Signaling Game Logic

Origin. Lewis's signaling games (1969). Spence's signaling (economics, 1973). Game-theoretic semantics of communication. Logic for strategic information transmission. Foundation for pragmatics and mechanism design.

Models. Strategic communication. Sender has private information (type). Sender sends signal/message. Receiver updates beliefs, takes action. Payoffs depend on type, signal, action. Equilibrium: consistent beliefs and strategies.

Formalism.

Signaling game: G = (T, M, A, p, uS, uR) where:

T: sender types (private info)
M: messages (signals)
A: receiver actions
p: prior over types
uS, uR: payoff functions

Strategies:

σ: T → Δ(M) (sender: type to message distribution)
ρ: M → Δ(A) (receiver: message to action distribution)

Beliefs: μ: M → Δ(T) (receiver's belief given message) Bayesian: μ(t|m) = p(t)σ(m|t) / Σₜ' p(t')σ(m|t')

Equilibrium types:

Separating: different types send different messages
Pooling: all types send same message
Semi-separating: partial separation

Logical expressions: K_R(t | m): receiver knows type t given message m Signal(m) → Action(a): message induces action Truthful: σ(m|t) > 0 only if m "means" t

Symbols.

Symbol	Unicode	Name	Meaning
T	—	Types	Private information
M	—	Messages	Signals
A	—	Actions	Receiver choices
σ	U+03C3	Sigma	Sender strategy, type to message
ρ	U+03C1	Rho	Receiver strategy, message to action
μ	U+03BC	Mu	Posterior belief over types
Δ	U+0394	Delta, capital	The simplex of distributions over a set
Σ	U+03A3	Sigma, capital	Summation in the Bayes normalization
→	U+2192	Function arrow	The type of a strategy
>	—	Greater than	A message sent with positive probability
p	—	Prior	Common distribution over types
u	—	Utility	Payoff to sender or receiver
K	—	Knowledge	Epistemic

Metatheory. The solution concept is perfect Bayesian or sequential equilibrium (Kreps and Wilson, 1982), which pairs strategies with beliefs required to satisfy Bayes' rule wherever the denominator is nonzero — and leaves beliefs off the equilibrium path unconstrained, which is why refinements exist at all. Cho and Kreps (1987) proposed the intuitive criterion, deleting equilibria supported by beliefs assigning weight to a type for which the deviation is equilibrium-dominated; in the Spence model with two types it selects the least-cost separating outcome. Banks and Sobel (1987) strengthened this to divinity and universal divinity. For costless signals Crawford and Sobel (1982) proved the central limitative result: when sender and receiver preferences diverge, every equilibrium of the cheap-talk game partitions the state space into finitely many intervals, with an upper bound on their number that falls as the bias grows, so full revelation is unattainable and babbling is always an equilibrium. Kamenica and Gentzkow (2011) showed that with sender commitment the optimal signal is given by concavifying the sender's value function.

Applies to. Economic signaling (education, advertising). Pragmatics (Gricean reasoning). Animal communication. Political communication. Mechanism design. Persuasion games.

Limitations. Equilibrium is massively non-unique — babbling always exists alongside informative equilibria — and the refinements that cut the set down are restrictions on beliefs at events the equilibrium says never occur, so they are not derived from the model but added to it, and different refinements select different outcomes. The framework presupposes a common prior, common knowledge of the payoff structure, and exact Bayesian updating, none of which is plausible for the communication it is applied to. Messages carry no meaning of their own: any permutation of the signals gives another equilibrium, so the account explains coordination on a code but not which code, and the equilibrium concept is silent on how agents reach one. Dynamics must be supplied separately, and evolutionary versions of the game converge to partial pooling under some priors rather than to full signaling.