Signaling games are dynamic games with two players, the sender (S) and the receiver (R). The sender has a certain type, t, which is given by nature. The sender observes his own type while the receiver does not know the type of the sender. Based on his knowledge of his own type, the sender chooses to send a message from a set of possible messages M = {m₁, m₂, m₃,..., m_j}. The receiver observes the message but not the type of the sender. Then the receiver chooses an action from a set of feasible actions A = {a₁, a₂, a₃,...., a_k}. The two players receive payoffs dependent on the sender's type, the message chosen by the sender and the action chosen by the receiver.

Perfect Bayesian equilibrium

The equilibrium concept that is relevant for signaling games is Perfect Bayesian equilibrium. Perfect Bayesian equilibrium is a refinement of Bayesian Nash equilibrium, which is an extension of Nash equilibrium to games of incomplete information. Perfect Bayesian equilibrium is the equilibrium concept relevant for dynamic games of incomplete information. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, a Bayesian game is one in which information is incomplete. ... In game theory, the Nash equilibrium (named after John Nash who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only their own strategy. ...

Finding the perfect Bayesian equilibrium of the signaling game

To find the perfect Bayesian equilibrium of the signaling game, we have to apply four requirements to the signaling game.

Requirement 1

The receiver must have a belief about which types can have sent message m. These beliefs can be described as a probability distribution μ(t_i | m), the probability that the sender has type t_i if he chooses message m. The sum over all types t_i of these probabilities has to be 1.

Requirement 2

The action the receiver chooses must maximize the expected utility of the receiver gives his beliefs about which type could have sent message m_j, μ(t_i | m_j). This means that the sum

is maximized. The action that maximizes this sum is a ^* .

Requirement 3

For each type, t_i, the sender may have, the sender chooses to send the message m ^* that maximizes the sender's utility U_S(t_i,m_j,a ^* ) given the action chosen by the receiver, a ^* .

Requirement 4

For each message m_j the sender can send, if there exists a type t_i such that m ^* = m_j (the sender will choose to send message m_j if he has type t_i), the belief the receiver has about the type of the sender if he observes message m_j, μ(t_i | m_j) satisfies the equation (Bayes rule)

The perfect Bayesian equilibria in such a game can be divided in two different categories, pooling equilibria and separating equilibria. A pooling equilibrium is an equilibrium where senders with different types all choose the same message. A separating equilibrium is an equilibrium where senders with different types choose different messages.

Applications of signaling games

The first application of signaling games to economic problems was Spence's model of job market signaling (1973). Spence describes a game where workers have a certain ability (high or low) that the employer does not know. The workers send a signal by their choice of education. The cost of the education is higher for a low ability worker than for a high ability worker. The employers observe the workers education but not their ability, and chooses to offer the worker a high or low wage. In this model it is assumed that the ability of the worker is independent of the education he has. Jump to: navigation, search 1973 was a common year starting on Monday. ...

Signaling games are important in several fields of economics. They describe situations where one player has information the other player does not have. These situations of asymmetric information are very common in economics. Situations with asymmetric information are common in other social sciences as well. Game theory can also be applied to evolutionary biology. In evolutionary biology one obvious application of signaling games is the signals sent by males of several species in the mating season to attract females and to send signals about strength to other males. The antlers of stags, the feathers of birds of paradise, the tail of the peacock and the song of the nightingale are all such signals. Crucially, however, the signal must distinguish types. For example, the size of a stag's antlers are indicative of its ability to fight. In the other examples, the costliness of the signal or its correlation with the aspect of the animal it emphasises (e.g. health or strength) is less clear. Thompson's gazelles are known sometimes to perform a 'stott', a jump into the air of several feet with the white tail showing, when they detect a predator. Alcock and others have suggested that this action is a signal of the gazelle's speed to the predator. This action successfully distinguishes types because it would be impossible or too costly for a sick creature to perform and hence the predator is deterred from chasing a stotting gazelle because it is obviously very agile and would prove hard to catch.

References

Robert Gibbons: A Primer in Game Theory, Harvester Wheatsheaf 1992 Jump to: navigation, search 1992 was a leap year starting on Wednesday. ...

See also

• Cheap talk
• Solution concept
• Extensive form game