# Relations among independent games with the same players and multiple equilibria

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?

• What are $Y_1$ and $Y_2$? Mar 8 '17 at 19:03
• Please explain what $\mathcal{A}\perp \mathcal{B}$ stands for.
– mlc
Mar 8 '17 at 22:46
• @PeteCaradonna: $Y_1$ and $Y_2$ are defined as the equilibria of $\mathcal{A}$
– mlc
Mar 8 '17 at 22:47
• @mlc So $Y_1$ is an $N$-tuple of strategies, and $Y_2$ is an $N$-tuple of strategies? Mar 8 '17 at 22:55
• @PeteCaradonna: Yes, this is my understanding
– mlc
Mar 8 '17 at 22:57

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 $A$ *andX_1$in$B\$.