Mixed strategies

Understand randomization as a probability distribution, not a predictable alternating script.

12 minutes · advanced · Original Poker4Quant lesson

What you will learn

Verify a matching-pennies equilibrium and distinguish robustness from exploitation.

Why a pure action can invite a response

In matching pennies, row receives +1 if both players choose the same symbol and −1 otherwise. Column receives the negative. If row always chooses H, column can choose T and make row lose. If row always chooses T, column chooses H.

A mixed strategy assigns probabilities to actions before the simultaneous move. With row choosing H and T equally, either pure column choice gives row expectation zero. The same holds for column’s equal mixture, so neither player can improve by a unilateral switch.

A robust mixture need not maximize value against a known bias

The equal equilibrium mixture protects value zero against every column mixture in this toy game. Against an opponent known to choose H 80% of the time, row could instead choose H always and earn expected payoff 0.8−0.2=0.6.

That exploit depends on the opponent behavior estimate and on whether it stays fixed. A deterministic H,T,H,T schedule is not the same information model as a fresh hidden random choice if the opponent can predict its next element. The timing and observability of the randomization matter.

Worked example

Row chooses H with probability 80% in matching pennies. What does a best-responding column player do?

If column chooses H, row earns 0.8×1+0.2×(−1)=0.6.
If column chooses T, row earns 0.8×(−1)+0.2×1=−0.6.
Column minimizes row payoff in the zero-sum model.

Column chooses T and gives row −0.6. An unbalanced mixture is exploitable here, even though neither pure row action is universally bad.

Common mistakes

Equating a visible alternating schedule with hidden randomization.
Claiming equilibrium always earns more against every fixed opponent.
Forgetting which player maximizes or minimizes the displayed payoff.

Keep these ideas

A mixture is a distribution over actions.
Equilibrium robustness and opponent-specific exploitation are distinct goals.
Randomness must respect the model’s information assumptions.