Mixed strategies: probability distributions and indifference in an abstract game

Define a distribution over the available actions

Two players each choose A or B simultaneously. Row receives one utility unit if the labels match and minus one if they differ; Column receives the opposite value. The Yale lecture defining mixed strategies explains that probabilities are nonnegative and sum to one. Let p be Row’s probability of A and q Column’s probability of A. Their probabilities of B are 1−p and 1−q. The utility units here are invented, with no monetary stakes or real-world performance interpretation.

Check why no pure profile is stable

If both choose the same label, Column prefers switching to create a mismatch. If they choose different labels, Row prefers switching to create a match. Thus every pure action pair offers one player an improving unilateral deviation. This exhausts the four possibilities, establishing that this particular game has no pure Nash equilibrium. It does not establish the absence of an equilibrium once probability distributions are available. Enlarging the strategy space changes what must be checked, while preserving the same underlying payoff table.

Calculate joint probabilities under independent mixing

The four probabilities sum to one. Row’s expected payoff is 1/4−1/4−1/4+1/4=0, and Column’s is also zero. Each realised payoff is still either one or minus one. An expected value of zero describes the probability-weighted average, not a fifth terminal outcome in which both players necessarily receive zero.

Use indifference to identify the candidate mix

Against Column’s probability q of A, Row’s expected utility from pure A is q−(1−q)=2q−1. From pure B it is (1−q)−q=1−2q. These are equal when q=1/2. Similarly, Column is indifferent between its two labels when p=1/2. At p=q=1/2 neither player can improve by changing their own distribution while the other distribution remains fixed. Because every mixture is a weighted average of the two pure-action expected utilities, equal zero values rule out any profitable unilateral mixture in this abstract model.

Know what the support condition requires

In Nash’s original equilibrium formulation, strategies used with positive probability at equilibrium must be best responses to the others. This is stronger than merely making some chosen actions equal in value. If an omitted third action gave a higher expected payoff, a mixture over two equally valued but inferior actions would not be a best response. Check both indifference within the positive-probability support and the absence of a better action outside it. A pure strategy is included as the special case assigning probability one to one action.

Distinguish probability from a visible schedule

Choosing A and B on a known alternating schedule can produce a fifty-fifty frequency over an even number of rounds, but it is not the same information structure as independent random choices. An opponent who observes the schedule can condition on the next label. Likewise, two players following a shared random signal can have correlated choices even if each marginal distribution is half A and half B. Marginal frequencies alone do not establish the four equal joint probabilities in the table. State the randomisation and information assumptions explicitly.

Keep formal equilibrium separate from observed behaviour

The Manchester course notes on game models distinguish mathematical representations and the reasoning applied to them. A mixed equilibrium is a property of specified utilities and available choices. It does not prove that people implement exact probabilities, that a short observed sequence matches theoretical frequencies or that every strategic interaction should be randomised. Empirical evaluation would require observations and a design appropriate to the question. The abstract calculation here establishes a model result while leaving those behavioural questions open.

Mixed-strategy calculation checklist

• Specify pure actions and all payoff pairs.
• Define probabilities that are nonnegative and sum to one.
• State whether players’ random choices are independent.
• Calculate expected utility for every pure action.
• Check supported actions are equally best, not merely equal.
• Test omitted actions and unilateral changes.
• Separate expected values from individual outcomes and observed frequencies.

The matching example is useful because every calculation fits in a small table. Retaining the general symbols p and q before substituting one half makes the logic inspectable: each player’s indifference is determined by the other player’s mixture. That distinction prevents a common error in which a player’s own probabilities are used to equalise the wrong payoff expressions.

Sources and verification

• Game Theory, Lecture 9: mixed strategies — Open Yale Courses — 2026-10-23
• Non-Cooperative Games, doctoral dissertation, 1950 — John Nash; Princeton University Library archive — 2026-10-23
• Games, Games, Games: notes for the game theory course — Andrea Schalk, University of Manchester — 2026-10-23