Incomplete Information and Bayesian Equilibrium

When the Opponent is a "Mystery"

In real games, information is often asymmetric: an insurance company does not know the health status of a client, an investor does not know the intentions of a manager, a buyer does not know the cost of goods. How do we analyze strategic interaction when players have private information?

Harsanyi (Nobel Prize 1994) proposed an elegant solution: introduce the notion of a type of a player—their private information—and model incomplete information as the random selection by nature of each player's type.

Type, Bayesian Game, and Bayesian Equilibrium

A type θᵢ of player i is his or her private characteristic (willingness to pay, production function, risk preferences). The type space is Θᵢ; nature "chooses" θ from the joint distribution μ(θ).

A Bayesian game in normal form: ⟨N, (Sᵢ), (Θᵢ), μ, (uᵢ)⟩. A strategy bᵢ: Θᵢ → Sᵢ is a rule of action for each possible type.

Bayesian Nash Equilibrium: A profile b* = (b₁*, ..., bₙ*) is a BNE if each type of each player maximizes expected payoff:

Eθ₋ᵢ[uᵢ(bᵢ*(θᵢ), b₋ᵢ*(θ₋ᵢ), θ) | θᵢ] ≥ Eθ₋ᵢ[uᵢ(sᵢ, b₋ᵢ*(θ₋ᵢ), θ) | θᵢ] for all sᵢ, θᵢ

Numerical Example: First-Price Auction

Two participants in a sealed-bid auction. Valuations θ₁, θ₂ ~ Uniform[0, 1]—independent, private. The winner pays their own bid.

We look for a symmetric linear equilibrium: b(θ) = αθ for some α.

Player 1, with value θ₁, chooses bid b₁ to maximize:

u₁ = (θ₁ − b₁) · P(b₁ > b₂)

With b₂ = αθ₂ and θ₂ ~ U[0,1]: P(b₁ > αθ₂) = P(θ₂ < b₁/α) = b₁/α (when b₁/α ≤ 1).

Problem: max_{b₁} (θ₁ − b₁)·b₁/α

First-order condition: (θ₁ − 2b₁)/α = 0 → b₁ = θ₁/2.

Equilibrium: b(θ) = θ/2* (bid exactly half your valuation).

Intuition: the player "shades" the bid below their valuation to leave themselves profit, balancing this against the risk of losing. With two participants, optimal "shading" is by a factor of two.

With n participants: b*(θ) = θ·(n−1)/n. As n→∞: b* → θ—the bids converge to the true value (competitive market!).

Signaling and Screening

Spence's signaling model (1973, Nobel 2001): The firm does not know the worker's productivity (high H or low L). The worker can obtain education (a costly signal).

Key point: for a high-productivity worker, the costs of education are lower (because it is easier for them to learn). Condition for a separating equilibrium:

• High-productivity worker gets educated: W_H − c_H·e ≥ W_L
• Low-productivity worker does not get educated: W_L ≥ W_H − c_L·e

Where e is the education level, c_H < c_L are the marginal costs. If e is chosen correctly, both conditions hold: workers are separated by education, even though education itself does not increase productivity (paradox!).

Separating vs pooling equilibrium: In a separating equilibrium, different types choose different strategies → the employer can identify the type. In a pooling equilibrium, all types behave identically → no information is revealed.

Akerlof's "Lemons" Market

The used car market (Akerlof, 1970, Nobel 2001). The seller knows the quality (good—"peach"—or bad—"lemon"). The buyer does not know. The equilibrium price is the average, based on expected quality. But at this price, "peach" sellers refuse to sell (as they are underpriced), and only "lemons" remain. The market shrinks, and in the limit may collapse completely.

This is adverse selection. Solutions: certification (eliminates asymmetry), warranties (signal from the seller), insurance (reduces buyer's risk).

Mechanisms to Combat Information Asymmetry

Certification: a third party verifies private information—rating agencies, medical licenses, audit opinions. Guarantees and sureties: the seller of a "good" product assumes the risk—a significant signal, as the cost of guarantees is prohibitive for the "bad" type. Screening contracts: insurance with a deductible encourages self-selection. High-risk clients prefer low deductibles (and pay a higher premium), low-risk clients choose high deductibles (and pay less). A well-designed menu of contracts allows partial elimination of information asymmetry without directly observing the agent's type, creating selection via incentive structures.

Bayesian Games in Finance and Corporate Governance

Models with incomplete information explain the key phenomena of financial markets. In the credit market, the bank does not know the borrower's creditworthiness—adverse selection arises: at a common interest rate, high-risk borrowers crowd out reliable ones, as they are willing to pay higher rates. The solution is screening through collateral requirements: reliable borrowers are prepared to provide large collateral, knowing the risk of loss is low. In corporate mergers, the buyer does not know the true value of the target—a "winner's curse" arises: the winner of the auction for the company has overpaid, since their estimate is above the average. In an initial public offering, companies intentionally underprice IPO shares as a costly signal of quality—only a strong company can "leave money on the table" and still remain profitable. Mechanism design theory generalizes all these results: the designer's goal is to create game rules so that private information is revealed in the course of strategic interaction.

Bayesian Games in Finance and Regulation

Bayesian equilibrium describes the strategies of players with different private information. In financial markets, insider trading is modeled as a Bayesian game: the market maker sets the bid-ask spread to compensate for expected losses from trading with informed participants. The higher the likelihood of insider presence, the wider the spread—a direct result of Bayesian updating of the market maker’s beliefs. In lending, the bank does not know the borrower’s type, and a menu of loan contracts is developed to separate borrowers: high-risk borrowers choose contracts with higher interest rates, low-risk borrowers with collateral security. This is a classic Bayesian screening mechanism. In the IPO process, investment bankers do not know the true market valuation and conduct "book building": they offer shares to institutional investors, collecting information through their bids. This is a Bayesian game between issuer and investors, where IPO underpricing compensates for the risk of adverse selection. Financial regulators use Bayesian models to detect manipulations and construct oversight systems that account for incomplete information.

Assignment: Construct a Bayesian game for the insurance market: the buyer knows their own type (high risk—probability of illness 0.8; low—0.2). The insurance company does not know the type. Show that if the company offers a single contract at the “average” price, high-risk types will crowd out low-risk ones. What mechanism allows types to be separated?