SMU H3 Notes Game TheoryGamesSMU H3

Zero-Sum Matching Pennies Game

Game theory analysis: Zero-Sum Matching Pennies Game.

1 May 2026

Content map: SMU H3 Game Theory Map

Setup

Definition:

Zero-Sum Matching Pennies Game

Players: Player 1 and Player 2.

Strategies: Each player chooses H or T.

Rules: Players move simultaneously; Player 1 wins on a match and Player 2 wins on a mismatch.

Payoff Matrix

	H	T
H	1, -1	-1, 1
T	-1, 1	1, -1

Derivation (Best Response Analysis)

No pure-strategy NE: at any pure profile, one player can profitably deviate.

Derivation (Mixed-Strategy Indifference)

Let Player 2 play H with probability $q$ . Player 1’s expected payoffs:

E[\pi_1(\text{H})]=q\cdot 1+(1-q)\cdot(-1)=2q-1,\qquad E[\pi_1(\text{T})]=q\cdot(-1)+(1-q)\cdot 1=1-2q.

Indifference for Player 1 requires $2q-1=1-2q \Rightarrow q=\tfrac{1}{2}$ .
By symmetry, Player 1 also mixes H with probability $\tfrac{1}{2}$ .

Nash Equilibrium

Result:

No pure-strategy NE. Unique mixed-strategy NE:

p_1(\text{H})=\tfrac{1}{2},\quad p_2(\text{H})=\tfrac{1}{2}

Insights

Insight:

In strictly competitive games without coordination, equilibrium often requires randomisation to make the opponent indifferent.

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