Lewis signaling game

An extensive form representation of a two-person Lewis signalling game

In game theory, the Lewis signaling game is a type of signaling game that features perfect common interest between players. It is named for the philosopher David Lewis who was the first to discuss this game in his Ph.D. dissertation, and later book, Convention.^[1]

The game

The underlying game has two players, the sender and the receiver. The world can be in any of a number of states and the sender is aware of that state. The sender has at his disposal a fixed set of signals that he can send to the receiver. The receiver can observe the signal sent, but not the state of the world, and must take some action. For each state, there is a unique correct action and both the sender and receiver prefer that the receiver take the correct action in every state. Because both the sender and receiver prefer the same outcomes as one another, this game is a game of pure common interest.

The simplest version of this game (pictured above) has two states, two signals, and two acts.

Equilibria

A signaling system equilibrium

A pooling equilibrium

A partial pooling equilibrium, dotted lines represent mixed strategies.

This game has many Nash equilibria. A few of them stand out where the sender sends a different signal in each state and the receiver takes the appropriate action in every state. Lewis dubbed these signaling systems.^[1] But there are also other equilibria. In some the sender sends the same signal in every state and the receiver takes the action that is best to take given no additional information about the state of the world (pooling equilibria).

Also, when there are more than two states, signals, and acts, there are partial pooling equilibria where some information is conveyed, but some states are also pooled.^[2]^[3]

References

1 2 Lewis, D.: 1969, Convention. A Philosophical Study, Harvard University Press, Harvard, Mass.
↑ Huttegger, Simon (2007) "Evolution and Explanation of Meaning", Philosophy of Science 74(1), 1–27.
↑ Pawlowitsch, Christina (2008) "Why evolution does not always lead to an optimal signaling system," Games and Economic Behavior 63: 203–226.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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