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Signaling Games

Signaling games are dynamic games where you have 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 choosen 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 equilibium. Perfect Bayesian equilibrium is a refinement of Bayesian Nash equilibrium, which again is a refinement of Nash equilibrium. Perfect Bayesian equilibrium is the equilibrium concept relevant for dynamic games of incomplete information.

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 beleif about which types can have sent message m. These beliefs can be described as a probability distribution \mu(t_i|m), the probability that the sender has type ti if he chooses message m. The sum over all types ti 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, \mu(t_i|m_j). This means that the sum

$\sum _{t_{i}}\mu (m_{j}|t_{i})U_{R}(t_{i},m_{j},a_{k})$

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

Requirement 3 For each type,ti, 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 mj the sender can send, if there exists a type ti such that m* = m_j (the sender will choose to send message mj if he has type ti), the belief the receiver has about the type of the sender if he observes message m_j, \mu(t_i|m_j) satisfies the equation (Bayes rule)

$\mu (t_{i}|m_{j})=p(t_{i})/\sum _{t_{i}}p(t_{i})$

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.

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 strenght to other males. The antlers of stags, the beautiful feathers of birds of paradise, the tail of the peacock and the song of the nightingale are all such signals.

References Robert Gibbons: A Primer in Game Theory, Harvester Wheatsheaf 1992