A mixed strategy equilibrium passes the best-response test. Learn which actions must tie, whose probability indifference determines and why omitted actions matter.
Mixed strategy equilibrium
A mixed strategy equilibrium is a Nash equilibrium in which players choose probability distributions over their available strategies and each distribution is a best response to the others. The term usually highlights an equilibrium where at least one player actually randomises; a pure strategy is also a probability distribution, concentrated on one choice.
For a quant modelling competing execution or quoting rules, the useful question is whether a proposed randomised policy survives a rival's best response. Adding randomness to a policy does not establish that consistency.
The actions you use must earn the same payoff
Consider a finite, simultaneous game with independent randomisation and expected-utility objectives. A player's support consists of the actions assigned positive probability. Hold the opponents' strategies fixed and write vi(a) for the expected utility of choosing action a. If the player's probabilities are xi(a), their utility is
Ui=a∑xi(a)vi(a).
This is a weighted average. Putting positive weight on a strictly worse action lowers the average: the player could transfer that weight to a better action. Therefore a best response satisfies
xi(a)>0xi(a)=0⇒vi(a)=Ui,⇒vi(a)≤Ui.
Used actions must tie at the maximum; unused actions cannot pay more. If this holds for every player, no unilateral change of probabilities can improve utility. This follows from payoff linearity and the pure-deviation test in MIT's equilibrium notes, Remark L1.3. Indifference need not extend to every available action.
A quick anchor: matching pennies
Row and Column simultaneously choose H or T. Row receives 1 token when they match and loses 1 when they differ; Column receives the opposite payoff. Both maximise expected tokens. This is a zero-sum game.
Let p be Row's probability of H and q Column's. Row's fixed-action payoffs are 2q−1 for H and 1−2q for T. They tie at q=1/2. Column's fixed-action payoffs are 1−2p and 2p−1, which tie at p=1/2.
At independent fair mixing, each of the four action pairs has probability 1/4 and each player's expected payoff is zero. Every own mixture earns zero against the opponent's fair mix, so neither can improve alone. Individual outcomes still pay +1 or −1. Magazinnik's lecture, slides 30–39, develops the underlying calculation.
Whose probability are you solving for?
Symmetry can hide the key step. Consider this original fictional one-round game, with entries (Row utility, Column utility) in arbitrary units:
Row chooses
Column: L
Column: R
A
(3, 0)
(0, 1)
B
(0, 2)
(1, 0)
Write p=P(Row chooses A) and q=P(Column chooses L). To make Row willing to use both actions,
3q=1−q⇒q=41.
Row's indifference determines Column's probability. It is Column's behaviour that changes the relative attractions of A and B. Column's indifference then determines Row's probability:
2(1−p)=p⇒p=32.
The independent outcome probabilities provide a separate check:
Outcome
Probability
Row's utility contribution
Column's utility contribution
A/L
1/6
1/2
0
A/R
1/2
0
1/2
B/L
1/12
0
1/6
B/R
1/4
1/4
0
Total
1
3/4
2/3
Each fixed Row action also earns 3/4; each fixed Column action earns 2/3. Consequently every alternative own mixture earns the same amount against the equilibrium opponent. There is no profitable unilateral deviation. None of the four pure profiles passes that test.
Now change only Column's payoff at B/L from 2 to 4. Row's equation still gives q=1/4, but Column's becomes 4(1−p)=p, giving p=4/5. Row must choose A more often to offset Column's stronger attraction to L when B occurs.
Row's A and B utilities intersect at Column probability q=1/4 with utility 3/4. Column's L and R utilities intersect at Row probability p=2/3. Raising Column's B/L payoff from 2 to 4 shifts that intersection to p=4/5; q remains 1/4.
Original fictional game: the left panel fixes Column's probability through Row's indifference; the right shows how changing Column's incentives shifts Row's probability. Lines show expected utilities, not realised paths.
Equalities find candidates; deviations certify them
Two checks prevent common mistakes:
Do not default to equal probabilities. In the original asymmetric game, independent 50/50 play gives Row utility 1 and Column utility 3/4. Row can switch to always A and earn 3/2; Column can switch to always L and earn 1. Both have profitable deviations.
Check actions outside the proposed support. Add a third Row action C paying 1 against either Column action. The old A/B indifference equation still holds at q=1/4, but C beats 3/4 by 1/4. The old candidate is no longer an equilibrium of the enlarged game, whatever Column's new C-row payoffs are.
For larger finite games, the same reasoning applies: propose supports, solve indifference and probability constraints, then check every omitted action. A negative probability or one above 1 invalidates that proposed support. A tie or a zero denominator needs separate analysis; it does not establish absence of equilibrium.
Interpret the randomisation carefully
Mixing over actions differs from holding a portfolio split across assets: a probability of choosing A is not a fractional implementation of A. It also differs from simply being unsure which fixed action an opponent chose. A mixed-strategy model specifies distributions and how their draws generate outcomes. Shared signals can correlate actions, so independence cannot be inferred from marginal frequencies; MIT's discussion of correlated play explains the distinction.
The public SelfMadeQuant market-making introduction, especially stage five, shows why competing quotes change who receives trades. That motivates specifying rivals' incentives before claiming equilibrium. It does not supply the payoff table above or solve a mixed quoting equilibrium. Real order timing, private information and inventory constraints require an appropriate richer game; observed randomness alone identifies none of its equilibrium probabilities.
Sources
Constantinos Daskalakis and Gabriele Farina, Setting and equilibria: the Nash equilibrium, MIT 6.7980 (2026), §L1.1 and Remark L1.3: distributions over actions, expected utilities and pure-deviation verification.
Asya Magazinnik, Lecture 3: Mixed Strategy Nash Equilibrium, MIT 17.810/17.811 (2021), slides 30–44: mixed best responses, payoff linearity, indifference and unequal mixing probabilities.
Constantinos Daskalakis and Gabriele Farina, Properties of Nash equilibrium, MIT 6.7980 (2026), §L3.2: independent mixed strategies versus correlated play.
