Strongly rational sets for normal-form games

Abstract

We introduce the concept of minimal strong curb sets which is a set-theoretic coarsening of the notion of strong Nash equilibrium. Strong curb sets are product sets of pure strategies such that each player’s set of recommended strategies contains all actions she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is consistent with the recommendations to the other players. Minimal strong curb sets are shown to exist and are compared with other well-known solution concepts. We provide a dynamic learning process leading the players to play strategies from a minimal strong curb set only.

Appendix

We now show that the existence of minimal strong curb sets holds in general. Let $\mathcal {G}$ be the class of normal-form games $G=\left\langle N,\left\{ A_{i}\right\} _{i\in N},\left\{ u_{i}\right\} _{i\in N}\right\rangle$ where for each player $i\in N=\left\{ 1,2,\ldots ,n\right\}$, $A_{i}$ is a compact subset of a metric space and $u_{i}:$ $A\rightarrow \mathbb {R}$ is a continuous von Neumann–Morgenstern utility function. Payoffs are extended to mixed strategies in the usual way. Let $\Delta (A_{i})$ be the set of Borel probability measures over $A_{i}$. If $B_{i}\subseteq A_{i}$ is a Borel set, then $\Delta (B_{i})$ denotes the set of Borel probability measures with support in $B_{i}$: $\Delta (B_{i})=\{\alpha _{i}\in \Delta (A_{i})\mid \alpha _{i}(B_{i})=1\}$. If $G\in \mathcal {G}$, that is, payoff functions are continuous and strategy sets compact, then each set $\mathrm{BR} ^{i}(\alpha _{-i})\subseteq A_{i}$ is nonempty and compact. A strong curb set is a product set $X=\times _{i\in N}X_{i}$ where (a) for each $i\in N$, $X_{i}\subseteq A_{i}$ is a nonempty, compact set of pure strategies; (b) $\forall J\subseteq N,\forall \mathbf {\alpha }=(\alpha _{-1},\ldots ,\alpha _{-n})$ with $\alpha _{-i}\in \times _{l\in N{\setminus } \{i\}}\Delta (X_{l})$, $i\in N$, $\mathrm{CBR}^{J}(\mathbf {\alpha })\subseteq \times _{j\in J}X_{j}$.

Theorem 1

Every game $G\in \mathcal {G}$ has a minimal strong curb set.

Proof

Let Q be the collection of all strong curb sets of G. A is a strong curb set of G since for every $J\subseteq N$ and $\mathbf {\alpha }=(\alpha _{-1},\ldots ,\alpha _{-N})$ with $\alpha _{-i}\in \times _{l\in N{\setminus } \{i\}}\Delta (A_{l})$, $i\in N$, we have $\mathrm{CBR} ^{J}(\mathbf {\alpha })\subseteq \times _{j\in J}A_{j}$. So Q is nonempty and partially ordered via set inclusion. According to the Hausdorff Maximality Principle, Q contains a maximal nested subset R. For each $i\in N$, let $X_{i}=\cap _{Y\in R}Y_{i}$ be the intersection of player i’s strategies in the nested set R. The set $X_{i}$ is nonempty since the conditions of the Cantor intersection principle are satisfied, i.e., (i) the collection $\{Y_{i}\mid Y\in R\}$ is nested and thus satisfies the finite intersection property and (ii) each $Y_{i}$ is nonempty and compact. It remains to prove that $X=\times _{i\in N}X_{i}$ is a minimal strong curb set. Take $\mathbf {\alpha }=(\alpha _{-1},\ldots ,\alpha _{-N})$ with $\alpha _{-i}\in \times _{l\in N{\setminus } \{i\}}\Delta (X_{l})$, $i\in N$. We have that $\mathrm{CBR}^{J}(\mathbf {\alpha })\cap \times _{j\in J}(A_{j}\backslash X_{j})=\emptyset$ for $J\subseteq N$ since $\mathrm{CBR}^{J}(\mathbf {\alpha } )\cap \times _{j\in J}(A_{j}\backslash X_{j})=\mathrm{CBR}^{J}(\mathbf {\alpha } )\cap (\times _{j\in J}\cup _{Y\in R}(A_{j}\backslash Y_{j}))\subseteq \mathrm{ CBR}^{J}(\mathbf {\alpha })\cap (\cup _{Y\in R}\times _{j\in J}(A_{j}\backslash Y_{j}))=\cup _{Y\in R}(\mathrm{CBR}^{J}(\mathbf {\alpha } )\cap \times _{j\in J}(A_{j}\backslash Y_{j}))$ and $\mathrm{CBR}^{J}(\mathbf { \alpha })\cap \times _{j\in J}(A_{j}\backslash Y_{j})=\emptyset$ for all $Y\in R$ (Y is a strong curb set). This establishes that X is a strong curb set. The fact that it is minimal follows directly from the fact that R is a maximal nested subset of Q. $\square$