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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.

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game theory
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
Components of a Game
  • Games have the following characteristics:
    • Players
    • Rules
    • Payoffs
      • Based on Information
    • Outcomes
    • Strategies
types of games
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
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
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
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
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
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
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
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
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
Some Simple Examples
  • Battle of the Bismark Sea
  • Prisoner’s Dilemma
  • Chicken
the battle of the bismarck sea
The Battle of the Bismarck Sea
  • Simple 2x2 Game
  • US WWII Battle
the battle of the bismarck sea examined
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 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
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
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
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
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
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 play24
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
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
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