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# 经济代写|博弈论代考Game theory代写|Rationalizability and Subjective Correlated Equilibria

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## 经济代写|博弈论代考Game theory代写|Rationalizability and Subjective Correlated Equilibria

In matching pennies (figure $1.10 \mathrm{a}$ ), rationalizability allows player 1 to be sure he will outguess player 2 , and player 2 to be sure he’ll outguess player 1; the players strategic beliefs need not be consistent. It is interesting to note that this kind of inconsistency in beliefs can be modeled as a kind of correlated equilibrium with inconsistent beliefs. We mentioned the possibility of inconsistent beliefs when we defined subjective correlated equilibrium, which generalizes objective corrclated equilibrium by allowing each player $i$ to have different belicfs $p_i(\cdot)$ over the joint recommendation $s \in S$. That notion is weaker than rationalizability, as is shown by figure 2.6 (which is drawn from Brandenburger and Dekel 1987). One subjective correlated equilibrium for this game has player l’s beliefs assign probability I 10 (U, L) and player 2 ‘s beliefs assign probability $\frac{1}{2}$ each to (L, L) and (I), L). Given his beliefs, player 2 is correct to play L. However, that strategy is deleted by iterated dominance, and so we see that subjective correlated equilibrium is less restrictive than rationalizability.

The point is that subjective correlated equilibrium allows each player’s beliefs about his opponents to be completely arbitrary, and thus cannot capture the restrictions implied by common knowledge of the payoffs. Brandenburger and Dekel introduce the idea of an a posteriori equilibrium, which does capture these restrictions.

Although this equilibrium concept, like correlated equilibrium, can be defined either with reference to explicit correlating devices or in a “direct version,” it is somewhat simpler here to make the correlating device explicit.

Given state space $\Omega$, partition $H_i$, and priors $p_i(\cdot)$, we now require, for each $\omega$ (even those with $\left.p_i(\omega)=0\right),{ }^5$ that player $i$ have well-defined conditional beliefs $p_i\left(\omega^{\prime} \mid h_i(\omega)\right)$, satisfying $p_i\left(h_i(\omega) \mid h_i(\omega)\right)=1$.

## 经济代写|博弈论代考Game theory代写|What Is a Multi-Stage Game?

Our first step is to give a more precise definition of a “multi-stage game with observed actions.” Recall that we said that this meant that (1) all players knew the actions chosen at all previous stages $0,1,2, \ldots, k-1$ when choosing their actions at stage $k$, and that (2) all players move “simultaneously” in each stage $k$. (We adopt the convention that the first stage is “stage 0 ” in order to simplify the notation concerning discounting when stages are interpreted as periods.) Players move simultaneously in stage $k$ if each player chooses his or her action at stage $k$ without knowing the stage- $k$ action of any other player. Common usage to the contrary. “simultaneous moves” does not exclude games where players move in alternation, as we allow for the possibility that some of the players have the one-clement choice set “do nothing.” For cxample, the Stackelberg game has two stages: In the first stage, the leader chooses an output level (and the follower “does nothing”). In the second stage, the follower knows the leader’s output and chooses an output level of his own (and the leader “does nothing”). Cournot and Bertrand games are one-stage games: All players choose their actions at once and the game ends. Dixit’s (1979) model of entry and entry deterrence (based on work by Spence (1977)) is a more complex example: In the first stage of this game, an incumbent invests in capacity; in the second stage, an entrant observes the capacity choice and decides whether to enter. If there is no entry, the incumbent chooses output as a monopolist in the third stage; if entry occurs, the two firms choose output simultaneously as in Cournot competition.

Often it is natural to identify the “stages” of the game with time periods. but this is not always the case. A counterexample is the Rubinstein-Ståhl model of bargaining (discussed in chapter 4), where each “time period” has two stages. In the first stage of each period, one player proposes an agreement; in the second stage, the other player either accepts or rejects the proposal. The distinction is that time periods refer to some physical measure of the passing of time, such as the accumulation of delay costs in the bargaining model, whereas the stages need not have a direct temporal interpretation.

In the first stage of a multi-stage game (stage 0 ), all players $i \in \mathscr{F}$ simultaneously choose actions from choice sets $A_i\left(h^0\right)$. (Remember that some of the choice sets may be the singleton “do nothing.” We let $h^0=\varnothing$ be the “history” at the start of play.) At the end of each stage, all players observe the stage’s action profile. Let $a^0 \equiv\left(a_1^0, \ldots, a_t^0\right)$ be the stage- 0 action profile. At the beginning of stage 1 , players know history $h^1$, which can be identified with $a^0$ given that $h^0$ is trivial. In general, the actions player $i$ has available in stage 1 may depend on what has happened previously, so we let $A_i\left(h^1\right)$ denote the possible second-stage actions when the history is $h^1$. Continuing iteratively, we define $h^{k+1}$, the history at the end of stage $k$, to be the sequence of actions in the previous periods,
$$h^{k+1}=\left(a^0, a^1, \ldots, a^k\right)$$
and we let $A_i\left(h^{k+1}\right)$ denote player $i$ ‘s feasible actions in stage $k+1$ when the history is $h^{k+1}$. We let $K+1$ denote the total number of stages in the game, with the understanding that in some applications $K=+\infty$, corresponding to an infinite number of stages; in this case the “outcome” when the game is played will be an infinite history, $h^{\infty}$. Since each $h^{K+1}$ by definition describes an entire sequence of actions from the beginning of the game on, the set $I^{K+1}$ of all “terminal histories” is the same as the set of possible outcomes when the game is played.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|What Is a Multi-Stage Game?

$$h^{k+1}=\left(a^0, a^1, \ldots, a^k\right)$$

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