经济代写|博弈论代写Game Theory代考|ECON2070

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Best Responses and Equilibrium

Consider the game in Figure 3.2(a). This is the Prisoner’s Dilemma game, with further graphical information added to help the analysis of the game. Namely, certain payoffs are surrounded by boxes that indicate that they are best-response payoffs.

A best response is a strategy of a player that is optimal for that player assuming it is known what all other players do. In a two-player game, there is only one other player. Hence, for the row player, a best response is found for each column. In the Prisoner’s Dilemma game, the best response of the row player against column $C$ is row $D$ because his payoff when playing $D$ is 3 rather than 2 when playing $C$ (so in the cell for row $D$ and column $C$, the payoff 3 to player $I$ is put in a box), and the best response against column $D$ is also row $D$ (with payoff 1 , in a box, which is larger than payoff 0 for row $C$ ). Similarly, the best response of the column player against row $C$ is column $D$, and against row $D$ is column $D$, as shown by the boxes around the payoffs for the column player. Because the game is symmetric, the best-response payoffs are also symmetric.

The game in Figure 3.2(b) has payoffs that are nearly identical to those of the Prisoner’s Dilemma game, except for the payoff to player II for the top right cell, which is changed from 3 to 1 . Because the game is not symmetric, we name the strategies of the players differently, here $T$ and $B$ for the row player (for “top” and “bottom”) and $l$ and $r$ for the column player (“left” and “right”). To distinguish them more easily, we often write the strategies of player I in upper case and those of player II in lower case.

经济代写|博弈论代写Game Theory代考|Games with Multiple Equilibria

We consider several well-known $2 \times 2$ games that have more than one equilibrium. The game of Chicken is shown in Figure 3.3. It is symmetric, as shown by the dotted line, and we have directly marked the best-response payoffs with boxes; recall that neither markup is part of the game description but already part of its analysis. The two strategies $A$ and $C$ stand for “aggressive” and “cautious” behavior, for example of two car drivers that drive towards each other on a narrow road. The aggressive strategy is only advantageous (with payoff 2 ) if the other player is cautious but leads to a crash (payoff 0 ) if the other player is aggressive, whereas a cautious strategy always gives payoff 1 to the player who uses it. This game has two equilibria, $(C, A)$ and $(A, C)$, because the best response to aggressive behavior is to be cautious and vice versa.

The game known as the Battle of the Sexes is shown in Figure $3.4(\mathrm{a})$; its rather antiquated gender stereotypes should not be taken too seriously. In this scenario, player I and player II are a couple who each decide (simultaneously and independently, which is rather unrealistic) whether to go to a concert ( $C$ ) or to a sports event (S). The players have different payoffs arising from which event they go to, but that payoff is zero if they have to attend the event alone. The game has two equilibria: $(C, C)$ where they both go to the concert, or $(S, S)$ where they both go to the sports event. Their preferences between these events differ, however.
The Battle of the Sexes game is not symmetric when written as in Figure 3.4(a), because the payoffs for the strategy pairs $(C, C)$ and $(S, S)$ on the diagonal are not the same for both players, which is clearly necessary for symmetry. However, changing the order of the strategies of one player, for example of player I as shown in Figure 3.4(b), makes this a game with a symmetric payoff structure; for a true symmetric game, one would also have to exchange the strategy names $C$ and $S$ of the strategies of player I, which would not represent the actions that player I takes. The game in Figure $3.4$ (b) is very similar to Chicken. However, because of the

meaning of the strategies, the Battle of the Sexes is a coordination game (both players benefit from choosing the same action), whereas Chicken is an “anti-coordination” game.

The Stag Hunt game in Figure $3.5$ models a conflict between cooperation and safety. Two hunters individually decide between $S$, to hunt a stag (a male deer), or $H$, to hunt a hare. Each can catch the hare, without the help of the other player, and then receives payoff 1 . In order to catch the stag they have to cooperate – if the other hunter goes for the hare then the stag hunter’s payoff is zero. If they both hunt the stag they succeed and each get payoff 3 .

经济代写|博弈论代写Game Theory代考|Dominated Strategies

In the Quality game in Figure $3.2(b)$, strategy B gives a better payoff to the row player than strategy $T$, no matter what the other player does. We say $B$ dominates $T$, by which we always mean $B$ strictly dominates $T$. The following definition states the concepts of strict dominance, weak dominance, and payoff equivalence for a general N-player game.

Definition 3.2. Consider an $N$-player game with strategy sets $S_{i}$ and payoff functions $u_{i}$ for each player $i=1, \ldots, N$, let $s_{i}$ and $t_{i}$ be two strategies of some player $i$, and let $S_{-i}=X_{j=1, j \neq i}^{N} S_{j}$ be the set of partial profiles of the other players. Then

• $t_{i}$ dominates (or strictly dominates) $s_{i}$ if
$$u_{i}\left(t_{i}, s_{-i}\right)>u_{i}\left(s_{i}, s_{-i}\right) \quad \text { for all } s_{-i} \in S_{-i}$$
• $t_{i}$ is payoff equivalent to $s_{i}$ if
$$u_{i}\left(t_{i}, s_{-i}\right)=u_{i}\left(s_{i}, s_{-i}\right) \quad \text { for all } s_{-i} \in S_{-i}$$
• $t_{i}$ weakly dominates $s_{i}$ if $t_{i}$ and $s_{i}$ are not payoff equivalent and
$$u_{i}\left(t_{i}, s_{-i}\right) \geq u_{i}\left(s_{i}, s_{-i}\right) \quad \text { for all } s_{-i} \in S_{-i}$$
The condition (3.4) of strict dominance is particularly easy to check in a twoplayer game. For example, if $i$ is the column player, then $s_{i}$ and $t_{i}$ are two columns, and $s_{-i}$ is an arbitrary row, and we look at the payoffs of the column player. Then (3.4) states that, row by row, each entry in column $t_{i}$ is greater than the respective entry in column $s_{i}$. For the columns $D$ and $C$ in the Prisoner’s Dilemma game in Figure 3.2(a), this can be written as $\left(\begin{array}{l}3 \ 1\end{array}\right)>\left(\begin{array}{l}2 \ 0\end{array}\right)$, which is true. However, for the Quality game in Figure 3.2(b), the two payoff columns for $r$ and $l$ are $\left(\begin{array}{l}1 \ 1\end{array}\right)$ and $\left(\begin{array}{l}2 \ 0\end{array}\right)$ where this does not hold, and neither $r$ dominates $l$ nor $l$ dominates $r$. However, for both games (which have the same payoffs to the row player) the bottom row dominates the top row because $\left(\begin{array}{ll}3 & 1\end{array}\right)>\left(\begin{array}{ll}2 & 0\end{array}\right)$ (we compare two vectors component by component).

经济代写|博弈论代写Game Theory代考|Dominated Strategies

• 吨一世支配（或严格支配）s一世如果
在一世(吨一世,s−一世)>在一世(s一世,s−一世) 对所有人 s−一世∈小号−一世
• 吨一世收益等于s一世如果
在一世(吨一世,s−一世)=在一世(s一世,s−一世) 对所有人 s−一世∈小号−一世
• 吨一世弱支配s一世如果吨一世和s一世不是收益等价的，并且
在一世(吨一世,s−一世)≥在一世(s一世,s−一世) 对所有人 s−一世∈小号−一世
严格支配的条件（3.4）在双人游戏中特别容易检查。例如，如果一世是列播放器，那么s一世和吨一世是两列，并且s−一世是任意行，我们看看列玩家的收益。然后（3.4）表明，逐行，列中的每个条目吨一世大于列中的相应条目s一世. 对于列D和C在图 3.2(a) 的囚徒困境博弈中，这可以写成(3 1)>(2 0), 这是真的。然而，对于图 3.2(b) 中的质量博弈，两个收益列r和l是(1 1)和(2 0)这不成立，也不成立r占主导地位l也不l占主导地位r. 然而，对于这两种游戏（对排玩家具有相同的收益），底行支配顶行，因为(31)>(20)（我们逐个比较两个向量）。

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