Games With Incomplete Information

You quote a two-sided market. Someone lifts your offer. Ordinary game theory assumes both players know the whole game — payoffs, rules, each other's options. But you don't know something crucial here: is this trader a retail order with no edge, or a sharp desk that just saw news you haven't? The same move — lifting your offer — means something completely different depending on who's doing it, and you can't see which one you're facing.

The analogy: playing poker where you can't see hands, but you know the deck

In ordinary poker, you don't see your opponent's cards, but you know the deck they were dealt from and roughly how a rational player with each possible hand would bet. A Bayesian game formalizes exactly this: nobody sees everyone's private information (their "hand," or in trading terms, whether they're informed or not), but everybody knows the probability distribution over what that hidden information could be, and everybody knows that everyone else is reasoning the same way. You can't read the specific card, but you can reason correctly about the odds of each card and how a rational holder of it would act.

The machinery, one piece at a time

A type is the piece of private information a player holds that others can't directly observe — in market making, a counterparty's type is roughly "informed" or "uninformed." Each player knows their own type but only a probability distribution over everyone else's.

A strategy in this setting is no longer just "an action" — it's a full plan mapping every possible type you could have to an action, because you have to commit to a rule before you learn how the game unfolds. Formally, a Bayesian Nash equilibrium requires that each type of each player is choosing the action that maximizes their expected payoff, averaging over their belief about the other player's type:

$\max_{a_i} \, E_{t_{-i}}\big[u_i(a_i, a_{-i}(t_{-i}), t_i, t_{-i})\big]$

In words: player $i$, holding type $t_i$, picks the action $a_i$ that gives the best payoff on average, where the average is taken over the possible types $t_{-i}$ the opponent might have and the actions $a_{-i}$ those types would rationally take. Nobody optimizes against a single guess of the opponent's type — everybody optimizes against the whole probability-weighted menu of who the opponent might be.

Worked example 1: pricing around an unknown counterparty

You quote a two-way market at 99–101 on a stock worth either 95 or 105 with equal probability, and only informed counterparties know which. Suppose 30% of your flow is informed (always trades correctly against your mispricing) and 70% is uninformed (trades randomly, 50/50 buy or sell regardless of true value). If someone lifts your 101 offer, what's your expected loss? An informed trader only lifts when the true value is 105 (paying 101 for something worth 105 is their profit — your loss of 4). An uninformed trader lifts with probability 0.5 regardless of true value. Weighting: informed traders are 30% of flow and always lift correctly when value is 105, contributing an expected loss of $0.30 \times 4 = 1.2$ from that segment alone (since informed flow only ever lifts against you). Uninformed flow, 70% of volume, is a coin flip between the two values, so half the time you lose 4 (value is 105) and half the time you gain 1 (value is 95, you sold at 101), and that segment's expected contribution is $0.70 \times (0.5 \times (-4) + 0.5 \times 1) = 0.70 \times (-1.5) = -1.05$. Total expected result per lift: $-1.2 - 1.05 = -2.25$. You lose money on average taking any offer lift at these quotes — which tells you 101 is too tight given 30% informed flow, and you need to widen.

Worked example 2: updating the type-belief with a signal

Suppose you additionally observe order size: informed traders send large orders 80% of the time, uninformed traders send large orders only 20% of the time. A large order arrives. Using Bayes' rule with the same 30/70 prior: $P(\text{informed} \mid \text{large}) = \dfrac{0.3 \times 0.8}{0.3 \times 0.8 + 0.7 \times 0.2} = \dfrac{0.24}{0.24 + 0.14} = \dfrac{0.24}{0.38} \approx 0.63$. Seeing a large order more than doubles your assessed probability that this is informed flow, from 30% to about 63% — and you should re-run the expected-loss calculation above with 0.63 in place of 0.30 before deciding whether to honor the same quote. This is the whole engine of a Bayesian game in miniature: observable signals shift the type-belief, and the belief shifts the optimal action.

What this means in practice

Every quoting decision on a real desk is implicitly a Bayesian game: you never know for certain who's on the other side, only a prior probability (informed vs. uninformed) that you update on signals like size, timing, and venue. Quote too tight and informed flow picks you off consistently; quote too wide and you lose the uninformed flow that was your actual profit source. The entire discipline of adverse-selection-aware quoting is Bayesian-game reasoning applied under a deadline.