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# 经济代写|博弈论代考Game theory代写|Applications of the Elimination of Dominated Strategies

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## 经济代写|博弈论代考Game theory代写|Applications of the Elimination of Dominated Strategies

In this subsection we present two classic games in which a single round of elimination of dominated strategies reduces the strategy set of each player to a single pure strategy. The first example uses the elimination of strictly dominated strategies, and the second uses the elimination of weakly dominated strategies.
Example 1.1: Prisoner’s Dilemma
One round of the elimination of strictly dominated strategies gives a unique answer in the famous “prisoner’s dilemma” game, depicted in figure 1.7. The story behind the game is that two people are arrested for a crime. The police lack sufficient evidence to convict either suspect and consequently need them to give testimony against each other. The police put each suspect in a different cell to prevent the two suspects from communicating with each other. The police tell each suspect that if he testifies against (doesn’t cooperate with) the other, he will be released and will receive a reward for testifying, provided the other suspect does not testify against him. If neither suspect testifies, both will be released on account of insufficient evidence, and no rewards will be paid. If onc testifies, the other will go to prison; if both testify, both will go to prison, but they will still collect rewards for testifying. In this game, both players simultaneously choose between two actions. If both players cooperate (C) (do not testify), they get 1 each. If they both play noncooperatively (D, for defect), they obtain 0 . If one cooperates and the other does not, the latter is rewarded (gets 2) and the former is punished (gets -1 ). Although cooperating would give each player a payoff of 1 , self-interest leads to an inefficient outcome with payoffs 0 . (To readers who feel this outcome is not reasonable, our response is that their intuition probably concerns a different game– perhaps one where players “feel guilty” if they defect, or where they fear that defecting will have bad consequences in the future. If the game is played repcatedly, other outcomes can be equilibria; this is discussed in chapters 4,5 , and 9.)

Many versions of the prisoner’s dilemma have appeared in the social sciences. One example is moral hazard in teams. Suppose that there are two workers, $i=1,2$, and that each can “work” $\left(s_i=1\right)$ or “shirk” $\left(s_i=0\right)$. The total output of the team is $4\left(s_1+s_2\right)$ and is shared equally between the two workers. Each worker incurs private cost 3 when working and 0 when shirking. With “work” identified with C and “shirk” with D, the payoff matrix for this moral-hazard-in-teams game is that of figure 1.7, and “work” is a strictly dominated strategy for each worker.

## 经济代写|博弈论代考Game theory代写|Definition of Nash Equilibrium

A Nash equilibrium is a profile of strategies such that each player’s strategy is an optimal response to the other players’ strategies.

Definition 1.2 A mixed-strategy profile $\sigma^$ is a Nash equilibrium if, for all players $i$, $$u_i\left(\sigma_i^, \sigma^\right) \geq u_i\left(s_i, \sigma_{-i}^\right) \text { for all } s_i \in S_i .$$
A pure-strategy Nash equilibrium is a pure-strategy profile that satisfies the same conditions. Since expected utilities are “linear in the probabilities,” if a player uses a nondegenerate mixed strategy in a Nash equilibrium (one that puts positive weight on more than one pure strategy) he must be indifferent between all pure strategies to which he assigns positive probability. (This linearity is why, in equation 1.2, it suffices to check that no player has a profitable pure-strategy deviation.)

A Nash equilibrium is strict (Harsanyi 1973b) if each player has a unique best response to his rivals’ strategies. That is, $s^*$ is a strict equi librium if and only if it is a Nash equilibrium and, for all $i$ and all $s_i \neq s_i^$, $$u_i\left(s_i^, s^{ }i\right)>u_i\left(s_i, s{-i}^\right)$$
By definition, a strict equilibrium is necessarily a pure-strategy equilibrium. Strict equilibria remain strict when the payoff functions are slightly perturbed, as the strict inequalities remain satisfied. 4.5

Strict equilibria may seem more compelling than equilibria where players are indifferent between their equilibrium strategy and a nonequilibrium response, as in the latter case we may wonder why players choose to conform to the equilibrium. Also, strict equilibria are robust to various small changes in the nature of the game, as is discussed in chapters 11 and 14. However, strict equilibria need not exist, as is shown by the “matching pennies” game of example 1.6 below: The unique equilibrium of that game is in (nondegenerate) mixed strategies, and no (nondegenerate) mixed-strategy equilibrium can be strict. ${ }^6$ (Even pure-strategy equilibria need not be strict; an example is the profile $(D, R)$ in figure 1.18 when $\lambda=0$.)

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Definition of Nash Equilibrium

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