Nash equilibrium means no profitable unilateral deviation. A coordination game shows two pure equilibria, a mixed equilibrium and the limits of efficiency and selection.
Nash equilibrium
A Nash equilibrium is a collection of strategies in which no player can improve their own expected payoff by changing strategy alone, while the other players keep theirs fixed. In a quant market model, this is a consistency test: a proposed quoting or trading policy must remain a best response to the other modelled participants’ policies.
The word alone does the work. Equilibrium does not mean that everyone gets the largest possible payoff, that a group cannot improve by changing together, or that actual traders will converge to the proposed strategies.
Hold the other strategies fixed
In game theory, a strategy profile lists one strategy for each player. Write player i’s strategy as si, everyone else’s as s−i, and their expected payoff as ui(si,s−i). A profile s∗ is a Nash equilibrium when
ui(si∗,s−i∗)≥ui(si,s−i∗)for every player i and every feasible si.
Each player’s strategy is a best response to the others. The best response need not be unique: a change that leaves payoff unchanged does not break equilibrium. The MIT notes, §L1.1.3, formalise this unilateral-deviation test.
This differs from a dominant strategy, which is best regardless of what the others do. In Nash equilibrium, the proposed opponents’ strategies matter. It also differs from an equilibrium point of a differential equation: here the defining condition concerns incentives, rather than a state with zero time derivative.
Two working conventions and one mixture
Consider a fictional, simultaneous, one-round coordination game. Row and Column each choose convention A or B without observing the other’s choice. The entries are (Row utility, Column utility) in arbitrary utility units; these are stipulated preferences, not estimated trading profits.
Row chooses
Column: A
Column: B
A
(3, 3)
(0, 0)
B
(0, 0)
(2, 2)
At (A, A), either player would lose 3 units by switching alone. At (B, B), either would lose 2. Both profiles are pure-strategy Nash equilibria. At either mismatch, each player can improve by matching the other, so neither mismatch is an equilibrium.
Yet both players prefer (A, A) to (B, B). B survives because, starting from (B, B), a player cannot make the other adopt A merely by changing their own action. A jointly attractive switch is different from a profitable unilateral switch.
Now allow each player to randomise independently. A mixed strategy is a probability distribution over actions, not an average action halfway between A and B. Let p be Row’s probability of A and q Column’s. Row’s expected utility from each fixed action is
uR(A,q)=3q,uR(B,q)=2(1−q).
For Row to use both actions with positive probability, neither can be strictly better. Indifference requires
3q=2(1−q)⇒q=52.
That equation determines Column’s probability, because Column’s mix makes Row indifferent. Applying the same calculation to Column gives p=2/5. Both players choose A with probability 0.4, and B with probability 0.6. Each earns
3(0.4)=2(0.6)=1.2
in expected utility. Against an opponent who mixes this way, any own mixture also earns 1.2; changing alone cannot help. However, only the mixture p=q=0.4 makes both players indifferent at once.
Left: action A pays 3q and B pays 2(1−q), meeting at q=0.4 and utility 1.2. Right: Row and Column best responses intersect at (q,p)=(0,0), (0.4,0.4) and (1,1), giving BB, the mixed equilibrium and AA.
Original calculation for the stipulated coordination game. The left panel compares fixed-action expected utilities; the right panel locates mutual best responses, including both pure equilibria and the independent mixed equilibrium.
The mixed outcome is worse in expectation for both players than either pure equilibrium. Independent mixing produces A/A with probability 0.16, B/B with probability 0.36, and a mismatch with probability 0.48. Direct enumeration gives 3(0.16)+2(0.36)=1.2 per player. Equilibrium is a statement about available deviations, not a guarantee of efficiency.
Nor does “randomise” mean “use 50/50”. If both use 50/50 here, each earns 1.25, but switching alone to A raises expected utility to 1.5. The 0.25 gain rules that profile out—even though its payoff exceeds the mixed equilibrium’s 1.2.
What existence does not settle
Nash’s 1950 paper establishes existence for games with finitely many players and finitely many pure strategies when mixed strategies are allowed. Some finite games have no pure equilibrium. The theorem does not say the equilibrium is unique, efficient or reached by a particular learning rule. Continuous action spaces require additional conditions; the finite-game result cannot simply be applied to arbitrary real-valued trading controls.
Our coordination game exposes the selection problem without any missing equilibrium: the model admits three. Predicting one requires further reasoning about the setting, such as communication, conventions, beliefs or adjustment dynamics. Those additions change what the model can claim; an equilibrium calculation by itself supplies none of them.
The consequence for a trading model
Suppose a market maker optimises a quote while treating competitors’ quotes and customer responses as fixed. That solves one player’s optimisation problem. To claim Nash equilibrium, the modeller must also specify the other strategic players’ objectives and feasible policies, then verify that each is best responding to the proposed profile.
The public market-making introduction, especially “Stage five: Why not quote a very wide spread?”, shows why a competitor’s tighter quote changes which dealer receives trades. It is a useful concrete motivation; that article does not solve a full Nash quoting model.
For execution research, the modelling question is whether other participants’ responses should be endogenous. If so, a strategy’s historical performance against a fixed environment does not establish equilibrium performance when rivals adapt. Information and timing also matter: private signals or decisions made after observing earlier trades call for a richer game and an appropriate equilibrium concept. Nash equilibrium remains an incentive test, not evidence that real markets obey a specified payoff matrix.
Sources
John F. Nash Jr., Equilibrium points in n-person games, PNAS 36(1), 48–49 (1950): finite games, mixed strategies, mutual best responses and existence.
Constantinos Daskalakis and Gabriele Farina, Setting and equilibria: the Nash equilibrium, MIT 6.7980 (2026), §§L1.1–L1.2: normal-form games, unilateral deviations, pure versus mixed strategies and existence.
