Signaling games - AbsoluteAstronomy.com

A signaling game is a dynamic, Bayesian game

Bayesian game

In game theory, a Bayesian game is one in which information about characteristics of the other players is incomplete. Following John C. Harsanyi's framework, a Bayesian game can be modelled by introducing Nature as a player in a game...

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. A related game is a screening game

Screening game

A screening game is a two-player principal–agent type game used in economic and game theoretical modeling. Principal–agent problems are situations where there are two players whose interests are not necessarily at ends, but where complete honesty is not optimal for one player...

where rather than choosing an action based on a signal, the receiver gives the sender proposals based on the type of the sender, which the sender has some control over.

Signaling games were introduced by In-Koo Cho and David M. Kreps

David M. Kreps

David Marc "Dave" Kreps is a game theorist and economist and professor at the Graduate School of Business at Stanford University. He is known for his analysis of dynamic choice models and non-cooperative game theory, particularly the idea of sequential equilibrium, which he developed with Stanford...

in a 1987 article.

Costly versus cost-free signaling

One of the major uses of signaling games both in economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

and biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

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

John Maynard Smith

John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....

regarding transfer between related individuals. Suppose a signaler can be either starving or just hungry, and she can signal that fact to another individual which has food. Suppose that she would like more food regardless of her state, but that the individual with food only wants to give her the food if she is starving. While both players have identical interests when the signaler is starving, they have opposing interests when she is only hungry. When the signaler is hungry she has an incentive to lie about her need in order to obtain the food. And if the signaler regularly lies, then the receiver should ignore the signal and do whatever he thinks 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.

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

Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a 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 his own strategy unilaterally...

to games of incomplete information. Perfect Bayesian equilibrium is the equilibrium concept relevant for dynamic games of incomplete information.

Definition of perfect Bayesian equilibrium of the signaling game

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.

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

, the probability that the sender has type

if he chooses message

. The sum over all types

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

. This means that the sum

is maximized. The action

that maximizes this sum is

Requirement 3

For each type,

, the sender may have, the sender chooses to send the message

that maximizes the sender's utility

given the strategy chosen by the receiver,

Requirement 4

For each message

the sender can send, if there exists a type

such that

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

satisfies the equation (Bayes rule)

The perfect Bayesian equilibria in such a game can be divided in three different categories, pooling equilibria, semi-pooling (also called semi-separating), and separating equilibria. A pooling equilibrium is an equilibrium where senders with different types all choose the same message. A semi-pooling equilibrium is an equilibrium where some types of senders choose the same message and other types choose different messages. A separating equilibrium is an equilibrium where senders with different types always choose different messages. Therefore, if there are more types of actors than there are messages, the equilibrium can never be a separating equilibrium (but may be semi-separating equilibria).

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. dissertation (and later book) Convention. Replying to W.V.O. Quine, Lewis attempts to develop a theory of convention

Convention (norm)

A convention is a set of agreed, stipulated or generally accepted standards, norms, social norms or criteria, often taking the form of a custom....

and meaning

Meaning (linguistics)

In linguistics, meaning is what is expressed by the writer or speaker, and what is conveyed to the reader or listener, provided that they talk about the same thing . In other words if the object and the name of the object and the concepts in their head are the same...

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.

The use of signaling games has been continued in the philosophical literature. Other have used evolutionary models

Evolutionary game theory

Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...

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

Norm (philosophy)

Norms are concepts of practical import, oriented to effecting an action, rather than conceptual abstractions that describe, explain, and express. Normative sentences imply “ought-to” types of statements and assertions, in distinction to sentences that provide “is” types of statements and assertions...

and descriptive language.

Economics

The first application of signaling games to economic problems was Michael Spence

Michael Spence

Andrew Michael Spence is an American economist and recipient of the 2001 Nobel Memorial Prize in Economic Sciences, along with George A. Akerlof and Joseph E. Stiglitz, for their work on the dynamics of information flows and market development. He conducted this research while at Harvard University...

's model of job market signaling

Signalling (economics)

In economics, more precisely in contract theory, signalling is the idea that one party credibly conveys some information about itself to another party...

. 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 choose to offer the worker a high or low wage. In this model it is assumed that the level of education does not cause the high ability of the worker, but rather, 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.

Biology

Valuable advances have been made by applying signaling games to a number of biological questions. Most notably, Alan Grafen

Alan Grafen

Alan Grafen is a Scottish ethologist and evolutionary biologist. He currently teaches and undertakes research at St John's College, Oxford. Along with regular contributions to scientific journals, Grafen is known publicly for his work as co-editor of the 2006 festschrift Richard Dawkins: How a...

's (1990) handicap

Handicap principle

The handicap principle is a hypothesis originally proposed in 1975 by biologist Amotz Zahavi to explain how evolution may lead to "honest" or reliable signaling between animals who have an obvious motivation to bluff or deceive each other...

model of mate attraction displays. The antlers of stags, the elaborate plumage of peacocks and birds of paradise, and the song

Bird song

Bird vocalization includes both bird calls and bird songs. In non-technical use, bird songs are the bird sounds that are melodious to the human ear. In ornithology and birding, songs are distinguished by function from calls.-Definition:The distinction between songs and calls is based upon...

of the nightingale

Nightingale

The Nightingale , also known as Rufous and Common Nightingale, is a small passerine bird that was formerly classed as a member of the thrush family Turdidae, but is now more generally considered to be an Old World flycatcher, Muscicapidae...

are all such signals. Grafen’s analysis of biological signaling is formally similar to the classic monograph on economic market signaling by Michael Spence

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

Charles Godfray

Hugh Charles Jonathan Godfray CBE FRS is a British zoologist. He is Hope Professor of Zoology at Jesus College, Oxford.-Life:...

(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

Stotting

Stotting is a gait of quadrupeds, particularly gazelles , involving jumping high into the air by lifting all four feet off the ground simultaneously. This may occur during pursuit by a predator...

', 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.

Costly versus cost-free signaling

Perfect Bayesian equilibrium

Definition of perfect Bayesian equilibrium of the signaling game

Requirement 1

Requirement 2

Requirement 3

Requirement 4

Applications of signaling games

Philosophy

Economics

Biology

See also