A Robust Characterization of Nash Equilibrium

TL;DR

This work provides an axiomatic, population-wise characterization of Nash equilibrium for finite normal-form games by positing three principles—Consequence-based cloning (Consequentialism), stability under mixing of games (Consistency), and minimal rationality (Rationality). It proves that Nash is the unique total solution satisfying these axioms and establishes a robust, approximate version: any solution concept that approximately satisfies the axioms is close to Nash, with extensions to natural game subclasses. The approach hinges on structural game manipulations (clones, blow-ups, convex combinations) and a linear-algebraic intuition to show that any deviation leads to a violation of Rationality. The results have implications for equilibrium refinements, the role of axioms, and the applicability to restricted game classes, while highlighting limitations in extending to correlated equilibria.

Abstract

We characterize Nash equilibrium by postulating coherent behavior across varying games. Nash equilibrium is the only solution concept that satisfies the following axioms: (i) strictly dominant actions are played with positive probability, (ii) if a strategy profile is played in two games, it is also played in every convex combination of these games, and (iii) players can shift probability arbitrarily between two indistinguishable actions, and deleting one of these actions has no effect. Our theorem implies that every equilibrium refinement violates at least one of these axioms. Moreover, every solution concept that approximately satisfies these axioms returns approximate Nash equilibria, even in natural subclasses of games, such as two-player zero-sum games, potential games, and graphical games.

Figures (3)

• Figure 1: Example for an application of \ref{['lem:linearalgebra']}. Here, one half of the second and third action of the first player are added to the fourth action. That is, $\hat{a}_1$ is the fourth action, $k_1 = (0,1,1,2)$, $\kappa_1 = 1/6$, and $x_1 = (0,1/6,1/6,1/3)$; $\hat{a}_2$ is arbitrary, say, the first action of player 2, $k_2 = (1,0,0)$, $\kappa_2 = 1$, and $x_2 = (1,0,0)$.
• Figure 2: Schematic depiction of the games $G$, $\tilde{G}$, and $\hat{G}$ constructed in the proof of \ref{['lem:linearalgebra']} (with $n = 2$, $|A_i| = 2$, $k_i = (2,2)$, and $\hat{a}_i\not\in A_i$ for $i = 1,2$). $\tilde{G}$ is obtained from $G$ by adding $k_i(a_i)$ clones of every action $a_i$ of player $i$. Then, an intermediate game $\bar{G}$ is constructed from $\tilde{G}$ by permuting the actions outside of $A_i$ and summing over the resulting games. The actions outside of $A_i$ are now clones obtained from a convex combination (with weights $k_i$) of actions in $A_i$. Removing all but one of these clones gives $\hat{G}$.
• Figure 3: The matrix $\tilde{M}$ in the proof of \ref{['lem:bvnslice']} for $n = 3$ and $m = 2$. $I_8$ denotes the identity matrix with $8 = m^n$ rows and columns. Each of the four pairs of columns separated by dashed lines corresponds to some $a_{-i}\in A_{-i}$. One can check that $\tilde{M}$ is totally unimodular by verifying the equivalent condition \ref{['eq:rowsum']}.

Theorems & Definitions (52)

• Definition 1: Consequentialism
• Definition 2: Consistency
• Definition 3: Rationality
• Theorem 1
• Lemma 1
• proof
• proof : Proof of \ref{['thm:nash']}
• Definition 4: $\varepsilon$-equilibrium
• Definition 5: $\delta$-consequentialism
• Definition 6: $\delta$-consistency