Signaling games

Signalling 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 [Gibbons, Robert (1992) "A Primer in Game Theory", Harvester Wheatsheaf ISBN 0-7450-1159-4] [Osborne, M.J. and Rubenstein, A. (1994) "A Course in Game Theory", MIT Press ISBN 0-262-65040-1] . A related game is a screening game where rather than choosing an action based on a signal, the receiver gives the sender proposals based on the type of the sender, which the sender has some control over.

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

$sum\_\{t\_i\}\; mu(t\_i|m)U\_R(t\_i,m,a)$

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 [Lewis, D.: 1969, "Convention. A Philosophical Study", Harvard University Press, Harvard, Mass.] ), "Convention". Replying to W.V.O. Quine (1936 [Quine, W.v.O (1936) "Truth by Convention" in "Philosophical Essays for Alfred North Whitehead" pp 90-124. Longmans, Green & Co. London (ISBN 0-8462-0970-5, for 1967 Russel and Russel Publishers reprinting)] , 1960 [Quine, W.v.O (1960) "Carnap and Logical Truth" "Synthese" 12(4):350-374.] ), 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 [Huttegger, S. M. (2005) "Evolution and the Explanation of Meaning." Forthcomming in Philosophy of Science] ), Grim, "et al." (2001 [Grim, P., T. Kokalis, A. Alai-Tafti, N. Kilb, and Paul St. Denis. (2001) "Making Meaning Happen." Technical Report #01-02, Stony Brook: Group for Logic and Formal Semantics, SUNY, Stony Brook.] ), Skyrms (1996 [Skyrms, B. (1996) "Evolution of the Social Contract." Cambridge: Cambridge University Press.] , 2000 [Skyrms, B. (2000) "Stability and Explanatory Significance of Some Simple Evolutionary Models." "Philosophy of Science" 67:94–113.] ), and Zollman (2005 [Zollman, K. J. S. (2005) "Talking to Neighbors: The Evolution of Regional Meaning." "Philosophy of Science" 72:69–85.] ). Harms (2000 [Harms, W. F. (2000) "Adaption and Moral Realism." Biology and Philosophy 15:699–712.] , 2004 [Harms, W. F. (2004) "Information and Meaning in Evolutionary Processes." Cambridge: Cambridge University Press.] ) and Huttegger (2005 [Huttegger, S. M. (2005) Evolutionary Explanations of Normative and Descriptive Statements, to appear in "Erkenntnis".] ) 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, A.M. (1973) "Job Market Signaling", Quarterly Journal of Economics. 87:355-374.
* ] . 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 choose to offer the worker a high or low wage. In this model it is assumed that the level of education does not cause the high ability of the worker, but rather, only workers with high ability are able to attain a specific level of education without it being more costly than their increase in wage. In other words, the benefits of education are only greater than the costs for workers with a high level of ability, so only workers with a high ability will get an education.

Biology

Valuable advances have been made by applying signaling games to a number of biological questions. Most notably, Alan Grafen's (1990 [Grafen, A. (1990) Biological signals as handicaps. "Journal of Theoretical Biology" 144:517-546.] ) 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.

Charles Godfray (1991 [Godfray, H.C.J. (1991) Signalling of need by offspring to their parents. "Nature" 352:328–330.] ) 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 [Yachi, S. (1995) How can honest signalling evolve? The role of the handicap principle. "Proceedings of the Royal Society of London", B262:283–288.] ). 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.

ee also

*Cheap talk
*Extensive form game
*Signalling (economics)
*Signalling theory
*Solution concept
*Game theory

References

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