# 经济代写|博弈论代写Game Theory代考|ECON6025

## 经济代写|博弈论代写Game Theory代考|Symmetric Games with Two Strategies per Player

In this section, we consider $N$-player games that, somewhat surprisingly, always have a pure equilibrium, namely symmetric games where each player has only two strategies.

An $N$-player game is symmetric if each player has the same set of strategies, and if the game stays the same after any permutation (shuffling) of the players and, correspondingly, their payoffs. For two players, this means that the game stays the same when exchanging the players and their payoffs, visualized by reflection along the diagonal.

We now consider symmetric $N$-player games where each player has two strategies, which we call 0 and 1 . Normally, any combination of these strategies defines a separate strategy profile, so there are $2^N$ profiles, and each of them specifies a payoff to each player, so the game is defined by $N \cdot 2^N$ payoffs. If the game is symmetric, vastly fewer payoffs are needed. Then a strategy profile is determined by how many players choose 1 , say $k$ players (where $0 \leq k \leq N$ ), and then the remaining $N-k$ players choose 0 , so the profile can be written as
$$(\underbrace{1, \ldots, 1}k, \underbrace{0, \ldots, 0}{N-k}) \text {. }$$
Because the game is symmetric, any profile where $k$ players choose 1 has to give the same payoff as (3.12) to any player who chooses 1 , and a second payoff to any player who chooses 0 . Hence, we need only two payoffs for these profiles (3.12) when $1 \leq k \leq N-1$. When $k=0$ then the profile is $(0, \ldots, 0)$ and all players play the same strategy 0 and only one payoff is needed, and similarly when $k=N$ where all players play 1 . Therefore, a symmetric $N$-player game with two strategies per player is specified by only $2 \mathrm{~N}$ payoffs.

Proposition 3.7. Consider a symmetric N-player game where each player has two strategies, 0 and 1. Then this game has a pure equilibrium. The strategy profile $(1,1, \ldots, 1)$ is the unique equilibrium of the game if and only if 1 dominates 0.

Proof. If 1 dominates 0 , then $(1,1, \ldots, 1)$ is clearly the unique equilibrium of the game, so suppose this not the case. Because 0 is not dominated and there are only two strategies, for some profile the payoff when playing 0 is greater than or equal to the payoft when playing 1 , that is, 0 is a best response to the remaining strategies. Consider such a profile (3.12) where 0 is a best response, with the smallest number $k$ of players who play 1 , where $0 \leq k<N$. Then this profile is an equilibrium: By assumption and symmetry, 0 is a best response to the strategies of the other players for every player who plays 0 . If 1 was not a best response for every player who plays 1 , then such a player would obtain a higher payoff when changing his strategy to 0 . This change would result in a profile where $k-1$ players play 1 , the remaining players play 0 , and 0 is a best response, which contradicts the smallest of choice of $k$. This shows that the game has an equilibrium.

The display of staggered payoffs in the lower left and upper right of each cell in the payoff table is due to Thomas Schelling. In 2005, he received, together with Robert Aumann, the Nobel memorial prize in Economic Sciences (officially: The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) “for having enhanced our understanding of conflict and cooperation through game-theory analysis.” According to Dixit and Nalebuff (1991, p. 90), he said with excessive modesty: “If I am ever asked whether I ever made a contribution to game theory, I shall answer yes. Asked what it was, I shall say the invention of staggered payoffs in a matrix.”

The strategic form is taught in every course on non-cooperative game theory (it is sometimes called the “normal form”, now less used in game theory because “normal” is an overworked term in mathematics). Osborne (2004) gives careful historical explanations of the games considered in this chapter, including the original duopoly model of Cournot (1838), and many others. Our Exercise $3.4$ is taken from that book. Gibbons (1992) shows that the Cournot game is dominance solvable, with less detail than our proof of Proposition 3.6. Both Osborne and Gibbons disregard Schelling and use comma-separated payoffs as in (3.1).

The Cournot game in Section $3.6$ is also a potential game with a strictly concave potential function, which has a unique maximum and therefore a unique equilibrium. Neyman (1997) showed that it is also a unique correlated equilibrium (see Chapter 12). Potential games (Monderer and Shapley, 1996) generalize games with a potential function such as the congestion games considered in Section 2.5.
A classic survey of equilibrium refinements is van Damme (1987). Proposition $3.7$ seems to have been shown first by Cheng, Reeves, Vorobeychik, and Wellman (2004, theorem 1).

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Symmetric Games with Two Strategies per Player

$$(\underbrace{1, \ldots, 1} k, \underbrace{0, \ldots, 0} N-k) .$$

van Damme (1987) 对均衡改进进行了经典调查。主张3.7似乎首先由 Cheng、Reeves、Vorobeychik 和 Wellman（2004 年，定理 1）证明。

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