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Best Response and Nash Equilibrium

Dominance settles the easy games. In the rest, your best action genuinely depends on theirs, and the question becomes which pairs of choices are stable.

Best response

Your best response to a particular action of theirs is whatever maximises your payoff given that action. It is a function of their choice, not a single answer, and writing it down for each of their options is the whole of the work.

In the quoting game with a wide half-spread of 2 ticks and a tight one of 1.5, your best response to wide is tight, and your best response to tight is also tight. Both point the same way, which is what dominance means. Change the tight half-spread to 0.75 and the picture changes: undercutting now buys the flow too cheaply, so your best response to wide is wide, and your best response to tight is tight. Your answer now mirrors theirs.

The quoting game, cell by cell

You, themThey WideThey Tight
You Wide120, 1200, 180
You Tight180, 090, 90equilibrium

Bold marks the best response: yours in flame, theirs in ink. A cell where both are bold is an equilibrium, because neither side wants to move. Tightening is better in both columns, so it wins without any view on what they will do. You both tighten and take 90 each, when staying wide would have paid 120. Nobody made a mistake.

Bold marks the best response: yours in flame, theirs in ink. A cell where both are bold is an equilibrium, because neither side wants to move. Tightening no longer wins in both columns: at 1.50 ticks against 2, undercutting buys flow too cheaply to be worth it. Dominance is gone and there are now two equilibria, so what you should do depends entirely on what you think they will do.

120 lots
2.00 ticks
1.50 ticks
Both quote wide
120
Both quote tight
90
Cost of competing
−30

Set the tight half-spread to 0.75 ticks. Two cells light up as equilibria, and neither of them is anyone's dominant choice.

Nash equilibrium

A Nash equilibrium is a profile of actions where every player is simultaneously playing a best response to the others. Nobody can improve by changing their own action alone.

Equilibrium condition
πi(ai∗,a−i∗)≥πi(ai,a−i∗)for every ai, for every player i\pi_i(a_i^*, a_{-i}^*) \geq \pi_i(a_i, a_{-i}^*) \quad \text{for every } a_i, \text{ for every player } i

Holding everyone else fixed, no unilateral deviation helps. That is the entire definition, and it is worth stating precisely because most misuse comes from adding to it.

What it does not promise

Three assumptions get smuggled in, and all three are false.

It is not the best joint outcome. Both firms tightening is an equilibrium and pays 90 each, when both quoting wide pays 120. Equilibrium means stable, not efficient.

It need not be unique. At 0.75 ticks the game has two: both wide, and both tight. Both are stable, they pay differently, and the theory does not tell you which one you are in. Only knowing your competitors does.

It need not be what a novice does. Equilibrium describes where experienced players settle after learning each other. Against someone playing their first hand, the equilibrium action is often not the profitable one.

Key takeaway

Equilibrium is a stability condition, not a recommendation. It answers "could this configuration persist?", not "what should I do this afternoon?". Those coincide only when everyone else is already playing well.

Coordination, and why it is a real trading problem

When a game has several equilibria, the difficulty stops being calculation and becomes expectation: you want to be in the same one as everyone else. Market conventions work like this. Which venue liquidity concentrates on, which contract month is the liquid one, which time of day everyone trades: none of these is uniquely correct, all of them are self-reinforcing, and being right about the mechanics while wrong about where everyone else went is still being wrong.

Tip

Asked to solve a game in an interview, find the best responses first and mark them on the matrix. Equilibria are then just the cells marked by both players, and you have shown the method rather than asserting an answer.

Test your knowledge

Both firms quoting tight is a Nash equilibrium paying 90 each, while both quoting wide would pay 120 each. Is the equilibrium therefore wrong?
A game has two Nash equilibria that pay differently. What does game theory tell you to do?
NextZero-Sum Games and Minimax

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