An extensive form representation of a signalling game

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 (Gibbons, 1992, Osborne & Rubenstein, 1994). Image File history File links Download high resolution version (1619x1194, 14 KB) Summary Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Signaling games User:Kzollman/Images ... Image File history File links Download high resolution version (1619x1194, 14 KB) Summary Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Signaling games User:Kzollman/Images ... It has been suggested that Game tree be merged into this article or section. ... In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ...

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, a Bayesian game is one in which information about characteristics of the other players (i. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...

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 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 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, μ(t | m). This means that the sum

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, μ(t | m) 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

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. disertation (and later book), Convention. Replying to W.V.O. Quine (1960, 1967), 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: David K. Lewis David Kellogg Lewis (September 28, 1941 - October 14, 2001) is considered by many to have been the leading Analytic philosopher of the latter half of the 20th century. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... A convention is a set of agreed, stipulated or generally accepted rules, norms, standards or criteria, often taking the form of a custom. ... In linguistics, meaning is the content carried by the words or signs exchanged by people when communicating through language. ...

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. Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ... In philosophy, normative is usually contrasted with positive, descriptive or explanatory when describing types of theories, beliefs, or statements. ... This page is a candidate to be moved to Wiktionary. ...

Economics

The first application of signaling games to economic problems was Michael 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. Michael Spence is a winner of Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, along with George A. Akerlof and Joseph E. Stiglitz, for their work on the dynamics of information flows and market development. ... In economics, more precisely in contract theory, signalling is the idea that one party (termed the agent) conveys some meaningful information about itself to another party (the principal). ... 1973 (MCMLXXIII) was a common year starting on Monday. ...

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. The handicap principle is an idea proposed by the Israeli biologist Amotz Zahavi. ... Peacock re-directs here; for alternate uses see Peacock (disambiguation). ... For the flowering plant of this name, see Strelitzia Genera Cicinnurus Diphyllodes Epimachus Lophorina Manucodia Paradisaea Parotia Ptiloris Seleucidis Lesser Bird of Paradise Paradisaea minor (c)Roderick Eime The birds of paradise are members of the family Paradisaeidae of the order Passeriformes, found in Oceania. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Binomial name Luscinia megarhynchos (Brehm, 1831) This article is about the bird. ...

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 benfit level of investment. The parents are trading off investment in the current nestlings against investment in future offspring.

Pusuit 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.

See also

• Cheap talk
• Extensive form game
• Signalling (economics)
• Solution concept

Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ... It has been suggested that Game tree be merged into this article or section. ... In economics, more precisely in contract theory, signalling is the idea that one party (termed the agent) conveys some meaningful information about itself to another party (the principal). ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ...

References

• Gibbons, Robert (1992) A Primer in Game Theory, Harvester Wheatsheaf
• Godfray, H.C.J. (1991) Signalling of need by offspring to their parents. Nature 352:328–330.
• Grafen, A. (1990) Biological signals as handicaps. Journal of Theoretical Biology 144:517-546.
• 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.
• Harms, W. F. (2000) "Adaption and Moral Realism." Biology and Philosophy 15:699–712.
• Harms, W. F. (2004) Information and Meaning in Evolutionary Processes. Cambridge: Cambridge University Press.
• Huttegger, S. M. (2005) "Evolution and the Explanation of Meaning." Forcomming in Philosophy of Science
• Huttegger, S. M. (2005) "Evolutionary Explanations of Normative and Descriptive Statements"
• Osborne, M.J. and Rubenstein, A. (1994) A Course in Game Theory, MIT Press
• Quine, W.v.O (1967) "Truth by Convention" in Philosophica Essays for A.N. Whitehead Russel and Russel Publishers. ISBN 0846209705
• Quine, W.v.O (1960) "Carnap and Logical Truth" Synthese 12(4):350-374.
• Skyrms, B. (1996) Evolution of the Social Contract. Cambridge: Cambridge University Press.
• Skyrms, B. (2000) "Stability and Explanatory Significance of Some Simple Evolutionary Models." Philosophy of Science 67:94–113.
• Spence, A.M. (1973) Job Market Signaling, Quarterly Journal of Economics. 87:355-374.
• Yachi, S. (1995) How can honest signalling evolve? The role of the handicap principle. Proceedings of the Royal Society of London, B262:283–288.
• Zollman, K. J. S. (2005) "Talking to Neighbors: The Evolution of Regional Meaning." Philosophy of Science 72:69–85.

Ariel Rubenstein (born April 13, 1951) is an economist who works in game theory. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... Brian Skyrms is a Distinguished Professor of Logic and Philosophy of Science and Economics at the University of California, Irvine. ... Michael Spence is a winner of Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, along with George A. Akerlof and Joseph E. Stiglitz, for their work on the dynamics of information flows and market development. ...