Game Theory
Signaling Games: When a Costly Signal Is More Honest Than Words
MIT diploma - $300K and 4 years of life. Spence's model 1973: this may not be Human Capital - it's a costly signal. IPO underpricing of 15% = 'money left on the table' as a company quality signal. Samsung Galaxy lifetime warranty = more expensive for defective products. Three examples of one principle: a costly signal is credible where words are free.
- **Spence signaling (1973, Nobel 2001):** education as a signal. ~50% of the degree return by econometric estimates is a signaling premium, not skills.
- **IPO underpricing:** in 2021 the average first trading day was +17% below the 'fair' price. Rock model: quality companies leave money as a costly signal. Those who cannot afford underpricing are not quality.
- **Financial ratings (AAA/BBB):** S&P, Moody's - partition equilibrium with N≈7 categories. Bias: agencies are paid by issuers (b > 0). Before 2008: b was too large - ratings as cheap talk.
Предварительные знания
MIT Diploma as a Costly Signal
An MIT degree costs $300,000 and 4 years of one's life. Michael Spence showed in 1973 that this cost may not create skills - it merely **signals** innate ability. Nobel Prize 2001 for this provocative claim.
The problem: an employer cannot directly observe a candidate's productivity. A candidate can say 'I'm smart' - cheap talk, anyone can say that. What's needed is a signal that is **costly for weak types and cheap for strong types** - only then is it credible.
Two types of signals: **costly signaling** (education, expensive advertising, warranty) - works through differential cost across types. **Cheap talk** - words without material consequences. Cheap talk is informative only when interests align (Crawford-Sobel 1982).
**IPO underpricing as a signal:** at IPO, companies deliberately underprice by 10-20%. Rock (1986) model: high-quality companies 'leave money on the table' as a costly signal. Bad companies cannot mimic - they need all IPO proceeds. Underpricing = costly but credible quality signal.
A company offers a lifetime warranty on its product. Why is this a credible quality signal even though legally any company can offer it?
Costly signaling: c(warranty, L) > benefit from price premium > c(warranty, H). A reliable producer (H) expects few claims - low cost. An unreliable producer (L) faces many claims - high cost. If c(L) > benefit > c(H): only H benefits from offering the warranty, L does not mimic. This is the single crossing condition: differential costs create a separating equilibrium.
Spence's Model: Education as a Signal
Spence's model: two worker types - high (H) and low (L). Productivity: H = 2, L = 1. The employer doesn't observe type, offers wage = E[productivity]. Education does not change productivity - it only signals.
**Spence's model is a provocation, not a fact.** Real education simultaneously: 1) creates human capital (skills), 2) serves as a signal. Econometric estimates (Acemoglu, Autor) show: ~50% of the return to education is skills, ~50% is signaling. But the proportion depends on the field and type of work.
In Spence's model, education creates no skills yet all H types rationally obtain it in the separating equilibrium. Why is this called 'socially wasteful'?
In the separating equilibrium, H types study for e* years but the model assumes education produces no skills. Resources (time, money) are spent without any increase in aggregate productivity. If types could be directly observed, nobody would study just for the signal. Deadweight loss = e*·N_H. Paradox: individually rational for H (gets w_H = 2 instead of average 1.5), but collectively wasteful.
Separating, Pooling, and PBE
In signaling games there are three types of equilibria. To analyze them a stronger concept than BNE is needed: **Perfect Bayesian Equilibrium (PBE)**, which requires consistency of strategies and beliefs.
**Cho-Kreps Intuitive Criterion (1987):** a refinement of PBE. If upon observing an off-path signal m the receiver thinks 'L would never send m under reasonable beliefs' - such a belief is inadmissible. This eliminates many implausible pooling equilibria. In Spence's model: the Intuitive Criterion leaves only the separating equilibrium with the smallest e*.
In Spence's pooling equilibrium H gets wage 1.5, but in the separating equilibrium H gets only 1.3. Why can't H unilaterally move to pooling?
If H deviates to e=0, the employer cannot distinguish H from L and pays average wage 1.5. But this assumes L also stays at e=0. In the separating equilibrium at e*=0.7: L doesn't mimic (cost 2·0.7=1.4 > benefit 0.0). If H moves to e=0, L follows - both pool. H's wage drops from 2.0 to 1.5, and the gain 1.5 > 1.3 is illusory because it depends on L remaining at e=0 (which L won't guarantee).
Cheap Talk: When Words Transmit Information
Crawford and Sobel (1982): even words without material consequences can convey information when interests **partially align**. With large divergence - babbling equilibrium. With full alignment - full information.
**Financial analysts (Buy/Hold/Sell):** three recommendation categories = partition equilibrium with N=3. Bias b: analyst earns commission from trades (incentive to Buy) or from the issuer company (incentive to Hold). Separation of research and trading businesses in banks (Glass-Steagall reformed 2003) = reducing b to increase N.
At bias b = 0.5 in the Crawford-Sobel model with θ ~ Uniform[0,1], how many information levels can cheap talk transmit?
The maximum N satisfies N(N+1) ≤ 1/(2b) = 1/(2·0.5) = 1. For N=1: 1·2 = 2 > 1, which fails. So N = 1 - only the trivial (babbling) equilibrium exists. At b = 0.5 the sender always wants to shift the receiver's action by 0.5 above the truth, so the receiver ignores all messages and acts on the prior. No information is transmitted.
Key Ideas
- **Signal is credible**: if c(m, L) > benefit for L > benefit for H > c(m, H). Single crossing property separates types.
- **Spence's model**: education as signal. e* ∈ (0.5, 1) = separating equilibrium. Socially wasteful, but individually rational.
- **Separating**: different types send different signals → type revealed. **Pooling**: all send same signal → no information.
- **PBE**: strategies + beliefs. Cho-Kreps Intuitive Criterion eliminates implausible pooling equilibria.
- **Cheap talk**: informative at small bias b. N_max ≈ 1/(2b) intervals in the Crawford-Sobel partition equilibrium.
Related Topics
Signaling games are a direct continuation of Bayesian games with dynamic communication:
- Bayesian Games — Signaling games are a special type of Bayesian game with dynamic communication (sender-receiver game). PBE = refinement of BNE.
- Repeated Games — Reputation in repeated games = type signal through action history. Grim Trigger = costly honesty signal.
Вопросы для размышления
- MIT diploma = costly signal. But what if education simultaneously creates skills (human capital) AND serves as a signal? How would the optimal e* and social losses change compared to pure Spence model?
- Ratings agencies are paid by issuers (issuer-pays model) - this creates bias b. Alternative: investor-pays model. How does changing the payment model affect N_max in Crawford-Sobel, and why wasn't this sufficient to prevent the 2008 crisis?
- LinkedIn endorsements and GitHub stars - cheap talk or costly signal? How could the platform change the mechanism to make signals more credible?