### 经济代写|博弈论代写Game Theory代考|ECON 3503

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Solution Concepts

Since EFGs can be converted into NFGs, the solution concepts introduced in Section 2.3.1 still apply. Also, there are other solution concepts specifically for EFGs. We briefly introduce some of them at a high level.

One solution concept is subgame perfect equilibrium (or subgame perfect Nash equilibrium, SPE in short). It is a refinement of NE. This means the set of SPE is a subset of NE. A strategy profile is an SPE equilibrium if, in every subgame of the original game, the strategies remain an NE. A subgame is defined as the partial tree consisting of a node and all its successors, with the requirement that the root node of the partial tree is the only node in its information set. This means if the players play a small game that is part of the original game, they still have no incentive to deviate from their current strategy. A subgame can be a subgame of another subgame rooted at its ancestor node. A finite EFG with perfect recall always has an SPE. To find an SPE in a finite game with perfect information, one can use backward induction as all the information sets are singletons. First, consider the smallest subgames rooted at a parent node of a terminal node. After solving these subgames by determining the action to maximize the acting player’s expected utility, one can solve a slightly larger subgame whose subgames have already been solved, assuming the player will play according to the smaller subgames’ solution. This process continues until the original game is solved. The resulting strategy is an SPE. For games of imperfect or incomplete information, backward induction cannot be applied as it will require reasoning about the information sets with more than one node. The SPE concept can also be defined through the one-shot deviation principle: a strategy profile is an SPE if and only if no player can gain any utility by deviating from their strategy for just one decision point and then reverting back to their strategy in any subgame.

In the example game in Figure $2.1$, if we remove the dashed box, the game becomes one with perfect information, and it has two subgames in addition to the original game, with the root nodes being nodes 2 and 3 respectively. In this new game, Player 1 choosing $L$ and Player 2 choosing $r$ at both nodes 2 and 3 is an NE but not an SPE, because, in the subgame rooted at node 3 , Player 2 choosing $r$ is not the best action. In contrast, Player 1 choosing $R$ and Player 2 choosing $r$ at node 2 , 1 at node 3 is an SPE.

Another solution concept is sequential equilibrium (Kreps and Wilson 1982), which is a further refinement of SPE. A sequential equilibrium consists of not only the players’ strategies but also the players’ beliefs for each information set. The belief describes a probability distribution on the nodes in the information set since the acting player cannot distinguish them when they play the game. Unlike SPE which only considers the subgames rooted from a node that is the only element in its information set, the sequential equilibrium requires that the players are sequentially rational and takes the best action in terms of expected utility concerning the belief at every information set even if it is not a singleton.

## 经济代写|博弈论代写Game Theory代考|Stackelberg Game

Consider a two-player NFG with finite actions. The two players choose an action in their action set simultaneously without knowing the other player’s strategy ahead of time. In an NE of the game, each player’s strategy is a best response to the other player’s strategy. But what if one player’s strategy is always known to other players ahead of time? This kind of role asymmetry is depicted in a Stackelberg game.

In a Stackelberg game, one player is the leader and chooses her strategy first, and the other players are followers who observe the leader’s strategy and then choose their strategies. The leader has to commit to the strategy she chooses, i.e. she cannot announce a strategy to the followers and play a different strategy when the game is actually played. She can commit to either a pure strategy or a mixed strategy. If she commits to a mixed strategy, the followers can observe her mixed strategy but not the realization of sampled action from this mixed strategy. A Stackelberg game can be described in the same way as before, with a tuple $(\mathcal{N}, \mathcal{A}, u)$ for games in normal form and $(\mathcal{N}, \mathcal{A}, \mathcal{H}, \mathcal{Z}, \chi, \rho, \sigma, u)$ for games in extensive form, with the only difference that one of the players (usually Player 1 ) is assumed to be the leader.

At first sight, one may think that the followers have an advantage as they have more information than the leader when they choose their strategies. However, it is not the case. Consider the Football vs. Concert game again. If the row player is the leader and commits to choosing the action $\mathrm{F}$, a rational column player will choose action $\mathrm{F}$ to ensure a positive payoff. Thus, the row player guarantees herself a utility of 2, the highest utility she can get in this game, through committing to playing pure strategy F. In the real world, it is common to see similar scenarios. When a couple is deciding what to do during the weekend, if one of them has a stronger opinion and insists on going to the activity he or she prefers, they will likely choose that activity in the end. In this example game, the leader ensures a utility that equals the best possible utility she can get in an NE. In some other games, the leader can get a utility higher than any NE in the game by committing to a good strategy. In the example game in Table $2.5$ (Up, Right) is the only NE, with the row player’s utility being 4 . However, if the row player is the leader and commits to a uniform random strategy, the column player will choose Left as a best response, yielding an expected utility of 5 for the row player. This higher expected utility for the row players shows the power of commitment.

## 经济代写|博弈论代写Game Theory代考|Solution Concept

As shown in the example, NEs are no longer well suited to prescribe the behavior of players in a Stackelberg game. Instead, the concept of Stackelberg equilibrium is used. A Stackelberg equilibrium in a two-player Stackelberg game is a strategy profile where the follower best responds to the leader’s strategy, and the leader commits to a strategy that can maximize her expected utility, knowing that the follower will best respond. One subtle part of this definition is the tie-breaking rule. There can be multiple best responses. Although they lead to the same expected utility for the follower, they can result in different expected utility for the leader. Therefore, to define a Stackelberg equilibrium, we need to specify a best response function that maps a leader’s strategy to the follower’s best response strategy.

Definition $2.5$ (Best Response Function): $f: S_{1} \mapsto S_{2}$ is a best response function if and only if $u_{2}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{2}\left(s_{1}, s_{2}\right), \forall s_{1} \in S_{1}, s_{2} \in S_{2}$

Since the expected utility of playing a mixed strategy is a linear combination of the expected utility of playing the pure strategies in its support, the constraint is equivalent to $u_{2}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{2}\left(s_{1}, a_{2}\right), \forall s_{1} \in S_{1}, a_{2} \in A_{2}$. The Stackelberg equilibrium is defined based on the best response function definition.

Definition 2.6 (Stackelberg Equilibrium): A strategy profile $s=\left(s_{1}, f\left(s_{1}\right)\right)$ is a Stackelberg equilibrium if $f$ is a best response function and $u_{1}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{1}\left(s_{1}^{\prime}, f\left(s_{1}^{\prime}\right)\right), \forall s_{1}^{\prime} \in S_{1}$.

For some best response functions, the corresponding Stackelberg equilibrium may not exist. A commonly used best response function is one that breaks ties in favor of the leader. The resulting equilibrium is called the strong Stackelberg equilibrium (SSE).

Definition $2.7$ (Strong Stackelberg Equilibrium (SSE)): A strategy profile $s=\left(s_{1}, f\left(s_{1}\right)\right.$ ) is a strong Stackelberg equilibrium if

• $u_{2}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{2}\left(s_{1}, s_{2}\right), \forall s_{1} \in S_{1}, s_{2} \in S_{2}$ (attacker best responds)
• $u_{1}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{1}\left(s_{1}, s_{2}\right), \forall s_{1} \in S_{1}, s_{2} \in B R\left(s_{1}\right)$ (tie-breaking is in favor of the leader)
• $u_{1}\left(s_{1}, f\left(s_{1}\right)\right) \geq u_{1}\left(s_{1}^{\prime}, f\left(s_{1}^{\prime}\right)\right), \forall s_{1}^{\prime} \in S_{1}$ (leader maximizes her expected utility)
SSE is guaranteed to exist in two-player finite games. The weak Stackelberg equilibrium (WSE) can be defined in a similar way, with a tie-breaking rule against the leader. However, WSE may not exist in some games. SSE in two-player normal-form Stackelberg games can be computed in polynomial time through solving multiple linear programs, each corresponding to a possible best response of the follower (Conitzer and Sandholm 2006).

## 经济代写|博弈论代写Game Theory代考|Solution Concept

• 在2(s1,F(s1))≥在2(s1,s2),∀s1∈小号1,s2∈小号2（攻击者最佳响应）
• 在1(s1,F(s1))≥在1(s1,s2),∀s1∈小号1,s2∈乙R(s1)（打破平局有利于领导者）
• 在1(s1,F(s1))≥在1(s1′,F(s1′)),∀s1′∈小号1（领导者最大化她的期望效用）
SSE 保证存在于两人有限博弈中。弱斯塔克伯格均衡（WSE）可以用类似的方式定义，对领导者有一个打破平局的规则。但是，某些游戏中可能不存在 WSE。两人范式 Stackelberg 游戏中的 SSE 可以通过求解多个线性程序在多项式时间内计算，每个线性程序对应于跟随者的可能最佳响应（Conitzer 和 Sandholm 2006）。

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