Signaling game
In game theory, a signaling game is a simple type of a dynamic Bayesian game. The essence of a signalling game is that one player takes an action, the signal, to convey information to another player, where sending the signal is more costly if they are conveying false information. A manufacturer, for example, might provide a warranty for its product in order to signal to consumers that its product is unlikely to break down. The classic example is of a worker who acquires a college degree not because it increases their skill, but because it conveys their ability  to employers. A simple signalling game would have two players, the sender and the receiver. The sender has one of two types that we might call "desirable" and "undesirable" with different payoff functions, where the receiver knows the probability of each type but not which one this particular sender has. The receiver has just one possible type. The sender moves first, choosing an action called the "signal" or "message" (though the term "message" is more often used in non-signalling "cheap talk" games where sending messages is costless). The receiver moves second, after observing the signal. The two players receive  payoffs dependent on the sender's type, the message chosen by the sender and the action chosen by the receiver. The tension in the game is that the sender wants to persuade the receiver that they have the desirable type, and they will try to choose a signal to do that. Whether this succeeds depends on whether the undesirable type would send the same signal, and how the receiver interprets the signal. Perfect Bayesian equilibrium. The equilibrium concept that is relevant for signaling games is Perfect Bayesian equilibrium, a refinement of Bayesian Nash equilibrium. Nature chooses the sender to have  type formula_1 with probability formula_2. The sender then chooses the probability with which to take signalling action formula_3, which we can write as formula_4 for each possible formula_5 The receiver observes the signal formula_6 but not formula_7, and chooses the probability with which to take response action formula_8, which we can write as formula_9 for each possible formula_10 The sender's payoff is formula_11 and the receiver's is formula_12 A perfect Bayesian equilibrium is a combination of beliefs and strategies for each player. Both players believe that the other will follow the strategies specified in the equilibrium, as in  simple Nash equilibrium, unless they observe something that has probability zero in the equilibrium. The receiver's beliefs also include a probability distribution formula_13 representing the probability put on the sender having type formula_7 if the receiver observes signal formula_15. The receiver's strategy is a choice of formula_16 The sender's strategy is a choice of formula_17. These beliefs and strategies must satisfy certain conditions: The kinds of perfect Bayesian equilibria that may arise can be divided in three different categories: pooling equilibria, separating equilibria and semi-separating. A given game may or may not have more than one  equilibrium. 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). There are also hybrid equilibria, in which the sender randomizes between pooling and separating. Examples. Reputation game. In this game, 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. Michael Spence's 1973 paper on education as a signal of ability is the start of the economic analysis of signalling. In this game, the senders are workers and the receivers are employers. The example below has two types of workers and a continuous signal level. The players are a worker and two firms. The worker chooses an education level formula_18 the signal, after which the firms simultaneously offer him a wage formula_19 and formula_20 and he accepts one or the other. The worker's type, known only to himself, is either high ability with formula_21 or low  ability with formula_22 each type having probability 1/2. The high-ability worker's payoff is formula_23 and the low-ability's is formula_24 A firm that hires the worker at wage formula_25 has payoff formula_26 and the other firm has payoff 0. In this game, the firms compete the wage down to where it equals the expected ability, so if there is no signal possible, the result would be formula_27 This will also be the wage in a pooling equilibrium, one where both types of worker choose the same signal, so the firms are left using their prior belief of  .5 for the probability he has High ability. In a separating equilibrium, the wage will be 0 for the signal level the Low type chooses and 10 for the high type's signal. There are many equilibria, both pooling and separating, depending on expectations. In a separating equilibrium, the low type chooses formula_28 The wages will be formula_29 and formula_30 for some critical level formula_31 that signals high ability. For the low type to choose formula_32 requires that formula_33 so formula_34 and we can conclude that formula_35 For the high type to choose formula_36 requires that formula_37 so formula_38 and  we can conclude that formula_39 Thus, any value of formula_31 between 5 and 10 can support an equilibrium. Perfect Bayesian equilibrium requires an out-of-equilibrium belief to be specified too, for all the other possible levels of formula_41 besides 0 and formula_42 levels which are "impossible" in equilibrium since neither type plays them. These beliefs must be such that neither player would want to deviate from his equilibrium strategy 0 or formula_31 to a different formula_44 A convenient belief is that formula_45 if formula_46 another, more realistic, belief that would support an equilibrium is formula_47 and formula_48 if formula_49  There is a continuum of equilibria, for each possible level of formula_50 One equilibrium, for example, is In a pooling equilibrium, both types choose the same formula_44 One pooling equilibrium is for both types to choose formula_53 no education, with the out-of-equilibrium belief formula_54 In that case, the wage will be the expected ability of 5, and neither type of worker will deviate to a higher education level because the firms would not think that told them anything about the worker's type. The most surprising result is that there are also pooling equilibria with formula_55 Suppose we  specify the out-of-equilibrium belief to be formula_56 Then the wage will be 5 for a worker with formula_57 but 0 for a worker with wage formula_58 The low type compares the payoffs formula_59 to formula_60 and if formula_61 he is willing to follow his equilibrium strategy of formula_62 The high type will choose formula_63 a fortiori. Thus, there is another continuum of equilibria, with values of formula_64 in [0, 2.5]. In the signalling model of education, expectations are crucial. If, as in the separating equilibrium, employers expect that high-ability people will acquire a certain level of education  and low-ability ones will not, we get the main insight: that if people cannot communicate their ability directly, they will acquire educations even if it does not increase productivity, just to demonstrate ability. Or, in the pooling equilibrium with formula_53 if employers do not think education signals anything, we can get the outcome that nobody becomes educated. Or, in the pooling equilibrium with formula_66 everyone acquires education that is completely useless, not even showing who has high ability, out of fear that if they deviate and do not acquire education, employers will think they have low ability. Beer -Quiche game. The Beer-Quiche game of Cho and Kreps 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 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 Replying to W.V.O. Quine, 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: 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, Grim, "et al.", Skyrms, and Zollman. Harms, and Huttegger, 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. 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. More recently, a series of papers by  Getty 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 . 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. 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. 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. 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.