Bounded rationality and correlated equilibria

Abstract

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality $\overline{p}$ that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.

Appendices

APPENDIX

1.1 A Proof of Lemma 1

• (i) The left implication is obvious, so let’s focus on the right one. By definition, $CB^{p}\left( R\right) \subseteq \bigcap _{i\in I}B_{i}^{p}\left( R\right)$, and therefore, for any $i\in I$, $CB^{p}\left( R\right) \subseteq B_{i}^{p}\left( R_{i}\right)$. Then, since $p>0$, and $\omega \in B_{i}^{p}\left( R_{i}\right)$, we have that $R_{i}\cap \Pi _{i}\left( \omega \right) \ne \emptyset$, and therefore, that $\Pi _{i}\left( \omega \right) \subseteq R_{i}$ and, in particular, $\omega \in R_{i}$. Thus, $B_{i}^{p}\left( R_{i}\right) \subseteq R_{i}$.

• (ii) It is immediate that $CB^{1}\left( R\right) =\Omega$ if and only of $\bigcap _{i\in I}B_{i}^{1}\left( R\right) =\Omega$, so it suffices to prove that $\bigcap _{i\in I}B_{i}^{1}\left( R\right) =\Omega$ if and only if $\bigcap _{i\in I}B_{i}^{1}\left( R_{-i}\right) =\Omega$. The right implication is immediate. For the proof of the left one, from part (i) of the lemma, it is enough to check that if $\bigcap _{i\in I}B_{i}^{1}\left( R_{-i}\right) =\Omega$ then $R=\Omega$. But this is immediate: take $i,j\in I$, $i\ne j$, then $B_{j}^{p_{j}}\left( R_{-j}\right) \subseteq B_{j}^{p_{j}}\left( R_{i}\right)$, and therefore, $B_{j}^{p_{j}}\left( R_{i}\right) =\Omega$. Hence $\mu \left( R_{i}\right) = \sum _{\omega \in \Omega } \mu \left( R_{i}\cap \Pi _{j}\left( \omega \right) \right) = \sum _{\omega \in \Omega } \mu \left( \Pi _{j}\left( \omega \right) \right) =1$. Since $\mu$ has full support on $\Omega$, the latter implies that $R_{i}=\Omega$. As the proof applies for any $i\in I$, we obtain that $R=\Omega$.

• (iii) If $\bigcap _{i\in I}B_{i}^{p}\left( R_{-i}\right) =\Omega$, then $R=\bigcap _{i\in I}\left( B_{i}^{p}\left( R_{-i}\right) \cap R_{i} \right) =\bigcap _{i\in I}B_{i}^{p}\left( R\right)$. Thus, R is p-evident belief and therefore, $R\subseteq CB^{p}(R)$. Now, since $\bigcap _{i\in I}B_{i}^{p}\left( R_{-i}\right) =\Omega$, we have both that $\mu \left[ R_{-i}\right] \ge p$ and $\mu \left[ R_{-i}\cap R_{i}\right] \ge p\cdot \mu \left[ R_{i}\right]$. The former implies that that $\mu [R_{j}]\ge p$ for any $j\ne i$; and thus, the same argument and some reindexing imply that $\mu [R_{i}]\ge p$. The fact that for any $j\ne i$, $\mu \left[ R\right] =\mu \left[ R_{-i}|R_{i}\right] \cdot \mu \left[ R_{i}\right] \ge p\cdot \mu \left[ R_{i}\right] \ge p^{2}$ completes the proof.

B Proofs of the characterization results

In this section we first prove a technical lemma. Then we prove Theorem 4, and then Theorem 1 as a special case of Theorem 4. The technical lemma is the following:

Lemma 2

Let $\left( G,S\right)$ be a Bayesian game, p a probability, $n\in \{2,3\}$, and $\pi ^{*}\in \left( A_{(k)}, p\right)$-$BCE\left( nG,S\right)$ for some $k\le n$. For game nG, let belief system $B=\left\langle \Omega ,\left( \Pi _{i}\right) _{i\in I},\mu ,\kappa ,\left( \alpha _{i}\right) _{i\in I},\left( \tau _{i}\right) _{i\in I}\right\rangle$ be given by: (i) $\Omega =\left\{ \left( t,a,\nu ,\theta \right) \in T\times A'\times \Theta \right. \left. \left| \pi ^{*}\left[ \left( a,\nu ,\theta \right) \right] >0\right. \right\}$, and for any $\left( t,a,\nu ,\theta \right) \in \Omega$, (ii) $\mu \left[ (t,a,\nu ,\theta )\right] =\pi ^{*}\left[ \left( t,a,\nu ,\theta \right) \right]$, (iii) $\kappa \left( t,a,\nu ,\theta \right) =\theta$, and for any $i\in I$, (iv) we have cells $\Pi _{i}\left( t,a,\nu ,\theta \right) =T_{-i}\times A'_{-i}\times \Theta \times \left\{ (t_{i},a_{i},\nu _{i})\right\}$, (v) $\alpha _{i}\left( t,a,\nu ,\theta \right) =a_{i}$, and (vi) $\tau _{i}\left( t,a,\nu ,\theta \right) =t_{i}$. Then, B is a belief system for (G, S) satisfying consistency, MB$^{p}$R and the common prior assumption, and induces $\pi ^{*}$ in $T\times A'\times \Theta$.

Proof

It is immediate that $\alpha _{i}$ and $\tau _{i}$ are measurable w.r.t. $\Pi _{i}$ for any $i\in I$. Take $\left( t,\theta \right) \in T\times \Theta$; then, we have $\mu \left[ \tau =t,\kappa =\theta \right] =\pi ^{*}\left[ \left( t,\theta \right) \right] =\psi \left[ \theta \right] \cdot \sigma \left[ t|\theta \right]$, and therefore, B satisfies consistency. Now, note first the fact that for any $\omega =(t,a,\nu ,\theta )\in \Omega$ and any $a'_{i}\in A_{i}$ it holds that,

$$\begin{aligned} \mathbb {E}(\omega )\left[ u_{i}\left( \alpha _{-i},a'_{i},\kappa \right) \right]&=\sum _{\left( t'_{-i},\left( a'_{-i},\nu '_{-i}\right) ,\theta '\right) }\pi ^{*}\left[ \left( t'_{-i},t_{i},(a'_{-i},\nu '_{-i}),(a_{i},\nu _{i}),\theta '\right) \right] \\&\quad \times u_{i}\left( ((a'_{-i},\nu '_{-i});(a_{i},\nu _{i})),\theta '\right) , \end{aligned}$$

together with the incentive constraints, implies that for any $i\in I$ we have that $T\times A_{-i}'\times (A_{i}\times \{k\})\times \Theta \subseteq R_{i}$, and therefore,²⁴ that $\mu \left[ \prod _{j\ne i}(A_{j}\times \{k\})\cap \Pi _{i}(\omega )\right] \le \mu \left[ R_{-i}\cap \Pi _{i}(\omega )\right]$ for any $i\in I$ and any $\omega \in \Omega$. Then, take $i\in I$ and $\omega =(t,a,\nu ,\theta )\in \Omega$ and note that:

$$\begin{aligned} \mu [\prod _{j\ne i}(A_{j}\times \{k\})\cap \Pi _{i}(\omega )]&=\pi ^{*}[\prod _{j\ne i}(A_{j}\times \{k\})\times \left\{ (t_{i},a_{i},\nu _{i})\right\} ],\\ \mu \left[ \Pi _{i}(\omega )\right]&=\pi ^{*}[A'_{-i}\times \left\{ (t_{i},a_{i},\nu _{i})\right\} ]. \end{aligned}$$

In consequence, due to the p-belief constraints, we have $\mu \left[ R_{-i}\cap \Pi _{i}(\omega )\right] \ge p\cdot \mu \left[ \Pi _{i}(\omega )\right]$, and therefore, that B satisfies MB$^{p}$R. $\square$

1.1 B.1 Proof of Theorem 4

For the right inclusion, pick $\pi \in p$-$RBO\left( G,S\right)$ and B, a belief model that induces $\pi$. Take distribution $\pi ^{*}\in \Delta \left( T\times A'\times \Theta \right)$ given by,

$$\begin{aligned} \pi ^{*}\left[ \left( t,a,\nu ,\theta \right) \right] =\mu \left[ \left[ \tau =t,\alpha =a,\kappa =\theta \right] \cap \bigcap _{i:v_{i}=0}R_{i}\cap \bigcap _{i:v_{i}=1}\lnot R_{i}\right] \end{aligned}$$

for any $(t,a,\nu ,\theta )\in T\times A'\times \Theta$. Then, $\pi ^{*}\in \Delta \left( T\times A'\times \Theta \right)$. The consistency constraint is satisfied, since,

$$\begin{aligned} \pi ^{*}\left[ (t,\theta )\right]&=\mu \left[ \left[ \tau =t,\kappa =\theta \right] \cap \bigcup _{\nu \in N^{I}}\left( \bigcap _{i:v_{i}=0}R_{i}\cap \bigcap _{i:v_{i}=1}\lnot R_{i}\right) \right] \\&=\mu \left[ \tau =t,\kappa =\theta \right] =\psi \left[ \theta \right] \cdot \sigma [t|\theta ] \end{aligned}$$

for any $\left( t,\theta \right) \in T\times \Theta$. Now, note that for any $i\in I$, any $a_{i},a'_{i}\in A_{i}$, any $t_{i}\in T_{i}$ and any $\nu _{i}\in N$ we have that,

$$\begin{aligned}&\sum _{\left( t_{-i},a_{-i},\nu _{-i},\theta \right) }\pi ^{*}[(t_{-i},t_{i},(a_{-i},\nu _{-i}),(a_{i},1),\theta )]\cdot u_{i}\left( (a_{-i},\nu _{-i}),(a_{i},1),\theta \right) \\&\qquad -\sum _{\left( t_{-i},a_{-i},\nu _{-i},\theta \right) }\pi ^{*}\left[ \left( t_{-i},t_{i},(a_{-i},\nu _{-i}),(a_{i},1),\theta \right) \right] \cdot u_{i}\left( (a_{-i},\nu _{-i}),(a'_{i},\nu _{i}),\theta \right) \\&\quad =\sum _{\omega \in R_{i}\cap \left[ \tau _{i}=t_{i},\alpha _{i}=a_{i}\right] }\left( \mathbb {E}(\omega )\left[ u_{i}\left( \left( \alpha _{-i},\alpha _{i}\left( \omega \right) \right) ,\kappa \right) \right] -\mathbb {E}(\omega )\left[ u_{i}\left( \left( \alpha _{-i},a'_{i}\right) ,\kappa \right) \right] \right) \ge 0. \end{aligned}$$

In addition, note that for any $(t_{i},a_{i},\nu _{i})\in T_{i}\times A'_{i}$,

$$\begin{aligned} \begin{aligned} \pi \left[ X_{-i}\times \left\{ \left( t_{i},a_{i},\nu _{i}\right) \right\} \right]&=\mu \left[ R_{-i}\cap \left[ \tau _{i}=t_{i},\alpha _{i}=a_{i}\right] \cap \left[ 1_{R_{i}}=2-\nu _{i}\right] \right] \\&=\sum _{\omega \in [\tau _{i}=t_{i},\alpha _{i}=a_{i}]\cap \left[ 1_{R_{i}}=2-\nu _{i}\right] }\mu \left[ R_{-i}\cap \Pi _{i}\left( \omega \right) \right] \\&\ge \sum _{\omega \in [\tau _{i}=t_{i},\alpha _{i}=a_{i}]\cap \left[ 1_{R_{i}}=2-\nu _{i}\right] }p\cdot \mu \left[ \Pi _{i}\left( \omega \right) \right] \\&=p\cdot \pi ^{*}\left[ A'_{-i}\times \left\{ \left( t_{i},a_{i},\nu _{i}\right) \right\} \right] . \end{aligned} \end{aligned}$$

Thus, both the incentive constraints and the p-belief constraints are satisfied. For the left inclusion, just apply Lemma 2 to $A_{i}'=A_{i}\times \{1,2\}$ and $k=1$.

1.2 B.2 Proof of Theorem 1

This theorem can be seen as a corollary of Theorem 4. To see this, note that if G is a game with complete information, we can define Bayesian game $\left( G',S\right)$, where we have (i) $G'=\left\langle I,\Theta ,\psi ,\left( A_{i}\right) _{i\in I},\left( u'_{i}\right) _{i\in I}\right\rangle$, with $\Theta =\left\{ \theta \right\}$, $\psi =1_{\left\{ \theta \right\} }$ and $u'_{i}(a,\theta )=u_{i}(a)$ for any $a\in A$ and $i\in I$, and (ii) $S=\left\langle \left( T_{i}\right) _{i\in I},\sigma \right\rangle$ with $T_{i}=\left\{ t_{i}\right\}$ for any $i\in I$ and $\sigma (t|\theta )=1$. It is immediate that, for any $X=\prod _{i\in I}X_{i}\subseteq A$ and any $p\in [0,1]$, we have that $\left( X,p\right) - CE\left( G\right) =\left( X,p\right) - BCE\left( G',S\right)$. But note also that if we take some list $B'=\left\langle \Omega ,\left( \Pi _{i}\right) _{i\in I},\left( \alpha _{i}\right) _{i\in I},\left( \tau _{i}\right) _{i\in I},\mu \right\rangle$, which is a candidate to be a belief system for $G'$, for any $i\in I$, forcefully $\tau _{i}=1_{\left\{ t_{i}\right\} }$; and therefore, $B'$ is a belief system for $G'$ if and only if $B=\left\langle \Omega ,\left( \Pi _{i}\right) _{i\in I},\left( \alpha _{i}\right) _{i\in I},\mu \right\rangle$ is a belief system for G. Thus, it is immediate that, for any $p\in [0,1]$, we have $p- RO\left( G\right) = p- RBO\left( G',S\right)$. So, let G be a game, and $p\in [0,1]$. Then, we just checked above that both $p- RO\left( G\right) =p- RBO\left( G',S\right)$ and $\left( A_{(1)},p\right) - CE\left( 2G\right) =\left( A_{(1)},p\right) - BCE\left( 2G',S\right)$, hold, so from Theorem 4 we conclude that $p- RO\left( G\right) = \left( A_{(1)},p\right) - CE\left( 2G\right)$.

1.3 B.3 Proof of Theorem 3

We prove the first statement; the next one concerning the p-rational expectations of rational types then follows directly. We suppose we are taking some player i’s expectation. For right inclusion, take p-rational belief system $B=\left\langle \Omega ,\left( \Pi _{i}\right) _{i\in I},\left( \alpha _{i}\right) _{i\in I},\mu \right\rangle$ and $\omega \in \Omega$. We define:

$$\begin{aligned} \pi _{i,\omega }^{*}\left[ (a,\nu )\right] =\mu \left[ \left[ \alpha =a\right] \cap W_{i}\cap \bigcap _{j\ne i,\nu _{j}=1}R_{j}\cap \bigcap _{j\ne i,v_{j}\ne 1}\lnot R_{j}\right] \end{aligned}$$

for any $(a,\nu )\in A'$, where $W_{i}=R_{i}{\setminus }\Pi _{i}\left( \omega \right)$ if $\nu _{i}=0$, $W_{i}=\lnot \left( R_{i}{\setminus }\Pi _{i}\left( \omega \right) \right)$ if $\nu _{i}=1$, and $W_{i}=\Pi _{i}\left( \omega \right)$ if $\nu _{i}=2$. It is immediate that $\pi _{i,\omega }^{*}\in \Delta \left( A'\right)$. By an argument similar to the one in the first part of the proof of Theorem 4, reduced to the degenerate case where $|\Theta |=1$, we can conclude that $\pi _{i,\omega }$ is a $\left( A_{(1)},p\right)$-correlated equilibrium of G; moreover, it is immediate that player i’s expectation conditional on playing $\left( \alpha _{i}\left( \omega \right) ,3\right)$ induced by $\pi _{i,\omega }$ is exactly $\mathbb {E}(\omega )\left[ u_{i}\left( \alpha _{-i},\alpha _{i}\left( \omega \right) \right) \right]$. For the left inclusion, take Lemma 2 for the case $A'_{i}=A_{i}\times \{1,2,3\}$, $k=1$, and $|\Theta |=1$.

C Proof of Theorem 2

Nonemptiness follows from the fact that correlated equilibria always exist for any finite game G and constitute p-rational outcomes for any $p\in [0,1]$. Given that the set of p-rational outcomes is a projection of the (X, p)-correlated equilibria of 2G, with $X=A_{(k)}$ a copy of the action space of the original game G, the remaining properties follow once they have been shown for the (X, p)-correlated equilibria of 2G. This is what we do next. For the given game G, define the (X, p)-correlated equilibrium correspondence, where $X=A _{(k)}$ with $k \in \{0, 1\}$, is fixed:

Clearly $\rho$ is convex- and compact-valued; it remains to be shown that it is also continuous. We do this by showing that it is upper- and lower-hemicontinuous (respectively, uhc and lhc) as a correspondence of p.

Since 2A is finite, $\Delta (2A)$ is compact, and hence upper-hemicontinuity is equivalent to showing that $\rho$ has a closed graph. But this is immediate from inspection of the inequalities defining the sets $(X, p)- CE\left( 2G\right)$. In particular, all the inequalities are all weak inequalities, linear in p. Moreover, the domain [0, 1] is compact.

Denote by $\Gamma _{\rho } \subset [0,1]\times \Delta (2A)$ the graph of the correspondence $\rho$. Fix $(p, \hat{\pi }) \in \Gamma _{\rho }$ and let $\left( p^n\right) _n \subset [0,1]$ be a sequence converging to p. We need to show that there exists a sequence $\left( (\hat{\pi })^n\right) _n$ converging to $\hat{\pi }$ such that $(\hat{\pi })^n \in \rho (p^n)$ for sufficiently large n. Take the point $(p, \hat{\pi })$. Clearly this satisfies all inequalities defining $\rho (p)$, in particular also the p-rationality constraints. Consider the following sequence $\left( p^n, \hat{\pi }\right) _n \subset [0,1] \times \Delta (2A)$. If for sufficiently large n the elements are contained in $\Gamma _{\rho }$ we are done. So consider the case where they are not. Consider the family of projections $\Pi _{\rho }: [0,1] \times \Delta (2A) \longrightarrow [0,1] \times \Delta (2A)$ that map, for fixed $\bar{p} \in [0,1]$, any element $(\bar{p}, \bar{\pi }) \in [0,1] \times \Delta (2A)$ to the point in the set $\{ \bar{p} \} \times \rho ( \bar{p} )$ that is closest to $(\bar{p}, \bar{\pi })$. Since the sets $\rho ( \cdot )$ are always nonempty, convex, compact polyhedra, we have that $\Pi _{\rho } \left( p^n, \hat{\pi } \right)$ is uniquely defined and moreover, $\Pi _{\rho } \left( p^n, \hat{\pi } \right) \in \Gamma _{\rho }$ for any point in the sequence $\left( p^n, \hat{\pi } \right) _n$. It remains to be shown that the sequence $\left( \Pi _{\rho } \left( p^n, \hat{\pi } \right) \right) _n$ converges to the point $(p, \hat{\pi })$. Apart from the p-belief constraints all other constraints defining $\rho (p)$ are independent of p. Hence, if $(p, \hat{\pi })$ satisfies those constraints, then so must any other point in the sequence $\left( p^n, \hat{\pi }\right) _n$. Therefore the only constraints that can be violated by elements of the sequence $\left( p^n, \hat{\pi }\right) _n$ are the p-belief constraints. Consequently, any point in the sequence $\left( \Pi _{\rho } \left( p^n, \hat{\pi } \right) \right) _n$ lies on the boundary of the polyhedra defined by the p-belief constraints. As mentioned, these constraints are linear in p, and since they also define nonempty, convex, compact polyhedra, the sequence $\left( \Pi _{\rho } \left( p^n, \hat{\pi } \right) \right) _n$ indeed converges to $(p, \hat{\pi })$. This shows the continuity of $\rho$ and hence also of p-RO(G) in p.

Finally, the claims that, for $p=0$, we have 0-$RO(G)=\Delta (A)$, and for $p=1$, we have 1-$RO(G)=CE(G)$, are immediate. To see that for any $p \in [0,1)$, we have dim[p-$RO(G)]=$ dim$[\Delta (A)]$, notice that the (X, p)-correlated equilibria with $X=A^1$ and $p<1$ entail distributions that put strictly positive weight on all strategies in $A^2$ as well as all convex combinations of such distributions. Projecting onto the original space $\Delta (A)$ implies distributions with strictly positive weights on all strategies in A as well as all possible convex combinations. This concludes the proof.

D Proof of Proposition 1

Fix G and let $A^n=\Pi _{i \in I} A_i^n$ denote the space of all pure strategy profiles that survive n rounds of iterated elimination of strictly dominated strategies in G, and similarly for the individual sets $A_i^n$. Let $G^n$ denote the subgame of G with strategies restricted to $A^n$. Because G is finite, the limit sets $A_i^{\infty }, A^{\infty }$, and $G^{\infty }$ are well defined (and are obtained after finitely many iterations). Also, for any subset $Y \subset A$, let $Y^c=A{\setminus } Y$ denote the complement of Y in A. For any given $p\in \left[ 0,1\right]$, take $p'\ge p$. We show that for p sufficiently close to 1, behavior is supported with high probability in $A^{\infty }$. Specifically, we construct a $\bar{p}<1$ such that for any $p \in [\bar{p}, 1]$, if $\pi \in p$-RO(G), then $\pi \left[ \left( A^{\infty }\right) ^c \right] \le 1-p$. Consider the game $G^0=G$ and pick some $p^1<1$. It immediately follows from p-rationality that for $p\in [p^1,1]$, if $\pi \in p$-RO(G), then we have $\pi \left[ \left( A^1\right) ^c \right] \le 1-p$. Suppose now that the above statement is true for $n-1$, namely there exists $p^{n-1}<1$ such that for $p \in [p^{n-1},1]$, if $\pi \in p$-RO(G), then we have $\pi \left[ \left( A^{n-1}\right) ^c \right] \le 1-p$. We show that the statement also holds for n. Fix game $G^{n-1}$. It follows from finiteness of G and continuity of the payoffs that there exists $p^n \in [p^{n-1}, 1)$ such a strategy in $A^{n-1}{\setminus } A^n$ that is strictly dominated in $G^{n-1}$ (by some strategy in $G^{n-1}$ and hence in G) is also strictly dominated in G (by the same strategy) given a p-rational belief system with $p \ge p^n$.²⁵ This implies that for any $p \in [p^n, 1]$ and any $\pi \in p$-RO(G), we also have $\pi \left[ \left( A^n\right) ^c\right] \le 1-p$. Finiteness of the game implies that the process ends after finitely many steps implying that indeed there exists $p^{\infty } <1$ such that for $p \in [p^{\infty }, 1]$ and any $\pi \in p$-RO(G), we have $\pi \left[ \left( A^{\infty }\right) ^c \right] \le 1-p$. Taking $\bar{p}=p^{\infty }$ proves the claim.

E Proof of Proposition 2

To see the if part, take $\pi \in \Delta (A)$ a p-rational outcome of G. By Theorem 1, there exists an $(A_{(1)}, p)$-correlated equilibrium $\hat{\pi } \in \Delta (A')$ of the doubled game 2G. Define $P \equiv \hat{\pi }$ and set

$$\begin{aligned} \alpha _{i} (t_{i})[a_{i}] = \left\{ \begin{array}{ll} 1 &{}\quad \text{ if } t_{i} = a_{i} \\ 0 &{}\quad \text{ else } , \end{array} \right. \end{aligned}$$

for all $t_{i} \in T_{i}$. Then, by definition of the type spaces $T_{i}=T_{i}^{s} \cup T_{i}^{c}$, we have:

• $P[ T^{s}_{-i} \, | \, t_{i}] \ge p$, for any $t_{i} \in T_{i}$;

• $\alpha _{i} (t_{i})[a_{i}]= 1$ if $t_{i} \in T^{c}_{i}$ and $t_{i}= a_{i}$

• $a_{i} \in$ argmax$_{a_{i}' \in A_{i}} \sum _{t_{-i} \in T_{-i}}P[t_{-i} \, | \, t_{i}]\cdot \sum _{a_{-i} \in A_{-i}}\alpha _{-i} (t_{-i})[a_{-i}]\cdot u_{i}(a_{-i};a_{i}')$, where $\alpha _{-i}(t_{-i})[a_{-i}]=\prod _{j\ne i}\alpha _{j}(t_{j})[a_{j}]$.

Let $\mu [a]=\sum _{t \in T}P(t)\cdot \alpha (t)[a]$, where $\alpha (t)[a]=\prod _{i\in I}\alpha _{i}(t_{i})[a_{i}]$; then, again by construction, and using Theorem 1, we have $\mu = \pi$.

To see the only if part, take $\mu \in \Delta (A)$ an EAD of (G, P) where P satisfies $P[ T^{s}_{-i} \, | \, t_{i}] \ge p$, for any $t_{i} \in T_{i}$, and for some equilibrium profile $\alpha$ satisfying conditions (i) and (ii). By definition, together with P, $\alpha$ can be seen as inducing a probability distribution on the outcomes of the doubled game $\hat{\mu } \in \Delta (2A)$, where types in $T_{i}^{s}$ and corresponding strategies chosen are associated to $A_{(2)}$ and types in $T_{i}^{c}$ are associated to strategies in $A_{(2)}$. As a consequence of (i) and (ii) and the definition of P, $\hat{\mu }$ will also satisfy the incentive constraints and the p-belief constraints for 2G. By Theorem 1, computing the EAD $\mu$ of $\hat{\mu }$ gives us, a p-rational outcome of G.

F Proof of result in Remark 2

Adapting the original definition by Aumann (1974), Aumann (1987), for any $X=\prod _{i\in I}X_{i}\subseteq A$, we say that the family $\left( \pi _{i}\right) _{i\in I}\subseteq \left( \Delta \left( A\right) \right) ^{I}$ is a $\left( X,p\right)$-subjective correlated equilibrium of G, if, for any $i\in I$ the following are satisfied:

• Incentive constraints For any $a_{i}\in X_{i}$, $a_{i}\in \text {argmax}_{a'_{i}\in A_{i}}\sum _{a_{-i}\in A_{-i}}\pi _{i}[\left( a_{-i};a_{i}\right) ]\cdot u_{i}\left( a_{-i};a'_{i}\right)$.

• p-Belief constraints For any $a_{i}\in A_{i}$, $\pi _{i}\left[ X_{-i}\times \{a_{i}\}\right] \ge p\cdot \pi _{i}\left[ A_{-i}\times \{a_{i}\}\right]$.

We denote the set of $\left( X,p\right)$-subjective correlated equilibria of game G by $\left( X,p\right) - SCE\left( G\right)$. Given $n\in \{2,3\}$ and nG, we have a map $\mathbf marg _{A^{I}}:\left( \Delta \left( A'\right) \right) ^{I}\rightarrow \left( \Delta \left( A\right) \right) ^{I}$, where for any $\left( \hat{\pi _{i}}\right) _{i\in I}\in \left( \Delta \left( A'\right) \right) ^{I}$, we have that $\mathbf{marg }_{A^{I}}\left( \left( \hat{\pi _{i}}\right) _{i\in I}\right) =\left( \text {marg}_{A}\left( \hat{\pi _{i}}\right) \right) _{i\in I}$. Then, the proof of the identity,

$$\begin{aligned} p- SRO\left( G\right) =\mathbf{marg }_{A^{I}}\left( \left( X, p\right) - SCE\left( 2G\right) \right) , \end{aligned}$$

where $X=\prod _{i\in I}\left( A_{i}\times \left\{ 1\right\} \right)$, is the same as the one for Theorem 1 after slight modifications (just add sub-indices where needed). To see that the above marginals constitute the whole space, take $\left( a^{i}\right) _{i\in I}\subseteq A$, and for any $i\in I$, $\pi _{i}=1_{\left\{ a^{i}\right\} }$. Fix $k\in \left\{ 1,2\right\}$, and define, for any $i\in I$, $\hat{\pi }_{i}=1_{\{((a_{j}^{i},k)_{j\ne i};(a_{i}^{i},2-k))\}}$. It is immediate that $\mathbf{marg }_{A^{I}}\left( \left( \hat{\pi }_{i}\right) _{i\in I}\right) =\left( \pi _{i}\right) _{i\in I}$. Now, take $i\in I$, then the incentive constraints are trivially satisfied, since $\hat{\pi }_{i}[A_{i}\times \left\{ 1\right\} ]=0$. Moreover, the p-belief constraint is also satisfied, because, regardless of i’s action, the sums are on both sides 0 or 1. We conclude that, again for $X=\prod _{i\in I}\left( A_{i}\times \left\{ 1\right\} \right)$, we have $\left( 1_{\{a^{i}\}}\right) _{i\in I}\in \mathbf{marg }_{A^{I}}((X ,p)- SCE(2G))$ for any $(a^{i})_{i\in I}\subseteq A$; thus, by convexity, $\mathbf{marg }_{A^{I}}((X ,p)- SCE(2G))=((\Delta \left( A\right) )^{I}$.