### 经济代写|博弈论代写Game Theory代考|ECON3301

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Games without a Pure-Strategy Equilibrium

Not every game has an equilibrium in pure strategies. Figure $3.12$ shows two well-known examples. In Matching Pennies, the two players reveal a penny which can show Heads $(H)$ or Tails $(T)$. If the pennies match, then player I wins the other player’s penny, otherwise player II. No strategy pair can be stable because the losing player would always deviate.

Rock-Paper-Scissors is a $3 \times 3$ game where both players choose simultaneously one of their three strategies Rock $(R)$, Paper $(P)$, or Scissors $(S)$. Rock loses to Paper, Paper loses to Scissors, and Scissors lose to Rock, and it is a draw otherwise. No two strategies are best responses to each other. Like Matching Pennies, this is a zero-sum game because the payoffs in any cell of the table sum to zero. Unlike Matching Pennies, Rock-Paper-Scissors is symmetric. Hence, when both players play the same strategy (the cells on the diagonal), they get the same payoff, which is zero because the game is zero-sum.

The game-theoretic recommendation is to play randomly in games like Matching Pennies or Rock-Paper-Scissors that have no equilibrium, according to certain probabilities that depend on the payoffs. As we will see in Chapter 6, any finite game has an equilibrium when players are allowed to use randomized strategies.

## 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.

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).

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

(1,…,1⏟ķ,0,…,0⏟ñ−ķ).

van Damme (1987) 是对均衡细化的经典调查。主张3.7Cheng, Reeves, Vorobeychik, and Wellman (2004, theorem 1) 似乎首先证明了这一点。

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