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Concept
Signaling game
Summary
In game theory, a signaling game is a simple type of a dynamic Bayesian game.
The essence of a signalling game is that one player takes an action, the signal, to convey information to another player, where sending the signal is more costly if they are conveying false information. A manufacturer, for example, might provide a warranty for its product in order to signal to consumers that its product is unlikely to break down. The classic example is of a worker who acquires a college degree not because it increases their skill, but because it conveys their ability to employers.
A simple signalling game would have two players, the sender and the receiver. The sender has one of two types that might be called "desirable" and "undesirable" with different payoff functions, where the receiver knows the probability of each type but not which one this particular sender has. The receiver has just one possible type.
The sender moves first, choosing an action called the "signal" or "message" (though the term "message" is more often used in non-signalling "cheap talk" games where sending messages is costless). The receiver moves second, after observing the signal.
The two players receive payoffs dependent on the sender's type, the message chosen by the sender and the action chosen by the receiver.
The tension in the game is that the sender wants to persuade the receiver that they have the desirable type, and they will try to choose a signal to do that. Whether this succeeds depends on whether the undesirable type would send the same signal, and how the receiver interprets the signal.
Perfect Bayesian equilibrium
The equilibrium concept that is relevant for signaling games is the perfect Bayesian equilibrium, a refinement of Bayesian Nash equilibrium.
Nature chooses the sender to have type with probability . The sender then chooses the probability with which to take signalling action , which can be written as for each possible The receiver observes the signal but not , and chooses the probability with which to take response action , which can be written as for each possible The sender's payoff is and the receiver's is
A perfect Bayesian equilibrium is a combination of beliefs and strategies for each player.
Official source
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