General incentives in finite game theory

Fryer, Dashiell

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https://hdl.handle.net/2142/29786 Copy

Description

Title

General incentives in finite game theory

Author(s)

Fryer, Dashiell

Issue Date

2012-02-06T20:16:42Z

Director of Research (if dissertation) or Advisor (if thesis)

Muncaster, Robert G.

Committee Member(s)

• Muncaster, Robert G.
• Rapti, Zoi
• Gaines, Brian J.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• game theory
• incentive
• Incentives
• Nash
• equilibrium
• Kullback-Leibler divergence
• evolutionary stable states (ESS)
• evolutionary game theory
• Prisoner's Dilemma
• Traveler's Dilemma
• Incentive stable states (ISS)

Abstract

A general framework for analyzing finite games will be introduced. The concept of an incentive function will be defined so that it is compatible with the updating protocol defined in Nash's proof of existence of equilibrium in all finite games. A general notion of incentive equilibrium will be defined as the fixed points of the updating protocol. It will be shown that given a continuous incentive, an incentive equilibrium will exist for any finite game. Specific examples will be given that have connections to canonical game dynamics and the incentive equilibrium are described fully. Non canonical examples will also be defined including an example based on simultaneous updating of strategies by the agents. A system of differential equations will be derived from the updating protocol, which have fixed points exactly at incentive equilibrium. It will be shown that the canonical dynamics can be achieved using the canonical incentives. Specifically the Brown-von Neumann-Nash dynamics, replicator equations, projection dynamics, logit equations, best-reply dynamics, and pairwise comparison dynamics will all be derived from their respective incentives. It will be shown that the incentive dynamics are fully general in the sense that it can be used to describe all possible game dynamics that preserve the strategy space. Incentive stable states, ISS, for general incentive dynamics will be defined as an analog to the concept of the evolutionary stable states, ESS, in the replicator dynamics. The connection between the replicator dynamics and information theory is discussed. Of particular interest is the use of the Kullback-Leibler divergence as a Lyapunov function to show that an ESS is asymptotically stable for the replicator dynamics. It will be shown that the Kullback-Liebler divergence is also a Lyapunov function for the incentive dynamics at ISS. An important example of ISS is given by realizing the uniform distribution as an interior ISS for an incentive based on simultaneous updating in a class of games that includes all variations of Rock-Paper-Scissors, RPS. The uniform distribution is the unique incentive (Nash) equilibrium for all of the canonical dynamics. However, for a specific choice of parameters in RPS if an orbit has an initial condition that is not the uniform distribution it will not converge to the unique fixed point in any of the canonical dynamics. In stark contrast, the ISS condition guarantees the uniform distribution is asymptotically stable for simultaneous updating. It will be shown that it is in fact globally asymptotically stable for the interior of the strategy space. A collection of numerical results will be given for a particular incentive. It will be shown in a number of distinct games that the incentive equilibrium is a better approximation to human behavior than the Nash equilibrium. This collection of games includes the Prisoner's Dilemma, Matching Pennies, Battle of the Sexes, the Traveler's Dilemma, Chicken, and other unnamed games of interest. Several different variations on each one of these games will be given to demonstrate the dependence on absolute differences in payoffs inherent in this model of incentive. This feature, known as cardinal dependence, is exhibited by human actors but is missing from the Nash model, which is preference, or ordinal, dependent. It will also be shown that under this incentive agents display a competitive nature as evidenced by certain games with `win-win' strategies having alternative equilibrium that are asymptotically stable. Specifically this behavior seems to appear when one agent has more opportunities to attain its maximum payoff than the other agents. This is also an observable behavior of human actors which is absent from the Nash model.

Graduation Semester

2011-12

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http://hdl.handle.net/2142/29786

Copyright and License Information

Copyright 2011 Dashiell Edward Alston Fryer

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