_n } )$

$$\begin{aligned} \text{ Pr }({\varvec{a}}=a\mid {\varvec{t}}=t)=\Pr \left( {\left( {\sigma _1 \left( {t_1^{j_1 } ,{\varvec{r}}} \right) ,\sigma _2 \left( {t_2^{j_2 } ,{\varvec{r}}} \right) ,\ldots ,\sigma _n \left( {t_n^{j_n } ,{\varvec{r}}} \right) } \right) =a} \right) . \end{aligned}$$

By (24) and the definition of ${\varvec{r}}$, the right-hand side is the $\mu$-measure of the set of all pure-strategy profiles $\left( {a_1^1 ,a_1^2 ,\ldots ;a_2^1 ,a_2^2 ,\ldots ;\ldots ;a_n^1 ,a_n^2 ,\ldots } \right)$ such that $(a_1^{j_1 } ,a_2^{j_2 } ,\ldots ,a_n^{j_n } )=a$, which by definition is equal to $\mu ^{t}\left( {\{a\}} \right)$. Thus, (25) holds, so that ${\varvec{m}}$ implements the CSD $\eta$.

To prove the necessity of the condition, consider a CSD $\eta$ that is equal to the joint distribution of a pair of random variables ${\varvec{t}}$ and ${\varvec{a}}$ such that (7) holds for some correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ with a mechanism that satisfies $O$. It has to be shows that there is a probability measure $\mu$ such that (14), or equivalently (25), holds. Fix a type profile $t^{{\prime }}$. The random variable

$$\begin{aligned}&\left( \sigma _1 \left( {t_1^1 ,{\varvec{m}}_1 \left( {t_1^1 ,t_{-1}^{\prime } } \right) } \right) ,\sigma _1 \left( {t_1^2 ,{\varvec{m}}_1 \left( {t_1^2 ,t_{-1}^{\prime } } \right) } \right) ,\ldots ;\right. \\&\qquad \left. \sigma _2 \left( {t_2^1 ,{\varvec{m}}_2 \left( {t_2^1 ,t_{-2}^{\prime } } \right) } \right) ,\sigma _2 \left( {t_2^2 ,{\varvec{m}}_2 \left( {t_2^2 ,t_{-2}^{\prime } } \right) } \right) ,\ldots ;\ldots \right) \end{aligned}$$

returns values in $A_1^{T_1 } \times A_2^{T_2 } \times \cdots \times A_n^{T_n }$, i.e., pure-strategy profiles. Its distribution $\mu$ is given by

$$\begin{aligned}&\mu \left( {\{(a_1^1 ,a_1^2 ,\ldots ;a_2^1 ,a_2^2 ,\ldots ;\ldots )\}} \right) \\&\qquad =\text{ Pr }\left( \sigma _1 \left( {t_1^1 ,{\varvec{m}}_1 \left( {t_1^1 ,t_{-1}^{\prime } } \right) } \right) =a_1^1 ,\sigma _1 \left( {t_1^2 ,{\varvec{m}}_1 \left( {t_1^2 ,t_{-1}^{\prime } } \right) } \right) =a_1^2 ,\ldots ;\right. \\&\qquad \qquad \qquad \qquad \left. \sigma _2 \left( {t_2^1 ,{\varvec{m}}_2 \left( {t_2^1 ,t_{-2}^{\prime } } \right) } \right) =a_2^1 ,\sigma _2 \left( {t_2^2 ,{\varvec{m}}_2 \left( {t_2^2 ,t_{-2}^{\prime } } \right) } \right) =a_2^2 ,\ldots ;\ldots \right) . \end{aligned}$$

For every type profile $t=\left( {t_1^{j_1 } ,t_2^{j_2 } ,\ldots ,t_n^{j_n } } \right) \in \text{ supp }(\eta _T )$ and action profile $a=(a_1 ,a_2 ,\ldots a_n )$,

$$\begin{aligned}&\mu \left( \left\{ \left( {a_1^1 ,a_1^2 ,\ldots ;a_2^1 ,a_2^2 ,\ldots ;\ldots ;a_n^1 ,a_n^2 ,\ldots } \right) \right. \right. \\&\qquad \left. \left. \in A_1^{T_1 } \times A_2^{T_2 } \times \cdots \times A_n^{T_n } \big |\left( {a_1^{j_1 } ,a_2^{j_2 } ,\ldots ,a_n^{j_n } } \right) =a \right\} \right) \\&\qquad =\Pr \left( {\sigma _1 \left( {t_1^{j_1 } ,{\varvec{m}}_1 \left( {t_1^{j_1 } ,t_{-1}^{\prime } } \right) } \right) =a_1 ,\sigma _2 \left( {t_2^{j_2 } ,{\varvec{m}}_2 \left( {t_2^{j_2 } ,t_{-2}^{\prime } } \right) } \right) =a_2 ,\ldots } \right) \\&\qquad =\Pr \left( {{\varvec{a}}=a|{\varvec{t}}=t} \right) , \end{aligned}$$

where the second equality uses (7) and the assumption that ${{\varvec{m}}}$ has property $O$. Thus, (25) holds, as had to be shown. $\square$

Proof of Proposition 3

In view of Lemma 4, it suffices to show that the default mechanism ${\varvec{m}}$ of a CSD $\eta$ has property $\tilde{O}$ if and only if the condition that defines the conditional independence property (Definition 3) holds for the default random action profile ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}} \right)$. That condition can be presented as follows: for every player $i$ and type $t_i$ for that player and all type profiles $t^{{\prime }}$ and $t^{\prime \prime }$ with $\left( {t_i ,t_{-i}^{\prime } } \right) ,\left( {t_i ,t_{-i}^{{\prime }{\prime }} } \right) \in \text{ supp }(\eta _T )$,

$$\begin{aligned} \Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}=\left( {t_i ,t_{-i}^{\prime } } \right) } \right) =\Pr \left( {{\varvec{a}}_i =a_i |{\varvec{t}}=\left( {t_i ,t_{-i}^{{\prime }{\prime }} } \right) } \right) ,\qquad a_i \in A_i . \end{aligned}$$

(26)

Since ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}} \right)$, (26) is equivalent to

$$\begin{aligned} {\varvec{m}}_i (t_i ,t_{-i}^{\prime } )\mathop {=}\limits ^{d} {\varvec{m}}_i (t_i ,t_{-i}^{{\prime }{\prime }} ). \end{aligned}$$

(27)

It follows from property (5) of the default mechanism that (27) holds for all type profiles $t^{{\prime }}$ and $t^{\prime \prime }$ with $\left( {t_i ,t_{-i}^{\prime } } \right) ,\left( {t_i ,t_{-i}^{{\prime }{\prime }} } \right) \in \text{ supp }(\eta _T )$ if and only if it holds for all $t^{{\prime }},t^{{\prime }{\prime }}\in T$. This is so for every player $i$ and type $t_i$ if and only if ${\varvec{m}}$ has property $\tilde{O}$.   $\square$

Proposition 4

A CSD $\eta$ is $I$-implementable if and only if the following holds for some (equivalently, every) random variable ${\varvec{a}}=\left( {{\varvec{a}}_1 ,{\varvec{a}}_2 ,\ldots ,{\varvec{a}}_n } \right)$ such that the joint distribution of ${\varvec{t}}$ and ${\varvec{a}}$ is equal to $\eta$:

• (i) ${\varvec{a}}_1 ,{\varvec{a}}_2 ,\ldots ,{\varvec{a}}_n$ are conditionally independent, given ${\varvec{t}}$.

A CSD is $D$-implementable if and only if it satisfies the stronger condition in which (i) is replaced by:

• (ii) The conditional distribution of ${\varvec{a}}$ given ${\varvec{t}}$ is degenerate.

Proof

In view of Lemma 4, it suffices to show that the default mechanism ${\varvec{m}}$ of $\eta$ has property $I$ or $D$ if and only if (i) or (ii), respectively, holds for the default random action profile ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}}\right)$. In other words, the condition that ${\varvec{m}}\left( t \right)$ satisfies (3) or the condition that its distribution is degenerate holds for all type profiles $t$ if and only if the condition holds for $t\in \text{ supp }(\eta _T )$. These equivalences are implied by properties (6) and (5), respectively, of the default mechanism. $\square$

1.2 Equivalences

This subsection identifies a number of equivalent formulations for various attributes of CSDs. The first result establishes the irrelevance in the present context of the properties of mechanisms that concern the connections between the messages they send to a player and the player’s own type, namely, properties $S$ and $\tilde{S}$.

Proposition 5

For CSDs, $\left\{ S \right\} \Leftrightarrow \{ {\tilde{S}} \}\Leftrightarrow \left\{ \right\}$, $\{S,\tilde{O}\}\Leftrightarrow \{\tilde{S},\tilde{O}\}\Leftrightarrow \{\tilde{O}\}$, $\{S,O\}\Leftrightarrow \{\tilde{S},O\}\Leftrightarrow \{O\}$, $\{ {S,I} \}\Leftrightarrow \{ {\tilde{S},I} \}\Leftrightarrow \left\{ I \right\}$ and $\left\{ {S,D} \right\} \Leftrightarrow \{ {\tilde{S},D} \}\Leftrightarrow \left\{ D \right\}$.

Proof

Property $S$ of mechanisms implies $\tilde{S}$, and therefore $\{S\}\Rightarrow \{\tilde{S}\}\Rightarrow \{\}$. Hence, to prove that the three attributes are equivalent it suffices to show that every CSD is $S$-implementable.

Consider the default mechanism ${\varvec{m}}$ of a CSD $\eta$. That mechanism does not necessarily have property $S$ (see Sect. 4.2.1). A mechanism $\vec {{\varvec{m}}}$ that does have that property is defined by

$$\begin{aligned} \vec {{\varvec{m}}}_i \left( t \right) =\left( {{\varvec{m}}_i \left( {t_i^1 ,t_{-i} } \right) ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i} } \right) ,\ldots } \right) ,\qquad i\in N. \end{aligned}$$

(28)

This mechanism is similar to that defined in (18) in that the message that each player receives is a pure strategy, but differs from it in that the partial type profile on the right-hand side is $t_{-i}$ rather than the constant one $t_{-i}^1$. Therefore, the mechanism $\vec {{\varvec{m}}}$ satisfies $S$ but not $O$. To prove that it implements $\eta$, it only needs to be noted that the random action profile of the correlated strategy $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$, with $\bar{\sigma }$ defined by (21), coincides with the default one:

$$\begin{aligned} \bar{\sigma }_i \left( {{\varvec{t}}_i ,\vec {{\varvec{m}}}_i \left( {\varvec{t}} \right) } \right) ={\varvec{m}}_i \left( {\varvec{t}} \right) ,\qquad i\in N. \end{aligned}$$

(29)

The proofs that $\{S,\tilde{O}\}\Leftrightarrow \{\tilde{S},\tilde{O}\}\Leftrightarrow \{\tilde{O}\}, \quad \{S,I\}\Leftrightarrow \{\tilde{S},I\}\Leftrightarrow \left\{ I \right\}$ and $\{S,D\}\Leftrightarrow \{\tilde{S},D\}\Leftrightarrow \{D\}$ are very similar, and only require the following additions to the above proof.

If the CSD $\eta$ is $\tilde{O}$-, $I$- or $D$-implementable, then by Lemma 4 the default mechanism ${\varvec{m}}$ has property $\tilde{O}$, $I$ or $D$, respectively. It has to be shown that the mechanism $\vec {{\varvec{m}}}$ also has the same property, in addition to $S$. When ${\varvec{m}}$ satisfies $\tilde{O}$, for every player $i$ and type profiles $t^{\prime }$ and $t^{\prime \prime }$ the equality (27) holds for all types $t_i$, which by (4) implies that

$$\begin{aligned} \left( {{\varvec{m}}_i \left( {t_i^1 ,t_{-i}^{\prime } } \right) ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i}^{\prime } } \right) ,\ldots } \right) \mathop {=}\limits ^{d} \left( {{\varvec{m}}_i \left( {t_i^1 ,t_{-i}^{{\prime }{\prime }} } \right) ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i}^{{\prime }{\prime }} } \right) ,\ldots } \right) . \end{aligned}$$

Thus, $\vec {{\varvec{m}}}$ has property $\tilde{O}$. When ${\varvec{m}}$ satisfies $I$ or $D$, (4) implies that the random variables

$$\begin{aligned} \left\{ {{\varvec{m}}_j \left( t \right) } \right\} _{j\in N,t\in T} \end{aligned}$$

(30)

are independent or have degenerate distributions, respectively. It follows that the same is true, for every type profile $t$, for the $n$ random variables

$$\begin{aligned} \left\{ {\vec {{\varvec{m}}}_j \left( t \right) } \right\} _{j\in N} , \end{aligned}$$

each of which is a vector whose entries are a subset of the random variables in (30), such that these $n$ subsets are disjoint. Thus, the mechanism $\vec {{\varvec{m}}}$ satisfies $I$ or $D$, respectively.

To prove that $\{S,O\}\Leftrightarrow \{\tilde{S},O\}\Leftrightarrow \{O\}$, it suffices to show that $\{O\}\Rightarrow \{S,O\}$. This is shown in the proof of Proposition 2, where it is proved that the existence of a measure $\mu$ as in that proposition, which is implied by $O$-implementability, in turn implies $S,O$-implementability. $\square$

Proposition 6

For CSDs, $\{S,O,I\}\Leftrightarrow \{\tilde{S},O,I\}\Leftrightarrow \{S,\tilde{O},I\}\Leftrightarrow \{\tilde{S},\tilde{O},I\}\Leftrightarrow \{O,I\}\Leftrightarrow \{\tilde{O},I\}\Leftrightarrow \left( {\{ {\tilde{O}} \}\wedge \left\{ I \right\} }\right)$ and $\{S,O,D\}\Leftrightarrow \{\tilde{S},O,D\}\Leftrightarrow \{S,\tilde{O},D\}\Leftrightarrow \{\tilde{S},\tilde{O},D\}\Leftrightarrow \{O,D\}\Leftrightarrow \{\tilde{O},D\}\Leftrightarrow \left( {\{ {\tilde{O}} \}\wedge \left\{ D \right\} }\right)$.

Proof

It clearly suffices to show that $\left( {\{\tilde{O}\}\wedge \{I\}} \right) \Rightarrow \left\{ {S,O,I} \right\}$ and $\left( {\{\tilde{O}\}\wedge \{D\}} \right) \Rightarrow \left\{ {S,O,D} \right\}$. As shown in the last part of the proof of Proposition 1, every CSD $\eta$ that is both $\tilde{O}$- and $I$-implementable, or both $\tilde{O}$- and $D$-implementable, is the joint distribution of ${\varvec{t}}$ and some random variable ${\varvec{a}}$ that satisfies condition (i) or (ii), respectively, in Proposition 1. Therefore, by that proposition, in the first case $\eta$ is also $S,O,I$-implementable, and in the second, it is $S,O,D$-implementable. $\square$

1.3 Implications

Propositions 5 and 6 identify seven attributes of correlated strategy distributions that are defined (in several equivalent ways) by subsets of the six fundamental properties of mechanisms. Fig. 1 shows these attributes as well as certain trivial implications between them, which all follow immediately from relations between properties of mechanisms. To prove that the figure presents a complete picture of the implication relations between attributes of CSDs, it remains to prove that implications additional to those shown do not hold, so that, in particular, none of the seven attributes is equivalent to another. For this, the following four propositions are required.

Proposition 7

For CSDs, $\{S,O,I\}\nRightarrow \{D\}$.

Proof

It suffices to consider any complete information game (that is, a game where every player has only one type) with a mixed-strategy profile that is not pure. $\square$

Proposition 8

For CSDs, $\{S,D\}\nRightarrow \{\tilde{O}\}$.

Proof

In a two-player Bayesian game in which player 1 has a single type and two actions and player 2 has a single action and two types, consider a correlated strategy distribution in which player 1 takes his first or second action if player 2 is of the first or second type, respectively. This CSD is implementable by mechanism that simply tells player 1 the type of player 2, and thus satisfies $S$ and $D$. However, the CSD is not $\tilde{O}$-implementable, since a mechanism with property $\tilde{O}$ cannot possibly provide player 1 with any information about player 2’s type. $\square$

Proposition 9

For CSDs, $\{\tilde{O}\}\nRightarrow \{O\}$.

Proof

By Example 1, there exists a CSD that has the conditional independence property but is not $O$-implementable. By Proposition 3, that CSD is $\tilde{O}$-implementable. $\square$

Proposition 10

For CSDs, $\{S,O\}\nRightarrow \{I\}$.

Proof

In a complete information game, properties $S$ and $O$ automatically hold for every mechanism, but a CSD is $I$-implementable only if the players’ actions are independent. $\square$

Proposition 7 proves that attribute VI only implies the other attributes in Fig. 1 that the diagram indicates it implies (in other words, it does not imply III or VII). Proposition 8 proves the same for attribute III. These two results prove that attribute II (which is implied by both III and VI) only implies attribute I, and therefore the latter does not imply IV. Proposition 9 proves that IV does not imply V. Proposition 10 proves that attribute V only implies the (two) attributes that the diagram indicates it implies, which establishes the same for attribute IV and for attribute I.

Since, for mechanisms, property $S$ implies $\tilde{S}$, property $O$ implies $\tilde{O}$, and $D$ implies $I$, there are only 27 relevant subsets of $\{S,\tilde{S},O,\tilde{O},D,I\}$, which all appear in Fig. 1. Therefore, there are no additional attributes of CSDs that can be described by single subsets of the six fundamental properties of mechanisms. The following lemma shows that the same is true for pairs (hence also triplets, etc.) of sets of properties of mechanisms: no additional attributes of CSDs can be defined by them. This is because, if a CSD is implementable both by a mechanism with one set of properties and by a mechanism with a second set of properties, then it is implementable by a single mechanism that has all the properties of the other two.

Lemma 5

For CSDs, for every two subsets $\mathcal P ,\mathcal Q \subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$,

$$\begin{aligned} \left( \mathcal{P \wedge \mathcal Q } \right) \Leftrightarrow \mathcal P \cup \mathcal Q . \end{aligned}$$

(31)

Proof

(an outline). Proposition 6 proves the two cases of (31) in which $\mathcal P =\{{\tilde{O}}\}$ and $\mathcal{Q}$ is $\left\{ I \right\}$ or $\left\{ D \right\}$. By inspection of Fig. 1, every other case follows from one of these two. $\square$

It is shown in Appendix B below that for correlated equilibrium distributions a similar result to Lemma 5 does not hold. In other words, the requirement of incentive compatibility may invalidate the equivalence (31).

Appendix B: Correlated equilibrium distributions

The analysis of correlated strategy distributions in the previous appendix is a first step in the analysis of correlated equilibrium distributions. The former concerns qualitative differences between distributions that reflect the limitations of the implementing mechanisms. The latter also incorporates the constraints inherent in the incentive compatibility requirement. Whereas in the case of CSDs the limiting factor is the mechanism’s ability to transmit information to players (about the other players’ types and outside random events), in the case of CEDs its ability to do so selectively also comes into play.

As for CSDs, each subset $\mathcal P$ of the six fundamental properties of mechanisms defines an attribute of correlated strategy distributions, namely, $\mathcal P$-implementability. A CED has this attribute if there is some mechanism with all the properties in $\mathcal P$ that implements it. However, in the present context, implementability has a different meaning than for CSDs. Namely, the correlated strategy involved is required to be a correlated equilibrium. Thus, an expression like $\mathcal P \Rightarrow \mathcal{Q}$ has a different meaning for CSDs and CEDs. Wherever confusion is possible, the generic implication sign may be replaced by the more explicit one $\mathop {\Rightarrow }\limits _{\mathrm{CSD}}$ or $\mathop {\Rightarrow }\limits _{\mathrm{CED}}$. As the following proposition shows, the second relation is in a sense stronger than the first one.

Proposition 11

For every two subsets $\mathcal P ,\mathcal{Q}\subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$,

$$\begin{aligned} \mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}\,\mathrm{implies}\,\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CSD}} \mathcal{Q}. \end{aligned}$$

(32)

The same is moreover true with $\mathcal P$ replaced by by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$, for any list $\mathcal P ^{{\prime }},\mathcal P ^{\prime \prime },\ldots$ of subsets of $\{S,\tilde{S},O,\tilde{O},V,I\}$.

Proof

It has to be shown that (i) $\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}$ and (ii) $\mathcal P \mathop {\nRightarrow }\limits _{\mathrm{CSD}} \mathcal{Q}$ are contradictory. Condition (i) means that, in every Bayesian game, every $\mathcal P$-implementable CED is also $\mathcal{Q}$-implementable. Condition (ii) means that there is some CSD in some Bayesian game that is $\mathcal P$- but not $\mathcal{Q}$-implementable. Without loss of generality, the payoff functions in that game (which are irrelevant for CSD implementability) are identically zero. However, this means that every correlated strategy in the game is a correlated equilibrium and vice versa, which contradicts (i).

The same argument applies virtually unchanged also to the more general version in which $\mathcal P$ is replaced by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$. $\square$

The converse of (32) does not generally hold. Consequently, not all attributes of CEDs that can be described in terms of the six fundamental properties of mechanisms correspond to attributes of CSDs. In other words, the former are not simply the restrictions of the latter to correlated equilibrium distributions. Rather, restriction is followed by refinement, which gives rise to additional attributes.

Some of the attributes of CEDs, including the majority of those inherited from CSDs, are presented in the following subsection. The subsequent subsection describes additional attributes, by specifically identifying instances in which the converse of (32) does not hold. The last subsection in this appendix completes the description of the implication relation $\mathop {\Rightarrow }\limits _{\mathrm{CED}}$ (henceforth written simply as $\Rightarrow$) by considering implications involving conjunctions of attributes of CEDs.

1.1 Equivalences

The following propositions identify equivalent formulations for several attributes of CEDs.

Proposition 12

For CEDs, $\{S,\tilde{O}\}\Leftrightarrow \{\tilde{S},\tilde{O}\}\Leftrightarrow \{\tilde{O}\}$.

Proof

In view of Lemma 4, it suffices to show that if the default mechanism ${\varvec{m}}$ of a CED $\eta$ has property $\tilde{O}$, then the correlated strategy $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ defined in the proof of Proposition 5 is a correlated equilibrium. As shown in that proof, if the default mechanism satisfies $\tilde{O}$ (or $I$ or $D)$, then $\vec {{\varvec{m}}}$ satisfies both $S$ and $\tilde{O}$ (or $I$ or $D$, respectively).

By Lemma 3, the default random action profile ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}} \right)$ satisfies (11). By (29) and the definition of correlated equilibrium (Definition 1), $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ is a correlated equilibrium if and only if

$$\begin{aligned} E\Big (u_i \left( {{\varvec{t}},{\varvec{a}}} \right) -u_i \left( {{\varvec{t}},\left( {a_i^{\prime } ,{\varvec{a}}_{-i} } \right) } \right) \mid {\varvec{t}}_i ,\vec {{\varvec{m}}}_i ({\varvec{t}})\Big )\ge 0,\qquad i\in N,a_i^{\prime } \in A_i. \end{aligned}$$

(33)

It follows that the latter condition holds if the conditional expectations in (11) and (33) are equal. The formal difference between the former and the latter is that player $i$’s action ${\varvec{a}}_i$, which coincides with the message ${\varvec{m}}_i ({\varvec{t}})$ he receives from the default mechanism, is replaced by $\vec {{\varvec{m}}}_i \left( {\varvec{t}} \right)$, which by (28) also specifies the messages the player would receive if his type were different. Therefore, the meaning of the above equality is that these messages do not provide player $i$ with any information that he could use for choosing a better action.

Since ${\varvec{a}}={\varvec{m}}\left( {\varvec{t}} \right)$, if the conditional expectations in (11) and (33) are not equal, then by (28) there is some type of player $i$, say the first one $t_i^1$, and some messages $m_i^1 ,m_i^2 ,\ldots$ such that

$$\begin{aligned}&E\left( {u_i \left( {{\varvec{t}},{\varvec{m}}\left( {t_i^1 ,{\varvec{t}}_{-i} } \right) } \right) -u_i \left( {{\varvec{t}},\left( {a_i^{\prime } ,{\varvec{m}}_{-i} \left( {t_i^1 ,{\varvec{t}}_{-i} } \right) } \right) } \right) \Big |{\varvec{t}}_i =t_i^1 ,{\varvec{m}}_i \left( {t_i^1 ,{\varvec{t}}_{-i} } \right) =m_i^1 } \right) \\&\qquad \ne E\left( u_i \left( {{\varvec{t}},{\varvec{m}}\left( {t_i^1 ,{\varvec{t}}_{-i} } \right) } \right) -u_i \left( {{\varvec{t}},\left( {a_i^{\prime } ,{\varvec{m}}_{-i} \left( {t_i^1 ,{\varvec{t}}_{-i} } \right) } \right) } \right) \Big |{\varvec{t}}_i =t_i^1 ,{\varvec{m}}_i \left( {t_i^1 ,{\varvec{t}}_{-i} } \right) =m_i^1,\right. \\&\qquad \left. {\varvec{m}}_i \left( {t_i^2 ,{\varvec{t}}_{-i} } \right) =m_i^2 ,\ldots \right) . \end{aligned}$$

The inequality implies that the pair of random variables ${\varvec{t}}$ and ${\varvec{m}}\left( {t_i^1 ,{\varvec{t}}_{-i} } \right)$ is not independent of ${\varvec{m}}_i \left( {t_i^2 ,{\varvec{t}}_{-i} } \right) ,{\varvec{m}}_i \left( {t_i^3 ,{\varvec{t}}_{-i} } \right) ,\ldots$. However, if the default mechanism ${\varvec{m}}$ has property $\tilde{O}$, then it follows from (4) that such independence does in fact hold, so that the above inequality cannot hold, which proves that $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ is a correlated equilibrium.$\square$

Proposition 13

For CEDs, $\{\tilde{S},O\}\Leftrightarrow \{O\}$.

Proof

Let $\left( {{\varvec{m}},\sigma } \right)$ be a correlated equilibrium with a mechanism that satisfies $O$. It has to be shown that there is some correlated equilibrium $\left( {\hat{{\varvec{m}}},\hat{\sigma }} \right)$ with a mechanism that satisfies $\tilde{S}$ and $O$ such that the two correlated equilibria have identical CEDs.

The construction of $\left( {\hat{{\varvec{m}}},\hat{\sigma }} \right)$ is based on the idea of encoding the messages that the mechanism ${\varvec{m}}$ sends to the players in a particular manner. Suppose, without loss of generality, that these messages are integers, more specifically, that the message space $M_i$ of each player $i$ consists of all integers between 1 and some number $K_i$. Since property $O$ of the mechanism implies $\tilde{O}$, the (random) message ${\varvec{m}}_i (t)$ that player $i$ receives has a distribution function $F_{i,t_i }$ that only depends on the player’s own type $t_i$. That is, for any $t_{-i}$,

$$\begin{aligned} F_{i,t_i } \left( s \right) =\Pr ({\varvec{m}}_i \left( {t_i ,t_{-i} } \right) \le s),\qquad -\infty <s<\infty . \end{aligned}$$

The mechanism $\hat{{\varvec{m}}}$ combines the message ${\varvec{m}}_i \left( {\varvec{t}} \right)$ with a random variable ${\varvec{r}}$ that is uniformly distributed on the half-open interval (0,1] and is independent of ${\varvec{m}}$. (The assumption of uniform distribution is actually inconsistent with the definition of random variable in footnote 4, which requires a finite probability space. However, this assumption is only temporary; it is removed in the last part of the proof.) Specifically, for every player $i$ and type profile $t$,

$$\begin{aligned} \hat{{\varvec{m}}}_i \left( t \right) ={\varvec{r}}F_{i,t_i } \left( {{\varvec{m}}_i \left( t \right) } \right) +(1-{\varvec{r}})F_{i,t_i } \left( {{\varvec{m}}_i \left( t \right) -1} \right) . \end{aligned}$$

(34)

It is not difficult to see that $\hat{{\varvec{m}}}_i \left( t \right)$ is uniformly distributed on the unit interval. Therefore, the mechanism $\hat{{\varvec{m}}}$ satisfies $\tilde{S}$ as well as $O$.

The next step is to define a strategy $\hat{\sigma }_i$ for each player $i$ by $\hat{\sigma }_i (t_i ,\hat{m}_i )=\sigma _i (t_i ,\psi _i \left( {t_i ,\hat{m}_i } \right) )$, where $\psi _i$ is a function that “decodes” the message $\hat{{\varvec{m}}}_i \left( t \right)$ and recovers the original message ${\varvec{m}}_i \left( t \right)$:

$$\begin{aligned} \psi _i \left( {t_i ,x} \right) =\min \left\{ {m_i \in M_i| F_{i,t_i } \left( {m_i } \right) \ge x} \right\} . \end{aligned}$$

By virtue of this decoding, $\hat{\sigma }_i$ always specifies the same action as $\sigma _i$. Since, in addition, the messages that the players receive from the mechanism $\hat{{\varvec{m}}}$ convey precisely the same information about the other players’ types and actions as those from ${\varvec{m}}$, this proves that $\left( {\hat{{\varvec{m}}},\hat{\sigma }} \right)$, like $\left( {{\varvec{m}},\sigma } \right)$, is a correlated equilibrium.

The above construction does not strictly conforms to the definition of mechanism since the message spaces in $\hat{{\varvec{m}}}$ are infinite. This problem may be overcome by replacing the uniformly-distributed random variable ${\varvec{r}}$ with one that has only finitely many possible values, specifically, a random variable of the form $\mu ({\varvec{r}})$, where $\mu$ is a real-valued function with a finite range. The first step is to consider the (finite) set

$$\begin{aligned} X=\left\{ {F_{i,t_i } \left( {m_i } \right) } \right\} _{i,t_i ,m_i } \end{aligned}$$

of all values that may appear in the first term in (34). The next step is to modify the definition of the mechanism $\hat{{\varvec{m}}}$ by changing the message that it sends to each player $i$ from $\hat{{\varvec{m}}}_i \left( t \right)$ (which is defined by (34)) to $\lambda \left( {\hat{{\varvec{m}}}_i \left( t \right) } \right)$, where $\lambda :\left[ {0,1} \right] \rightarrow [0,1]$ is the non-decreasing left continuous function defined by $\lambda \left( x \right) =\min \left( {\left\{ {x^{{\prime }}\in X\,|\,x^{{\prime }}\ge x} \right\} } \right)$. This change is inconsequential. Since always $\hat{{\varvec{m}}}_i \left( t \right) \le \lambda \left( {\hat{{\varvec{m}}}_i \left( t \right) } \right) \le F_{i,t_i } \left( {{\varvec{m}}_i \left( t \right) } \right)$, applying the decoder $\psi _i$ to the modified message $\lambda \left( {\hat{{\varvec{m}}}_i \left( t \right) } \right)$ still recovers ${\varvec{m}}_i \left( t \right)$. Let the function $\mu :(0,1]\rightarrow (0,1]$ be defined by

$$\begin{aligned} \mu \left( r \right) =\max \left\{ {0<r^{{\prime }}\le 1 {\Bigg |}{\begin{array}{l} {\lambda \left( {r^{{\prime }}F_{i,t_i } \left( {m_i } \right) +\left( {1-r^{{\prime }}} \right) F_{i,t_i } \left( {m_i -1} \right) } \right) }\\ {=\lambda \left( {r\,F_{i,t_i } \left( {m_i } \right) +\left( {1-r} \right) F_{i,t_i } \left( {m_i -1} \right) } \right) } \\ { \text{ for } \text{ all } i,t_i ,m_i } \\ \end{array} }} \right\} . \end{aligned}$$

It is not difficult to see that the function $\mu$ is well defined and, as required, has only finitely many possible values. The final step is to replace ${\varvec{r}}$ in (34) with $\mu \left( {\varvec{r}} \right)$. By definition of $\mu$, this replacement does not change the message $\lambda \left( {\hat{{\varvec{m}}}_i \left( t \right) } \right)$. $\square$

Proposition 14

For CEDs, $\{S,I\}\Leftrightarrow \{\tilde{S},I\}$ and $\{S,D\}\Leftrightarrow \{\tilde{S},D\}$.

Proof

To prove that $\{\tilde{S},I\}\Rightarrow \{S,I\}$, it has to be shown that every CED implementable by a mechanism ${\varvec{m}}$ with properties $\tilde{S}$ and $I$ is also implementable by a mechanism with properties $S$ and $I$.

Property $\tilde{S}$ of ${\varvec{m}}$ means that for every type profile $t$ and player $i$ the distribution of ${\varvec{m}}_i \left( t \right)$ does not change when only player $i$’s type $t_i$ changes. In other words, the distribution only depends on $i$ and on the partial type profile $t_{-i}$. Therefore, it is possible to construct a family $\left\{ {{\varvec{r}}^{i,t_{-i} }} \right\} _{i,t_{-i} }$ of independent random variables, indexed by the players and partial type profiles, such that each entry ${\varvec{r}}^{i,t_{-i} }$ has the distribution described above. Define a mechanism $\tilde{{\varvec{m}}}$ by

$$\begin{aligned} \tilde{{\varvec{m}}}\left( t \right) =\left( {{\varvec{r}}^{1,t_{-1} },{\varvec{r}}^{2,t_{-2} },\ldots ,{\varvec{r}}^{n,t_{-n} }} \right) ,\qquad t\in T. \end{aligned}$$

Thus,

$$\begin{aligned} \tilde{{\varvec{m}}}_i \left( t \right) \mathop {=}\limits ^{d} {\varvec{m}}_i \left( t \right) ,\qquad i\in N,t\in T. \end{aligned}$$

(35)

The mechanism $\tilde{{\varvec{m}}}$ has properties $S$ and $I$ by construction. Since ${\varvec{m}}$ also has property $I$, it follows from (35) that

$$\begin{aligned} \tilde{{\varvec{m}}}\left( t \right) \mathop {=}\limits ^{d} {\varvec{m}}\left( t \right) ,\qquad t\in T. \end{aligned}$$

(36)

It is not difficult to see that any correlated strategy of the form $\left( {{\varvec{m}},\sigma } \right)$ is a correlated equilibrium if and only if this is so for $\left( {\tilde{{\varvec{m}}},\sigma } \right)$. Therefore, the two mechanisms implement precisely the same CEDs.

An almost identical proof shows that $\{\tilde{S},D\}\Rightarrow \{S,D\}$. The only required change it to assume that the mechanism ${\varvec{m}}$ has properties $\tilde{S}$ and $D$. This assumption implies that for every $t$ the distribution of ${\varvec{m}}\left( t \right)$ is degenerate, which by (36) implies the same for $\tilde{{\varvec{m}}}\left( t \right)$. Thus, $\tilde{{\varvec{m}}}$ satisfies $D$. $\square$

Proposition 15

For CEDs, $\{S,O,I\}\Leftrightarrow \{\tilde{S},O,I\}\Leftrightarrow \{S,\tilde{O},I\}\Leftrightarrow \{\tilde{S},\tilde{O},I\}\Leftrightarrow \{O,I\}\Leftrightarrow \{\tilde{O},I\}\Leftrightarrow \left( {\{\tilde{O}\}\wedge \{I\}} \right)$ and $\{S,O,D\}\Leftrightarrow \{\tilde{S},O,D\}\Leftrightarrow \{S,\tilde{O},D\}\Leftrightarrow \{\tilde{S},\tilde{O},D\}\Leftrightarrow \{O,D\}\Leftrightarrow \{\tilde{O},D\}\Leftrightarrow \left( {\{\tilde{O}\}\wedge \{D\}} \right)$.

Proof

In view of Lemma 4, it suffices to show that every CED $\eta$ whose default mechanism ${\varvec{m}}$ has property $\tilde{O}$ and, in addition, property $I$ or $D$ is $S,O,I$- or $S,O,D$-implementable, respectively. It is shown by Proposition 6 that such a CED is indeed thus implementable as a CSD. The proof of that proposition refers to the proof of Proposition 1, where it is shown that the mechanism $\bar{{\varvec{m}}}$ defined by (18) has the required three properties, and the correlated strategy $\left( {\bar{{\varvec{m}}},\bar{\sigma }} \right)$, where $\bar{\sigma }$ is defined in (21), has the distribution $\eta$. Therefore, it only remains to show that $\left( {\bar{{\varvec{m}}},\bar{\sigma }} \right)$ is in fact a correlated equilibrium.

As shown in proof of Proposition 12, the mechanism $\vec {{\varvec{m}}}$ defined by (28) satisfies $\tilde{O}$ and $I$, and the correlated strategy $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ is a correlated equilibrium. Therefore, it suffices to prove that

$$\begin{aligned} \bar{{\varvec{m}}}\left( t \right) \mathop {=}\limits ^{d} \vec {{\varvec{m}}}\left( t \right) ,\qquad t\in T. \end{aligned}$$

(37)

Since both mechanisms satisfy $I$ and $\tilde{O}$, (37) is equivalent to

$$\begin{aligned} \bar{{\varvec{m}}}_i \left( {t_i ,t_{-i}^1 } \right) \mathop {=}\limits ^{d} \vec {{\varvec{m}}}_i \left( {t_i ,t_{-i}^1 } \right) ,\qquad i\in N,t_i \in T_i . \end{aligned}$$

This condition holds by definition of the two mechanisms. $\square$

1.2 Implications

By Proposition 11, an implication relation that does not hold for CSDs also does not hold for CEDs. Therefore, an immediate corollary of Propositions 7, 8, 9 and 10 is the following result.

Proposition 16

For CEDs, $\{S,O,I\}\nRightarrow \{D\}$, $\{S,D\}\nRightarrow \{\tilde{O}\}$, $\{\tilde{O}\}\nRightarrow \{O\}$ and $\{S,O\}\nRightarrow \{I\}$.

The next three propositions identify implication relations that do hold for CSDs but do not hold for CEDs.

Proposition 17

For CEDs, $\{D\}\nRightarrow \{\tilde{S}\}$.

Proof

This is demonstrated by Example 2. $\square$

Proposition 18

For CEDs, $\{\tilde{S}\}\nRightarrow \{S\}$.

Proof

This is demonstrated by Example 3. $\square$

Proposition 19

For CEDs, $\{\tilde{S},O\}\nRightarrow \{S,O\}$.

Proof

This is demonstrated by Example 5. $\square$

Propositions 12, 13, 14 and 15 identify six attributes of correlated equilibrium distributions that are defined by subsets of the fundamental properties of mechanisms. These attributes plus $S$-implementability are shown in Fig. 2 as attributes $\text{ I }_{\mathrm{a}}$, $\text{ II }_{\mathrm{a}}$, $\text{ III }_{\mathrm{a}}$, IV, V, VI and VII. The implication relations that are specified by the Hasse diagram among these attributes all hold trivially since they follow immediately from relations between properties of mechanisms. It follows from Proposition 16 that additional implications among the seven attributes do not hold, and in particular, none of them is equivalent to any of the others. (The more detailed argument given in Appendix A also applies here, mutatis mutandis.) Three more attributes are defined by $\left\{ \right\}$, $\{\tilde{S}\}$ and $\{D\}$ (I, $\text{ I }_{\mathrm{b}}$ and III in Fig. 2). The implication relations shown in Fig. 2 among these attributes and between them and the other seven all hold trivially. It follows from Propositions 17 and 18, and from $\{S,O,I\}\nRightarrow \{D\}$ in Proposition 16, that additional such implications do not hold. Two more attributes are defined by $\{I\}$ and $\{S,O\}$ (II and $\text{ V }_{\mathrm{a}}$). It follows from Proposition 19, and from $\{S,O\}\nRightarrow \{I\}$ in Proposition 16, that the implication relations shown in Fig. 2 between each of these attributes and each of the other ten are the only ones holding. This proves that there are precisely twelve distinct attributes of CEDs that can be defined by single subsets of the fundamental properties.

Theorem 1 in Sect. 6.1 can now be proved. It follows immediately from it that attributes I, II, III, IV, V, VI and VII of CEDs are obtained from the similarly numbered attributes of CSDs by restriction. That is, a CED has any of these attributes if and only if it has the corresponding attribute as a CSD.

Proof of Theorem 1

Suppose first that $\mathcal P \subseteq \{\tilde{O},D,I\}$, and let $\eta$ be a CED that is $\mathcal P$-implementable as a CSD. By Lemma 4, the default mechanism of $\eta$ has all the properties in $\mathcal P$. By Lemma 3, the default correlated strategy is a correlated equilibrium. Therefore, $\eta$ is $\mathcal P$-implementable also as a CED.

Next, consider the case $\mathcal P =\{O\}$. Let $\eta$ be a CED that is $O$-implementable as a CSD. By Lemma 3, the default correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ of $\eta$ is a correlated equilibrium. However, as remarked, the default mechanism ${\varvec{m}}$ may not satisfy $O$. On the other hand, by assumption, $\eta$ is the CSD of some correlated strategy $\left( {{\varvec{m}}^{{\prime }},\sigma ^{{\prime }}} \right)$ with a mechanism that satisfies $O$, but that correlated strategy may not be a correlated equilibrium. Consider the mechanism $\tilde{{\varvec{m}}}$ defined by

$$\begin{aligned} \tilde{{\varvec{m}}}_i \left( t \right) =\sigma _i^{\prime } \left( {t_i ,{\varvec{m}}_i^{\prime } (t)} \right) ,\qquad i\in N,t\in T. \end{aligned}$$

It clearly satisfies $O$. In addition, by (15),

$$\begin{aligned} \tilde{{\varvec{m}}}\left( t \right) \mathop {=}\limits ^{d} {\varvec{m}}\left( t \right) ,\qquad t\in \text{ supp }(\eta _T ). \end{aligned}$$

These equalities imply that the correlated strategy obtained from the default one $\left( {{\varvec{m}},\sigma } \right)$ by replacing ${\varvec{m}}$ with a mechanism $\tilde{{\varvec{m}}}$ is also a correlated equilibrium, with the same CED. This proves that $\eta$ is $O$-implementable also as a CED.

To complete the proof of the theorem it remains to note that, by Propositions 6 and 15, for both CSDs and CEDs, $O,I$-implementability is equivalent to $\tilde{O},I$-implementability, and the same is true for $O,D$- and $\tilde{O},D$-implementability. $\square$

1.3 Conjunction of attributes

The next step is to consider attributes of CEDs that are defined by pairs (or possibly triplets, etc.) of subsets of fundamental properties of mechanisms, that is, by conjunction of two (or more) of the twelve attributes identified in the previous subsection. Unlike for CSDs (see Lemma 5), genuinely new attributes can be defined this way. For example, it follows from the next example that the conjunction of $S$-implementability and $D$-implementability is a new attribute.

1.4 Example 6 A correlated equilibrium distribution that is $S$- as well as $D$-implementable but not $S,D$ -implementable. In a two-player Bayesian game, player 1 has two types, $t_1^{{\prime }}$ and $t_1^{\prime \prime }$, and two actions, $L$ and $R$. Player 2 has three types, $t_2^{{\prime }}$, $t_2^{\prime \prime }$ and $t_2^{\prime \prime \prime }$, and only one action, $L$. All type profiles except $(t_1^{{\prime }} ,t_2^{{\prime }} )$ may occur, and they have the same probability (1/5). If player 1 plays $R$, the payoff to both players is 0.5. If he plays $L$, the payoff vector is determined by the type profile according to the following table:

The lowest possible expected payoff for player 2 in this game is 0.1. For this payoff to be reached, player 1 should play $R$ if and only if the type profile is $(t_1^{{\prime }} ,t_2^{\prime \prime \prime } )$. This is in fact a correlated equilibrium distribution, which is implementable by a mechanism that sends to player 1 the message $R$ if the type profile is $(t_1^{{\prime }} ,t_2^{\prime \prime \prime } )$ and otherwise sends $L$. Acting according to the message is incentive compatible for player 1 since it always gives maximum payoff to type $t_1^{{\prime }}$ and gives $t_1^{\prime \prime }$ (who is always instructed to play $L$) an expected payoff of 1, which is greater than the 0.5 he would receive from playing $R$. An alternative implementing mechanism sends to player 1 the message $L$ or $R$ if player 2 has type $t_2^{\prime \prime }$ or $t_2^{\prime \prime \prime }$ respectively, and sends either message with probability 1/2 if the type is $t_2^{{\prime }}$. The correlated strategy with this mechanism that instructs player 1 to follow the mechanism’s instructions if his type is $t_1^{{\prime }}$ but play $L$ if the type is $t_1^{\prime \prime }$ is a correlated equilibrium. This is because the message that type $t_1^{\prime \prime }$ of player 1 receives never changes the probability he assigns to player 2’s type being $t_2^{{\prime }}$, which remains 1/3 regardless of the received message.

There is no implementing mechanism that, like the first mechanism above, does not randomize and, like the second mechanism, sends to player 1 a message that only depends on 2’s type, say $m_1^{{\prime }}$, $m_1^{\prime \prime }$ or $m_1^{\prime \prime \prime }$ if the type is $t_2^{{\prime }}$, $t_2^{\prime \prime }$ and $t_2^{\prime \prime \prime }$, respectively. To see this, note that since the action that the given correlated equilibrium distribution specifies for type $t_1^{{\prime }}$ of player 1 depends on whether 2’s type is $t_2^{\prime \prime }$ or $t_2^{\prime \prime \prime }$, the message $m_1^{\prime \prime }$ must be different from $m_1^{\prime \prime \prime }$. Therefore, one of them, say $m_1^{\prime \prime }$, must also be different from $m_1^{{\prime }}$. However, this means that the mechanism effectively tells player 1 whether or not 2’s type is $t_2^{\prime \prime }$. It follows that, for player 1, choosing $L$ when the type profile is $(t_1^{\prime \prime } ,t_2^{\prime \prime } )$ is not incentive compatible: $R$ would yield a higher payoff.

Proposition 20

For CEDs, $\left( {\{S\}\wedge \{D\}} \right) \nRightarrow \{S,D\}$ but $\left( {\{S\}\wedge \{D\}} \right) \Leftrightarrow \left( {\{\tilde{S}\}\wedge \{D\}} \right) \Rightarrow \{S,I\}$.²⁵

Proof

The first part of the proposition is proved by Example 6. To prove the second part, it suffices to show that $\left( {\{\tilde{S}\}\wedge \{D\}} \right) \Rightarrow \{S,I\}$; the equivalence then follows immediately from the trivial implications $\left\{ {S,I} \right\} \Rightarrow \{S\}\Rightarrow \{\tilde{S}\}$.

Consider a CED $\eta$ that is both $\tilde{S}$- and $D$-implementable. It has to be shown that $\eta$ is also $S,I$-implementable. By the assumption of $D$-implementability and the second part of Proposition 4, there is a mapping $\phi =(\phi _1 ,\phi _2 ,\ldots ,\phi _n ):T\rightarrow A$ such that $\eta \left( {\left\{ {\left( {t,\phi (t)} \right) } \right\} } \right) =\eta _T (\{t\})$ for all type profiles $t$. By the assumption of $\tilde{S}$-implementability, there is a correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ with a mechanism that satisfies $\tilde{S}$ such that $\eta$ is equal to the joint distribution of the random type profile ${\varvec{t}}$ and the random action profile ${\varvec{a}}$ defined by (7). In particular, for every $t\in \text{ supp }\left( {\eta _T } \right)$,

$$\begin{aligned} \Pr (\sigma _i \left( {t_i ,{\varvec{m}}_i \left( t \right) } \right) =\phi _i \left( t \right) \text{ for } \text{ all } i)=\Pr ({\varvec{a}}=\phi \left( t \right) \mid {\varvec{t}}=t)=\frac{\eta \left( {\left\{ {\left( {t,\phi \left( t \right) } \right) } \right\} } \right) }{\eta _T \left( {\left\{ t \right\} } \right) }=1. \end{aligned}$$

Therefore,

$$\begin{aligned} {\varvec{a}}_i =\sigma _i \left( {{\varvec{t}}_i ,{\varvec{m}}_i \left( {\varvec{t}} \right) } \right) =\phi _i \left( {\varvec{t}} \right) ,\qquad i\in N. \end{aligned}$$

(38)

Let the mechanism $\tilde{{\varvec{m}}}$ be as in the proof of Proposition 14. By (35) and (38),

$$\begin{aligned} \sigma _i \left( {{\varvec{t}}_i ,\tilde{{\varvec{m}}}_i \left( {\varvec{t}} \right) } \right) =\phi _i \left( {\varvec{t}} \right) ,\qquad i\in N. \end{aligned}$$

Therefore, using the mechanism $\tilde{{\varvec{m}}}$ instead of ${\varvec{m}}$ gives a correlated strategy $\left( {\tilde{{\varvec{m}}},\sigma } \right)$ whose CSD is also $\eta$. Moreover, if $\tilde{{\varvec{m}}}$ is used and a single player $i$ changes his strategy from $\sigma _i$ to some other strategy $\sigma _i^{\prime }$, the player’s expected payoff changes to $E\left( {u_i \left( {{\varvec{t}},\left( {\sigma _i^{\prime } \left( {{\varvec{t}}_i ,\tilde{{\varvec{m}}}_i \left( {\varvec{t}} \right) } \right) ,\phi _{-i} \left( {\varvec{t}} \right) } \right) } \right) } \right)$. By (35), this new payoff is equal to

$$\begin{aligned} E\left( {u_i \left( {{\varvec{t}},\left( {\sigma _i^{\prime } \left( {{\varvec{t}}_i ,{\varvec{m}}_i \left( {\varvec{t}} \right) } \right) ,\phi _{-i} \left( {\varvec{t}} \right) } \right) } \right) } \right) , \end{aligned}$$

which by (38) is also $i$’s expected payoff if he unilaterally changes his strategy from $\sigma _i$ to $\sigma _i^{\prime }$ in the correlated strategy $\left( {{\varvec{m}},\sigma } \right)$ (rather than $\left( {\tilde{{\varvec{m}}},\sigma } \right) )$. Since the latter is a correlated equilibrium, $i$’s change of strategy cannot increase his expected payoff. This proves that $\left( {\tilde{{\varvec{m}}},\sigma } \right)$ is also a correlated equilibrium, with a mechanism that by construction satisfies $S$ and $I$. $\square$

It follows from the next example that the conjunction of $S$-implementability and $I$-implementability is also a new attribute of CEDs.

Example 7 A correlated equilibrium distribution that is $S$- as well as $I$-implementable but not $S,I$-implementable. The game structure and common prior are as in Examples 1, 2 and 3. The payoff matrices of player 1 — one for each type profile — are shown in Table 2. Player 2 receives a constant payoff of zero. A mechanism randomly chooses for each type profile an action for each player according to the (marginal) probabilities shown in Table 2 in such a way that these eight choices are independent. It then informs each player of the action that was chosen for him for the actual type profile. The players’ strategy of always taking the indicated action is a correlated equilibrium. This is because a change of action by player 1 may affect his payoff only if his type is +1 and, in addition, (i) player 2 has type +1 and he plays $L$, (ii) player 2 has type +1 and he plays $R$, or (iii) player 2 has type $-1$ and he plays $L$. The effect in case (i) has the opposite sign and twice the magnitude of that in the other two cases. Since (i), (ii) and (iii) always have equal conditional probabilities, given that the type of player 1 is +1 and given his action, this relation between the effects means that the conditional expectation of the gain from changing action is always zero. Thus, the probabilities in Table 2 define a correlated equilibrium distribution, which the mechanism described above implements. By construction, the mechanism has property $I$.The same correlated strategy distribution is also implementable by the following mechanism, which has property $S$. The mechanism first chooses two pairs of actions, $a^{+}=(a^{++},a^{-+})$ and $a^{-}=(a^{+-},a^{--})$, independently of one another. The probability that the pair $a^{+}$ equals $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$ is 1/6, 1/2, 1/6 and 1/6, respectively, and for $a^{-}$ the corresponding probabilities are 1/3, 1/3, 1/6 and 1/6. Then, for each type profile $t=(t_1 ,t_2 )$, the mechanism chooses an action $a_2^t$ for player 2 with probabilities (for $L$ and $R$) that depend on (both $t$ and) $a^{-}$ (that was chosen in the first stage). Specifically, the probability that $a_2^t =L$ is 1/2 unless $t=(+1,-1)$ and, in addition, (i) $a^{-}=(L,L)$, in which case the probability is 1/4, or (ii) $a^{-}=(L,R)$, in which case the probability is 3/4. Finally, the mechanism sends messages to the players, which depend on the choices made in the first two stages as well as the players’ actual type profile $t=(t_1 ,t_2 )$. The message to player 1 is $a^{+}$ or $a^{-}$ if 2’s type is +1 or $-1$, respectively, and the message to player 2 is the pair of actions $(a_2^{(t_1 ,+1)} ,a_2^{(t_1 ,-1)} )$. Thus, neither message is affected by the player’s own type. It is not very difficult to check that the correlated strategy with this mechanism that instructs each player to choose the first or second action in his message if his type is +1 or $-1$, respectively, has the distribution in Table 2. For example, if $t= (+1,+1)$, the action profile is $(a^{++},a_2^{\left( {+1,+1} \right) } )$, which is $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$ with probability 1/3, 1/3, 1/6 and 1/6, respectively. Thus, the players’ actions are independent and are distributed as specified at the margins of the top-left box in Table 2.

To show that the above correlated strategy is a correlated equilibrium, it suffices to prove that, given that the type of player 1 is +1 and given the message he receives (which can be $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$), the conditional probabilities of the following three events are equal: (i) player 2 has type +1 and he plays $L$, (ii) player 2 has type +1 and he plays $R$, and (iii) player 2 has type $-1$ and he plays $L$. As indicated above, such equality means that player 1 is indifferent between his two actions. The equality can be viewed as the conjunction of two equalities: (a) events (i) and (ii) have equal conditional probabilities, which are necessarily one-half the conditional probability that $t_2 =+1$, and (b) the latter is also equal to twice the conditional probability of (iii). To prove (a), it suffices to note that, given that $t=\left( {+1,+1} \right)$, the message $a^{+}$ that player 1 receives and the action $a_2^{\left( {+1,+1} \right) }$ that player 2 takes are conditionally independent, and the probability that the latter is $L$ is 1/2. To prove (b), note, first, that by the specification of the mechanism and Bayes’ rule, the conditional probability that $t_2 =+1$, given that player 1’s type is +1 and that he receives the message $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$, is equal to 1/3, 3/5, 1/2 or 1/2, respectively. It is therefore sufficient to show that the conditional probability, given the same information, that $t_2 =-1$ and $a_2^{\left( {+1,-1} \right) } =L$ is 1/6, 3/10, 1/4 or 1/4, respectively. This conditional probability is equal to the product of two terms: the condition probability that $t_2 =-1$, given that $t_1 =+1$ and player 1’s message has the specified value, and the condition probability that $a_2^{\left( {+1,-1} \right) } =L$, given that $t=\left( {+1,-1} \right)$ and $a^{-}$ has that value. The first term is the complement of the conditional probability that $t_2 =+1$, and is hence 2/3, 2/5, 1/2 or 1/2 if the message is $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$, respectively; and by the specification of the mechanism, the second term is 1/4, 3/4, 1/2 or 1/2, respectively. Therefore, the product of the two terms is 1/6, 3/10, 1/4 or 1/4, respectively, as had to be shown.

Table 2 A correlated equilibrium distribution for Example 7. The four type profiles are equally probable. For each of them, the actions that players 1 and 2 take are independent. The probabilities that these actions are $L$ or $R$ are given at the margins of the corresponding box. The numbers inside the box are player 1’s payoffs. The payoff of player 2 is always 0

There does not exist any implementing mechanism for the CED specified by Table 2 that has both properties $S$ and $I$. Note that this is so despite the fact that, according to Table 2, for each type profile, the two players’ actions are independent. To see this, suppose that such a mechanism does exist. Consider a correlated equilibrium with that mechanism that has the above CED. Partition all the messages that player 1 may receive from the mechanism into four groups, $(L,L)$, $(L,R)$,$(R,L)$ and $(R,R)$, according to the actions that player 1’s strategy prescribes to him when he receives the message and his type is +1 (first entry) or $-1$ (second entry). Since the mechanism satisfies $S$, and hence also $\tilde{S}$, the probability of receiving a message that belongs to a particular group when player 2 has type +1 is the same for both types of player 1. Denote these probabilities by $p_{LL}^+$, $p_{LR}^+$, $p_{RL}^+$ and $p_{RR}^+$. Let $p_{LL}^-$, $p_{LR}^-$, $p_{RL}^-$ and $p_{RR}^-$ be the corresponding probabilities for the case in which player 2’s type is $-1$. Since for each type profile the probability that player 1 plays $L$ is as specified by Table 2, the following equalities hold:

$$\begin{aligned}&p_{LL}^+ +p_{LR}^+ =\frac{2}{3},\qquad p_{LL}^- +p_{LR}^- =\frac{2}{3}, \qquad (L_+ ) \\&p_{LL}^+ +p_{RL}^+ =\frac{1}{3},\qquad p_{LL}^- +p_{RL}^- =\frac{1}{2}. \qquad (L_- ) \end{aligned}$$

By definition, in a correlated equilibrium, taking the prescribed action is always incentive compatible. In particular, type +1 of player 1 cannot increase the conditional expectation of his payoff when the message he receives belongs to group $(L,L)$, $(L,R)$,$(R,L)$ or $(R,R)$ by playing $R$, $R$, $L$ or $L$, respectively (instead of the opposite action he is supposed to take). Since the mechanism satisfies $I$, for every type profile the message that player 1 receives is independent of player 2’s message, and hence of his action: player 2 always plays $L$ with probability 1/2. The above incentive compatibility condition is therefore expressed by the following four inequalities:

$$\begin{aligned}&p_{LL}^+ \left( {-\frac{1}{2}\cdot 2+\frac{1}{2}\cdot 1} \right) +p_{LL}^- \left( {\frac{1}{2}\cdot 1+\frac{1}{2}\cdot 0} \right) \le 0,\\&p_{LR}^+ \left( {-\frac{1}{2}\cdot 2+\frac{1}{2}\cdot 1} \right) +p_{LR}^- \left( {\frac{1}{2}\cdot 1+\frac{1}{2}\cdot 0} \right) \le 0,\\&p_{RL}^+ \left( {\frac{1}{2}\cdot 2-\frac{1}{2}\cdot 1} \right) +p_{RL}^- \left( {-\frac{1}{2}\cdot 1-\frac{1}{2}\cdot 0} \right) \le 0,\\&p_{RR}^+ \left( {\frac{1}{2}\cdot 2-\frac{1}{2}\cdot 1} \right) +p_{RR}^- \left( {-\frac{1}{2}\cdot 1-\frac{1}{2}\cdot 0} \right) \le 0. \end{aligned}$$

All inequalities must in fact hold as equalities. If any of the first two inequalities or any of the last two were strict, then $-\left( {p_{LL}^+ +p_{LR}^+ } \right) +\left( {p_{LL}^- +p_{LR}^- } \right) <0$ or $\left( {p_{RL}^+ +p_{RR}^+ } \right) -\left( {p_{RL}^- +p_{LR}^- } \right) <0$ would hold. These two inequalities are equivalent (since the probabilities in each quartet sum up to 1), and they contradict ($L_{+}$). Therefore, in particular, the first and third equalities above hold as equalities, which implies that $-\left( {p_{LL}^+ +p_{RL}^+ } \right) +\left( {p_{LL}^- +p_{RL}^- } \right) =0$. This equation contradicts ($L_{-}$). The contradiction proves that no implementing mechanism for the CED specified in Table 2 has both property $S$ (or even only $\tilde{S})$ and $I$.

Proposition 21

For CEDs, $\left( {\{S\}\wedge \{I\}} \right) \nRightarrow \{S,I\}$.

Proof

This is demonstrated by Example 7. $\square$

Whether the conjunction of $\tilde{S}$-implementability and $I$-implementability is also a new attribute of CEDs is not known. It depends on the answer to the following question.

Open Question. For CEDs, does $\left( {\{\tilde{S}\}\wedge \{I\}} \right) \Rightarrow \{S\}$?

This question corresponds to the question mark in Fig. 2. The marked attribute is different from the one below it (attribute $\text{ II }_{\mathrm{b}})$ if and only if the answer is negative, that is, if there exists a CED in some Bayesian game that is both $\tilde{S}$- and $I$-implementable but not $S$-implementable. If the answer is affirmative, the two attributes of CEDs are actually one and the same, that is, they are equivalent.

Depending on the answer to the Open Question, there are two or three attributes of CEDs that can be defined as the conjunction of a pair of incomparable attributes of the twelve ones presented in the previous subsection. Thus, there are in total 14 or 15 attributes of CEDs, which are related to one another as in Fig. 2. The following lemma shows that this list is complete in that there are no additional, nonequivalent attributes that can be defined as the conjunction of two or more of those in Fig. 2. This result holds regardless of the answer to the Open Question.

Lemma 6

The conjunction of any number of the attributes of CEDs in Fig. 2 is equivalent to one of the attributes in the same figure.

Proof

(an outline). It has to be shown that for every list $\mathcal P ^{{\prime }},\mathcal P ^{\prime \prime },\!\ldots \! \!\subseteq \! \{S,\tilde{S},O,\tilde{O},D,I\}$ the conjunction $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$ is equivalent to one of the attributes in Fig. 2. It suffices to consider lists with three or fewer entries, since in any longer list at least two elements represent comparable attributes. Proposition 15 proves the two cases of the conjunction of $\tilde{O}$-implementability and either $D$- or $I$-implementability. All the other cases follow quite easily from these two. $\square$

Appendix C: Communication equilibrium distributions

As for correlated strategy distributions and correlated equilibrium distributions, different kinds of mechanisms implement different kinds of communication equilibrium distributions. Specifically, for each subset $\mathcal P$ of the six fundamental properties of mechanisms, a MED is $\mathcal P$-implementable if it is the CSD of some communication equilibrium with a mechanism that has all the properties in $\mathcal P$. This appendix, like the previous two, is mainly concerned with the implication relation between these attributes, and conjunctions of several attributes. Implication is denoted by the generic symbol $\Rightarrow$ when it is clear from the context that it refers to attributes of MEDs. Otherwise, the more explicit symbol $\mathop {\Rightarrow }\limits _{\mathrm{MED}}$ is used.

The following proposition shows that a necessary condition for the last relation to hold is that a similar relation holds for CEDs. The propositions in the next subsection prove that this condition is also sufficient, as long as only attributes that are defined by single sets of properties of mechanisms are involved. Thus, the reverse of implication (39) below holds too. However, this result does not extend to attributes that are defined by conjunction (compare Proposition 33 below with the second part of Proposition 20). Hence the differences between the Hasse diagram of the implications relations between attributes of MEDs (Fig. 3) and the corresponding diagram for CEDs (Fig. 2).

Proposition 22

For every two subsets $\mathcal P ,\mathcal{Q}\subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$,

$$\begin{aligned} \mathcal P \mathop {\Rightarrow }\limits _{\mathrm{MED}} \mathcal{Q} \,\mathrm{implies }\,\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}. \end{aligned}$$

(39)

Moreover, the same is true with $\mathcal P$ replaced by by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$, for any list $\mathcal P ^{{\prime }},\mathcal P ^{\prime \prime },\ldots$ of subsets of $\{S,\tilde{S},O,\tilde{O},V,I\}$.

The proof of Proposition 22, which is given at the end of this appendix, uses the results in the following two subsections.

1.1 Equivalences

The following propositions parallel those in Appendix B.

Proposition 23

For MEDs, $\{S,\tilde{O}\}\Leftrightarrow \{\tilde{S},\tilde{O}\}\Leftrightarrow \{\tilde{O}\}$.

Proof

It has to be shown that the MED $\eta$ of any communication equilibrium $\left( {{\varvec{m}},\sigma } \right)$ with a mechanism that has property $\tilde{O}$ is $S,\tilde{O}$-implementable. Unlike in the proof of Proposition 12, it cannot be assumed that the mechanism ${\varvec{m}}$ is the default one. Nevertheless, without loss of generality, it may be assumed that it satisfies (4). Otherwise, ${\varvec{m}}$ could be replaced by any mechanism $\tilde{{\varvec{m}}}$ satisfying (4) such that

$$\begin{aligned} \tilde{{\varvec{m}}}\left( t \right) \mathop {=}\limits ^{d} {\varvec{m}}\left( t \right) ,\qquad t\in T. \end{aligned}$$

These equalities mean that, for any profile of reported types $t$, the messages that $\tilde{{\varvec{m}}}$ sends are indistinguishable from those of ${\varvec{m}}$. This implies that $\tilde{{\varvec{m}}}$ also has property $\tilde{O}$, and $\left( {\tilde{{\varvec{m}}},\sigma } \right)$ is also a communication equilibrium.

Consider the correlated strategy $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ defined by (21) and the following generalization of (28):

$$\begin{aligned} \vec {{\varvec{m}}}_i \left( t \right) =\left( {\sigma _i \left( {t_i^1 ,{\varvec{m}}_i \left( {t_i^1 ,t_{-i} } \right) } \right) ,\sigma _i \left( {t_i^2 ,{\varvec{m}}_i \left( {t_i^2 ,t_{-i} } \right) } \right) ,\ldots } \right) ,\qquad i\in N,t\in T. \end{aligned}$$

Arguments similar to those used in the proof of Proposition 5 show that $\vec {{\varvec{m}}}$ has properties $S$ and $\tilde{O}$. To prove that $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ is a communication equilibrium, is has to be shown that, for every player $i$, type $t_i^{\prime }$ for that player and strategy $\bar{\sigma }_i^{\prime } :T_i \times A_i^{T_i } \rightarrow A_i$,

$$\begin{aligned} E\left( {u_i \left( {{\varvec{t}},{\varvec{a}}} \right) -u_i \left( {{\varvec{t}},{\varvec{a}}^{{\prime }}} \right) |{\varvec{t}}_i } \right) \ge 0, \end{aligned}$$

(40)

where ${\varvec{a}}$ is the random action profile corresponding to $\bar{\sigma }$, that is,

$$\begin{aligned} {\varvec{a}}_j =\bar{\sigma }_j \left( {{\varvec{t}}_j ,\vec {{\varvec{m}}}_j \left( {\varvec{t}} \right) } \right) =\sigma _j \left( {{\varvec{t}}_j ,{\varvec{m}}_j \left( {\varvec{t}} \right) } \right) ,\qquad j\in N, \end{aligned}$$

(41)

and ${\varvec{a}}^{{\prime }}=\left( {{\varvec{a}}_1^{\prime } ,{\varvec{a}}_2^{\prime } ,\ldots ,{\varvec{a}}_n^{\prime } } \right)$ is defined by

$$\begin{aligned} {\varvec{a}}_i^{\prime }\!&= \!\bar{\sigma }_i^{\prime } \left( {{\varvec{t}}_i ,\vec {{\varvec{m}}}_i \left( {t_i^{\prime } ,{\varvec{t}}_{-i} } \right) } \right) \!=\!\bar{\sigma }_i^{\prime } \left( {{\varvec{t}}_i ,\left( {\sigma _i \left( {t_i^1 ,{\varvec{m}}_i \left( {t_i^1 ,{\varvec{t}}_{-i} } \right) } \right) \!,\!\sigma _i \left( {t_i^2 ,{\varvec{m}}_i \left( {t_i^2 \!,\!{\varvec{t}}_{-i} } \right) } \right) \!,\!\ldots } \right) } \right) , \\ {\varvec{a}}_j^{\prime }&= \bar{\sigma }_j \left( {{\varvec{t}}_j ,\vec {{\varvec{m}}}_j \left( {t_i^{\prime } ,{\varvec{t}}_{-i} } \right) } \right) =\sigma _j \left( {{\varvec{t}}_j ,{\varvec{m}}_j \left( {t_i^{\prime } ,{\varvec{t}}_{-i} } \right) } \right) ,\qquad j\ne i. \end{aligned}$$

Suppose that, for example with $i=1$ and $t_i^{\prime } =t_1^1$, (40) does not hold. This implies that there are some $t_1^{\prime \prime }$ and $m_1^2 ,m_1^3 ,\ldots$ such that

$$\begin{aligned}&E\left( {u_1 \left( {{\varvec{t}},{\varvec{a}}} \right) |{\varvec{t}}_1 =t_1^{\prime \prime } } \right) \\&\qquad <E\left( {u_1 \left( {{\varvec{t}},{\varvec{a}}^{{\prime }}} \right) |{\varvec{t}}_1 =t_1^{\prime \prime } ,{\varvec{m}}_1 \left( {t_1^2 ,{\varvec{t}}_{-1} } \right) =m_1^2 ,{\varvec{m}}_1 \left( {t_1^3 ,{\varvec{t}}_{-1} } \right) =m_1^3 ,\ldots } \right) . \end{aligned}$$

It follows from properties $\tilde{O}$ and (4) of ${\varvec{m}}$ that the pair of random variables ${\varvec{t}}$ and ${\varvec{m}}\left( {t_1^1 ,{\varvec{t}}_{-1} } \right)$ is independent of ${\varvec{m}}_1 \left( {t_1^2 ,{\varvec{t}}_{-1} } \right) ,{\varvec{m}}_1 \left( {t_1^3 ,{\varvec{t}}_{-1} } \right) ,\ldots$. Therefore, the last inequality is equivalent to

$$\begin{aligned}&E\left( {u_1 \left( {{\varvec{t}},{\varvec{a}}} \right) |{\varvec{t}}_1 =t_1^{\prime \prime } }\right) \nonumber \\&\qquad <E\left( u_1 \left( {\varvec{t}},\left( \sigma _1^{\prime } \left( {{\varvec{t}}_1 ,{\varvec{m}}_1 \left( {t_1^1 ,{\varvec{t}}_{-1} } \right) } \right) ,\sigma _2 \left( {{\varvec{t}}_2 ,{\varvec{m}}_2 \left( {t_1^1 ,{\varvec{t}}_{-1} } \right) } \right) ,\ldots ,\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. \sigma _n \left( {{\varvec{t}}_n ,{\varvec{m}}_n \left( {t_1^1 ,{\varvec{t}}_{-1} } \right) } \right) \right) \right) \Big |{\varvec{t}}_1 =t_1^{\prime \prime } \right) , \end{aligned}$$

(42)

where $\sigma _1^{\prime } :T_1 \times M_1 \rightarrow A_1$ is the function defined by

$$\begin{aligned} \sigma _1^{\prime } \left( {t_1 ,m_1 } \right) =\bar{\sigma }_1^{\prime } \left( {t_1 ,\left( {\sigma _1 \left( {t_1^1 ,m_1 } \right) ,\sigma _1 \left( {t_1^2 ,m_1^2 } \right) ,\sigma _1 \left( {t_1^3 ,m_1^3 } \right) ,\ldots } \right) } \right) . \end{aligned}$$

However, in conjunction with (41), inequality (42) contradicts the assumption that $\left( {{\varvec{m}},\sigma } \right)$ is a communication equilibrium, since it shows that when player 1’s type is $t_1^{\prime \prime }$, he can gain from misreporting it as $t_1^1$ and switching from $\sigma _1$ to $\sigma _1^{\prime }$. The contradiction proves that (40) must in fact hold, so that $\left( {\vec {{\varvec{m}}},\bar{\sigma }} \right)$ is a communication equilibrium. $\square$

Proposition 24

For MEDs, $\{\tilde{S},O\}\Leftrightarrow \{O\}$.

Proof

Identical to the proof of Proposition 13. $\square$

Proposition 25

For MEDs, $\{S,I\}\Leftrightarrow \{\tilde{S},I\}$ and $\{S,D\}\Leftrightarrow \{\tilde{S},D\}$.

Proof

Identical to the proof of Proposition 14. $\square$

Proposition 26

For MEDs, $\{S,O,I\}\Leftrightarrow \{\tilde{S},O,I\}\Leftrightarrow \{S,\tilde{O},I\}\Leftrightarrow \{\tilde{S},\tilde{O},I\}\Leftrightarrow \{O,I\}\Leftrightarrow \{\tilde{O},I\}\Leftrightarrow \left( {\{\tilde{O}\}\wedge \{I\}} \right)$ and $\{S,O,D\}\Leftrightarrow \{\tilde{S},O,D\}\Leftrightarrow \{S,\tilde{O},D\}\Leftrightarrow \{\tilde{S},\tilde{O},D\}\Leftrightarrow \{O,D\}\Leftrightarrow \{\tilde{O},D\}\Leftrightarrow \left( {\{\tilde{O}\}\wedge \{D\}} \right)$.

Proof

It suffices to show that $\left( {\{\tilde{O}\}\wedge \{I\}} \right) \Rightarrow \{S,O,I\}$ and similarly with $I$ replaced by $D$. By Proposition 15, both implications hold for CEDs. The result that they also hold for MEDs follows immediately from the fact that an $S,O$-implementable CED is automatically a MED. $\square$

1.2 Implications

Implication (39) in Proposition 22 can equivalently be expressed by its counterpositive: if an example of a $\mathcal P$-implementable CED that is not $\mathcal{Q}$-implementable exists, then a similar counterexample can be found for MEDs. Finding the latter may or may not be easy. The former holds if the CED example employs a correlated equilibrium (with a mechanism with the properties in $\mathcal P$) that is also a communication equilibrium, that is, players have no incentive to lie about their types. In this case, the same example can be used for MEDs, since a CED that is not $\mathcal{Q}$-implementable a fortiori does not have that attribute as a MED. The proofs of the following two propositions use this simple observation.

Proposition 27

For MEDs, $\{S,O,I\}\nRightarrow \{D\}$, $\{S,D\}\nRightarrow \{\tilde{O}\}$, $\{\tilde{O}\}\nRightarrow \{O\}$ and $\{S,O\}\nRightarrow \{I\}$.

Proof

Proposition 16, which establishes the same for CEDs, relies on Proposition 11. Therefore, it suffices to show that a result similar to the latter holds with communication equilibrium (distribution) replacing correlated equilibrium (distribution). This can be shown by simply making this replacement throughout the proof of Proposition 11.   $\square$

Proposition 28

For MEDs, $\{D\}\nRightarrow \{\tilde{S}\}$.

Proof

The correlated equilibrium with the mechanism with property $D$ that is described in Example 2 is in fact a communication equilibrium. If player 1 lies about his type, player 2 gets as a message an incorrect type profile and will consequentially choose an action with which a positive payoff for 1 is impossible. For a similar reason, player 2 cannot gain from lying; in this case, the lie will only affect type +1 of player 2. The MED of this communication equilibrium in not $\tilde{S}$-implementable since, as shown, it does not have that attribute even as a CED. $\square$

Even if the correlated equilibrium that is used for demonstrating that a certain implication does not hold for CEDs is not a communication equilibrium, not all hope is necessarily lost. It may be possible to make truthful type reports incentive compatible by augmenting the original game with a suitable auxiliary game and modifying the correlated strategy accordingly.

Suppose, for example, that each of the two players in a Bayesian game can have type +1 or $-1$, and all four type profiles are equally probable. The game can then be modified by adding to it an auxiliary game that requires each player to push one of three buttons, $B_1$, $B_2$ or $B_3$. Depending on both players’ choice of button and on their types, a very large number $K>0$ is either added to or subtracted from their payoffs in the original game. Specifically, the change in payoffs ($+K$ or $-K$) is determined according to the following table, where the rows and columns correspond to the choices of player 1 and 2, respectively, and $\tau =t_1 t_2$ is the product of their types:

Thus, for both $\tau =+1$ and $-1$, three cells in the table represent reward and six cells represent punishment. Any mechanism in the original game can be turned into a mechanism in the augmented game by appending to the message it sends to each player, which pertains to the original game, a recommendation of button in the auxiliary game. The recommendation is determined in the following way. The mechanism attempts to identify the “rewarding” cells by calculating the product of the players’ reported types. It then randomly selects one of these cells, each with probability 1/3, and recommends its row and column to player 1 and 2, respectively. As detailed below, the feature of the mechanism that encourages truth telling is that misreporting will result in misidentification of the rewarding cells. Note that, for any pair of reported types, the recommendation to each player is equally likely to be $B_1$, $B_2$ or $B_3$. Therefore, the modified mechanism has property $\tilde{S}$ or $\tilde{O}$ if the original mechanism has the same property. It cannot, however, have any of the other four fundamental properties. (However, with a somewhat more complicated auxiliary game, it is possible to also retain property $S$.)

To any correlated equilibrium in the original game, there corresponds a communication equilibrium in the augmented game. In that equilibrium, the mechanism appends recommendations as described above, and then each player pushes the recommended button and plays in the original game according to the original correlated equilibrium. To see that truthful type reports are incentive compatible, suppose that, for example, button $B_1$ is recommended to type +1 of player 1. The player can infer from the recommendation that, if both players reported their types truthfully and they will follow the mechanism’s recommendations, player 2 will choose $B_1$ or $B_2$ if his type is +1 or $-1$, respectively, and in both cases, the players will get the reward of $K$. However, if (only) he, player 1, misreported his type as $-1$, then player 2’s choice of button will have the opposite relation with his type. Consequently, player 1 can get a reward rather than a penalty of $K$ only by choosing $B_2$ or $B_3$ if 2’s type is +1 or $-1$, respectively. However, since the players’ types are independent, this means that player 1 cannot get more than zero in expectation. Hence, misreporting the type does not pay.

Proposition 29

For MEDs, $\{\tilde{S}\}\nRightarrow \{S\}$.

Proof

Consider the Bayesian game and the $\tilde{S}$- but not $S$-implementable CED presented in Example 3. That CED is not a MED. However, a communication equilibrium with a mechanism that has property $\tilde{S}$ can be obtained by modifying the game and the correlated equilibrium described in the original example by adding an auxiliary game as above. The corresponding MED is not $S$-implementable even as CED. It is not difficult to see that, if it were $S$-implementable, the same would be true for the original CED. $\square$

The proofs of the next two propositions involve more special modifications of the original counterexamples, i.e., those pertaining to CEDs.

Proposition 30

For MEDs, $\{\tilde{S},O\}\nRightarrow \{S,O\}$.

Proof

Consider the following modification of the game and CED in Example 5. Both players can have type +1 or $-1$, and all four type profiles are equally probable. If the players’ types are different or identical, respectively, they both receive the payoff specified by the matrix

$$\begin{aligned} {\begin{array}{cc} &{} {{\begin{array}{c@{\quad }c} {L}&{} R \\ \end{array} }} \\ {{\begin{array}{l} L \\ R \\ \end{array} }}&{} {\left( {{\begin{array}{c@{\quad }c} 0&{} 0 \\ 0&{} {1.5} \\ \end{array} }} \right) } \\ \end{array} } \text{ or } {\begin{array}{cc} &{} {{\begin{array}{c@{\quad }c} {L}&{} R \\ \end{array} }} \\ {{\begin{array}{l} L \\ R \\ \end{array} }}&{} {\left( {{\begin{array}{c@{\quad }c} 1&{} 0 \\ 0&{} 1 \\ \end{array} }} \right) } \\ \end{array} }. \end{aligned}$$

The correlated strategy described in the original example is a correlated equilibrium also in the modified game. For a player of any type who receives the message $L$ and takes that action, the expected payoff is $1/2 \times 0+1/2 \times 1=0.5$, whereas playing $R$ instead would only yield $1/2 \times 1/2 \times 1.5+1/2 \times 0=0.375$. If the message is $R$, taking that action yields 0.875, while playing $L$ would yield 0. This correlated equilibrium is moreover a communication equilibrium. If a player misreports his type, he will maximize his payoff by taking the recommended action, since this is also the action that the other player will take if the (real) types differ (and if the types are identical, then the expected payoff from any action is 0.5). Thus, a dishonest player cannot get more than $1/2 \times 1/2 \times 1.5+1/2 \times 0.5=0.625$, which is less than the $1/2 \times 0.5+1/2 \times 0.875=0.6875$ that a truthful report would yield him.

It remains to show that the corresponding MED is different from the distribution of any communication (or even correlated) equilibrium $\left( {{\varvec{m}},\sigma } \right)$ with a mechanism that has properties $S$ and $O$. The messages that ${\varvec{m}}$ sends to the players can be written as ${\varvec{m}}_1 (t^{{\prime }})$ and ${\varvec{m}}_2 (t^{{\prime }})$, for arbitrary type profile $t^{{\prime }}$. Since the players’ actions are identical if their types are identical, necessarily

$$\begin{aligned} \left( {\sigma _1 \left( {+1,{\varvec{m}}_1 \left( {t^{{\prime }}} \right) } \right) ,\sigma _1 (-1,{\varvec{m}}_1 \left( {t^{{\prime }}} \right) } \right) \big )=\left( {\sigma _2 \left( {+1,{\varvec{m}}_2 \left( {t^{{\prime }}} \right) } \right) ,\sigma _2 (-1,{\varvec{m}}_2 \left( {t^{{\prime }}} \right) } \right) \big ). \end{aligned}$$

(43)

If ${\varvec{m}}_1 \left( {t^{{\prime }}} \right)$ is such that the left- (and, hence, also the right-) hand side equals $\left( {L,R} \right)$ or $\left( {R,L} \right)$, respectively, then type $+1$ or $-1$ of player 1 will get $1/2 \times 1$ from taking the action $L$ he is supposed to take but $1/2 \times 1.5$ from playing $R$. Therefore, the probability that the four actions in (43) are not all the same must be zero, which shows that the above MED, in which the players’ actions may differ, cannot be attained. $\square$

Proposition 31

For MEDs, $\left( {\{S,I\}\wedge \{D\}} \right) \nRightarrow \{S,D\}$.

Proof

Consider the following modification of the game and CED in Example 6. Unlike in the original example, player 2 has the constant payoff 0, and he is allowed to choose action $R$ as well as $L$. Choosing $R$ rather than $L$ reduces by 3 the payoff of type $t_1^{\prime \prime }$ of player 1 but has no effect on the other payoffs. The two mechanisms considered in the original example and the corresponding correlated equilibria are modified as follows. Both mechanisms instruct player 2 to play $R$ if player 1 reports the type $t_1^{{\prime }}$ and to play $L$ otherwise, and player 2 obeys. Clearly, this means that type $t_1^{\prime \prime }$ of player 1 has an incentive to report his type truthfully. The same is true for type $t_1^{{\prime }}$, for whom the CED gives the highest possible payoff. $\square$

An alternative proof for the last proposition can be obtained by using the following simple and generally applicable modification of the game and correlated equilibria in the original example. Instead of changing the players’ action spaces, a new player is added to the game. This “player 0” has a single type, and his action space is the collection $T$ of all type profiles of the original players. If the action that player 0 chooses coincides with the other players’ actual type profile, everyone gets a huge bonus. Any correlated equilibrium in the original game can be modified as follows. The mechanism sends to player 0 the type reports of the other players, and he chooses the corresponding action. This obviously creates an incentive for the players to report their types truthfully, and thus turns the correlated equilibrium into a communication equilibrium (in the modified game). If property $S$, $\tilde{S}$, $I$ or $D$ holds for the original mechanism, the modified mechanism also has the same property.

The following proposition uses this construction to show that if the answer to the Open Question presented in Appendix B is negative, then the same is true for MEDs. Note that if the answer will turn out to be affirmative, the proposition is technically correct but uninformative, since its assertion holds vacuously.

Proposition 32

If, for MEDs, $\left( {\{\tilde{S}\}\wedge \{I\}} \right) \Rightarrow \{S\}$, then the same is true for CEDs.

Proof

Suppose that, in some Bayesian game, there is a CED $\eta$ that is $\tilde{S}$-implementable and $I$-implementable but not $S$-implementable. It has to be shown that a MED with similar properties also exists.

Consider two correlated equilibria whose CED is $\eta$, one with a mechanism that satisfies $\tilde{S}$ and the other with a mechanism that satisfies $I$, and modify the game and the two correlated equilibria by adding a new player, player 0, as detailed above. The modification turns the two correlated equilibria into communication equilibria, whose common MED assigns nonzero probability only to pairs of type and action profiles in which the former coincides with the action of player 0, and the probability in this case is equal to that assigned by $\eta$ to the pair obtained by omitting player 0’s action. Any communication, or even just correlated, equilibrium whose distribution is this MED can be turned into a correlated equilibrium in the original game (in which $\eta$ is a CED) simply by omitting the message to player 0 and that player’s strategy. If the mechanism in that equilibrium had property $S$, the same would be true after the omission. It therefore follows from the assumption concerning the CED $\eta$ that a mechanism implementing the MED cannot in fact have property $S$. $\square$

Proposition 32 precludes the possibility that the implication under consideration holds for MEDs but not for CEDs. However, it is mute about the opposite possibility. The following example, by contrast, considers a (different) implication for which the latter definitely holds, as comparison with the second part of Proposition 20 shows.

Example 8 A communication equilibrium distribution that is $S$- as well as $D$-implementable but not $S,I$-implementable. In a three-player Bayesian game, each player has two types: $t_1^{{\prime }}$ and $t_1^{\prime \prime }$ for player 1, and $t^{{\prime }}$ and $t^{\prime \prime }$ for players 2 and 3. All type profiles except $(t_1^{{\prime }} ,t^{\prime },t^{\prime })$ may occur, and they have the same probability (1/7). Each player can choose to play $L$ or $R$. Player 1’s payoff depends only on the type profile $t$ and on the other players’ actions. Specifically, the payoff is 0 if $t\ne \left( {t_1^{\prime \prime } ,t^{{\prime }},t^{\prime }} \right)$, and if an equality holds, it is given by the following symmetric matrix, where the rows and columns correspond to the actions of player 2 and 3:

$$\begin{aligned} {\begin{array}{cc} &{} {{\begin{array}{c@{\quad }c} {L}&{} R \\ \end{array} }} \\ {{\begin{array}{l} L \\ R \\ \end{array} }}&{} {\left( {{\begin{array}{c@{\quad }c} 0&{} {-1} \\ {-1}&{} 6 \\ \end{array} }} \right) } \\ \end{array} }. \end{aligned}$$

For players 2 and 3 the payoff is the sum of two numbers. The first number, which is the same for both players, is: (i) 4 if player 1 plays $L$ and players 2 and 3 have identical types, (ii) 4 also if player 1 plays $R$ and players 2 and 3 have different types, and (iii) 0 otherwise. The second number depends on whether the player’s own action is $R$ or $L$. In the first case, it is equal to 1/2, and in the second, it is given by the following table, in which the rows correspond to the player’s type and the columns correspond to the types of the other two players:

Consider the mechanism with property $D$ that, for each type profile $t$, instructs player 1 to play $L$ or $R$ if the (reported) types of players 2 and 3 are identical or different, respectively, and instructs players 2 and 3 to take the actions specified by the following table, in which the rows and columns correspond to the player’s own type and to that of the other player, respectively:

This mechanism and the strategies of following the instructions together constitute a communication equilibrium. Player 1 cannot increase his payoff of 0 since there is no way he can make players 2 and 3 play $R$ when they both have type $t^{\prime }$. And for these players, a truthful type report is incentive compatible, since if (only) one of them lies, they both lose the 4 they would get from a match between their types (either identical or different) and player 1’s action ($L$ or $R$, respectively). In addition, for players 2 and 3, acting according to their strategy is incentive compatible. For a player of type $t^{{\prime }}$, doing so always guarantees maximum payoff, and for type $t^{\prime \prime }$, playing $R$ instead of the action $L$ he is instructed to take would decrease the expected payoff by $(1/4\times 3-1/2=) 1/4$.

The MED of the above communication equilibrium is also implementable by a mechanism with property $S$. That mechanism sends to player 1 the same messages as the mechanism described above, and sends to each of the other two players $i\,(=2,3)$ a message that depends on the others’ types according to the table

where $({\varvec{Z}}_2 ,{\varvec{Z}}_3 )$ is a pair of (dependent) random variables that equals $(L,R)$ with probability 0.5 and $(R,L)$ with probability 0.5. A communication equilibrium with this mechanism that has the same MED as the previous one is defined as follows. The strategy of each player is to play according to the message he receives, unless he is of type $t^{\prime \prime }$, in which case he plays $L$. For a player of type $t^{\prime \prime }$, playing $R$ would not increase the conditional expectation of the payoff, regardless of the message he receives. This is because, given that the received message is $L$ or $R$, the conditional probability that the other players have types $t_1^{{\prime }}$ and $t^{\prime }$ is 1/3 or 1/5, respectively. Since both $1/3\times 3$ and $1/5\times 3$ are greater than 1/2, deviations to $R$ are unprofitable. The incentive compatibility of truthful type reports is proved by arguments similar to those used for the previous equilibrium.

There is no communication equilibrium with a mechanism with properties $S$ and $I$ that has the above MED. To see this, suppose that such a communication equilibrium exists. Since property $S$ implies $\tilde{S}$, the distribution of the mechanism’s messages to player 3 only depends on the other players’ types, so that it can be described by the table

where $P^{1}$, $P^{2}$, $P^{3}$, $P^{4}$ are four probability measures on player 3’s message space $M_3$. If the type of player 3 is $t^{\prime }$, he is supposed to play $L$ or $R$if he receives any message in $\text{ supp }\left( {P^{2}} \right)$ or in $\text{ supp }\left( {P^{3}} \right) \cup \text{ supp }\left( {P^{4}} \right)$, respectively. Therefore, these two subsets of $M_3$ must be disjoint. If the type of player 3 is $t^{\prime \prime }$, he is supposed to play $L$ regardless of the message $m_3$ he receives. Deviation to $R$ should not increase the conditional expectation of the player’s payoff, which means that

$$\begin{aligned} \left( {\frac{1}{2}-3} \right) P^{1}\left( {\left\{ {m_3 } \right\} } \right) +\frac{1}{2}P^{2}\left( {\left\{ {m_3 } \right\} } \right) +\frac{1}{2}P^{3}\left( {\left\{ {m_3 } \right\} } \right) +\frac{1}{2}P^{4}\left( {\left\{ {m_3 } \right\} } \right) \le 0. \end{aligned}$$

Summing over all $m_3 \in \text{ supp }\left( {P^{3}} \right) \cup \text{ supp }\left( {P^{4}} \right)$ gives

$$\begin{aligned} -\frac{5}{2}P^{1}\left( {\text{ supp }(P^{3})\cup \text{ supp }(P^{4})} \right) +0+\frac{1}{2}+\frac{1}{2}\le 0. \end{aligned}$$

It follows that if the type profile is $(t_1^{{\prime }} ,t^{\prime },t^{\prime })$, the probability that player 3 plays $R$ is at least 2/5. The same is true for player 2. Therefore, by the assumed independence of the messages (property $I$), the probability that both 2 and 3 play $R$ when the type profile is $(t_1^{{\prime }} ,t^{{{\prime }}},t^{{{\prime }}})$ is at least 4/25. Since $4/25\times 6+12/25\times (-1)>0$, this means that player 1 has an incentive to misreport his type as $t_1^{{\prime }}$ when it is really $t_1^{\prime \prime }$, which contradicts the equilibrium assumption.

Proposition 33

For MEDs, $\left( {\{S\}\wedge \{D\}} \right) \nRightarrow \{S,I\}$.

Proof

This is demonstrated by Example 8. $\square$

Propositions 23, 24, 25 and 26 identify six attributes of communication equilibrium distributions that are defined by subsets of the fundamental properties of mechanisms. Figure 3 shows these attributes, marked $\text{ II }_{\mathrm{a}}$, $\text{ III }_{\mathrm{a}}$, IV, V, VI and VII, as well as eleven additional ones. The implication relations that are specified by the Hasse diagram among these 17 attributes all hold trivially, since they follow immediately from relations between properties of mechanisms. For two of the implications, it is not known whether the reverse implication also holds. The uncertainty is indicated in Fig. 3 by a question mark. If the reverse implication does hold, then the marked box and the one below it should be coalesced, as they represent equivalent attributes. The following arguments show that none of the other attributes in Fig. 3 are equivalent, and more generally, that the diagram shows all the implication relations between the attributes.

If attributes that involve conjunctions were removed from Figs. 2 and 3, the two Hasse diagrams would become identical. In Appendix B it is shown that, among the remaining twelve attributes of CEDs, the implications shown in the diagram are the only ones holding. Essentially the same arguments prove the same for MEDs, except that Propositions 27, 28, 29 and 30 replace 16, 17, 18 and 19, respectively. For each of the attributes in Fig. 3 that does involve conjunction, it follows from Propositions 31 and 33 that the only other attributes that imply or are implied by it are those indicated as such by the Hasse diagram. This proves that the diagram is complete in terms of implication relations.

Like the Hasse diagram for CEDs (Fig. 2), that for MEDs (Fig. 3) is complete also in that it is closed under conjunctions. The proof is similar to that of Lemma 6 except that is uses Proposition 26 instead of 15. Since it follows from Lemma 5 that closedness under conjunctions also holds for CSDs (Fig. 1), this gives Theorem 2. Thus, for any given collection of attributes in one of the three Hasse diagrams, there is an attribute in the same diagram that that is equivalent to their conjunction. That attribute can easily be identified. Since it clearly implies each of the attributes and it is implied by every other attribute with the same property, it must be the meet, or greatest lower bound, of the given attributes (see Sect. 8). For example, Fig. 3 shows that the conjunction of $O$-implentability and $I$-implementabiltiy is equivalent to $S,O,I$-implentability. In other words, the only MEDs with both attributes are the mixed-equilibrium distributions.

It is now possible to give the proof of the result presented at the beginning of this appendix, namely, that the implication relation between attributes of MEDs is in a sense stronger than that for CEDs.

Proof of Proposition 22

As indicated, if attributes that involve conjunctions were removed from Figs. 2 and 3, they would become identical. This means that (39) as well as the reverse implication hold for all $\mathcal P$ and $\mathcal{Q}$ that belong to the collection of 27 subsets shown in these diagrams, which clearly implies the same for all subsets of the six fundamental properties of mechanisms.

To prove the more general implication in which $\mathcal P$ is replaced by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }$, it suffices to consider the case in which $\mathcal P ^{{\prime }}$- and $\mathcal P ^{\prime \prime }$-implementability (of CEDs, or equivalently MEDs) are incomparable; the case in which they are comparable reduces to the version just analyzed. A straightforward examination shows that, with a single possible exception, for all $\mathcal{Q}\subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$, if the meet of $\mathcal P ^{{\prime }}$-implementability and $\mathcal P ^{\prime \prime }$-implementability implies $\mathcal{Q}$-implementability in Fig. 3, this is so also in Fig. 2 (but not conversely). The possible exception is the case where $\mathcal P ^{{\prime }}=\{{\tilde{S}}\}$, $\mathcal P ^{\prime \prime }=\{I\}$ and $\mathcal{Q}=\{S\}$, which is however covered by Proposition 32. This proves the generalization of (39) in which $\mathcal P$ is replaced by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }$.

To prove the further generalization in which the list $\mathcal P ^{{\prime }},\mathcal P ^{\prime \prime },\ldots$ has three or more elements it again suffices to consider the case in which no two elements describe comparable attributes. However, it is not difficult to check that this means that, in both diagrams, $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$ is (equivalent to) attribute VII. Therefore, the version of (39) in which $\mathcal P$ is replaced by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$ holds trivially. $\square$

Appendix D: Correlated strategy, correlated equilibrium and communication equilibrium payoffs

Correlated strategy payoffs (CSPs), correlated equilibrium payoffs (CEPs) and communication equilibrium payoffs (MEPs) in Bayesian games can be classified in a manner similar to the classification of CSDs (Fig. 1), CEDs (Fig. 2) and MEDs (Fig. 3). Each subset $\mathcal P$ of the fundamental properties of mechanisms defines an attribute of CSPs, CEPs and MEPs, namely, $\mathcal P$-implementability. A payoff vector $v=\left( {v_1 ,v_2 ,\ldots ,v_n } \right) \in \mathbb{R }^{n}$ in a specified $n$-player Bayesian game is $\mathcal P$-implementable if it coincides with the players’ (expected) payoffs in some correlated strategy, correlated equilibrium or communication equilibrium with a mechanism that has all the properties in $\mathcal P$ (equivalently, if $v$ is obtained in some CSD, CED or MED, respectively, that is implementable by such a mechanism). For two subsets $\mathcal P ,\mathcal{Q}\subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$, $\mathcal P$-implementability of CSPs implies $\mathcal{Q}$-implementability if in every Bayesian game every CSP that is implementable by some mechanism with the properties in $\mathcal P$ is also implementable by a mechanism with the properties in $\mathcal{Q}$. This relation is written as $\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CSP}} \mathcal{Q}$. For CEPs and MEPs, the relations $\mathop {\Rightarrow }\limits _{\mathrm{CEP}}$ and $\mathop {\Rightarrow }\limits _{\mathrm{MEP}}$ are defined similarly.

The main result concerning implementability of payoff vectors is Theorem 3 in Sect. 8.2, which asserts that the effects of the properties of the implementing mechanisms on the payoffs mirror the effects on the joint distributions of types and actions. Thus, there is similarity in this respect between these two possible notions of “outcome” in a Bayesian game.

Proof of Theorem 3

The proofs for correlated equilibria and for communication equilibria are nearly identical. Only the former is presented below; the latter can be obtained from it essentially by replacing ‘correlated’ with ‘communication’ throughout. The proof for correlated strategies can also be easily obtained from the proof below by simplifying it in the obvious manner.

It has to be shown that, for every $\mathcal P ,\mathcal{Q}\subseteq \{S,\tilde{S},O,\tilde{O},D,I\}$,

$$\begin{aligned} \mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CEP}} \mathcal{Q} \text{ if } \text{ and } \text{ only } \text{ if } \mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}. \end{aligned}$$

(44)

One direction of (44) (“if”) is easy. $\mathcal P \mathop {\nRightarrow }\limits _{\mathrm{CEP}} \mathcal{Q}$ and $\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}$ are contradictory, since the former means that, in some Bayesian game, there is a $\mathcal P$-implementable CED $\eta$ with a payoff vector that is different from that of every $\mathcal{Q}$-implementable CED in the same game, whereas the latter implies that $\eta$ itself is $\mathcal{Q}$-implementable.

To prove the nontrivial direction of (44) (“only if”), define the extension of a Bayesian game as the game obtained by the addition of dummy players—one for each element of $T\times A$. A dummy player has only one possible type and one action, which are therefore insignificant in that they cannot affect the payoff of any player. In the following, the types and actions of the dummy players are ignored, and the collections of type profiles and action profiles in the extended game are thus identified with those in the original game, namely, $T$ and $A$, respectively. The significance of the dummy players lies in their payoff functions. The payoff function $u_{t,a} :T\times A\rightarrow \mathbb{R }$ of the dummy player representing the types-actions pair $\left( {t,a} \right) \in T\times A$ is defined as the indicator function $1_{\left\{ {\left( {t,a} \right) } \right\} }$. It returns $1$ if the argument is equal to $(t,a)$ and 0 otherwise. Thus, the dummy players’ payoffs indicate the types and actions of the original, real players. In particular, for every correlated equilibrium distribution $\eta$ and every element $\left( {t,a} \right)$ of $T\times A$, the expected payoff of the corresponding dummy player is equal to $\eta (\left\{ {\left( {t,a} \right) } \right\} )$. It follows that two CEDs in the extended game, $\eta$ and $\tilde{\eta }$, give the same CEP if and only if they are equal, $\eta =\tilde{\eta }$.

Every mechanism in the original game can be extended in a natural way to a mechanism in the extended game by sending arbitrary constant messages to the dummy players. The original and the extended mechanisms have the exact same fundamental properties, and in the following, they are identified. Using this identification, every correlated strategy in the original game can be extended in a natural way to a correlated strategy with the same mechanism in the extended game by assigning to each of the dummy players his single possible strategy. Observe that:

• 1. the original correlated strategy has the same distribution as the extended one (recall the above comment regarding the identification of profiles in the original and the extended games), and

• 2. one of them is a correlated equilibrium if and only if this is so for the other.

Moreover, every CED in the extended game can be obtained in the above manner from some CED in the original game. The former may be the distribution of a correlated strategy with a mechanism that sends variable messages to some dummy players. However, these messages are inconsequential (since a dummy player has only one possible action) and hence can be replaced by constant messages. Such replacement preserves each of the fundamental properties.

Suppose now that $\mathcal P \mathop {\Rightarrow }\limits _{\mathrm{CEP}} \mathcal{Q}$. Then, for every $\mathcal P$-implementable CED $\eta$ in the extended game there is a $\mathcal{Q}$-implementable CED $\tilde{\eta }$ in the same game with an identical payoff vector. As indicated, necessarily $\tilde{\eta }=\eta$, so that $\eta$ is also $\mathcal{Q}$-implementable. It follows, by Observations 1 and 2 above, that every $\mathcal P$-implementable CED in the original game is also $\mathcal{Q}$-implementable. This proves that $\mathcal P \mathop { \Rightarrow }\limits _{\mathrm{CED}} \mathcal{Q}$. $\square$

Note that the proof of the “only if” direction of (44) applies virtually unchanged also to the more general version in which $\mathcal P$ is replaced by $\mathcal P ^{{\prime }}\wedge \mathcal P ^{\prime \prime }\wedge \cdots$, for any list $\mathcal P ^{{\prime }},\mathcal P ^{\prime \prime },\ldots$ of subsets of $\{S,\tilde{S},O,\tilde{O},V,I\}$.