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GT ISYE 6230 - Introduction to Game Theory

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11Introduction to Game TheoryA collection of tools for predicting outcomes for a group of interacting agents where an action of a single agent directlyeffects the payoffs of the other participating agents.2What is a game?¡ Many types of games: board games, card games, video games, field games (e.g., football), etc. l A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses. l A game is non-zero sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain.¡ We focus on games where: l There are 2 or more players. l There is some choice of action where strategy matters. l The game has one or more outcomes, e.g. someone wins, someone loses. l The outcome depends on the strategies chosen by all players; there is strategic interaction. ¡ What does this rule out? l Games of pure chance, e.g., lotteries, slot machines. (Strategies don't matter). l Games without strategic interaction between players, e.g. Solitaire.23Elements of a game¡ The playersl how many players are there? l does nature/chance play a role? ¡ A complete description of what the players can do – the set of all possible actions. ¡ A description of the payoffconsequences for each player for every possible combination of actions chosen by all players playing the game. 4Example: NBC vs. ABC45% , 55%50% , 50%Sitcom52% , 48%55% , 45%NewsSitcomNewsNBCABC¡ Players: NBC and ABC¡ Set of all possible actions: sitcom, news¡ Payoffs: market shares for each outcome35Normal form game: Notationoutcome in theexcept players all of actions thedenotes},...,,,...,,{outcome game theis},...,,{ where payoff a hasplayer Each by action particular a is such that },...,,{set action an has player Each 1 players ofset A 11212121iaaaaaaaaaa(a) iiAaaaaANi, ..., N}{INNiiiNiiiikiiii+−−=•=•∈=∈•=•π6Assumptions¡ Payoffs are known and fixed. ¡ Players are risk neutral, i.e., maximize expected payoffs.l Example: a risk neutral person is indifferent between ¡ $25 for certain or ¡ a 25% chance of earning $100 and a 75% chance of earning 0. ¡ All players behave rationally. They understand and seek to maximize their own payoffs. ¡ The rules of the game are common knowledgel Each player knows the set of players, strategies and payoffs from all possible combinations of strategies: call this information “X.” Common knowledge means that each player knows that all players know X, that all players know that all players know X, that all players know that all players know that all players know X and so on,..., ad infinitum.47Princess Bride¡ An exceptionally-told fairy-tale. One scene humorously highlights both strategic manipulation of the rules of the game and the unrealistic assumption of common knowledge(Comedy, 1987) --gametheory.net8Classroom game, LC2, 25, 0Red0, 53, 3BlackRedBlackPlayer 1Player 259Classroom game, HC2, 210, 0Red0, 108, 8BlackRedBlackPlayer 1Player 210Game Lessons611Examples of Game12Prisoner’s Game: First Period¡ Playing a red card increases your earnings by $2 l (makes no change to partner’s earnings)¡ Playing a black card increases your partner’s earnings by $3 l (makes no change to yours)713Prisoner’s Game: Second Period¡ Red: increases your earnings by $2l (makes no change to partner’s earnings)¡ Black: increases partner’s earnings by $8 l (makes no change to yours)14Prisoner’s Game: Periods 3 – 5¡ Partner matchings will be the same for the final three rounds¡ One row may discuss their planned strategies815Questions¡ How did results change in second period?¡ How can we represent this game mathematically?¡ In which cases is there an incentive to deviate?¡ Effect of 1stperiod versus 2ndperiod?16Questions¡ What is the effect of repetition in periods 3 and 4?l (Ask people who played red in 1 and black later.)¡ What happens in the final period and why?917Notes¡ Static versus dynamic games¡ Dominated strategies¡ (Arms race, free-rider problem)¡ Trigger strategies l One switch, or tit-for-tat18Applications¡ Final period: tourist purchasing a good¡ Price-cutting; offering low-quality goodsl More demand if players collude on high price l More demand if players agree to all provide high quality¡ Provision of public goods¡ What characterizes all of these


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GT ISYE 6230 - Introduction to Game Theory

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