A Bayesian game models strategic interaction when players have private information about payoffs or other decision-relevant facts. Each player chooses using what they know and beliefs about what others know. For a quant, this matters when a counterparty's hidden cost, valuation or information changes whether it trades at your quote.
A type is information available to the player
In game theory, a type records a player's private information. A simple type might be a supplier's cost or a bidder's valuation. More general types can also encode beliefs. The modeller specifies players, actions, types, payoffs and beliefs; a hidden fact alone does not specify a game.
In the common-prior formulation used here, a joint probability distribution describes the players' types. Each player knows their own realised type and forms beliefs about the others by conditioning that distribution. Types can be correlated: knowing your own information can change your beliefs about someone else's. The type spaces, payoff rules and prior are assumed to be common knowledge, even though realised types are private. This is a substantive modelling assumption, not a finding about actual traders. Acemoglu and Ozdaglar's notes, slides 9–12, develop this construction.
A pure strategy is a complete rule mapping own type to action:
Here is player 's type set and its action set. The rule cannot depend directly on another player's unobserved type. Randomised strategies can instead specify a distribution over actions at each own type. Manea's MIT slides, slides 2–3, distinguish these ingredients.
One offer, two private costs
Consider an original fictional procurement example, with all amounts in currency units. A buyer values one service at 4 and offers a fixed price of 2. The provider alone observes its cost : 1 or 3, each with stipulated probability one-half. Both maximise expected monetary gain. Acceptance delivers the service and payment; rejection gives both zero. There are no other costs, bargaining rounds or competing providers.
At the offered price, the provider's payoffs are:
| Private cost | Accept: | Reject | Best action |
|---|---|---|---|
| Accept | |||
| Reject |
The response strategy is therefore accept at cost 1; reject at cost 3. It is one deterministic rule with two contingencies. There are four possible pure response rules, because each of two types can accept or reject.
Original enumeration of the four pure response rules in the fictional fixed-price example. Equal acceptance rates can hide opposite selection across private types.
Before the cost is revealed, the acceptance probability is . The provider's expected profit is
The buyer receives if the offer is accepted, so its expected surplus is . These are ex ante averages, taken before learning the provider's type. After observing cost 1, the provider knows its profit from accepting is 1; after observing cost 3, it rejects and earns zero. Expected profit conditional on a trade is 1, rather than 0.5.
Replacing the provider by its average cost, , loses the decision that private information enables. An average-cost calculation gives zero profit from accepting, but the actual provider can accept only when acceptance is profitable. Likewise, a fair coin independent of cost accepts half the time yet earns zero in expectation: it sometimes accepts the loss-making high-cost case. The observed 50% acceptance rate does not establish mixed-strategy play.
The price 2 is an input to this example. Checking the provider's response and calculating the buyer's surplus does not show that 2 is the buyer's optimal offer. That requires specifying the buyer's feasible prices and evaluating the provider's response to each one.
Equilibrium tests incentives at each type
A Bayesian Nash equilibrium extends the Nash equilibrium best-response condition to type-contingent strategies. Each player's action must maximise its conditional expected payoff, given its own type and the other players' strategy rules.
For a pure-strategy equilibrium with finite type sets, write the type profile as . For every player , own type with positive probability, and feasible alternative action , the condition is
Here is player 's payoff and collects the other players' rules. The expectation uses beliefs about others' types conditional on the player's own information. Hold their type-dependent rules fixed when testing a deviation; do not let the deviating player choose their hidden types. Manea, slides 6–9, explains ex ante and interim payoffs and the equilibrium condition.
“Bayesian” does not mean that the game is merely a Bayes' theorem calculation. Beliefs describe uncertainty; equilibrium also requires each player to best respond. In a signalling game, observable actions can themselves reveal private information, requiring further attention to beliefs and decisions after those observations.
Why selection matters for a trading model
In the procurement example, accepting providers are cheaper than the population average. In a market model, a counterparty's decision to trade can similarly select information or valuations that differ from the unconditional population. That is why a quote's apparent margin cannot be evaluated using an unconditional asset-value forecast alone.
The SelfMadeQuant Order Flow and Adverse Selection lesson develops the trading consequence: a passive fill can earn the initial spread while suffering a worse subsequent mark. Its public preview outlines the lesson; the full teaching requires membership. The lesson measures selected fill outcomes rather than solving a Bayesian equilibrium. A strategic extension must additionally specify private information, feasible trading rules and each participant's incentives.
Finally, incomplete information concerns uncertainty about payoffs or types. Imperfect information concerns what a player observes when moving, including earlier moves. A model can have both: private types can be represented by an unobserved initial chance move, as in Yildiz, §1.3. Neither term means incomplete financial markets, where available securities cannot span all contingent claims.
Sources
- Mihai Manea, Bayesian Games and Auctions, MIT 14.16, 22 February 2016, slides 2–3 and 6–9: private types, strategies, ex ante/interim representations and Bayesian Nash equilibrium.
- Daron Acemoglu and Asu Ozdaglar, Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning, MIT 6.207/14.15, November 2009, slides 3–12: incomplete information, the common-prior specification, conditional beliefs and type-dependent strategies.
- Muhamet Yildiz, Graduate Game Theory, Spring 2022 notes hosted with MIT 14.126 Spring 2024, §§1.1 and 1.3: information sets and the representation of private payoff information by a chance move.