Chapter 14 – Game Theory 14.1 Nash Equilibrium 14.2 Repeated Prisoners’ Dilemma 14.3 Sequential-Move Games and Strategic Moves 1 Game Theory and Life You are on a first date with the love of your dreams. You can propose 2 activities: 1) Safe activity (Coffee) 2) Exciting Activity (Waterpark) Your date could either want a safe activity or an exciting activity. There are different results if your ideas match up or clash: 2 First Date Game What is the outcome of this game? Payoff format is (Left, Top) You Mr/Miss Right Coffee Waterpark Coffee 10,10 0,-5 Waterpark -5,0 20,20 3 Chapter Fourteen Game Theory Components Players: agents participating in the game (You and Your Date Strategies: Actions that each player may take under any possible circumstance (Coffee, Waterpark) Outcomes: The various possible results of the game (four, each represented by one cell of the payoff matrix) Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the payoff matrix) 4 Chapter Fourteen Best Responses In all game theory games, players choose strategies without knowing with certainty what the opposing player will do. Players construct BEST RESPONSES -optimal actions given all possible actions of other players 5 First Date Game Best Responses If you know your date will pick coffee, you should pick coffee, since 10 > -5 If you know your date will pick waterpark, you should pick waterpark, since 20 > 0 You Mr/Miss Right Coffee Waterpark Coffee 10,10 0,-5 Waterpark -5,0 20,20 6 Chapter Fourteen First Date Game Best Responses If your date knows you will pick coffee, they should pick coffee, since 10 > -5 If your date knows you will pick waterpark, they should pick waterpark, since 20 > 0 You Mr/Miss Right Coffee Waterpark Coffee 10,10 0,-5 Waterpark -5,0 20,20 Note that this game is SYMMETRICAL 7 Chapter Fourteen Nash Equilibrium Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest") Nash Equilibria occur when best responses line up The Date Game: Nash equilibria: Each proposes coffee or each proposes waterpark. 8 Chapter Fourteen Game Theory •A special kind of Best Response: DOMINANT STRATEGY •Strategy that is best no matter what the other player does. 9 Advertising B’s STRATEGY Don’t advertise Advertise A’s STRATEGY Don’t advertise Advertise A’s profit= $50 000 B’s profit = $50 000 A’s loss = $25 000 B’s profit = $75 000 A’s profit= $75 000 B’s loss = $25 000 A’s profit = $10 000 B’s profit = $10 000 10 Dominant Strategy B’s dominant strategy is advertise Don’t advertise Advertise A’s dominant strategy is advertise Don’t advertise Advertise A’s profit= $50 000 B’s profit = $50 000 A’s loss = $25 000 B’s profit = $75 000 A’s profit= $75 000 B’s loss = $25 000 A’s profit = $10 000 B’s profit = $10 000 11 Prisoner’s Dilemma • This is an example of a prisoner’s dilemma type of game. – There is dominant strategy. – The dominant strategy does not result in the best outcome for either player. – It is hard to cooperate even when it would be beneficial for both players to do so – Cooperation between players is difficult to maintain because cooperation is individually irrational. • eg., The dominant strategy: advertise 12 Classic Prisoners’ Dilemma Rocky’s strategies Deny Deny Ginger’s strategies Confess 1 year Prison 1 year Prison 7 years Prison Go free Confess Go free 7 years Prison 5 years Prison Dominant strategy: confess, even though they would both be better off if they both kept their mouths shut. 5 years Prison 13 Dominant Strategy Equilibrium Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy. Toyota Honda Build a new plant Do not Build Build a new plant 16,16 20,15 Do not Build 15,20 18,18 Dominated Strategy Definition: A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does. Toyota Build a Don’t New Plant Build Honda Build a 12,4 New Plant 20,3 Don’t Build 18,5 5,6 15 Chapter Fourteen Dominant or Dominated Strategy Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier. 16 Chapter Fourteen Dominated Strategy Toyota Build Large Build Small Do Not Build Build Large 0,0 12,8 18,9 Build Small 8,12 16,16 20,15 Do Not Build 9,18 15,20 18,18 Honda "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce 17 the game matrix. Finding Nash Equilibrium Cases 1) Nash Equilibrium where Dominant Strategies overlap 2) Nash Equilibrium with one Dominant Strategy 3) Nash Equilibrium by eliminating Dominated Strategy 4) Nash Equilibrium through Best Responses 18 Chapter Fourteen Nash Equilibrium – Dominant Overlap Professor Easy Exam Hard Exam No Exam 100,10 80,80 50,0 Don’t Study 50,5 30,60 20,0 Drop Out 50,30 0,20 Study Student 30,10 19 Nash Equilibrium – One Dominant Professor Short Exam Long Exam No Exam 100,10 80,80 50,0 Don’t Study 50,40 30,10 20,0 Drop Out 50,30 0,20 Study Student 30,10 20 Nash Equilibrium – Eliminate Dominated Professor Short Exam Long Exam Test Bank 100,10 80,80 50,0 Don’t Study 50,40 30,10 20,0 Cheat 0,30 0,0 Study Student 0,10 21 Nash Equilibrium – Best Responses Professor Open Book Closed Book No Exam 20,10 80,80 50,0 Don’t Study 50,80 30,10 70,70 Drop Out 50, 0 0,20 Study Student 10,10 22 Nash Equilibrium • However it is found, a Nash Equilibrium ALWAYS occurs where Best Responses line up • If Multiple Nash Equilibria exist, we can’t conclude WHICH outcome will occur, only the possible outcomes that can occur • Also, it is often APPEARS that no Nash Equilibria exist: 23 No Nash Equilibrium Fred Rock Paper Scissors Rock 0,0 -1 , 1 1, -1 Paper -1 , 1 0, 0 -1, 1 Scissors -1, 1 1, -1 0, 0 Barney 24 Mixed Strategies Pure Strategy – A specific choice of a strategy from the player’s possible strategies in a game. (ie: Rock) Mixed Strategy – A choice among two or more pure strategies according to pre-specified probabilities. (ie: Rock, Paper or Scissors each 1/3rd of the time) If Pure Strategies can’t produce a Nash Equilibrium, Mixed Strategies can: If both players randomize each choice 1/3rd of the 25 time, nether have an incentive to deviate.