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What Is a Strategic Game

A dice question hands you a world that is not paying attention to you. The die does not care what you bet, and the probabilities are the same whether you are clever or not. Almost nothing in trading works like that.

Three ingredients

A game is three things, and naming them is most of the work.

Players. Everyone whose choice affects your payoff. In a quoting problem that is you and the other market makers; in an interview game it is you and the interviewer.

Actions. What each player can do. Quote wide or tight, bid or pass, raise or fold.

Payoffs. What each player gets for every combination of actions. Not for your action alone, which is the whole point.

A payoff

\pi_i = \pi_i(a_i, a_{-i})

Your payoff depends on the profile of actions, not just your own. Dropping the second argument is the mistake the rest of this course exists to prevent.

Here $a_i$ is your action and $a_{-i}$ is everyone else's. An expected value calculation that conditions only on $a_i$ has quietly assumed the other side is a coin flip.

A quoting game

Two market makers quote the same contract. Each chooses a wide market, earning 2 ticks of half-spread, or a tight one, earning 1.5 ticks. There are 120 lots of flow a day. Whoever is tighter takes all of it; equal widths split it.

Work the four cells. Both wide: 60 lots each at 2 ticks, so 120 each. Both tight: 60 lots each at 1.5 ticks, so 90 each. If you tighten alone you take all 120 lots at 1.5 ticks, which is 180, and your competitor gets nothing.

Your payoff, theirs	They quote wide	They quote tight
You quote wide	120, 120	0, 180
You quote tight	180, 0	90, 90

The quoting game, cell by cell

You, them	They Wide	They Tight
You Wide	120, 120	0, 180
You Tight	180, 0	90, 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.

Daily flow120 lots

Wide quote: half-spread2.00 ticks

Tight quote: half-spread1.50 ticks

Both quote wide

120

Both quote tight

Cost of competing

−30

Bold is the best response. Drag the tight half-spread below half the wide one and the dominance argument disappears.

Read your own row against each of their columns. If they quote wide, you make 120 by matching them and 180 by undercutting. If they quote tight, you make 0 by staying wide and 90 by tightening. Tightening wins in both columns, so it wins full stop, without you needing any view on what they will do.

The uncomfortable part

Both of you reason that way, both tighten, and you each end the day on 90 when quoting wide would have paid 120. Nobody made a mistake. Each player's best action made the pair worse off, and no amount of individual cleverness escapes it.

Key takeaway

Spread compression in a competitive market is not a failure of discipline. It is the equilibrium of a game where undercutting is individually correct at every width. This is why the number of market makers on an instrument predicts its spread better than anyone's intentions do.

Why this framing earns its keep

Three questions fall out of the matrix that never come up in a probability question. Does one action beat the others no matter what they do? If not, what does my best action depend on? And is there a pair of choices where neither of us wants to move?

Those are the next three lessons. The habit to build now is smaller: before computing anything, write down who else is choosing, and what they get.

Tip

In an interview, saying "this depends on what the other player does, so let me set up the payoffs" is worth real credit. It signals you have noticed the problem is strategic, which is the distinction the question was built to test.

Test your knowledge

In the two-market-maker quoting game, both firms end up quoting tight and earning 90 each, when both quoting wide would have paid 120 each. What does this show?

One of the two firms has miscalculated, since no rational player accepts 90 when 120 is available

Each firm's individually best action leads to an outcome that is worse for both of them

The payoffs must be wrong, because a game cannot have an outcome that both players dislike

The firms should agree to quote wide, which is what the equilibrium of the game predicts they will do

A candidate computes the expected value of their own action without reference to what the counterparty will do. What has this assumed?

Nothing problematic, since expected value is always computed over the candidate's own action

That the counterparty's payoffs are the same as the candidate's

That the counterparty acts independently of the candidate's own choice

That the game has no equilibrium, so no prediction about the counterparty is possible

NextDominance and Iterated Elimination

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