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

[edit]

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 its disposal a fixed set of signals that it 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 perfect common interest.

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

Equilibria

[edit]

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

[edit]

^ ^a ^b 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.

v t e Topics of game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game Mechanism design
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Coalition-proof Nash equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand equilibrium
Strategies	Appeasement Backward induction Bid shading Collusion Cheap talk De-escalation Deterrence Escalation Forward induction Grim trigger Markov strategy Pairing strategy Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat
Classes of games	Auction Bargaining problem Global game Intransitive game Mean-field game n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Strictly determined game Stochastic game Symmetric game Zero-sum game
Games	Go Chess Infinite chess Checkers All-pay auction Prisoner's dilemma Gift-exchange game Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Lewis signaling game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Electronic mail game Rock paper scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Bertrand competition Cournot competition Stackelberg competition Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Induction puzzles Trust game Princess and monster game Rendezvous problem
Theorems	Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Search optimizations	Alpha–beta pruning Aspiration window Principal variation search max^n algorithm Paranoid algorithm Lazy SMP
Miscellaneous	Bounded rationality Combinatorial game theory Confrontation analysis Coopetition Evolutionary game theory Glossary of game theory List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons