经济代写|博弈论代写Game Theory代考|ECON90022

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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|Normal-Form Games

In this section, we introduce normal-form games (NFGs) and formally introduce the solution concepts we mentioned in Section 2.2. Most of the notations and definitions are adapted from Leyton-Brown and Shoham 2008. The normal form is the most basic form of games to represent players’ interactions and strategy space. An NFG captures all possible combinations of actions or strategies for the players and their corresponding payoffs in a matrix, or multiple matrices for more than two players. A player can choose either a pure strategy that deterministically selects a single strategy or play a mixed strategy that specifies a probability distribution over the pure strategies. The goal for all players is to maximize their expected utility. Formally, a finite, $N$-person NFG is described by a tuple $(\mathcal{N}, \mathcal{A}, u)$, where:

• $\mathcal{N}={1, \ldots, N}$ is a finite set of $N$ players, indexed by $i$.
• $\mathcal{A}=\mathcal{A}{1} \times \cdots \times \mathcal{A}{N}$ is a set of joint actions of the players, where $\mathcal{A}{i}$ is a finite set of actions available to player $i . a=\left(a{1}, \ldots, a_{N}\right) \in \mathcal{A}$ is called an action profile with $a_{i} \in \mathcal{A}_{i}$.
• $u=\left(u_{1}, \ldots, u_{N}\right)$ where $u_{i}: \mathcal{A} \mapsto \mathbb{R}$ is a utility (or payoff) function for player $i$. It maps an action profile $a$ to a real value. An important characteristic of a game is that player $i$ ‘s utility depends on not only his own action $a_{i}$ but also the actions taken by other players; thus, the utility function is defined over the space of $\mathcal{A}$ instead of $\mathcal{A}_{i}$.

A player can choose a mixed strategy. We use $S_{i}=\Delta^{\left|\mathcal{A}{i}\right|}$ to denote the set of mixed strategies for player $i$, which is the probability simplex with dimension $\left|\mathcal{A}{i}\right|$. Similarly, $S=S_{1} \times \cdots \times S_{N}$ is the set of joint strategies and $s=\left(s_{1}, \ldots, s_{N}\right) \in S$ is called a strategy profile. The support of a mixed strategy is defined as the set of actions that are chosen with a nonzero probability. An action of player $i$ is a pure strategy and can be represented by a probability distribution with support size 1 (of value 1 in one dimension and 0 in all other dimensions). The utility function can be extended to mixed strategies by using expected utility. That is, if we use $s_{i}\left(a_{i}\right)$ to represent the probability of choosing action $a_{i}$ in strategy $s_{i}$, the expected utility for player $i$ given strategy profile $s$ is $u_{i}(s)=\sum_{a \in \mathcal{A}} u_{i}(a) \prod_{\eta^{n}=1}^{n} s_{i^{\prime}}\left(a_{\eta}\right)$. A game is zero-sum if the utilities of all the players always sum up to zero, i.e. $\sum_{i} u_{i}(s)=0, \forall s$ and is nonzero-sum or general-sum otherwise.

Many classic games can be represented in normal form. Table $2.3$ shows the game Prisoner’s Dilemma (PD). Each player can choose between two actions, Cooperate (C) and Defect (D). If they both choose $C$, they both suffer a small loss of $-1$. If they both choose $D$, they both suffer a big loss of $-2$. However, if one chooses $\mathrm{C}$ and the other chooses $\mathrm{D}$, the one who chooses $\mathrm{C}$ suffers a huge loss while the other one does not suffer any loss. If the row player chooses a mixed strategy of playing C with probability $0.4$ and D $0.6$, while the column player chooses the uniform random strategy, then the row player’s expected utility is $-1.4=(-1) \cdot 0.4 \cdot 0.5+(-3) \cdot 0.4 \cdot 0.5+(-2) \cdot 0.6 \cdot 0.5$.

经济代写|博弈论代写Game Theory代考|Solution Concepts

We will now introduce the formal definition of maxmin, minmax, and Nash equilibrium.
Definition 2.2 (Maxmin Strategy): The maxmin strategy for player $i$ is a strategy that maximizes the worst-case expected utility for player $i$, i.e. $\operatorname{arg~max}{s{i}} \min {s{-i}} u_{i}\left(s_{i}, s_{-i}\right)$.

Definition 2.3 (Minmax Strategy): The minmax strategy for player $i$ against player $i^{\prime}$ is the strategy player $i$ uses when she coordinates with all other players except player $i^{\prime}$ to minimize the best possible expected utility player $i^{\prime} s$ can get, i.e. player $i^{\prime}$ ‘s component in the strategy profile $\arg \min {s{-l}} \max {s{i}} u_{i^{\prime}}\left(s_{-i^{\prime}}, s_{i^{\prime}}\right)$.

The minmax strategy is defined in this way because when there are more than two players, a player’s utility depends on the strategy used by all other players. When there are only two players, the minmax strategy defined in Definition $2.3$ becomes $\arg ^{2} \min {s{1}} \max {s{-i}} u_{-i}\left(s_{i}, s_{-i}\right)$. The maximum and minimum value achieved by maxmin and minmax strategies is called the maxmin value and minmax value for the player. Similar to best responses, there can be more than one maxmin and minmax strategy for a player.

Definition 2.4 (Nash Equilibrium): A strategy profile $s=\left(s_{1}, \ldots, s_{N}\right)$ is a Nash equilibrium if $s_{i} \in B R\left(s_{-i}\right), \forall i .$

There can be multiple NEs in a game, and the strategies in an NE can either be a pure strategy or a mixed strategy. If all players use pure strategies in an NE, we call it a pure strategy NE (PSNE). If at least one player randomly chooses between multiple actions, i.e. the support set of their strategy has a size larger than 1 , we call it a mixed strategy NE.

In the PD game, the strategy profile (D,D) is an NE because when one player chooses D, the other player’s best response is D. Thus, both players are best responding to the other player’s strategy.

经济代写|博弈论代写Game Theory代考|Extensive-Form Games

Extensive-form games (EFGs) represent the sequential interaction of players using a rooted game tree. Figure $2.1$ shows a simple example game tree. Each node in the tree belongs to one of the players and corresponds to a decision point for that player. Outgoing edges from a node represent actions that the corresponding player can take. The game starts from the root node, with the player corresponding to the root node taking an action first. The chosen action brings the game to the child node, and the corresponding player at the child node takes an action. The game continues until it reaches a leaf node (also called a terminal node), i.e. each leaf node in the game tree is a possible end state of the game. Each leaf node is associated with a tuple of utilities or payoffs that the players will receive when the game ends in that state. In the example in Figure 2.1, there are three nodes. Node 1 belongs to Player 1 (P1), and nodes 2 and 3 belong to Player 2 (P2). Player 1 first chooses between action L and R, and then Player 2 chooses between action 1 and r. Player l’s highest utility is achieved when Player 1 chooses L, and Player 2 chooses $1 .$

There is sometimes a special fictitious player called Chance (or Nature), who takes an action according to a predefined probability distribution. This player represents the stochasticity in many problems. For example, in the game of Poker, each player gets a few cards that are randomly dealt. This can be represented by having a Chance player taking an action of dealing cards. Unlike the

other real players, the Chance player does not rationally choose an action to maximize his utility since he does not have a utility function. In the special case where Nature only takes an action at the very beginning of the game, i.e. the root of the game tree, the game is essentially a Bayesian game, as we will detail in Section 2.7.

In a perfect information game, every player can perfectly observe the action taken by players in the previous decision points. For example, when two players are playing the classic board game of Go or Tic-Tac-Toe, each player can observe the other players’ previous moves before she decides her move. However, it is not the case in many other problems. An EFG can also capture imperfect information, i.e. a game where players are sometimes uncertain about the actions taken by other players and thus do not know which node they are at exactly when they take actions. The set of nodes belonging to each player is partitioned into several information sets. The nodes in the same information set cannot be distinguished by the player that owns those nodes. In other words, the player knows that she is at one of the nodes that belong to the same information set, but does not know which one exactly. For example, in a game of Poker where each player has private cards, a player cannot distinguish between certain nodes that only differ in the other players’ private cards. Nodes in the same information set must have the same set of actions since otherwise, a player can distinguish them by checking the action set. It is possible that an information set only contains one node, i.e. a singleton. If all information sets are singletons, the game is a perfect information game. The strategy of a player specifies what action to take at each information set. In the example game in Figure 2.1, the dashed box indicates that nodes 2 and 3 are in the same information set, and Player 2 cannot distinguish between them. Thus, nodes 2 and 3 have the same action set. This information set effectively makes the example game a simultaneous game as Player 2 has no information about Player $1 s$ previous actions when he makes a move.

经济代写|博弈论代写Game Theory代考|Normal-Form Games

• ñ=1,…,ñ是一个有限集ñ球员，由一世.
• 一个=一个1×⋯×一个ñ是玩家的一组联合动作，其中一个一世是玩家可用的一组有限动作一世.一个=(一个1,…,一个ñ)∈一个被称为动作配置文件一个一世∈一个一世.
• 在=(在1,…,在ñ)在哪里在一世:一个↦R是玩家的效用（或收益）函数一世. 它映射了一个动作配置文件一个到一个真正的价值。游戏的一个重要特征是玩家一世的效用不仅仅取决于他自己的行动一个一世还有其他玩家采取的行动；因此，效用函数定义在一个代替一个一世.

经济代写|博弈论代写Game Theory代考|Solution Concepts

minmax 策略是这样定义的，因为当有两个以上的玩家时，一个玩家的效用取决于所有其他玩家使用的策略。当只有两个玩家时，定义中定义的 minmax 策略2.3变为 $\arg ^{2} \min {s {1}} \max {s {-i}} u_{-i}\left(s_{i}, s_{-i}\right)$。maxmin 和 minmax 策略达到的最大值和最小值称为玩家的 maxmin 值和 minmax 值。与最佳响应类似，玩家可以有不止一个 maxmin 和 minmax 策略。

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