Game theory

Game theory,branchmathematics, operations researcheconomics, isanalysisinteractionsformalized incentive structures ("games"). The predictedactual behaviorindividualsthese gamesstudied, as well as optimal strategies. Seemingly different typesinteractions can be characterized as having similar incentive structures, thus all being examplesone particular game.

Game theoryclosely relatedeconomicsthatseeksfind rational strategiessituations whereoutcome depends not only on one's own strategy"market conditions", but uponstrategies chosen by other playerspossibly different or overlapping goals. It also finds wider applicationfields such as political sciencemilitary strategy.

The results can be appliedsimple gamesentertainment ormore significant aspectslifesociety. An example ofapplicationgame theoryreal life isprisoner's dilemma as popularized by mathematician Albert W. Tucker;has many implications fornaturehuman cooperation. Biologists have used game theoryunderstandpredict certain outcomesevolution, such asconceptevolutionarily stable strategy introduced by John Maynard Smithhis essay Game Theory andEvolutionFighting. See also Maynard Smith's book Evolution andTheoryGames.

Other branchesmathematics,particular probability, statisticslinear programming,commonly usedconjunctiongame theoryanalyse games.

Tablecontents

1 Typesgamesexamples 2 Risk aversion 3 Gamesnumbers 4 History 5 External linksreferences

Typesgamesexamples

Game theory classifies games into many categories that determine which particular methods can be appliedsolving them (and indeed how one defines "solved" forparticular category). Some common categories are:

Zero-sum gamesthosewhichtotal benefitall players ingame addszero (or more informally put, that each player benefits only atexpenseothers). ChessPokerzero-sum games, because one wins exactlyamount one's opponents lose. Business, politics andPrisoner's dilemma,example,non-zero-sum games because some outcomesgoodall players or badall players. Iteasier, however,analyzezero-sum game,it turns outbe possibletransform any game intozero-sum game by adding an additional dummy player often called "the board," whose losses compensateplayers' net winnings.

A convenient wayrepresentgamegiven by its payoff matrix. Considerexampletwo-player zero-sum game withfollowing matrix:

                                Player 2 

                    Action A    Action B    Action C

          Action 1    30         -10          20 
Player 1
          Action 2    10          20         -20

This gameplayed as follows:first player chooses one oftwo actions 1 or 2, andsecond player, unaware offirst player's choice, chooses one ofthree actions A, B or C. Once these choices have been made,payoffallocated according totable;instance, iffirst player chose action 2 andsecond player chose action B, thenfirst player gains 20 points andsecond player loses 20 points. Both players knowpayoff matrixattemptmaximizenumbertheir points. What shoulddo?

Player 1 could reason as follows: "with action 2, I could lose up20 pointscan win only 20, whileaction 1 I can lose only 10 but can win up30, so action 1 lookslot better." With similar reasoning, player 2 would choose action C (negative numbers intablegoodhim). If both players take these actions,first player will win 20 points. But how about if player 2 anticipatesfirst player's reasoningchoiceaction 1,deviously goesaction B, so aswin 10 points? Or iffirst playerturn anticipates this devious trickgoesaction 2, so aswin 20 points after all?

The fundamentalsurprising insight by John von Neumann was that probability providesway outthis conundrum. Insteaddeciding ondefinite actiontake,two players assign probabilitiestheir respective actions,then userandom device which, accordingthese probabilities, chooses an actionthem. The probabilitiescomputed so asmaximizeexpected point gain independent ofopponent's strategy; this leads tolinear programming problem withunique solutioneach player. This method can compute provably optimal strategiesall two-player zero-sum games.

Forexample given above,turns out thatfirst player should chose action 1probability 57%action 243%, whilesecond player should assignprobabilities 0%, 57%43% tothree actions A, BC. Player one will then win 2.85 points on average per game.

Non Zero-Sum game The most famous example ofnon-zero-sum gamethe Prisoner's dilemma, as mentioned above. Any gain by one player does not necessarily correspond withloss by another player. Most real-world situationsnon zero-sum games. For example,business contract ideally ispositive-sum game, where each sidebetter off than ifdidn't havecontract. Most games that people playrecreationzero-sum.

Cooperative gamesthosewhichplayers may freely communicate among themselves before making game decisionsmay make bargainsinfluence those decisions. Monopoly can becooperative game, whilePrisoner's dilemmanot. However, Monopoly iszero-sum game as there can be only one winner, whereasPrisoner's dilemma isnon-zero-sum game.

Complete information gamesthosewhich each player hassame game-relevant information as every other player. Chess andPrisoner's dilemmacomplete-information games, while Pokernot. Complete information gamesrare inreal world, andusually used only as approximations ofactual game being played.

Risk aversion

Forabove examplework,participants ingame havebe assumedbe risk neutral. This means that,example,would valuebet with50% chancereceiving 20 'points' and50% chancepaying nothing as being worth 10 points. However,reality peopleoften risk averseprefermore certain outcome -will only takerisk ifexpectmake money on average. Subjective expected utility theory explains howmeasureutility can be derived which will always satisfycriterionrisk neutrality,hencesuitable asmeasure forpayoffgame theory.

One examplerisk aversion can be seen on Game Shows. For example, ifperson has13 chancewinning $50,000, or can takesure $10,000, many people will takesure $10,000.

Gamesnumbers

John Conway developednotationcertain gamesdefined several operations on those games, originallyorderstudy Go endgames. Insurprising connection, he found thatcertain subclassthese games can be used as numbers, leading tovery general classsurreal numbers.

History

Though touched on by earlier mathematical results, modern game theory becameprominent branchmathematics in1940s, especially after1944 publicationThe TheoryGamesEconomic Behavior by John von NeumannOskar Morgenstern. This profound work containedmethodfinding optimal solutionstwo-person zero-sum games alludedabove.

Around 1950, John Nash developeddefinitionan "optimum" strategymulti player games where no such optimum was previously defined, known as Nash equilibrium. This concept was further refined by Reinhard Selten. These men were awarded The BankSweden PrizeEconomic SciencesMemoryAlfred Nobel1994their work on game theory, alongJohn Harsanyi who developedanalysisgamesincomplete information.

Conway's number-game connection was found inearly 1970s.

See also Mathematical game; Artificial intelligence; Newcomb's paradox; game classification.

External linksreferences

• Paul Walker, An Outline ofHistoryGame Theory.
• Oskar Morgenstern, John von Neumann: The TheoryGamesEconomic Behavior, 3rd ed., Princeton University Press 1953
• Alvin Roth: Game TheoryExperimental Economics page, http://www.economics.harvard.edu/~aroth/alroth.html Comprehensive listlinksgame theory information onWeb
• Mike Shor: Game Theory .net, http://www.gametheory.net Lecture notes, interactive illustrationsother information.
• Maynard Smith: Evolution andTheoryGames, Cambridge University Press 1982
• Don Ross: Review Of Game Theory.