Create Presentation
Download Presentation

Download Presentation
## Chapter 14 – Game Theory

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Chapter 14 – Game Theory**14.1 Nash Equilibrium 14.2 Repeated Prisoners’ Dilemma 14.3 Sequential-Move Games and Strategic Moves**Game Theory and Life**You are on a first date with the love of your dreams. You can propose 2 activities: • Safe activity (Coffee) • 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:**You**Mr/Miss Right First Date Game What is the outcome of this game? Payoff format is (Left, Top) 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) 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**You**Mr/Miss Right 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 Chapter Fourteen**You**Mr/Miss Right 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 Note that this game is SYMMETRICAL 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. Chapter Fourteen**Game Theory**• A special kind of Best Response: DOMINANT STRATEGY • Strategy that is best no matter what the other player does.**Advertising**A’s STRATEGY Don’t advertise Advertise B’sSTRATEGY A’s profit= $50 000 A’s profit= $75 000 Don’t advertise B’s profit = $50 000 B’s loss = $25 000 A’s loss = $25 000 A’s profit = $10 000 Advertise B’s profit = $10 000 B’s profit = $75 000**Dominant Strategy**A’s dominant strategy is advertise B’s dominant strategy is advertise Don’t advertise Advertise A’s profit= $50 000 A’s profit= $75 000 Don’t advertise B’s profit = $50 000 B’s loss = $25 000 A’s loss = $25 000 A’s profit = $10 000 Advertise B’s profit = $10 000 B’s profit = $75 000**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**1 yearPrison**Go free 1 year Prison 7 years Prison 5 years Prison 7 years Prison 5 years Prison Go free Classic Prisoners’ Dilemma Rocky’s strategies Deny Confess Dominant strategy: confess, even though they would both be better off if they both kept their mouths shut. Deny Ginger’s strategies Confess**Dominant Strategy Equilibrium**Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy. Toyota Honda**Dominated Strategy**Definition: A player has a dominated strategywhen the player has another strategy that gives it a higher payoff no matter what the other player does. Toyota Honda 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. Chapter Fourteen**Dominated Strategy**Toyota Honda "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce the game matrix.**Finding Nash Equilibrium Cases**• Nash Equilibrium where Dominant Strategies overlap • Nash Equilibrium with one Dominant Strategy • Nash Equilibrium by eliminating Dominated Strategy • Nash Equilibrium through Best Responses Chapter Fourteen**Nash Equilibrium – Dominant Overlap**Professor Student**Nash Equilibrium – One Dominant**Professor Student**Nash Equilibrium – Eliminate Dominated**Professor Student**Nash Equilibrium – Best Responses**Professor Student**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:**No Nash Equilibrium**Fred Barney**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 time, nether have an incentive to deviate.