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经济代写|博弈论代考Game theory代写|ECON7062 Perfect folk theorem for limiting means

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经济代写|博弈论代考Game theory代写|Perfect folk theorem for limiting means

The following theorem is due to Aumann and Shapley [6], as well as Rubinstein [30].
Theorem 7.6. For every feasible, strictly enforceable payoff profile $w$ there exists a subgame perfect Nash equilibrium of $G$ with limiting means utilities whose associated payoff profile is $w$.

The idea behind these equilibria is still of punishing, but just for some time rather than for all infinity. Since lower stage utilities are only attained in finitely many periods there is no loss in the limit.

Proof of Theorem 7.6. As in the proof of Theorem 7.3, let $w_i=\sum_{a \in A} \alpha_a \cdot u_i(a)$, and let $\left(a^1, a^2, \ldots\right)$ be a sequence in $A$ such that $m\left(\mathbb{1}{\left{a^1=a\right}}, \mathbb{1}{\left{a^2=a\right}}, \ldots\right)=\alpha_a$. Likewise, for each player $i$ let $s_i^*$ be the strategy in which she chooses $a_i^t$, unless in some previous period $\tau$ some player $j$ deviated and did not choose $a_j^\tau$. In the latter case, we find for each such $\tau$ and $j$ a $\tau^{\prime}$ large enough so that, if all players but $j$ play
$$b_{-j} \in \underset{a_{-i}}{\operatorname{argmin}} \max {a_i} u_i\left(a{-1}, a_i\right)$$
in time periods $\left(\tau+1, \ldots, \tau^{\prime}\right)$ then the average of player $j$ ‘s payoffs in periods $\left(\tau, \tau+1, \ldots, \tau^{\prime}\right)$ is lower than $w_j$. Such a $\tau^{\prime}$ exists since the payoffs in periods $\left(\tau+1, \ldots, \tau^{\prime}\right)$ will all be at most $u_j^{m m}$, and since $w_j>u_j^{m m}$. We let all players but $j$ play $b_{-j}$ in time periods $\tau+1, \ldots, \tau^{\prime}$. We do not consider these punishments as punishable themselves, and after period $\tau^{\prime}$ all players return to playing $a_i^t$ (until the next deviation).

经济代写|博弈论代考Game theory代写|Perfect folk theorems for discounting

We next turn to proving perfect folk theorems for discounted utilities. An early, simple result is due to Friedman [12].

Theorem 7.7. Let $G_0$ have a pure Nash equilibrium $s^*$ with payoff profile $z$. Let $w$ be a payoff profile of some strategy profile a $\in A$ of $G_0$ such that $w_i>z_i$ for all $i \in \mathbb{N}$. Then there is a $\delta_0>0$ such that for all $\delta>\delta_0$ there exists a subgame perfect equilibrium of $G$ under discounting, with payoff profile $w$.

The idea behind this result is simple: the players all play a unless someone deviates. Once anyone has deviated, they all switch to playing $s^$. Given that $\delta$ is close enough to 1 , the deviant’s utility from playing $s^$ henceforth will trump any gains from the deviation. Since $s^*$ is an equilibrium, there is no reason for the punishers to deviate from the punishment.

A harder result is due to Fudenberg and Maskin [13] who, for the two player case, extend the Nash folk theorem 7.4 to a perfect Nash folk theorem.

博弈论代写

经济代写博恋论代考Game theory代写|Perfect folk theorem for limiting means

$$b_{-j} \in \underset{a-i}{\operatorname{argmin}} \max a_i u_i\left(a-1, a_i\right)$$

经济代写|博恋论代考Game theory代写|Perfect folk theorems for discounting

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