Example — Odds or evens

A zero-sum game with no pure-strategy equilibrium: the only stable choice is to randomise in equal parts. It is the basic model of mixed-strategy equilibrium.

Example — Zero-sum game: odds or evens

Two players simultaneously show $1$ or $2$ fingers. G1 wins $1$ EUR if the sum is even; G2 wins $1$ EUR if it is odd. Payoff matrix (for G1):

G1 / G2	1	2
1	$+1$	$-1$
2	$-1$	$+1$

(G2’s payoffs are the opposites: zero-sum game.)

Pure strategies: none dominant; no pure-strategy equilibrium (if G1 fixes a choice, G2 deduces their own winning move).

Mixed strategies: G1 plays “1” with probability $p$, G2 with probability $q$. G1’s expected winnings: $\bar V_1(p,q) = pq - p(1-q) - (1-p)q + (1-p)(1-q) = (2p-1)(2q-1).$ G2 chooses $q$ to minimise $\bar V_1$; if $p\neq 1/2$, G2 chooses $q$ at the extremes to push $\bar V_1$ negative. The only robust choice for G1 is $p=1/2$ (G1 indifferent to the opponent). By symmetry $q=1/2$.

Mixed-strategy Nash equilibrium: both play $1/2$-$1/2$. Expected winnings of each: $\bar V = 0$ (fair game).

Links

Topics: Probabilita
Concepts: Equilibrio di nash · Gioco a somma zero · Strategia mista
Methods: Equilibrio nash · Gioco somma zero · Strategia mista
Skills: Calcolo probabilita · Modellizzare