In game theory, a signaling game is a simple type of a dynamic Bayesian game.
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.
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.
|Sane, Prey||P1+P1, D2||P1+M1, 0|
|Sane, Accommodate||D1+D1, D2||D1+M1, 0|
|Crazy, Prey||X1, P2||X1, 0|
In this game, 326–329 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.
This game was first presented by Michael Spence. 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.
The Beer-Quiche game of Cho and Kreps 14–18 for more details.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 :
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.
The first signaling game was the Lewis signaling game, which occurred in David K. Lewis' Ph. D. dissertation (and later book) Convention. SeeReplying 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.
The first application of signaling games to economic problems was Michael Spence's Education game. A second application was the Reputation game.
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.
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.
In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.
In game theory, cheap talk is communication between players that does not directly affect the payoffs of the game. Providing and receiving information is free. This is in contrast to signaling in which sending certain messages may be costly for the sender depending on the state of the world.
In game theory, a player's strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.
An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".
In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information. It is a refinement of Bayesian Nash equilibrium (BNE). A PBE has two components - strategies and beliefs:
In game theory, a Bayesian game is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.
In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies not only a strategy for each of the players but also a belief for each of the players. A belief gives, for each information set of the game belonging to the player, a probability distribution on the nodes in the information set. A profile of strategies and beliefs is called an assessment for the game. Informally speaking, an assessment is a perfect Bayesian equilibrium if its strategies are sensible given its beliefs and its beliefs are confirmed on the outcome path given by its strategies. The definition of sequential equilibrium further requires that there be arbitrarily small perturbations of beliefs and associated strategies with the same property.
Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.
Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.
In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.
The intuitive criterion (IC) is a technique for equilibrium refinement in signaling games. It aims to reduce possible outcome scenarios by first restricting the type group to types of agents who could obtain higher utility levels by deviating to off-the-equilibrium messages and second by considering in this sub-set of types the types for which the off-the-equilibrium message is not equilibrium dominated.
Jean-François Mertens was a Belgian game theorist and mathematical economist.
Network games of incomplete information represent strategic network formation when agents do not know in advance their neighbors, i.e. the network structure and the value stemming from forming links with neighboring agents. In such a setting, agents have prior beliefs about the value of attaching to their neighbors; take their action based on their prior belief and update their belief based on the history of the game. While games with a fully known network structure are widely applicable, there are many applications when players act without fully knowing with whom they interact or what their neighbors’ action will be. For example, people choosing major in college can be formalized as a network game with imperfect information: they might know something about the number of people taking that major and might infer something about the job market for different majors, but they don’t know with whom they will have to interact, thus they do not know the structure of the network.
The Divinity Criterion or Divine Equilibrium or Universal Divinity or D1-Criterion is a refinement of Perfect Bayesian equilibrium in a signaling game.
M equilibrium is a set valued solution concept in game theory that relaxes the rational choice assumptions of perfect maximization and perfect beliefs. The concept can be applied to any normal-form game with finite and discrete strategies. M equilibrium was first introduced by Jacob K. Goeree and Philippos Louis.