Mixed Equilibria
In this chapter, we expand the range of possible actions that a player can take. Rather than committing to a single action (in a pure strategy), players can now use mixed strategies where they assign a probability to each action. Essentially, they commit to a probabilistic distribution of actions.
We can formalize this concept of a mixed strategy with some notation:
- Let $\alpha_i$ (an alpha, not an a) represent a mixed strategy profile.
- Let $\alpha_i(a_i)$ represent player $i$’s mixed strategy, a function that assigns a probability to each pure strategy $a_i.$
- For convenience like before, $\alpha_{-i}$ denotes ever other player’s mixed strategy.
- For convenience, we can express $\alpha_i$ as an ordered list of probabilities that correspond to actions.
- Let $U_i(\alpha)$ (with a capital U) represent the expected payoff, or Bernoulli payoff function for player $i$ of the mixed strategy profile $\alpha.$ Symbolically, it is defined as $$U_i(\alpha) = \sum_{a_i \in A_i} \alpha_i (a_i) E(a_i, a_{-i}),$$ where $E(a_i, \alpha_{-i})$ represents the expected payoff for player $i$ if they play pure strategy $a_i$ and everyone else plays the mixed profile $\alpha_{-i}.$