Unilaterally competitive games with more than two players

Abstract

We prove some interesting properties of unilaterally competitive games when there are more than two players. We show that such games possess: (1) a Nash equilibrium, (2) maximin-solvability, (3) strong solvability in the sense of Nash, and (4) weak acyclicity, all in pure strategies of finite or infinite games.

Appendix

1.1 A proof of Theorem 5

In this proof, we take a viewpoint that a UC game can be seen as a collection of two-person zero-sum games, to which we then apply Theorem 1.

Proof

Suppose that G is an n-person UC game with $n \ge 2$ and $E(G) \ne \emptyset$. By Lemma 1, we have that player i chooses $s_i$ to maximize $u_i(s_i,s_{-i})$ given $s_{-i}$ and player $-i$ (“other than i”) chooses $s_{-i}$ to minimize $u_i(s_i,s_{-i})$ given $s_i$, and this is true for any $i \in N$. Since minimizing $u_i(s_i,\cdot )$ is the same as maximizing $-u_i(s_i,\cdot )$, we can think of player i as playing a two-person zero-sum game $G_i=(\{i,-i\},(S_i,S_{-i}),(u_i,-u_i))$ with player $-i$, for any $i \in N$. Let $E(G_i)$ and $M(G_i)$ be the set of equilibria of $G_i$ and the set of maximin profiles of $G_i$, for each $i \in N$.

Now, let $s \in E(G)$. It is not difficult to see that $s=(s_i,s_{-i}) \in E(G_i)$ for every $i \in N$ (use Lemma 1 to see the optimality of $s_{-i}$ against $s_i$). Then, by Theorem 1, $s=(s_i,s_{-i}) \in M(G_i)$ for every $i \in N$. By the definition of maximin strategy in G, $s_i$ is also a maximin strategy of player i in G, for any $i \in N$. Hence $s \in M(G)$.

Let then $s \in M(G)$. By the definition of maximin strategy in G, we have $s=(s_i,s_{-i}) \in M(G_i)$ for any $i \in N$. Then, since $E(G) \ne \emptyset$, and hence $E(G_i) \ne \emptyset$ for any $i \in N$, we have $E(G_i)=M(G_i)$ for any $i \in N$ by Theorem 1. Hence $s=(s_i,s_{-i}) \in E(G_i)$ for any $i \in N$, and $s \in E(G)$, for obvious reasons. $\square$