A signalling game is a sequential interaction in which a privately informed sender chooses an observable message or action, and a receiver responds after seeing it. The receiver sees the signal, not the sender’s type. For a quant interpreting a counterparty’s behaviour, the question is what that action reveals once the counterparty’s incentive to influence the response is included.
This is a dynamic Bayesian game. The sender’s strategy maps private types to signals; the receiver’s strategy maps observed signals to responses. An equilibrium must also specify beliefs about the type behind each observation. Manea’s MIT notes, slides 3 and 23, set out this structure.
A signal need not be a sentence, and a signalling game need not require costly messages. Costly signalling is an important case because different types may face different costs of taking the same visible action. It differs from cheap talk, where the message itself has no direct payoff effect. An expensive action still needs an incentive explanation; expense alone does not prove that it reveals quality.
A credential that changes beliefs, not output
Consider an original fictional hiring game. A worker privately knows its productivity: H produces 4 payoff units; L produces 1. Each type has prior probability one-half. The worker chooses either no education or one observable education programme. The programme does not change productivity, but costs H 1 unit and L 4 units. No education costs zero.
Assume risk-neutral workers, no financing constraints and competitive risk-neutral employers who pay a wage equal to expected productivity conditional on the observed education choice. Every worker is hired. Worker utility is wage minus education cost, with all amounts on the same undiscounted payoff scale. Competition supplies the wage rule; a single employer freely choosing its preferred wage would be a different game. The educational role of this construction is discussed in Manea, slide 31.
Test this candidate outcome: H chooses education; L chooses none. Because the signals differ, this is a separating strategy profile. Bayes’ rule then gives
The wage rule therefore pays 4 after education and 1 without it. Check both types’ incentives against those wages:
| Private type | Education: wage minus cost | No education | Preferred choice |
|---|---|---|---|
| H | Education | ||
| L | None |
H would lose 2 by abandoning education; L would lose 1 by imitating H. Thus both workers choose optimally, and the wages match the productivity implied by their choices. The strategies and beliefs support a separating equilibrium in this stipulated binary model. The employer never needs to observe the worker’s cost directly.
The useful inequality compares the wage premium with the costs of imitation. If denotes the wage with education minus the wage without it, these sender incentive constraints are
Here , and . Strict inequalities give both types strict preferences; equality at either boundary makes that type indifferent. If both types faced the same cost, there would be no interval giving these two strict preferences under the same wage premium.
Original incentive comparison with education costs held fixed. The shaded interval tests sender incentives; it does not independently establish belief-consistent receiver wages at every trial premium.
The output assumption matters. Education here earns H a higher wage by revealing an existing productivity difference. It does not create the extra output. This isolates the information mechanism; it is not an empirical claim that actual education has no productive value.
The same costs can also support pooling
Pooling means that different types choose the same signal. Suppose both choose no education. Observing no education then leaves the posterior probability of H at one-half, giving a wage equal to expected productivity:
Education is now an off-path observation: neither type is supposed to choose it. Bayes’ rule alone cannot condition the equilibrium distribution on this zero-probability event. To complete the candidate assessment, suppose employers assign probability one to L after education, and hence offer wage 1 there. Deviating to education would then give H and L , both below the pooling payoff 2.5. Neither deviates, so these beliefs support a pooling equilibrium under the stated wage rule and the basic sequential incentive and on-path Bayes tests.
This is why the cost comparison alone does not predict a unique outcome. Beliefs determine wages; wages determine which signal is worth buying. Stronger equilibrium refinements examine the plausibility of off-path beliefs. Yildiz’s Chapter 4 and §4.1 explain sequential rationality and belief consistency; Manea, slides 34–36, illustrates pooling and refinements in the richer education model. Our binary example does not inherit a uniqueness result from that model.
In the separating example, both available signals occur with positive probability, so both receiver beliefs are determined on the equilibrium path. Allowing many education levels would introduce unused signals and additional beliefs to check.
Reading an observed action
For a trading or financing application, identify the privately known fact, the observable action, its cost to each type and the receiver’s subsequent choice. Then test whether a type that lacks the claimed information would profit by imitating the action. A costly trade can be easier for some types to bear, but also reflect liquidity needs, constraints or hedging. The hiring numbers above provide no estimates of those market incentives.
Finally, a strategic signal differs from a statistical alpha signal: a forecasting variable need not have been deliberately chosen by an informed sender. Likewise, a signal can reveal a type without making any future promise binding. Belief revision, predictive association and enforceable commitment require separate explanations.
Sources
- Mihai Manea, Equilibrium Refinements, MIT 14.16, 13 April 2016; slides 3 and 23: sender–receiver structure and beliefs; slides 31–36: education costs, competitive wages, separating and pooling equilibria, and refinements. The binary choices and numerical comparisons here are original teaching examples.
- Muhamet Yildiz, Graduate Game Theory, Spring 2022 notes hosted with MIT 14.126 Spring 2024; Chapter 4 introduction, pp. 101–104, and §4.1, definitions 4.2–4.6, pp. 105–107: explicit beliefs, sequential rationality, on-path Bayesian updating and consistency requirements.