Part 5: Playing in the Dark – Bayesian Games and Incomplete Information

If you missed the previous chapter, start here: Part 4: Extensive Form Games and Backward Induction

Welcome back! In Parts 2–4, we assumed players know each other’s payoffs — they understand exactly what motivates their opponents. In reality, this is rarely true. A firm doesn’t know its competitor’s costs. A bidder doesn’t know how much a painting is worth to others. A general doesn’t know the enemy’s strength.

This private information is the defining feature of Bayesian games, and it leads to a fundamentally different kind of strategic reasoning.

REFERENCE NOTE

The concepts and mathematics in this post are drawn from Chapters 9–10 of An Introduction to Game Theory by Martin J. Osborne (Oxford University Press, 2003).

1. The Problem of Incomplete Information

Incomplete information means players do not know the payoff functions of their opponents (as opposed to imperfect information, which means they don’t observe past moves).

The key insight, due to John Harsanyi (1967), is that incomplete information can be modeled as a game where each player has a privately known type that determines their payoffs. All players share a common prior belief about the distribution of types.

Example: In a bilateral trade, the buyer’s private valuation \(v_b\) is drawn from \([0,1]\). The seller’s private cost \(c_s\) is also drawn from \([0,1]\). Neither knows the other’s number. Their types determine whether trade is mutually beneficial.

2. Formal Definition: Bayesian Game

A Bayesian game (or game of incomplete information) is defined by:

A set of players \(N = \{1, \ldots, n\}\).
For each player \(i\): an action set \(A_i\), and a type space \(T_i\).
A joint prior distribution \(p \in \Delta(T_1 \times \cdots \times T_n)\) over type profiles, assumed to be common knowledge.
For each player \(i\): a payoff function \(u_i : A \times T \rightarrow \mathbb{R}\), which depends on all actions and all types.

The timing of a Bayesian game:

Nature draws a type profile \((t_1, \ldots, t_n)\) from \(p\).
Each player \(i\) privately observes their own type \(t_i\) only.
Players simultaneously choose actions based on their types.

graph LR
    N[Nature] -->|Draws types| T1[Player 1 observes t_1]
    N -->|Draws types| T2[Player 2 observes t_2]
    T1 -->|Chooses a_1| G((Payoffs))
    T2 -->|Chooses a_2| G

    classDef default fill:#1a1a1a,stroke:#888,stroke-width:2px,color:#fff;
    classDef nature fill:#9C27B0,stroke:#fff,stroke-width:2px,color:#fff;
    classDef player fill:#2196F3,stroke:#fff,stroke-width:2px,color:#fff;
    class N nature;
    class T1,T2 player;

3. Strategies and Bayesian Nash Equilibrium

In a Bayesian game, a strategy for player \(i\) is a function \(\sigma_i : T_i \rightarrow A_i\) (or \(\Delta(A_i)\) for mixed strategies) — a mapping from each possible type to an action.

Player \(i\) of type \(t_i\) evaluates strategy profile \(\sigma\) by taking the expectation over the other players’ types:

\[U_i(\sigma \mid t_i) = \mathbb{E}_{t_{-i} \mid t_i}\left[ u_i(\sigma_i(t_i), \sigma_{-i}(t_{-i}), t_i, t_{-i}) \right]\]

Definition (Bayesian Nash Equilibrium): A strategy profile \(\sigma^* = (\sigma_1^*, \ldots, \sigma_n^*)\) is a Bayesian Nash Equilibrium (BNE) if, for every player \(i\) and every type \(t_i \in T_i\):

\[U_i(\sigma^* \mid t_i) \geq U_i(\sigma_i', \sigma_{-i}^* \mid t_i) \quad \forall \sigma_i'\]

THE CORE CONCEPT

In a BNE, each player’s strategy is a best response to the other players’ strategies, type-by-type. A player of type \(t_i\) acts optimally given their type, knowing the type distribution and the equilibrium strategies of others.

4. Example: A Simple Two-Type Game

Consider a game where Player 2 can be one of two types: Strong (S) with probability \(p\), or Weak (W) with probability \(1-p\). Player 1 does not know the type. The payoff matrix (Player 2 is either type S or type W):

If Player 2 is type S:

	L	R
U	(1, 2)	(0, 0)
D	(0, 0)	(2, 1)

If Player 2 is type W:

	L	R
U	(1, 0)	(0, 1)
D	(0, 2)	(2, 0)

Player 2 knows their type and picks accordingly. Player 1 must pick U or D before knowing the type. Player 1’s expected payoff of choosing U:

\[\mathbb{E}[U|U, \sigma_2] = p \cdot u_1(U, \sigma_2^S) + (1-p) \cdot u_1(U, \sigma_2^W)\]

A BNE specifies: what Player 2 does for each type and what Player 1 does — such that no one wants to deviate given correct beliefs.

5. Auctions: The Canonical Bayesian Game

Auctions are the richest application of Bayesian game theory. In a first-price sealed-bid auction:

\(n\) bidders each have a private valuation \(v_i\) drawn independently from a common distribution \(F\) on \([0, \bar{v}]\).
Each bidder simultaneously submits a bid \(b_i\).
The highest bidder wins and pays their own bid. If bidder \(i\) wins, their payoff is \(v_i - b_i\); all others get 0.

Strategy: \(b_i(v_i)\) — a mapping from valuation to bid.

BNE Analysis (symmetric, \(n\) bidders, uniform \(v_i \sim U[0,1]\)):

In a symmetric BNE, all bidders use the same bidding function \(b(v)\). Bidder \(i\) with valuation \(v\) solves:

\[\max_{b} \; (v - b) \cdot P(\text{win} \mid b)\] \[P(\text{win} \mid b) = P(b_j(v_j) < b \; \forall j \neq i) = \left[F(b^{-1}(b))\right]^{n-1}\]

With \(F = U[0,1]\), letting \(x = b^{-1}(b)\) (the valuation that bids exactly \(b\)), \(P(\text{win}) = x^{n-1}\). The first-order condition and boundary condition \(b(0) = 0\) yield:

\[b^*(v) = \frac{n-1}{n} \cdot v\]

THE CORE CONCEPT

In the symmetric BNE of a first-price auction with \(n\) bidders, each bidder shades their bid below their valuation by a factor of \(\frac{n-1}{n}\). With more competition (larger \(n\)), the shade factor approaches 1 — bids approach valuations.

Revenue Equivalence Theorem: Under mild conditions (independent private values, symmetric bidders, risk neutrality), all standard auction formats (first-price, second-price, English, Dutch) yield the same expected revenue for the seller in their BNE. This is one of the most elegant and counterintuitive results in economics.

6. Second-Price Auctions and Dominant Strategies

In a second-price (Vickrey) auction, the highest bidder wins but pays the second-highest bid. Remarkably, bidding your true valuation is a weakly dominant strategy:

Proof: Fix any bids of the other players, and let \(m = \max_{j \neq i} b_j\) be the highest competitor bid.

If \(b_i > v_i\): If \(b_i > m > v_i\), you win but pay \(m > v_i\) — a negative payoff. Bidding \(v_i\) would have lost (better, as \(m > v_i\)). Overbidding can only hurt.
If \(b_i < v_i\): If \(v_i > m > b_i\), you lose. Bidding \(v_i\) would have won with payoff \(v_i - m > 0\). Underbidding can only hurt.
Bidding \(b_i = v_i\): Never wins when \(v_i < m\), wins and earns \(v_i - m > 0\) when \(v_i > m\).

This makes the second-price auction strategy-proof — a highly desirable property for mechanism design.

Summary

Concept	Key Idea
Incomplete Information	Players have private types unknown to others
Bayesian Game	Models incomplete info via type spaces and a common prior
BNE	Each type plays a best response given others’ type-contingent strategies
Bid Shading	First-price equilibrium bids strictly below valuations
Revenue Equivalence	Standard auction formats yield equal expected revenue in BNE
Second-Price Auction	Dominant strategy to bid true valuation (strategy-proof)

What’s Next?

We’ve now mastered strategic interaction with private information. In Part 6, the final chapter, we turn to a different kind of game: Cooperative Game Theory. When players can make binding agreements, how should they split the gains? We’ll derive the Nash Bargaining Solution and the Shapley Value — two of the most beautiful results in all of game theory.

REFERENCE NOTE