## Strategies in Normal Games

Today I’m going to go over the terminology for different strategies of normal games. You can find the information I’m about to write in Essentials of Game Theory, but I will be elaborating on some important concepts and hopefully making these concepts easier to understand. I will be using some symbols from my previous post. See here if you want a refresher.

1. Pure strategy: A strategy that is a single action chosen by a player to use throughout the entire course of the game.
2. Mixed strategy: A strategy that is a combination of different pure strategies where each pure strategy is played with a fixed probability (or a random probability) throughout the game. Defining $\Pi(X)$ to be the probability distribution over any set $X$, the set of all mixed strategies, $S_i$ available to player $i$ in a normal game is $S_i = \Pi(A_i)$. The set of the mixed strategies for all players in a game is the set of combinations of possible mixed strategies for each player (assuming we have $n$ players in a game): $S_1 \times ... \times S_n$.
• Expected utility: The expected utility for player $i$ of a mixed strategy game is the sum of the expected utilities of each strategy multiplied by the probability of using that strategy. Given a mixed strategy $s_i$, let $s_i(a_i)$ be the probability action $a_i$ will be played. Let $u_i(a)$ be the utility obtained by taking an action $a$. Therefore, the expected utility, $u_i$ of a player $i$ with a mixed strategy profile $s = (s_1, ..., s_n)$ composed of all the strategies for every player in the game can be defined as:
$u_i(s) = \displaystyle\sum\limits_{a \epsilon A_i} u_i(a) \prod\limits_{j=1}^n s_j(a_j)$

I will be talking about optimal utility functions and equilibrium in my next post.