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 with probability . The sender then chooses the probability with which to take signalling action , which we can write as for each possible The receiver observes the signal but not , and chooses the probability with which to take response action , which we can write as for each possible The sender's payoff is and the receiver's is 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 representing the probability put on the sender having type if the receiver observes signal . The receiver's strategy is a choice of The sender's strategy is a choice of . These beliefs and strategies must satisfy certain conditions: *Sequential rationality: each strategy should maximize a player's expected utility, given his beliefs. *Consistency: each belief should be updated according to the equilibrium strategies, the observed actions, and Bayes' rule on every path reached in equilibrium with positive probability. On paths of zero probability, known as off-equilibrium paths, the beliefs must be specified but can be arbitrary. 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. * In a pooling equilibrium, senders of different types all choose the same signal. This means that the signal does not give any information to the receiver, so the receiver's beliefs are not updated after seeing the signal. * In a separating equilibrium, senders of different types always choose different signals. This means that the signal always reveals the sender's type, so the receiver's beliefs become deterministic after seeing the signal. * In a semi-separating equilibrium (also called partial-pooling), some types of senders choose the same message and other types choose different messages. 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 sender can be one of two types: ''Sane'' or ''Crazy''. A sane sender can send one of two messages: ''Prey'' and ''Accommodate''. A crazy sender can only Prey. * The receiver can do one of two actions: ''Stay'' or ''Exit''. The payoffs are given by the table at the right. We assume that: * M1>D1>P1, i.e., a sane sender prefers to be a monopoly (M1), but if it is not a monopoly, it prefers to accommodate (D1) than to prey (P1). Note that the value of X1 is irrelevant since a Crazy firm has only one possible action. * D2>0>P2, i.e., the receiver prefers to stay in a market with a sane competitor (D2) than to exit the market (0), but prefers to exit than to stay in a market with a crazy competitor (P2). * Apriori, the sender has probability ''p'' to be sane and 1-''p'' to be crazy. We now look for perfect Bayesian equilibria. It is convenient to differentiate between separating equilibria and pooling equilibria. * A separating equilibrium, in our case, is one in which the sane sender always accommodates. This separates it from a crazy sender. In the second period, the receiver has complete information: their beliefs are "If Accommodate then the sender is sane, otherwise the sender is crazy". Their best-response is: "If Accommodate then Stay, if Prey then Exit". The payoff of the sender when they accommodate is D1+D1, but if they deviate to Prey their payoff changes to P1+M1; therefore, a necessary condition for a separating equilibrium is D1+D1≥P1+M1 (i.e., the cost of preying overrides the gain from being a monopoly). It is possible to show that this condition is also sufficient. * A pooling equilibrium is one in which the sane sender always preys. In the second period, the receiver has no new information. If the sender preys, then the receiver's beliefs must be equal to the apriori beliefs, which are, the sender is sane with probability ''p'' and crazy with probability 1-''p''. Therefore, the receiver's expected payoff from staying is: 'p'' D2 + (1-''p'') P2 the receiver stays if-and-only-if this expression is positive. The sender can gain from preying, only if the receiver exits. Therefore, a necessary condition for a pooling equilibrium is ''p'' D2 + (1-''p'') P2 ≤ 0 (intuitively, the receiver is careful and will not enter the market if there is a risk that the sender is crazy. The sender knows this, and thus hides their true identity by always preying like a crazy). But this condition is not sufficient: if the receiver exits also after Accommodate, then it is better for the sender to Accommodate, since it is cheaper than Prey. So it is necessary that the receiver stays after Accommodate, and it is necessary that D1+D1Education 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 the signal, after which the firms simultaneously offer him a wage and and he accepts one or the other. The worker's type, known only to himself, is either high ability with or low ability with each type having probability 1/2. The high-ability worker's payoff is and the low-ability's is A firm that hires the worker at wage has payoff 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 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 The wages will be and for some critical level that signals high ability. For the low type to choose requires that so and we can conclude that For the high type to choose requires that so and we can conclude that Thus, any value of 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 besides 0 and 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 to a different A convenient belief is that if another, more realistic, belief that would support an equilibrium is if and if . There is a continuum of equilibria, for each possible level of One equilibrium, for example, is : In a pooling equilibrium, both types choose the same One pooling equilibrium is for both types to choose no education, with the out-of-equilibrium belief 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 Suppose we specify the out-of-equilibrium belief to be Then the wage will be 5 for a worker with but 0 for a worker with wage The low type compares the payoffs to and if he is willing to follow his equilibrium strategy of The high type will choose a fortiori. Thus, there is another continuum of equilibria, with values of in , 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 if employers do not think education signals anything, we can get the outcome that nobody becomes educated. Or, in the pooling equilibrium with 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 ''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 inEconomics
The first application of signaling games to economic problems was Michael Spence'sBiology
Valuable advances have been made by applying signaling games to a number of biological questions. Most notably,Costly versus cost-free signaling
One of the major uses of signaling games both inSee also
* Cheap talk *References
{{DEFAULTSORT:Signaling Game Game theory game classes Asymmetric information