For two player games, with payoff matrices $(A,B)$, let $x \in \Delta_x$ denote the mixed strategy of player $1$, and $y \in \Delta_y$ denote the mixed strategy of player $2$.
Then the payoff function for player $1$ is written as,
$\mathcal{U}^1(x,y) = x^TAy$
and for player $2$ is
$\mathcal{U}^2(x,y) = y^TBx$
See definition 4:
What would the payoff matrix look like for games with more than two players? Can someone provide a simple example?
$\endgroup$1 Answer
$\begingroup$In definition 4, the expected payoff for Player $i$ could instead be written as$$
U_i(\boldsymbol{p},\boldsymbol{q})=\sum_{(s,t)\in S\times T}u_i(s,t)p_sq_t.
$$This say the utility of a pair of mixed strategies $(\boldsymbol{p},\boldsymbol{q})$ for Player $i$ is
the SUM of the utility of a pair of pure strategies $(s,t)$ for Player $i$ MULTIPLIED BY the probability $s$ is chosen MULTIPLIED BY the probability $t$ is chosen
ranging over all pairs of strategies $(s,t)$ in $S\times T$.
For three players, where for the third player their set of strategies is $U$, the definition is then$$
U_i(\boldsymbol{p},\boldsymbol{q},\boldsymbol{r})=\sum_{(s,t,u)\in S\times T\times U}u_i(s,t,u)p_sq_tr_u.
$$Here's an example (ignore the part on finding equilibria):
