11Recap of Jan 9, 2007¡ Course goals and objectives¡ Examples of “strategic” business interactions¡ Classroom game2Today—Jan 11, 2007¡ Intro to Game Theoryl Elements of a game (normal form)l Dominant actionsl Best responsel Nash Equilibriuml Mixed Strategiesl Pareto dominance23Introduction 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.4What 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.35Elements 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. 6Example: 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 outcome47Normal 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+−−=•=•∈=∈•=•π8Assumptions¡ 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.59Princess 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.net10Example: NBC vs. ABC45% , 55%50% , 50%Sitcom52% , 48%55% , 45%NewsSitcomNewsNBCABC¡ What should ABC do given each of NBC’s decisions?¡ What should NBC do given each of ABC’s decisions?611Example: NBC vs. ABC45% 55%50% 50%Sitcom52% 48%55% 45%NewsSitcomNewsNBCABC¡ Regardless of ABC’s action, NBC chooses sitcom - Sitcom is a dominant action for NBC¡ Regardless of NBC’s action, ABC chooses news - News is a dominant action for ABC12Definition: Dominant and dominated actions. oneleast at for strict is inequality theand inequalitywith weak holdsequation thesuch that ~an exists thereif player for action dominated (weakly) a is ~action particularA allfor ,,~payoff s' maximizes ~ playing playing, are playersother allt matter wha no if, player for actiondominant a is ~action particularA iiiiiiiiiiiiiiiaaiAaAa)a(a)aa(iaiAa−−−∈•∈>∈•ππ713Dominant Actions¡ Rational players do not play strategies that are dominated.¡ (All players must know this.)¡ Allows some strategies to be eliminated from final outcomes.14Example: NBC vs. ABC45% , 55%50% , 50%Sitcom52% , 48%55% , 45%NewsSitcomNewsNBCABC¡ Sitcom is a dominant action for NBC¡ News is a dominant action for ABC¡ (News,Sitcom) is an equilibrium in dominant actions815Definition: Equilibrium in dominant actions player each for actiondominant a is ~ ifactionsdominant in mequilibriuan is}~,...,~,~{ outcomeAn 21iAaaaaaiiN∈=•16Example: Old vs. new technology0, 0-a, aCurrenta, -a0, 0NewCurrentNewFirm 2Firm 1¡ Decisions for Firm 1 and 2?917Example: Old vs. new technology0, 0-a, aCurrenta, -a0, 0NewCurrentNewFirm 2Firm 1¡ Regardless of Firm 1’s action, Firm 2 chooses new technology – Dominant action for firm 2 is to choose new technology18Example: Old vs. new technology0, 0-a, aCurrenta, -a0, 0NewCurrentNewFirm 2Firm 1¡ Regardless of Firm 2’s action, Firm 1 chooses new technology – Dominant action for firm 1 is to choose new technology1019Example: Old vs. new technology0, 0-a, aCurrenta, -a0, 0NewCurrentNewFirm 1¡ New technology is a dominant action for both players¡ (New,New) is an equilibrium in dominant actions¡ What about (Current, Current)?Firm 220Classroom game, LC2, 25, 0Red0, 53, 3BlackRedBlackPlayer 1Player 21121Classroom game, New Valuesl What will happen here? 2, 2302, 0Red0, 302300, 300BlackRedBlackPlayer 1Player 222Another game¡ Is there a dominant action for player 2? For player 1?¡ What do you think will happen?6, 57, 6Y5, 68, 7XYXPlayer 1Player 21223Definition: Best responsei)(aR, ..., n, i, ..., i-, aaa)(aRiiiiiiiiiplayer fromaction -rebest theis ,1121 players of actions given the s,other wordIn ),(maxargsuch that isplayer of function responsebest the game,player Nan In −−−−+=•π24Example1, 20, 0Y0, 02, 1XYXPlayer 1¡ Player 1 plays X → Player 2’s best response: R2(X)=X ¡ Player 1 plays Y → Player 2’s best response: R2(Y)=Y ¡ Player 2 plays X → Player 1’s best response: R1(X)=X ¡ Player 2 plays Y → Player 1’s best response: R1(Y)=Y¡ What is the dominant action for player 1? Player 2?Player 21325Example (cont.)1, 20, 0Y0, 02, 1XYXPlayer 1¡ There is no equilibrium in dominant actions¡ What is the likely outcome of this game?Player 226Nash EquilibriumiiiiiiiiNAaa( aaa( aaaa∈≥=•−− allfor