Section 23.2

Mixed Strategies and Equilibria

Simultaneous-move games: zero-sum games on intervals or integers, rock-paper-scissors variants, duels, equilibria in auctions and redraw games.

Toolkit

A mixed-strategy Nash equilibrium requires each player to randomise so that every pure strategy in the support of the equilibrium mixture yields the same expected payoff against the opponent’s strategy; otherwise a player would deviate to the best pure response. For two-player zero-sum games this is equivalent to minimax: a strategy is optimal iff it guarantees the value against every pure reply. Useful facts:

Strict or weak dominance lets you delete strategies before solving; the reduced game has the same equilibria.
Indifference equations: if player uses mixture $p$ over support $S$ , then $\mathbb{E}[\text{payoff}\mid s]$ is constant over $s\in S$ and at least as large for $s\notin S$ .
On a continuum, best-response payoffs are often piecewise linear or smooth in the opponent’s parameter; equilibrium pins down where derivatives vanish or boundary values tie.
For games with continuous mixtures over $\{1,2,\dots\}$ or an interval, a density/pmf proportional to a tractable kernel (e.g. $1/n^2$ ) can equalise payoffs against all pure replies, certifying minimax optimality via $\sum$ or $\int$ comparisons.
In duels/auctions, equilibrium conditions often reduce to a threshold where a marginal player is exactly indifferent between two actions, giving an algebraic (often quadratic) equation for the cutoff.

Problems