Game Theory

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## Game Theory

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**Game Theory**• Game theory was developed by John Von Neumann and Oscar Morgenstern in 1944 - • Economists! • One of the fundamental principles of game theory, the idea of equilibrium strategies was developed by John F. Nash, Jr. (A Beautiful Mind), a Bluefield, WV native. • Game theory is a way of looking at a whole range of human behaviors as a game.**Components of a Game**• Games have the following characteristics: • Players • Rules • Payoffs • Based on Information • Outcomes • Strategies**Types of Games**• We classify games into several types. • By the number of players: • By the Rules: • By the Payoff Structure: • By the Amount of Information Available to the players**Games as Defined by the Number of Players:**• 1-person (or game against nature, game of chance) • 2-person • n-person( 3-person & up)**Games as Defined by the Rules:**• These determine the number of options/alternatives in the play of the game. • The payoff matrix has a structure (independent of value) that is a function of the rules of the game. • Thus many games have a 2x2 structure due to 2 alternatives for each player.**Games as Defined by the Payoff Structure:**• Zero-sum • Non-zero sum • (and occasionally Constant sum) • Examples: • Zero-sum • Classic games: Chess, checkers, tennis, poker. • Political Games: Elections, War , Duels ? • Non-zero sum • Classic games: Football (?), D&D, Video games • Political Games: Policy Process**Games defined by information**• In games of perfect information, each player moves sequentially, and knows all previous moves by the opponent. • Chess & checkers are perfect information games • Poker is not • In a game of complete information, the rules are known from the beginning, along with all possible payoffs, but not necessarily chance moves**Strategies**• We also classify the strategies that we employ: • It is natural to suppose that one player will attempt to anticipate what the other player will do. Hence • Minimax - to minimize the maximum loss - a defensive strategy • Maximin - to maximize the minimum gain - an offensive strategy.**Iterated Play**• Games can also have sequential play which lends to more complex strategies. • Tit-for-tat - always respond in kind. • Tat-for-tit - always respond conflictually to cooperation and cooperatively towards conflict.**Game or Nash Equilibria**• Games also often have solutions or equilibrium points. • These are outcomes which, owing to the selection of particular reasonable strategies will result in a determined outcome. • An equilibrium is that point where it is not to either players advantage to unilaterally change his or her mind.**Saddle points**• The Nash equilibrium is also called a saddle point because of the two curves used to construct it: • an upward arching Maximin gain curve • and a downward arc for minimum loss. • Draw in 3-d, this has the general shape of a western saddle (or the shape of the universe; and if you prefer). .**Some Simple Examples**• Battle of the Bismark Sea • Prisoner’s Dilemma • Chicken**The Battle of the Bismarck Sea**• Simple 2x2 Game • US WWII Battle**The Battle of the Bismarck Sea - examined**• This is an excellent example of a two-person zero-sum game with a Nash equilibrium point. • Each side has reason to employ a particular strategy • Maximin for US • Minimax for Japanese). • If both employ these strategies, then the outcome will be Sail North/Watch North.**The Prisoners Dilemma**• The Prisoner’s dilemma is also 2-person game but not a zero-sum game. • It also has an equilibrium point, and that is what makes it interesting. • The Prisoner's dilemma is best interpreted via a “story.”**Alternate Prisoner’s Dilemma Language**Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.**What Characterizes a Prisoner’s Dilemma**Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.**What makes a Game a Prisoner’s Dilemma?**• We can characterize the set of choices in a PD as: • Temptation (desire to double-cross other player) • Reward (cooperate with other player) • Punishment (play it safe) • Sucker (the player who is double-crossed) • A game is a Prisoner’s Dilemma whenever: • T > R > P > S • Or Temptation > Reward > Punishment > Sucker**What is the Outcome of a PD?**• The saddle point is where both Confess • This is the result of using a Minimax strategy. • Two aspects of the game can make a difference. • The game assumes no communication • The strategies can be altered if there is sufficient trust between the players.**Solutions to PD?**• The Reward option is the joint optimal payoff. • Can Prisoner’s reach this? • Minimax strategies make this impossible • Are there other strategies?**Iterated Play**• The PD is a single decision game in which the Nash equilibrium results from a dominant strategy. • In iterated play (a series of PDs), conditional strategies can be selected**Chicken**• The game that we call chicken is widely played in everyday life • bicycles • Cars • James Dean – variant • Mad Max • Interpersonal relations • And more…**Chicken is an Unstable game**• There is no saddle point in the game. • No matter what the players choose, at least one player can unilaterally change for some advantage. • Chicken is therefore unstable. • We cannot predict the outcome