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# 经济代写|博弈论代考Game theory代写|Existence of Nash Equilibrium in Infinite Games with Continuous Payoffs

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## 经济代写|博弈论代考Game theory代写|Existence of Nash Equilibrium in Infinite Games with Continuous Payoffs

lconomists often use models of games with an uncountable number of actions (as in the Cournot game of example 1.3 and the Hotelling game of example 1.4). Some might argue that prices or quantities are “really” infinitely divisible, while others might argue that “reality” is discrete and the continuum is a mathematical abstraction, but it is often easier to work with a continuum of actions rather than a large finite grid. Moreover, as I)asgupta and Maskin (1986) argue, when the continuum game does not have a Nash equilibrium, the equilibria corresponding to fine, discrete grids (whose existence was proved in subsection 1.3.1) could be very sensitive to exactly which finite grid is specified: If there were equilibria of the finite-grid version of the game that were fairly insensitive to the choice of the grid, one could take a sequence of finer and finer grids “converging” to the continuum, and the limit of a convergent subsequence of the discreteaction-space equilibria would be a continuum equilibrium under appropriate continuity assumptions. (To put it another way, one can pick equilibria of the discrete-grid version of the game that do not fluctuate with the grid if the continuum game has an equilibrium.)

Theorem 1.2 (Debreu 1952; Glicksberg 1952; Fan 1952) Consider a strategic-form game whose strategy spaces $S_i$ are nonempty compact convex subsets of an Euclidean space. If the payoff functions $u_i$ are continuous in $s$ and quasi-concave in $s_i$, there exists a pure-strategy Nash equilibrium. ${ }^{22}$
Proof The proof is very similar to that of Nash’s theorem: We verify that continuous payoffs imply nonempty, closed-graph rcaction correspondences, and that quasi-concavity in players’ own actions implies that the reaction correspondences are convex-valued.

## 经济代写|博弈论代考Game theory代写|Iterated Strict Dominance: Definition and Properties

Definition 2.1 The process of iterated deletion of strictly dominated strategies proceeds as follows: Set $S_i^0 \equiv S_i$ and $\Sigma_i^0 \equiv \Sigma_i$. Now define $S_i^n$ recursively by
$S_i^n=\left{s_i \in S_i^{n-1} \mid\right.$ there is no $\sigma_i \in \Sigma_i^{n-1}$ such that
$$\left.u_i\left(\sigma_i, s_{-i}\right)>u_i\left(s_i, s_{-i}\right) \text { for all } s_{-i} \in S_{-i}^{n-1}\right}$$
and define
$$\Sigma_i^n=\left{\sigma_i \in \Sigma_i \mid \sigma_i\left(s_i\right)>0 \text { only if } s_i \in S_i^n\right} .$$
Set
$$S_i^x=\bigcap_{n=0}^1 S_i^n .$$
$S_i^x$ is the set of player $i$ ‘s pure strategies that survive iterated deletion of strictly dominated strategies. Set $\Sigma_i^{\infty}$ to be all mixed strategies $\sigma_i$ such that there is no $\sigma_i^{\prime}$ with $u_i\left(\sigma_i^{\prime}, s_{-i}\right)>u_i\left(\sigma_i, s_{-i}\right)$ for all $s_{-i} \in S_{-i}^x$. This is the set of player $i$ ‘s mixed strategies that survive iterated strict dominance.

In words, $S_i^n$ is the set of player $i$ ‘s strategies that are not strictly dominated when players $j \neq i$ are constrained to play strategies in $S_j^{n-1}$ and $\Sigma_i^n$ is the set of mixed strategies over $S_i^n$. Note, however, that $\Sigma_i^{\infty}$ may be smaller than the set of mixed strategies over $S_i^{\infty}$. The reason for this, as was shown in figure 1.3 , is that some mixed strategies with support $S_i^x$ can be dominated. (In that example, $S_i^x=S_i$ for both players $i$ because no pure strategy is eliminated in the first round of the process.)

Note that in a finite game the sequence of iterations defined above must cease to delete further strategies after a finite number of steps. The intersection $S_i^{\prime}$ is simply the final set of surviving strategics. Note also that each step of the iteration requires one more level of the assumption “I know that you know… that I know the payoffs.” For this reason, conclusions based on a large number of iterations tend to be less robust to small changes in the information players have about one another.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Iterated Strict Dominance: Definition and Properties

2.1严格支配策略的迭代删除过程如下:设置$S_i^0 \equiv S_i$和$\Sigma_i^0 \equiv \Sigma_i$。现在递归地定义$S_i^n$
$S_i^n=\left{s_i \in S_i^{n-1} \mid\right.$没有$\sigma_i \in \Sigma_i^{n-1}$这样的
$$\left.u_i\left(\sigma_i, s_{-i}\right)>u_i\left(s_i, s_{-i}\right) \text { for all } s_{-i} \in S_{-i}^{n-1}\right}$$

$$\Sigma_i^n=\left{\sigma_i \in \Sigma_i \mid \sigma_i\left(s_i\right)>0 \text { only if } s_i \in S_i^n\right} .$$

$$S_i^x=\bigcap_{n=0}^1 S_i^n .$$
$S_i^x$是参与人$i$剔除严格劣势策略后幸存下来的纯策略集合。将$\Sigma_i^{\infty}$设置为所有混合策略$\sigma_i$，这样对于所有$s_{-i} \in S_{-i}^x$就不会有$\sigma_i^{\prime}$和$u_i\left(\sigma_i^{\prime}, s_{-i}\right)>u_i\left(\sigma_i, s_{-i}\right)$。这是玩家$i$在严格优势迭代中幸存下来的混合策略集合。

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