Game Theory

Signaling Games

Information asymmetry pervades markets: a worker knows their ability, a seller knows product quality, a borrower knows their creditworthiness. Signaling games formalize when and how costly actions transmit credible information - and why this can be socially wasteful.

  • **Labor markets:** diplomas, certifications, and university prestige as ability signals under Spence's model
  • **Financial markets:** dividend payments as quality signals (Bhattacharya 1979) - only firms with strong cash flows can sustain them
  • **IPO underpricing:** intentionally low IPO prices signal issuer quality - only strong firms can afford to leave money on the table
  • **Advertising:** uninformative but expensive advertising signals the firm's long-term commitment to staying in the market

Предварительные знания

  • Bayesian games
  • Nash equilibrium in mixed strategies
  • Bayes' theorem
  • Correlated equilibrium

Signaling games: separating equilibria

Michael Spence in 1973 proposed education as a signal: a diploma need not raise productivity but serves as a costly signal of ability. The 2001 Nobel Prize followed. When education costs for a high-productivity worker satisfy c_H < w_H - w_L and for a low-productivity worker c_L > w_H - w_L, education becomes credible precisely because of the cost difference between types.

What characterises a separating equilibrium?

In a separating equilibrium each type theta_i chooses a unique signal m_i ≠ m_j (for different types). The receiver fully infers the type. Spence (1973) example: high-ability workers get an MBA as an ability signal; low-ability types find the MBA cost exceeds the wage premium and abstain.

Pooling equilibria

The single-crossing condition: indifference curves of types theta_H and theta_L cross exactly once in (signal, wage) space. Formally, the marginal rate of substitution of signal for wage decreases in type. This condition is necessary for a separating equilibrium to exist in the Spence model.

What distinguishes a pooling equilibrium?

In a pooling equilibrium the signal carries no type information: P(m | theta_high) = P(m | theta_low). The receiver cannot distinguish types from signals and responds as before the game. Common with cheap talk or with identical signal costs across types.

Equilibrium refinements: Intuitive Criterion

Social inefficiency of signaling: in a separating equilibrium, education is pure cost with no informational value (employers already know which education level matches which type). Total welfare is below the first-best. Spence called this a 'rat race': everyone signals, no one gains in aggregate.

Why are PBE refinements like Cho-Kreps's Intuitive Criterion needed?

Perfect Bayesian Equilibrium does not pin down receiver beliefs for signals that no type sends (off-path). This admits many equilibria with arbitrary beliefs. The Cho-Kreps Intuitive Criterion (1987): reject an equilibrium if there exists a type for which deviating to an off-path signal is strictly beneficial under any reasonable belief.

Connections to other fields

Signaling games explain information economics, financial markets, and contract design.

  • Bayesian Games and Incomplete Information — Signaling games are a two-stage special case of Bayesian games
  • Signaling Games: When a Costly Signal Is More Honest Than Words — Applied chapter on how costly signals shape hiring, education and advertising markets
  • Global Games and Coordination — Global games are signaling games with a continuous noisy signal of the fundamental

Итоги

  • Signaling game: informed sender sends a signal, uninformed receiver updates beliefs via Bayes and responds
  • Separating PBE: different types choose different signals; beliefs are precise; requires single-crossing of signal costs in type
  • Pooling PBE: all types choose the same signal; beliefs not updated; sustained by harsh off-path beliefs
  • D1 criterion (Cho-Kreps): eliminates unreasonable off-path beliefs - if a deviation benefits only one type, the receiver must believe that type deviates
  • Social inefficiency: in separating equilibrium, education is pure waste; total welfare is below first-best

Вопросы для размышления

  • Under what conditions does a pooling equilibrium survive the D1 criterion?
  • Why is Zahavi's handicap principle in biology exactly the same idea as in Spence's model?
  • How can mechanism design reduce the welfare losses from signaling - for example through direct disclosure of information?
Signaling Games

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