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	Sender strategy	Type to message
ρ	U+03C1	Receiver strategy	Message to action
μ	U+03BC	Beliefs	Posterior on types
K	—	Knowledge	Epistemic

Metatheory. Perfect Bayesian Equilibrium solution concept. Refinements: intuitive criterion, divine equilibrium. Cheap talk: costless signals, equilibria exist. Costly signaling: separation possible. Information design: sender's optimization.

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

Limitations. Equilibrium selection problem. Multiple equilibria common. Rationality assumptions strong. Dynamic extensions complex. Boundedly rational agents not modeled. Evolutionary alternatives exist.