Outline What is game theory? History of game theory. Application of game theory. Key elements of game theory. Types of game. Nash Equilibrium(NB) Pure strategies and mixed strategies. 2 players Zero- sum game.
Game theoryGame theoryi s the analysis of conflict and cooperationamong intelligent rational decision makers.A decision maker is said to be rational if he makesdecisions consistently in a pursuit of his own objectives.In a game, two or more individuals make decisions thatinuence each others expected utility.The decision makers are called the players.The decision objects of players are generally calledstrategies.
Game theory uses two basic toolsModels offers two games.There is a variety of models that represent dierentscenarios that might show up in real-life situations-Solution concepts.Solution concepts are predictions about what rational intelligenplayers should play.
History of game theory Von Neumann wrote a key paper in 1928 1944:”Theory of games and Economic Behaviour "by von Neumann and Morgenstern 1950:Nash invent concepts of Nash equilibrium Game theory booms after this…. 1944:Harsani,Nash and Selten win Nobel prize in economics for game theory work
Application of Game Theory Mathematics Psychology Computer science Law Biology Military Strategy Economics Sports Political science Game Playing International Relation Philosophy
Key elements of a Game Players: Who is interacting? Strategies: What are their options? Payoffs: What are their incentives? Information: What do they work? Rationality: How do they think?
Types of games Cooperative or non-cooperztive Zero sum and non –zero sum Simultaneous and sequential Perfect information and imperfect information Finite and Infinite Strategies
Pure Strategies The upper value of the game is equel to the minimum of the maximum values in the columns The lower value of the game is equalto the maximum of the minimum values in the rows.
An Example AB Y1 Y2 MinimumX1 10 6 6X2 -12 7 -12Maximum 10 6 6 7
Mixed Strategies A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy
Nash Eqilibrium(NE) A player’s best strategy is that strategy that maximizes that player’s payoff(utility)’knowing the strategy’s of the other players.
Penny Matching: Each of the two players has a penny. 2 players must simultaneously choose whwther to show the Head or the Tail. Both players know the following rules: If two pennies match (both heads or both tails ) then players 2 wins payer1’s penny. Otherwise,player 1 wins player 2’s penny Player 2 Head Tail Head -1 , 1 1 , -1 Player 1 Tail 1 , -1 -1 , 1
Prisoner’s Dilemma No communication: Strategies must be undertaken without the full knowledge of what the the other players (prisoners)will do. Players (prisoners) develop dominant strategies but are not necessarily the best one.
Payoff Matrix for Prisoner’s Dilemma Ted Confess Not confess Confess Both get 5 years 1 year for Bill 10 years for tedBill 10 years for Bill Both get 3 years Not Confess 1 year for Ted
An Example of Mixed Startegygame A B believes B doesn’t believe B A bluffs 1 0 A doesn’t bluff 0 0.5
Nash Equilibrium This equilibrium occurswhen player’s strategy is optimal,knowing the strategy’s of the other player’s. A player’s best strategy is thet startegy that maximize that player’s payoff(utility), knowing the strategy’s of the other players. So when each player within a game follows their best strategy, a Nash Equilibrium will occur.