11Recap of Jan 10, 2006¡ Course goals and objectives¡ Examples of “strategic” business interactions¡ Classroom game2Today—Jan 12, 2006¡ Intro to Game Theoryl Elements of a game (normal form)l Dominant actionsl Best responsel Nash Equilibriuml Mixed Strategiesl Pareto dominance23Example: 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?4Example: 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 ABC35Definition: 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−−−∈•∈>∈•ππ6Dominant Actions¡ Rational players do not play strategies that are dominated.¡ (All players must know this.)¡ Allows some strategies to be eliminated from final outcomes.47Example: 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 actions8Definition: Equilibrium in dominant actions player each for actiondominant a is ~ ifactionsdominant in mequilibriuan is}~,...,~,~{ outcomeAn 21iAaaaaaiiN∈=•59Example: Old vs. new technology0, 0-a, aCurrenta, -a0, 0NewCurrentNewFirm 2Firm 1¡ Decisions for Firm 1 and 2?10Example: 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 technology611Example: 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 technology12Example: 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 actionsFirm 2713Classroom game¡ Is there a dominant action for player 2? For player 1?¡ What do you think will happen?6, 57, 6Y5, 68, 7XYXPlayer 1Player 214Definition: 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 −−−−+=•π815Example1, 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 216Example (cont.)1, 20, 0Y0, 02, 1XYXPlayer 1¡ There is no equilibrium in dominant actions¡ What is the likely outcome of this game?Player 2917Nash EquilibriumiiiiiiiiNAaa( aaa( aaaa∈≥=•−− allfor )ˆ,)ˆ,ˆ:extremeNash at the played strategies their from deviatenot doplayersother all that provided -strategy a fromdeviate tobeneficialit find ldplayer wou no if(NE) mEquilibriuNash a is}ˆ,...,ˆ,ˆ{ strategies ofset An 21ππ18Finding Nash Equilibrium¡ Calculate the best response functions of every player ¡ Check which outcomes lie (if any) on the best response function of every player1019Example1, 20, 0Y0, 02, 1XYXPlayer 1¡ The outcomes (X,X) and (Y,Y) are Nash equilibriumPlayer 220Classroom game, LC2, 25, 0Red0, 53, 3BlackRedBlackPlayer 1Player 21121Observation¡ An equilibrium in dominant actions is a NE, but the converse is not necessarily true¡ There will be only one equilibrium in dominant actions while there can be multiple NEs¡ What is the relationship between an equilibrium in dominant actions and a NE?22Existence of mixed strategy equilibrium¡ Theorem: Every finite normal-form game has an equilibrium¡ It may result from a “mixed”strategy rather than a pure one.Nash (1950)1223Normal form game: Notation iplayer every for and,10 whereset,action over the },...,,{ ondistributiy probabilit a isplayer for strategy mixedA },...,,{ can take, player at action tha specific a define strategies ure k2121∑≤≤=•=∈•ikiikiiiikiiiiiiippppppiaaaAaPl Can be used if player is indifferent between several pure strategiesl Can model the other player’s uncertainty about the current player’s intentions24Example2, 00, 2Y0, 22, 0XYXPlayer 1¡ There are neither equilibrium in dominant actions nor (pure strategy) Nash equilibriumPlayer 21325Example: Mixed strategy equilibrium2, 00, 2Y0, 22, 0XYXPlayer 1Player 2¡ Player 2: Play X with probability 0.5 and Y with probability 0.5l Player 1 (expected) payoff if plays X: (0.5)(2)+(0.5)(0)=1l Player 1 (expected) payoff if plays Y: (0.5)(0)+(0.5)(2)=1l Player 1 is indifferent between playing X and Y!¡ Player 1: Play X with probability 0.5 and Y with probability 0.5l Player 2 is indifferent between playing X and Y¡ Neither player has an incentive to deviate from the strategy of randomly selecting between X and Y26Example: Mixed strategies1, 31, 11, 0R1, 01, 11, 4LZYXPlayer 1Player 2¡ Are there any dominant or dominated pure strategies for player 2?¡ Player 2 mixed strategy: l Play X with probability 3/7, Z with probability 4/7¡ Player 1’s expected payoff is 1 whether he plays L or R¡ Expected payoff of player 2l Player 1 plays L: (3/7)(4)+(4/7)(0)=12/7l Player 1 plays R: (3/7)(0)+(4/7)(3)=12/7¡ The mixed strategy dominates the pure strategy Y!1427Finding the mixed strategy equilibrium1, 20, 0Football0, 02, 1BalletFootballBalletAliceBob¡ Prob of playing (B, F): Alice: (q,1-q) Bob: (p,1-p)¡ Alice’s expected payoff¡ Bob’s expected payoff ¡ How many equilibria? 28Finding the mixed strategy equilibrium1, 20, 0Football0, 02, 1BalletFootballBalletAliceBob¡ Alice: (q,1-q) Bob: (p,1-p)¡ Alice’s expected payoffl If Alice chooses ballet: 2p If Alice chooses football:1-pl 2p=1-p → p=1/3 → Bob chooses football with probability 2/3¡