Poisson game - Misplaced Pages

Game theory model of voting

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In game theory and political science, Poisson games are a class of games often used to model the behavior of large populations. One common application is determining the strategic behavior of voters with imperfect information about each other's preferences. Poisson games are most often used to model strategic voting in large electorates with secret and simultaneous voting.

A Poisson game consists of a random population of players of various types, the size of which follow a Poisson distribution. This can occur when voters are not sure what the relative turnout of each party will be, or when they have imperfect polling information. For example, a model of the 1992 United States presidential election might include 4 types of voters: Democrats, Republicans, and two classes of Reform voters (those with second preferences of either Bill Clinton or George H.W. Bush).

Main assumptions

The first assumption of the model is that the total number of players of each type follows a Poisson distribution. In other words, the probability of $n$ voters turning out to support a given candidate is given by:
$P(N=k)=e^{-n}{\frac {n^{k}}{k!}}$

More important is the assumption that voters are only interested in securing the best possible election outcome for themselves, and are motivated only by the possibility of casting the deciding vote. In other words, voters are assumed not to care about expressing their true opinions; about showing support for a minor party, even if they do not win; or about allowing other voters' voices to be heard. All of these effects tend to produce more honest voting in real elections than would be found in the Poisson model.

In the model, all information is publicly-available, meaning that every voter can estimate the probability that each pair of candidates will be tied. An example of this would be an election with public opinion polling.

Results

The Poisson voting model generates several key results.

Approval and score

Under the Poisson model, approval voting and score voting behave identically, as each voter's best strategy involves casting a ballot that assigns every candidate either the maximum or minimum score.

Plurality

Under plurality, sincere voting is never a stable equilibrium with more than two candidates, i.e. many voters are incentivized to lie about their favorite candidate and vote for the lesser of two evils. For example, in the 2016 United States presidential election, some polls suggest that Gary Johnson was the majority-preferred winner. However, Johnson ultimately received only a small fraction of the vote because voters expected him to lose, creating a self-fulfilling prophecy.

References

^ Myerson, Roger (1998). "Population Uncertainty and Poisson games". International Journal of Game Theory. 27 (3): 375–392. CiteSeerX 10.1.1.21.9555. doi:10.1007/s001820050079.
Potthoff, Richard F.; Munger, Michael C. (November 2021). "Condorcet Loser in 2016: Apparently Trump; Condorcet Winner: Not Clinton?". American Politics Research. 49 (6): 618–636. doi:10.1177/1532673X211009499. ISSN 1532-673X.

Myerson, Roger B. (2000). "Large Poisson Games". Journal of Economic Theory. 94 (1): 7–45. doi:10.1006/jeth.1998.2453.
Myerson, Roger B. (1998). "Population Uncertainty and Poisson Games". International Journal of Game Theory. 27 (3): 375–392. CiteSeerX 10.1.1.21.9555. doi:10.1007/s001820050079.
De Sinopoli, Francesco; Pimienta, Carlos G. (2009). "Undominated (and) perfect equilibria in Poisson games". Games and Economic Behavior. 66 (2): 775–784. CiteSeerX 10.1.1.549.9282. doi:10.1016/j.geb.2008.09.029.

v t e Topics of game theory
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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
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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

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Main assumptions

Results

Approval and score

Plurality

See also

References