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Consider two games with the same number $n$ of players.

Let $\mathcal{A}:=\{Y_1,Y_2\}$ be the set of equilibria of game 1. $Y_i$ is an $n\times 1$ vector reporting the action of each player, for $i=1,2$.

Let $\mathcal{B}:=\{X_1,X_2\}$ be the set of equilibria of game 2. $X_i$ is an $n\times 1$ vector reporting the action of each player, for $i=1,2$.

The selection rule is defined as "the rule according to which players pick the outcome to play among the predicted equilibria".

(*) Suppose that "the selection rule of game 1 is independent of the selection rule of game 2". In other words, how players pick the outcome to play in game 1 does not affect how players pick the outcome to play in equilibrium 2.

Now, let's attach some probabilities (denoted by "Pr") to these objects and consider

$$ Pr(\text{players pick $Y_1$ in game 1}| \text{the set of equilibria of game 1 is $\mathcal{A}$}) $$ and $$ Pr(\text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 2 is $\mathcal{B}$}) $$ Does (*) imply that $$ Pr(\text{players pick $Y_1$ in game 1}| \text{the set of equilibria of game 1 is $\mathcal{A}$})\times Pr(\text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 2 is $\mathcal{B}$})= Pr(\text{players pick $Y_1$ in game 1}, \text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 1 is $\mathcal{A}$}, \text{the set of equilibria of game 2 is $\mathcal{B}$}) $$ ? Or do we need other conditions?

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  • $\begingroup$ What are $Y_1$ and $Y_2$? $\endgroup$ Mar 8 '17 at 19:03
  • $\begingroup$ Please explain what $\mathcal{A}\perp \mathcal{B}$ stands for. $\endgroup$
    – mlc
    Mar 8 '17 at 22:46
  • $\begingroup$ @PeteCaradonna: $Y_1$ and $Y_2$ are defined as the equilibria of $\mathcal{A}$ $\endgroup$
    – mlc
    Mar 8 '17 at 22:47
  • $\begingroup$ @mlc So $Y_1$ is an $N$-tuple of strategies, and $Y_2$ is an $N$-tuple of strategies? $\endgroup$ Mar 8 '17 at 22:55
  • $\begingroup$ @PeteCaradonna: Yes, this is my understanding $\endgroup$
    – mlc
    Mar 8 '17 at 22:57
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A random selection rule is not usual in game theory, but let's go with it.

I find your notation a tad confusing, because it suggests that "the set of equilibria of game 1" is a conditioning event and hence that it should be probabilised.

My guess (and of course I may be wrong) is that you mean to say that game $A$ has two equilibria and game $B$ has two equilibria. Let $p$ the probability of pick $Y_1$ in game $A$ and $q$ the probability to pick $X_1$ in game $B$. Then independent implies that $pq$ is the probability to pick $Y_1$ in $A$ *and$ $X_1$ in $B$.

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