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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
    • 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:
games as defined by the number of players
Games as Defined by the Number of Players:
  • 1-person (or game against nature)
  • 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
      • Non-zero sum
        • Classic games: Football (?), D&D, Video games
        • Political Games: Policy Process
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 outcomes.
  • 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 an 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.”
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
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?
the theory of metagames
The Theory of Metagames
  • Metagames step back from the game and look at the other players strategy
  • Strategic choice is based upon opponents choice.
  • For instance, we could adopt the following strategies:
    • Tit-for-tat
    • Tat-for-tit
    • Choose Confess regardless
    • Choose ~Confess regardless
the full pd metagame
The Full PD Metagame
  • Using the Metagame strategy, we get three possible equilibria
    • One the original both confess
    • The other two, both ~confess (a cooperative solution)
chicken
Chicken
  • The game that we call chicken is widely played in everyday life
    • bicycles
    • Cars
    • Interpersonal relations
chicken is an unstable game
Chicken is an Unstable game
  • There is no saddle point in the game.
  • No matter what one player chooses, the other player can unilaterally change for some advantage.
  • Chicken is therefore unstable.
  • We cannot predict the outcome
national missile defense
National Missile Defense
  • Let’s pick a current problem
  • National Missile Defense
  • Structure this as a game
calculating expected utility of nmd
Calculating Expected Utility of NMD
  • E(Build)=pA(B-C)+p~A(B-C)
  • E(~Build)=pA(B-C)+p~A(B-C)
  • E(Build)=pA(0-60)+p~A(0-60)
  • E(~Build)=pA(0-1000)+p~A(0-0)
  • Build NMD if E(Build)>E(~Build)
spreadsheet
Spreadsheet
  • Open Excel table
tragedy of the commons
Tragedy of the Commons
  • First observed during the British Enclosure movement
  • Describes the problem of the unregulated use of a public good
  • Take a commons – e.g. a common pasture for grazing of cattle in a village
an example
An example
  • Take a village with 10 families
  • Each family has 10 cows which just exactly provide the food they need.
  • The village commons has a carrying capacity of 100 cows
cow carrying capacity
Cow Carrying Capacity
  • Each cow produces 500 lbs of meat & dairy per year up to or at carrying capacity of the pasture.
  • 10 families X 10 Cows X 500 lbs = 50,000 lbs of food at carrying capacity
  • …and then Farmer Symthe’s wife has triplets…
one more cow
One more cow
  • So Farmer Smythe decides he really needs one more cow.
  • And there is no one to tell him no because the commons is an unregulated public good
    • Like
      • Air
      • Water
      • Security ?
reduced capacity
Reduced Capacity
  • With the overgrazing, each cow will now produce onl;y 490 lbs of food.
  • 10 families X 10 Cows X 490 lbs = 49,000 lbs of food at carrying capacity
  • Each family gets 4900 lbs of meat & dairy, instead of 5000.
  • Except Farmer Smythe, who gets 5390 lbs
  • Even with the reduced carrying capacity, it is still to his advantage to add the extra cow
look familiar
Look Familiar?
  • Look at the situation
  • N players
  • Equilibrium solution is to ~ cooperate
  • Joint optimal outcome is to cooperate,
  • This is an n-person Prisoner’s dilemma
thinking strategically
Thinking Strategically
  • Some simple concepts that are strategic in nature.
  • They are worth reviewing, as they all demonstrate some particular strategic choice.
the hot hand
The Hot Hand
  • Are ‘hot hands’ just random sequences in a long series of trials?
  • Probabilistic analysis suggests that what sports observers claim as periods of exceptional performance are not statistically excessive
  • But hot hands may be masked by team responses.
  • Take tennis:
    • If your backhand is weak, your opponent will play to it.
    • Improve your backhand, and you get to use your better forehand more
to lead or not to lead
To Lead or not to Lead
  • Front-runner strategy
  • The leading sailboat copies the strategy of the trailing boat.
  • Doesn’t work in 3 boat races
  • Applicable to election?
    • When to go negative?
here i stand
Here I stand
  • Taking an irrevocable stand may change your opponent’s strategies.
  • A public statement makes a committement (and thereby changes payoffs) in ways that may dictate an outcome.
  • Opponent has to “take it or leave it.
  • May be costly next time!
belling the cat
Belling the Cat
  • Is the individual willing to assume the risks of the group
  • This is a “Hostages Dilemma”
  • Note the reference to plane full of passengers powerless before a hijacker with a gun.
    • Is this likely to be the case after 9/11?
    • What does this say about this strategic decision?
mix your plays
Mix your plays
  • Rely on you best strategy – strongest asset
  • But not exclusively
  • If you run the football every play, the defensive backfield will pull in and you will be less effective.
  • The pass sets up the run.
never give a sucker an even bet
Never give a sucker an even bet
  • When someone offers to bet you, they often know the odds – don’t bet.
  • Such as appliance warranties
game theory can be dangerous to your health
Game theory can be dangerous to your health
  • Check your bargaining position before you negotiate.
  • Do you negotiate first or afterwards?
  • Does your physical setting influence strategy?