Signaling games

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,..., mj}. 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,...., ak}. 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.

Definition of perfect Bayesian equilibrium of the signaling game
A sender of type ,$$t_j$$ sends a message $$m^*(t_j)$$ in the set of probability distributions over M (a mixed message!). ($$m(t_j)$$ represents the probabilities that type $$t_j$$ will take any of the messages in M.) The receiver observing the message m takes an action $$a^*(m)$$ in the space of probability distributions over A.

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 $$\mu(t_i|m)$$, 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 conditional on any message m.

Requirement 2
The action the receiver chooses must maximize the expected utility of the receiver given his beliefs about which type could have sent message $$m$$, $$\mu(t | m)$$. This means that the sum

$$\sum_{t_i} \mu(m|t_i)U_R(t_i,m,a)$$

is maximized. The action a that maximizes this sum is $$a^*(m)$$.

Requirement 3
For each type, $$t$$, the sender may have, the sender chooses to send the message $$m^*$$ that maximizes the sender's utility $$U_S (t, m,a^*(m))$$ given the strategy chosen by the receiver, $$a^*$$.

Requirement 4
For each message $$m$$ the sender can send, if there exists a type $$t$$ such that $$m^*(t)$$ assigns strictly positive probability to m (i.e. for each message which is sent with positive probability), the belief the receiver has about the type of the sender if he observes message $$m$$, $$\mu(t|m)$$ satisfies the equation (Bayes rule)

$$\mu(t|m) = p(t)/\sum_{t_i} p(t_i)$$

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
Signaling games describe situations where one player has information the other player does not have. These situations of asymmetric information are very common in economics and behavioral biology.

Philosophy
The first known use of signaling games occurs in David K. Lewis' Ph. D. dissertation (and later book ), Convention. Replying to W.V.O. Quine (1936, 1960 ), Lewis attempts to develop a theory of convention and meaning using signaling games. In his most extreme comments, he suggests that understanding the equilibrium properties of the appropriate signaling game captures all there is to know about meaning:


 * I have now described the character of a case of signaling without mentioning the meaning of the signals: that two lanterns meant that the redcoats were coming by sea, or whatever. But nothing important seems to have been left unsaid, so what has been said must somehow imply that the signals have their meanings (Lewis 1969, 124).

The use of signaling games has been continued in the philosophical literature. Other have used evolutionary models of signaling games to describe the emergence of language. Work on the emergence of language in simple signaling games includes models by Huttegger (2005 ), Grim, et al. (2001 ), Skyrms (1996, 2000 ), and Zollman (2005 ). Harms (2000, 2004 ) and Huttegger (2005 ) have attempted to extend the study to include the distinction between normative and descriptive language.

Economics
The first application of signaling games to economic problems was Michael Spence's model of job market signaling. 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.

Biology
Valuable advances have been made by applying signaling games to a number of biological questions. Most notably, Alan Grafen's (1990 ) handicap model of mate attraction displays. The antlers of stags, the elaborate plumage of peacocks and birds of paradise, and the song of the nightingale are all such signals. Crucially, however, the signal must distinguish types.

Godfray (1991 ) modeled the begging behavior of nestling birds as a signaling game. The nestlings begging not only informs the parents that the nestling is hungry, but also attracts predators to the nest. The parents and nestlings are in conflict. The nestlings benefit if the parents work harder to feed them than the parents ultimate benefit level of investment. The parents are trading off investment in the current nestlings against investment in future offspring.

Pursuit deterrent signals have been modeled as signaling games (Yachi, 1995 ). 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.