Signaling game

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An extensive form representation of a signaling game Signaling Game.svg
An extensive form representation of a signaling game

In game theory, a signaling game is a simple type of a dynamic Bayesian game. [1]

Contents

It is a game with two players, called the sender (S) and the receiver (R):

The game has two steps:

The two players receive payoffs dependent on the sender's type, the message chosen by the sender and the action chosen by the receiver. [2] [3]

Perfect Bayesian equilibrium

The equilibrium concept that is relevant for signaling games is Perfect Bayesian equilibrium—a refinement of both Bayesian Nash equilibrium and subgame-perfect equilibrium.

A sender of type sends a message in the set of probability distributions over M. ( represents the probabilities that type will take any of the messages in M.) The receiver observing the message m takes an action in the space of probability distributions over A.

A game is in perfect Bayesian equilibrium if it meets all four of the following requirements:

The perfect Bayesian equilibria in such a game can be divided in three different categories: pooling equilibria, separating equilibria and semi-separating

Note that, if there are more types of senders than there are messages, the equilibrium can never be a separating equilibrium (but may be semi-separating equilibria). There are also hybrid equilibria, in which the sender randomizes between pooling and separating.

Examples

Reputation game

Receiver
Sender
StayExit
Sane, PreyP1+P1, D2P1+M1, 0
Sane, AccommodateD1+D1, D2D1+M1, 0
Crazy, PreyX1, P2X1, 0

In this game, [1] :326–329 [4] the sender and the receiver are firms. The sender is an incumbent firm and the receiver is an entrant firm.

The payoffs are given by the table at the right. We assume that:

We now look for perfect Bayesian equilibria. It is convenient to differentiate between separating equilibria and pooling equilibria.

To summarize:

Education game

This game was first presented by Michael Spence. [5] [1] :329–331 In this game, the sender is a worker and the receiver is an employer.

In this model it is assumed that the level of education does not influence the productivity of the worker; it is used only as a signal regarding the worker's talent.

To summarize: 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.

Beer-Quiche game

The Beer-Quiche game of Cho and Kreps [6] draws on the stereotype of quiche eaters being less masculine. In this game, an individual B is considering whether to duel with another individual A. B knows that A is either a wimp or is surly but not which. B would prefer a duel if A is a wimp but not if A is surly. Player A, regardless of type, wants to avoid a duel. Before making the decision B has the opportunity to see whether A chooses to have beer or quiche for breakfast. Both players know that wimps prefer quiche while surlies prefer beer. The point of the game is to analyze the choice of breakfast by each kind of A. This has become a standard example of a signaling game. See [7] :14–18 for more details.

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 signaling game was the Lewis signaling game, which occurred in David K. Lewis' Ph. D. dissertation (and later book) Convention. See [8] Replying to W.V.O. Quine, [9] [10] 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. [11]

The use of signaling games has been continued in the philosophical literature. Others 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, [12] Grim, et al., [13] Skyrms, [14] [15] and Zollman. [16] Harms, [17] [18] and Huttegger, [19] 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 Education game. A second application was the Reputation game.

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. [20] The antlers of stags, the elaborate plumage of peacocks and bird-of-paradise, and the song of the nightingale are all such signals. Grafen's analysis of biological signaling is formally similar to the classic monograph on economic market signaling by Michael Spence. [21] More recently, a series of papers by Getty [22] [23] [24] [25] shows that Grafen's analysis, like that of Spence, is based on the critical simplifying assumption that signalers trade off costs for benefits in an additive fashion, the way humans invest money to increase income in the same currency. This assumption that costs and benefits trade off in an additive fashion might be valid for some biological signaling systems, but is not valid for multiplicative tradeoffs, such as the survival cost – reproduction benefit tradeoff that is assumed to mediate the evolution of sexually selected signals.

Charles Godfray (1991) modeled the begging behavior of nestling birds as a signaling game. [26] 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. [27] 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.

The concept of information asymmetry in molecular biology has long been apparent. [28] Although molecules are not rational agents, simulations have shown that through replication, selection, and genetic drift, molecules can behave according to signaling game dynamics. Such models have been proposed to explain, for example, the emergence of the genetic code from an RNA and amino acid world. [29]

Costly versus cost-free signaling

One of the major uses of signaling games both in economics and biology has been to determine under what conditions honest signaling can be an equilibrium of the game. That is, under what conditions can we expect rational people or animals subject to natural selection to reveal information about their types?

If both parties have coinciding interest, that is they both prefer the same outcomes in all situations, then honesty is an equilibrium. (Although in most of these cases non-communicative equilbria exist as well.) However, if the parties' interests do not perfectly overlap, then the maintenance of informative signaling systems raises an important problem.

Consider a circumstance described by John Maynard Smith regarding transfer between related individuals. Suppose a signaler can be either starving or just hungry, and they can signal that fact to another individual who has food. Suppose that they would like more food regardless of their state, but that the individual with food only wants to give them the food if they are starving. While both players have identical interests when the signaler is starving, they have opposing interests when the signaler is only hungry. When they are only hungry, they have an incentive to lie about their need in order to obtain the food. And if the signaler regularly lies, then the receiver should ignore the signal and do whatever they think is best.

Determining how signaling is stable in these situations has concerned both economists and biologists, and both have independently suggested that signal cost might play a role. If sending one signal is costly, it might only be worth the cost for the starving person to signal. The analysis of when costs are necessary to sustain honesty has been a significant area of research in both these fields.

See also

Related Research Articles

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