Game theory with translucent players

Abstract

A traditional assumption in game theory is that players are opaque to one another—if a player changes strategies, then this change in strategies does not affect the choice of other players’ strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appropriate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i believe that everyone else would still be rational even if i were to switch strategies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax-dominated strategies, where a strategy $\sigma$ for player i is minimax dominated by $\sigma '$ if the worst-case payoff for i using $\sigma '$ is better than the best possible payoff using $\sigma$.

Proofs

In this appendix, we prove the results whose proofs were omitted in the main text. We repeat the statements of the results for the reader’s convenience.

Proposition 2.2

If $M = (\Omega , \mathbf {s},f, \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$ is a counterfactual structure with a probabilistic closest-state function f, then there exists a counterfactual structure $M' = (\Omega ',\mathbf {s}',f',\mathcal {PR}_1',\ldots ,\Pr _n')$ with a deterministic closest-state function $f'$ and a surjective mapping $G: \Omega ' \rightarrow \Omega$ such that for all formulas $\varphi \in \mathcal {L}$, we have

$$\begin{aligned} (M',\omega ') \models \varphi \text{ iff } (M, G(\omega ')) \models \varphi . \end{aligned}$$

Moreover, if M is strongly appropriate then so is $M'$.

Proof

We change f player by player, state by state, and strategy by strategy. More precisely, we construct a sequence $M^h = (\Omega ^h, \mathbf {s}^h,f^h,\mathcal {PR}_1^h, \ldots , \mathcal {PR}_n^h)$ of counterfactual structures and surjective mappings $G^h: \Omega ^h \rightarrow \Omega$ for $h = 0, \ldots , |\Omega |(|\Sigma _1| + \cdots + |\Sigma _n|)$ such that (a) for all formulas $\varphi \in \mathcal {L}^0$, we have $(M^h,\omega ') \models \varphi \text{ iff } (M, G^h(\omega ')) \models \varphi$, (b) $M^0 = M$, and (c) there exist h distinct tuples $(i,\omega ,\sigma _i)$ with $\omega \in \Omega$ and $\sigma _i \in \Sigma _i$ such that if $G^h(\omega ') = \omega$, then $f^h(i,\omega ',\sigma _i)$ is deterministic. Clearly, $f^{|\Omega |(|\Sigma _1| + \cdots + |\Sigma _n|)}$ must be deterministic, so $M^{|\Omega |(|\Sigma _1| + \cdots + |\Sigma _n|)}$ is the desired structure $M'$. We proceed as follows.

Set $M^0 = M$. Suppose that we have constructed $M^0, \ldots , M^h$ and $h < |\Omega |(|\Sigma _1| + \cdots + |\Sigma _n|)$. If $f^h$ is deterministic, we can take $M^{h+1} = M^h$. If not, there is some tuple $(i,\omega ',\sigma _i)$ such that $f^h(i,\omega ', \sigma _i)$ is not deterministic. Suppose that $G^h(\omega ') = \omega$ and the support of $f^h(i,\omega ',\sigma _i) = \{\omega ^1, \ldots , \omega ^N\}$. Note that $\mathbf {s}(\omega ^j) = \sigma _i$ for $j = 1, \ldots , N$ and $\mathbf {s}(\omega ') \ne \sigma _i$ (for otherwise $f(i,\omega ',\sigma _i)$ would be $\omega '$, and f would be deterministic.) Let $\Omega ^{h+1}$ be the result of adding $N-1$ new states, call them $\omega ^*_2, \ldots , \omega ^*_N$, to $\Omega ^h$ for each state $\omega ^* \in \Omega ^h$ such that $G^h(\omega ^*) = \omega$. (Implicitly, we identify $\omega ^*$ with $\omega ^*_1$ in the construction below.) Define $f^{h+1}(i,\omega ^*_{\ell },\sigma _i) = \omega ^{\ell }$ for $\ell = 1, \ldots , N$. (Since we are identifying $\omega ^{*}_1$ with $\omega ^{*}$, $f^{h+1}(i,\omega ^*,\sigma _i) = \omega ^1$.) Thus, $f^{h+1}(i,\omega ^*_{\ell },\sigma _i)$ is deterministic. Define $f^{h+1}(j,\omega ^*_{\ell },\sigma _j) = f^h(j,\omega ^*,\sigma _j)$ if $i \ne j$ or $\sigma _j \ne \sigma _i$. Finally, $f^{h+1}(j,\omega '',\sigma _j) = f^{h}(j,\omega '',\sigma _j)$ for all players j and strategies $\sigma _j$ if $G(\omega '') \ne \omega$. Thus, $f^{h+1}$ agrees with $f^h$ except on inputs of the form $(i,\omega ^*_{\ell },\sigma _i)$. Define $G^{h+1}(\omega '') = G^h(\omega '')$ if $G^h(\omega '') \ne \omega$ and $G^{h_1}(\omega ^*_{\ell '}) = G^h(\omega ^*) = \omega$ for all the states $\omega ^*_{\ell '} \in \Omega ^{h+1} - \Omega ^h$. Thus, $f^{h+1}$ is deterministic on all the tuples for which $f^h$ is deterministic, and, in addition, is deterministic on tuples $(i,\omega ^*,\sigma _i)$ such that $G^{h+1}(\omega ^*) = \omega$. Define $\mathbf {s}^{h+1}(\omega ') = \mathbf {s}(G(\omega '))$ for all $\omega ' \in \Omega ^{h+1}$. Finally, define $\mathcal {PR}^{h+1}_j(\omega '')(\omega ''') = \mathcal {PR}^h_j(\omega '')(\omega ''')$ for all players j if $G^h(\omega ''') \ne \omega$ and $\omega '' \in \Omega ^h$; $\mathcal {PR}^{h+1}(\omega '')(\omega '''_{\ell }) = (\mathcal {PR}_j^h(\omega '')(\omega '''))(f^h(i,\omega '',\sigma _i')(\omega '''))$; and $\mathcal {PR}^{h+1}(\omega ''_{\ell }) = \mathcal {PR}^{h+1}(\omega '')$ for $\ell = 2, \ldots , N$. Let $M^{h+1} = (\Omega ^{h+1},f^{h+1},\Pr _1^{h+1}, \ldots , \mathcal {PR}^{h+1}_n)$.

With these definitions, we leave it to the reader to check that $(M^h,\omega ') \models \varphi \text{ iff } (M, G^h(\omega ')) \models \varphi$, and that $M^h$ is strongly appropriate if M is. This completes the proof. $\square$

Proposition 3.3

For every $\varphi \in \mathcal {L}^0$, there exists a finite probability structure M appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \varphi$ iff there exists a finite counterfactual structure $M'$ (strongly) appropriate for $\Gamma$ that respects unilateral deviations, and a state $\omega '$ such that $(M',\omega ') \models \varphi$.

Proof

For the “if” direction, let $M' = (\Omega , f, \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$ be a finite counterfactual structure that is counterfactually appropriate for $\Gamma$ (but not necessarily strongly counterfactually appropriate) and respects unilateral deviations. Define $M= (\Omega , \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$. Clearly M is a finite probability structure appropriate for $\Gamma$; it follows by a straightforward induction on the length of $\varphi$ that $(M',\omega ) \models \varphi$ iff $(M,\omega ) \models \varphi$.

For the “only-if” direction, let $M = (\Omega , \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$ be a finite probability structure, and let $\omega \in \Omega$ be a state such that $(M,\omega ) \models \varphi$. We assume without loss of generality that for each strategy profile $\vec {\sigma }'$ there exists some state $\omega _{\vec {\sigma }'} \in \Omega$ such that $\mathbf {s}(\omega _{\vec {\sigma }'}) = \vec {\sigma }'$ and for each player i, $\mathcal {PR}_i(\omega _{\vec {\sigma }'})(\omega _{\vec {\sigma }'}) = 1$. (If such a state does not exist, we can always add it.)

We define a finite counterfactual structure $M' = (\Omega ', f', \mathcal {PR}'_1, \ldots , \mathcal {PR}'_n)$ as follows:

• $\Omega ' = \{ (\vec {\sigma }',\omega '): \vec {\sigma }' \in \Sigma (\Gamma ), \omega ' \in \Omega \}$;

• $\mathbf {s}'( \vec {\sigma }', \omega ') = \vec {\sigma }'$;

• $f( (\vec {\sigma }',\omega '), i, \sigma ''_i) = ((\sigma ''_i, \sigma '_{-i}), \omega ')$;

• $\mathcal {PR}'_i(\sigma ',\omega ')(\sigma '',\omega '') = \left\{ \begin{array}{lll} \mathcal {PR}_i(\omega ')(\omega '') &{}\text{ if } \sigma ' = \mathbf {s}(\omega '),\,\sigma '' = \mathbf {s}(\omega '') \\ 1 &{}\text{ if } \sigma ' \ne \mathbf {s}(\omega '),\,(\sigma '',\omega '') = (\sigma ', \omega _{\vec {\sigma }'})\\ 0 &{}\text{ otherwise. } \end{array}\right.$

It follows by construction that $M'$ is strongly appropriate for $\Gamma$ and respects unilateral deviations. Furthermore, it follows by an easy induction on the length of the formula $\varphi '$ that for every state $\omega \in \Omega$, $(M,\omega ) \models \varphi '$ iff $(M', (\mathbf {s}(\omega ), \omega )) \models \varphi '$. $\square$

Theorem 3.4

The following are equivalent:

• (a) $\vec {\sigma }$ is rationalizable in $\Gamma$;

• (b) there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and respects unilateral deviations, and a state $\omega$ such that $(M,\omega ) \models play (\vec {\sigma }) \wedge _{i=1}^n {\textit{WRAT}}_i^k$ for all $k \ge 0$;

• (c) there exists a finite counterfactual structure M that is strongly appropriate for $\Gamma$ and respects unilateral deviations and a state $\omega$ such that all $k \ge 0$;

• (d) there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and respects unilateral deviations and a state $\omega$ such that $(M,\omega ) \models play (\vec {\sigma }) \wedge _{i=1}^n {\textit{SRAT}}_i^k$ for all $k \ge 0$;

• (e) there exists a finite counterfactual structure M that is strongly appropriate for $\Gamma$ and respects unilateral deviations and a state $\omega$ such that $(M,\omega ) \models play (\vec {\sigma }) \wedge _{i =1}^n {\textit{SRAT}}_i^k$ for all $k \ge 0$.

Proof

The equivalence of (a), (b), and (c) is immediate from Theorem 3.2, Theorem 3.3, and Proposition 2.1. We now prove the equivalence of (b) and (d). Consider an counterfactual structure M that is appropriate for $\Gamma$ and respects unilateral deviations. The result follows immediately once we show that for all states $\omega$ and all $i \ge 0$, $(M,\omega ) \models {\textit{WRAT}}^k_i$ iff $(M,\omega ) \models {\textit{SRAT}}^k_i$. An easy induction on k shows that ${\textit{SRAT}}^k_i \Rightarrow {\textit{WRAT}}^k_i$ is valid in all counterfactual structures, not just ones that respect unilateral deviations. We prove the converse in structures that respect unilateral deviations by induction on k. The base case holds trivially. For the induction step, suppose that $(M,\omega ) \models {\textit{WRAT}}^k_i$; that is, $(M,\omega ) \models RAT _i \wedge B_i (\wedge _{j \ne i} {\textit{WRAT}}^{k-1}_{j})$. Thus, for all $\omega ' \in Supp (\mathcal {PR}_i(\omega ))$, we have that $(M,\omega ') \models \wedge _{j \ne i} {\textit{WRAT}}^{k-1}_{j}$. Thus, by the induction hypothesis, $(M,\omega ') \models \wedge _{j \ne i} {\textit{SRAT}}^{k-1}_{j}$. Since, as we have observed, the truth of a formula of the form $B^*_j\varphi$ at a state $\omega ''$ depends only on j’s beliefs at $\omega ''$ and the truth of $RAT _j$ depends only on j’s strategy and beliefs at $\omega ''$, it easily follows that, if j has the same beliefs and plays the same strategy at $\omega _1$ and $\omega _2$, then $(M,\omega _1) \models {\textit{SRAT}}^{k-1}_j$ iff $(M,\omega _2) \models {\textit{SRAT}}^{k-1}_j$. Since $(M,\omega ') \models \wedge _{j \ne i} {\textit{SRAT}}^{k-1}_{j}$ and M respect unilateral deviations, for all strategies $\sigma '_i$, it follows that $(M,f(\omega ', i, \sigma _i')) \models \wedge _{j \ne i} {\textit{SRAT}}^{k-1}_{j}$. Thus, $(M,\omega ) \models RAT _i \wedge B^*_i( \wedge _{j \ne i} {\textit{SRAT}}^{k-1}_{j})$, as desired. The argument that (c) is equivalent to (e) is identical; we just need to consider strongly appropriate counterfactual structures rather than just appropriate counterfactual structures. $\square$

Proposition 3.8

No (finite) set of strategy profiles has ambiguous terminating sets.

Proof

Let T be a set of strategy profiles of least cardinality that has ambiguous terminating deletion sequences $\vec {S} = (T, S_1, \ldots , S_m)$ and $\vec {S}'= (T, S'_1, \ldots , S'_{m'})$, where $S_m \ne S'_{m'}$. Let $T'$ be the set of strategies that are not minimax dominated with respect to T. Clearly $T' \ne \emptyset$ and, by definition, $T' \subseteq S_1 \cap S_1'$. Since $T'$, $S_1$, and $S_1'$ all have cardinality less than that of T, they must all have unique terminating sets; moreover, the terminating sets must be the same. For consider a terminating deletion sequence starting at $T'$. We can get a terminating deletion sequence starting at $S_1$ by just appending this sequence to $S_1$ (or taking this sequence itself, if $S_1 = T'$). We can similarly get a terminating deletion sequence starting at $S_1'$. Since all these terminating deletion sequences have the same final element, this must be the unique terminating set. But $(S_1, \ldots , S_m)$ and $(S_1', \ldots , S'_{m'})$ are terminating deletion sequences with $S_m \ne S'_{m'}$, a contradiction. $\square$

Theorem 3.13

The following are equivalent:

• (a) $\vec {\sigma } \in NSD ^{\infty }(\Gamma )$;

• (b) $\vec {\sigma }$ is minimax rationalizable in $\Gamma$;

• (c) there exists a finite counterfactual structure M that is strongly appropriate for $\Gamma$ and a state $\omega$ such that all $k \ge 0$;

• (d) for all players i, there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models play _i(\sigma _i) \wedge {\textit{SRAT}}_i^k$ for all $k \ge 0$.

Proof

We prove that (a) implies (b) implies (c) implies (d) implies (a). We first introduce some helpful notation. Recall that $\arg \max _x g(x) = \{y: \text{ for } \text{ all } z, g(z) \le g(y)\}$; $\arg \min _x g(x)$ is defined similarly. For us, x ranges over pure strategies or pure strategy profiles, and we will typically be interested in considering some element of the set, rather than the whole set. Which element we take does not matter. For definiteness, we define a tie-breaking rule by assuming that there is some order on the set of pure strategies and strategy profiles, and take the $\arg \max ^*_x g(x)$ to be the maximum element of $\arg \max _x g(x)$ with respect to this order; $\arg \min ^*_x g(x)$ is defined similarly.

(a) $\Rightarrow$ (b): Let K be an integer such that $NSD ^K(\Gamma ) = NSD ^{K+1}(\Gamma )$; such a K must exist since the game is finite. It also easily follows that for each player j, $NSD ^{K}_j(\Gamma )$ is non-empty: in iteration $k+1$, no $NSD ^k_j$-maximin strategy, that is, no strategy in $\arg \max _{\sigma '_j \in NSD ^{k}_j(\Gamma )} \min _{\tau _{-j} \in NSD ^k_j(\Gamma )} u_j(\sigma '_j, \tau _{-j})$, is deleted, since no maximin strategy is minimax dominated by a strategy in $NSD ^k_j(\Gamma )$ (recall that by Remark 3.10, it suffices to consider domination by strategies in $NSD ^{k}_j(\Gamma )$). Let $\mathcal{Z}'_j = NSD ^K_j(\Gamma )$. It immediately follows that the sets $\mathcal{Z}'_1, \ldots , \mathcal{Z}'_n$ satisfy the conditions of Definition 3.12.

(b) $\Rightarrow$ (c): Suppose that $\vec {\sigma }$ is minimax rationalizable. Let $\mathcal{Z}_1, \ldots , \mathcal{Z}_n$ be the sets guaranteed to exist by Definition 3.12. Let $\mathcal{W}^i = \{(\vec {\sigma }, i) \; | \; \vec {\sigma } \in \mathcal{Z}_{-i} \times \Sigma _{i}\}$, and let $\mathcal{W}^0 = \{(\vec {\sigma }, 0) \; | \; \vec {\sigma } \in \mathcal{Z}_1 \times \ldots \times \mathcal{Z}_n\}$. Think of $\mathcal{W}^0$ as states where common counterfactual belief of rationality holds, and of $\mathcal{W}^i$ as “counterfactual” states where player i has changed strategies. In states in $\mathcal{W}^0$, each player j assigns probability 1 to the other players choosing actions that maximize j’s utility (given his action). On the other hand, in states in $\mathcal{W}^i$, where $i \ne 0$, player i assigns probability 1 to the other players choosing actions that minimize i’s utility, whereas all other player $j \ne i$ still assign probability 1 to other players choosing actions that maximize j’s utility.

Define a structure $M = (\Omega ,f,\mathbf {s}, \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$, where

• $\Omega = \cup _{i \in \{0, 1, \ldots , n\}} \mathcal{W}^i$;

• $\mathbf {s}(\vec {\sigma }',i) = \vec {\sigma }'$;

• $\mathcal {PR}_j(\vec {\sigma }',i)(\vec {\sigma }'',i')$ $= \left\{ \begin{array}{lll} 1 &{}\text{ if } i =j = i', \sigma '_i = \sigma ''_i,\hbox { and }\sigma ''_{-i} = \arg \min ^*_{\tau _{-i} \in \mathcal{Z}_{-i}}u_j(\sigma '_i,\tau _{-i}), \\ 1 &{}\text{ if } i\ne j, i' = 0,\hbox { and }\sigma '_j = \sigma ''_j,\hbox { and }\sigma ''_{-j} = \arg \max ^*_{\tau _{-j} \in \mathcal{Z}_{-j}}u_j(\sigma '_j,\tau _{-j}), \\ 0 &{}\text{ otherwise; } \end{array} \right.$

• $f((\vec {\sigma }',i),j,\sigma ''_j)$ $= \left\{ \begin{array}{lll} (\vec {\sigma }',i) &{}\text{ if } \sigma '_j = \sigma ''_j,\\ ((\sigma ''_j,\tau '_{-j}), j) &{}\text{ otherwise, } \text{ where } \tau '_{-j} = \arg \min ^*_{\tau _{-j} \in \mathcal{Z}_{-j}} u_j(\sigma '_j,\tau _{-j}).\\ \end{array}\right.$

It follows by inspection that M is strongly appropriate for $\Gamma$. We now prove by induction on k that, for all $k \ge 1$ all $i \in \{0,1, \ldots , n\}$, and all states $\omega \in \mathcal{W}^i$, $(M,\omega ) \models \wedge _{j \ne i} {\textit{SRAT}}^k_j$.

For the base case $(k=1)$, since ${\textit{SRAT}}^1_j$ is logically equivalent to $RAT _j$, we must show that if $\omega \in \mathcal{W}^i$, then $(M,\omega ) \models \wedge _{j \ne i} RAT _j$. Suppose that $\omega = (\vec {\sigma }',i) \in \mathcal{W}^i$. If $i \ne j$, then at $\omega$, player j places probability 1 on the true state being $\omega ' = (\vec {\sigma }'',0)$, where $\sigma ''_j = \sigma '_j$ and $\sigma ''_{-j} = \arg \max ^*_{\tau _{-j} \in \mathcal{Z}_{-j}} u_j(\sigma '_j,\tau _{-j})$. Player j must be rational, since if there exists some strategy $\tau '_j$ such that $u_j(\vec {\sigma }'') < \sum _{\omega ' \in \Omega } \mathcal {PR}^c_{j,\tau '_j}(\omega )(\omega ') u_j( \tau '_j, \mathbf {s}_{-j}(\omega '))$, then the definition of $\mathcal {PR}_j$ guarantees that $u_j(\vec {\sigma }'') < u_j(\tau _j',\vec {\tau }''_{-j})$, where $\tau ''_j = \arg \min ^*_{\tau _{-j} \in \mathcal{Z}_{-j}} u_j(\sigma '_j,\tau _{-j})$. If this inequality held, then $\tau '_j$ would minimax dominate $\sigma '_j$, contradicting the assumption that $\sigma '_j \in \mathcal{Z}_j$. For the induction step, suppose that the result holds for k; we show that it holds for $k+1$. Suppose that $\omega \in \mathcal{W}^i$ and $j \ne i$. By construction, the support of $\mathcal {PR}_{j}(\omega )$ is a subset of $\mathcal{W}^0$; by the induction hypothesis, it follows that $(M,\omega ) \models B_j (\wedge _{j' =1}^n {\textit{SRAT}}^{k}_{j'})$. Moreover, by construction, it follows that for all players j and all strategies $\sigma '_j \ne \mathbf {s}_i(\omega )$, the support of $\mathcal {PR}^c_{j,\sigma '_j}(\omega )$ is a subset of $\mathcal{W}^j$. By the induction hypothesis, it follows that for all $j \ne i$, $(M,\omega ) \models B^*_j (\wedge _{j' \ne j} {\textit{SRAT}}^{k}_{j'})$. Finally, it follows from the induction hypothesis that for all $j \ne i$, $(M,\omega ) \models {\textit{SRAT}}^{k}_{j}$. Since ${\textit{SRAT}}^k_j$ implies $RAT _j$, it follows that for all $j \ne i$, $(M,\omega ) \models RAT _j \wedge B^*_j (\wedge _{j' \ne j} {\textit{SRAT}}^{k}_{j'})$, which proves the induction step.

(c) $\Rightarrow$ (d): The implication is trivial.

(d) $\Rightarrow$ (a): We prove an even stronger statement: For all $k\ge 0$, if there exists a finite counterfactual structure $M^k$ that is appropriate for $\Gamma$ and a state $\omega$ such that $(M^k,\omega ) \models play _i(\sigma _i) \wedge {\textit{SRAT}}^k_i$, then $\sigma _i\in NSD _i ^k(\Gamma )$.¹⁴ We proceed by induction on k. The result clearly holds if $k=0$. Suppose that the result holds for $k-1$ for $k \ge 1$; we show that it holds for k. Let $M^k=(\Omega ,f,\mathbf {s}, \mathcal{P}_1, \ldots , \mathcal{P}_n)$ be a finite counterfactual structure that is appropriate for $\Gamma$ and a state $\omega$ such that $(M^k,\omega ') \models play _i(\sigma _i) \wedge {\textit{SRAT}}_i^{k}$. Replacing ${\textit{SRAT}}_i^k$ by its definition, we get that

$$\begin{aligned} (M^k,\omega ') \models play _i(\sigma _i) \wedge RAT _i\wedge B^*_i (\wedge _{j \ne i} {\textit{SRAT}}_j^{k-1}). \end{aligned}$$

By definition of $B^*_i$, it follows that for all strategies $\sigma '_i$ for player i and all $\omega ''$ such that $\mathcal {PR}^c_{i,\sigma '_i}(\omega ')(\omega '') > 0$,

$$\begin{aligned} (M^k,\omega '') \models \wedge _{j \ne i} {\textit{SRAT}}_j^{k-1}, \end{aligned}$$

so by the induction hypothesis, it follows that for all $\omega ''$ such that $\mathcal {PR}^c_{i,\sigma '_i}(\omega ')(\omega '') > 0$, we have $\mathbf {s}_{-i}(\omega '') \in NSD _{-i}^{k-1}(\Gamma )$. Since $(M^k,\omega ') \models play _i(\sigma _i) \wedge RAT _i$, it follows that $\sigma _i$ cannot be minimax dominated with respect to $NSD _{-i}^{k-1}(\Gamma )$. Since, for all $j'>1$, $NSD _{-i}^{j'}(\Gamma ) \subseteq NSD _{-i}^{j'-1}(\Gamma )$, it follows that, for all $k' < k$, $\sigma _i$ is not minimax dominated with respect to $NSD _{-i}^{k'}(\Gamma )$. Thus, $\sigma _i \in NSD _i^{k}(\Gamma )$. $\square$

Theorem 4.4

The following are equivalent:

• (a) $\vec {\sigma } \in \textit{IR}( NSD ^{\infty }(\Gamma ), \Gamma )$;

• (b) $\vec {\sigma } \in \textit{IR}'( NSD ^{\infty }(\Gamma ), \Gamma )$;

• (c) $\vec {\sigma }$ is minimax rationalizable and $\vec {\sigma } \in \textit{IR}'(\mathcal{Z}_1 \times \ldots \times \mathcal{Z}_n,\Gamma )$, where $\mathcal{Z}_1, \ldots , \mathcal{Z}_n$ are the sets of strategies guaranteed to exists by the definition of minimax rationalizability;

• (d) there exists a finite counterfactual structure M that is strongly appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \mathrm {KW}\wedge play (\vec {\sigma }) \wedge _{i=1}^n {\textit{SRAT}}_i^k$ for every $k \ge 0$;

• (e) $\vec {\sigma }$ is a translucent equilibrium; that is, there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \mathrm {KS}\wedge play (\vec {\sigma }) \wedge _{i =1}^n {\textit{SRAT}}_i^k$ for every $k \ge 0$.

Proof

Again, we prove that (a) implies (b) implies (c) implies (d) implies (e) implies (a). (a) $\Rightarrow$ (b): We show that if $\vec {\sigma } \in \textit{IR}( NSD ^k(\Gamma ), \Gamma )$ then $\vec {\sigma } \in \textit{IR}'( NSD ^k(\Gamma ), \Gamma )$. The implication then follows from the fact that since the game is finite there exists some K such that $NSD ^K(\Gamma ) = NSD ^{\infty }(\Gamma )$.

Assume by way of contradiction that $\vec {\sigma } \in \textit{IR}( NSD ^k(\Gamma ), \Gamma )$ but $\vec {\sigma } \notin \textit{IR}'( NSD ^k(\Gamma ), \Gamma )$; that is, there exists a player i and a strategy $\sigma '_i \notin NSD _i^{k}(\Gamma )$ such that

$$\begin{aligned} \min _{\tau _{-i}\in NSD _{-i}^k(\Gamma )} u_i(\sigma '_i,\tau _{-i}) > u_i(\vec {\sigma }). \end{aligned}$$

By the argument in Remark 3.10, there exists a strategy $\sigma ''_i \in NSD ^k_i(\Gamma )$ such that $u_i(\sigma ''_i, \tau ''_{-i}) > u_i(\sigma '_i, \tau '_{-i})$ for all $\tau ''_{-i}, \tau '_{-i} \in NSD_{-i}^k(\Gamma )$. It follows that

$$\begin{aligned} \min _{\tau _{-i}\in NSD _{-i}^k(\Gamma )} u_i(\sigma ''_i,\tau _{-i}) > u_i(\vec {\sigma }). \end{aligned}$$

Thus, $\vec {\sigma } \notin \textit{IR}( NSD ^k(\Gamma ), \Gamma )$.

(b) $\Rightarrow$ (c): The implication follows in exactly the same way as in the proof that (a) implies (b) in Theorem 3.13.

(c) $\Rightarrow$ (d): Suppose that $\vec {\sigma }$ is minimax rationalizable. Let $\mathcal{Z}_1, \ldots , \mathcal{Z}_n$ be the sets guaranteed to exist by Definition 3.12, and suppose that $\vec {\sigma } \in \textit{IR}'(\mathcal{Z}_1 \times \mathcal{Z}_n, \Gamma )$. Define the sets $\mathcal{W}^i$ as in the proof of Theorem 3.13. Define the structure M just as in the proof of Theorem 3.13, except that for all players i, let $\mathcal {PR}_i((\vec {\sigma },0))((\vec {\sigma }',i')) = 1$ in case $\vec {\sigma }' = \vec {\sigma }$ and $i' = 0$. Clearly $(M,(\vec {\sigma },0)) \models \mathrm {KW}$. It follows using the same arguments as in the proof of Theorem 3.13 that M is strongly appropriate and that $(M,(\vec {\sigma },0) \models play (\vec {\sigma }) \wedge _{i=1}^n {\textit{SRAT}}_i^k$ for every $k \ge 0$; we just need to rely on the (strong) IR property of $\vec {\sigma }$ to prove the base case of the induction.

(d) $\Rightarrow$ (e): The implication is trivial. (e) $\Rightarrow$ (a): Recall that since the game is finite, there exists a constant K such that $NSD^{K-1}(\Gamma ) = NSD^{K}(\Gamma ) = NSD ^{\infty }(\Gamma )$. We show that if there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \mathrm {KS}\wedge play (\vec {\sigma }) \wedge _{i=1}^n {\textit{SRAT}}_i^K$, then $\vec {\sigma } \in \textit{IR}( NSD ^K(\Gamma ), \Gamma )$.

Consider some state $\omega$ such that $(M,\omega ) \models \mathrm {KS}\wedge play (\vec {\sigma }) \wedge _{i =1}^n {\textit{SRAT}}_i^K$. By Theorem 3.13, it follows that $\vec {\sigma } \in NSD ^K(\Gamma )$. For each player i, it additionally follows that $(M,\omega ) \models play (\vec {\sigma }) \wedge {\textit{EB}}( play (\vec {\sigma })) \wedge RAT _i \wedge B^*_i (\wedge _{j \ne i} {\textit{SRAT}}_j^{K-1})$. By Theorem 3.13, it follows that for every strategy $\sigma '_i$ for i, the support of the projection of $\mathcal {PR}^c_{i,\sigma '_i}(\omega )$ onto strategies for players $-i$ is a subset of $NSD _{-i}^{K-1}(\Gamma ) = NSD _{-i}^{K}(\Gamma )$. Thus, we have that for every $\sigma '_i$, there exists $\tau _{-i} \in NSD ^{K}_{-i}(\Gamma )$ such that $u_i(\vec {\sigma }) \ge u_i(\sigma '_i, \tau _{-i})$, which means that $\vec {\sigma }$ is IR in the subgame induced by restricting the strategy set to $NSD ^K(\Gamma )$. $\square$

Theorem 4.5

The following are equivalent:

• (a) $\vec {\sigma } \in \textit{IR}(\Gamma )$;

• (b) there exists a finite counterfactual structure M that is strongly appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \mathrm {KW}\wedge play (\vec {\sigma }) \wedge {\textit{CB}}( RAT )$;

• (c) there exists a finite counterfactual structure M that is appropriate for $\Gamma$ and a state $\omega$ such that $(M,\omega ) \models \mathrm {KS}\wedge play (\vec {\sigma }) \wedge {\textit{CB}}( RAT )$.

Proof

Again, we prove that (a) implies (b) implies (c) implies (a).

(a) $\Rightarrow$ (b): Define a structure $M = (\Omega ,f,\mathbf {s}, \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$, where

• $\Omega = \Sigma (\Gamma )$;

• $\mathbf {s}(\vec {\sigma }') = \vec {\sigma }'$;

• $\mathcal {PR}_j(\vec {\sigma }')(\vec {\sigma }') = 1$.

• $f(\vec {\sigma }',i,\sigma ''_j) = \left\{ \begin{array}{lll} \vec {\sigma }' &{}\text{ if } \sigma '_j = \sigma ''_j,\\ (\sigma ''_j,\tau '_{-j}) &{}\text{ otherwise, } \text{ where } \tau '_{-j} = \arg \min ^*_{\tau _{-j} \in \Sigma _{-j}(\Gamma )} u_j(\sigma '_j,\tau _{-j}). \end{array}\right.$

It follows that M is strongly appropriate for $\Gamma$ and that $(M,\vec {\sigma }) \models \mathrm {KW}$. Moreover, $(M,\vec {\sigma }) \models RAT$ since $\vec {\sigma }$ is individually rational; furthermore, since each player considers only the state $\vec {\sigma }$ possible at $\vec {\sigma }$, it follows that $(M,\vec {\sigma }) \models {\textit{CB}}(RAT)$.

(b) $\Rightarrow$ (c): The implication is trivial.

(c) $\Rightarrow$ (a): Suppose that $M = (\Omega ,f, \mathbf {s}, \mathcal {PR}_1, \ldots , \mathcal {PR}_n)$ is a finite counterfactual structure appropriate for $\Gamma$, and $(M,\omega ) \models \mathrm {KW}\wedge play (\vec {\sigma })\wedge {\textit{CB}}( RAT )$. It follows that for each player i, $(M,\omega ) \models play (\vec {\sigma }) \wedge {\textit{EB}}( play (\vec {\sigma })) \wedge RAT _i$. Thus, we have that for all strategies $\sigma '_i$, there exists $\tau _{-i} \in \Sigma _{-i}(\Gamma )$ such that $u_i(\vec {\sigma }) \ge u_i(\sigma '_i, \tau _{-i})$, which means that $\vec {\sigma }$ is IR. $\square$