Strategy

EC487_Advanced_Microeconomics

Players choose their strategies. In a game, they lottery over their action sets:

$$S^j=\Delta (A^j)=\left\{s^j:A^j\to {\mathbb{R}}_{+}\text{s.t.}\sum _{a^j\in A^j}{s}^j(a^j)=1\right\}$$

and $S={\times }_{j=1}^N{S}^j$.

The easiest way to understand $s^j$ is that it is a probability distribution over the action set $A^j$.

Knowing everyone’s strategy (and actions), the expected payoff for player $j$ is:

$$v^j(s^j,s^{-j})=\sum _{a^1\in A^1}\cdots \sum _{a^N\in A^N}{u}^j(a^1,\dots ,a^N)s^1(a^1)\cdots s^N(a^N)$$

Implications: Given a strategy profile $(s^j,s^{-j})$, individual beliefs and other players’ strategies coincide, “no surprise”!