Maximal number of mixed Nash equilibria in generic games where each player has two pure strategies

TL;DR

This paper analyzes the maximal number of mixed Nash equilibria in generic finite games where each of the $m$ players has two pure strategies. It derives a near-optimal lower bound $\frac{1}{2}(V(m) + !m)$ for the maximal count $\mu(m;2,...,2)$, where $V(m)$ is Vidunas' upper bound and $!m$ the derangement count, showing the bound is strikingly close to $V(m)$. The authors construct maximal product two-action games, establish a detailed combinatorial framework for equilibrium candidates via derangements and increment maps, and prove that these games achieve the lower bound. They conjecture equality $\mu(m;2,...,2) = \frac{1}{2}(V(m) + !m)$ and discuss a semicontinuity principle across strata of Nash equilibria, with $m=3$ already settled in the literature.

Abstract

The number of Nash equilibria of the mixed extension of a generic finite game in normal form is finite and odd. This raises the question how large the number can be, depending on the number of players and the numbers of their pure strategies. Here we present a lower bound for the maximal possible number in the case of m-player games where each player has two pure strategies. It is surprisingly close to a known upper bound.

Figures (2)

• Figure 1: Left: $\mathop{\mathrm{Gr}}\nolimits(r^3)$ in (i). Right: $\mathop{\mathrm{Gr}}\nolimits(r^3)$ in (ii).
• Figure 2: (iii) The upper picture shows the signs of $\lambda^1,\lambda^2,\lambda^3$. The lower picture shows the hyperplanes $\{\gamma^j-a^i_j=0\}\subset G\cong[0;1]^3$, the $8+0+6+2$ equilibrium candidates and the $4+0+3+2$ equilibria.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Definition 3.4
• Theorem 3.7
• Definition 4.1
• Lemma 4.2