5/16/20151 Game Theory Game theory was developed by John Von Neumann and Oscar Morgenstern in 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.A Beautiful Mind Game theory is a way of looking at a whole range of human behaviors as a game.

5/16/20152 Components of a Game Games have the following characteristics: Players Rules Payoffs Based on Information Outcomes Strategies

5/16/20153 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

5/16/20154 Games as Defined by the Number of Players: 1-person (or game against nature, game of chance) 2-person n-person( 3-person & up)

5/16/20155 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.

5/16/20156 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

5/16/20157 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

5/16/20158 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.

5/16/20159 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.

5/16/ 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.

5/16/ 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)..shape of the universeif you prefer

5/16/ Some Simple Examples Battle of the Bismark Sea Prisoner’s Dilemma Chicken

5/16/ The Battle of the Bismarck Sea Simple 2x2 Game US WWII Battle Japanese Options Sail North Sail South US Options Recon North 2 Days Recon South 1 Day3 Days

5/16/ The Battle of the Bismarck Sea Japanese Options Sail North Sail South Minima of Rows US Options Recon North 2 Days 2 Recon South 1 Day3 Days1 Maxima of Columns23

5/16/ 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.

5/16/ Decision Tree

5/16/ 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.”

5/16/ A Simple Prisoner’s Dilemma Prisoner A ~ ConfessConfess Prisoner B ~ Confess Confess

5/16/ Alternate Prisoner’s Dilemma Language Prisoner A CooperateDefect Prisoner B Cooperate Defect Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.

5/16/ What Characterizes a Prisoner’s Dilemma Prisoner A CooperateDefect Prisoner B CooperateReward Tempt Sucker DefectSucker Tempt Punish Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.

5/16/ 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

5/16/ 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.

5/16/ 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?

5/16/ 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

5/16/ Chicken The game that we call chicken is widely played in everyday life bicycles Cars James Dean – variant James Dean Mad Max Interpersonal relations And more… And more

5/16/ The Game of Chicken Driver A ~ SwerveSwerve Driver B ~ Swerve Swerve 4 2 3

5/16/ 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

5/16/ Chicken is Nuclear Deterrence