Game Theory

Game theory is a way of looking at a whole range of human behaviors as a game.

Games have the following characteristics:

• Players
• Rules
• Payoffs
• Strategies

We classify games into several types.
 

• By the number of players:
  • 1-person (or game against nature)
  • 2-person
  • n-person( 3-person & up)
• 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.
• By 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

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.

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 or cooperatively towards conflict.

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.

It is called a saddlepoint 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). Also called a Nash equilibrium.

Classic Game theory Examples

The Battle of the Bismarck Sea

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. It is not to
 
 
 
  [No picture of payoff matrix available]
 
 
 
 

A simple game

The Prisoners Dilemma

The Prisoners dilemma is also 2-person game but not a zero-sum game.

It does have an equilibrium point, and that is what makes it interesting.

The Prisoner's dilemma is best interpreted via a "story:
 

In empirical examination of this game, the seeming paradoxical solution does indeed occur. It does so in the absence of trust and communication. In iterated play, the Prisoners dilemma has been shown to develop a cooperative outcome through player communication via a tit-for-tat strategy.

The Prisoners dilemma has a clear analog in Arms race models
 

And the Tragedy of the commons!

PS160 - skip over this !!! Go to Chicken
 

Can the Prisoner's Dilemma be "solved"?  Nigel Howard's theory of metagames provides us a 'solution' by stepping back from the situation.

The Prisoner's Dilemma starts with the assumption of two strategies

1. Choose C, or
2. Choose D

 This gives rise to four metastrategies

1. Choose C regardless    (Martyr)
2. Choose D regardless    (Cynic)
3. Choose C when the other player chooses C, and D when he/she chooses D. (Tit-for-tat)
4. Choose D when the other player chooses C, and C when he/she chooses D. (Tat-for-tit)

Chicken

Chicken is the game of Nuclear Deterence.

How do you win the game of chicken?
    Convince your opponent that you are crazy enough not to swerve!
    Act "irrationally"

    Reagan and the "rationality of irrationality"

    What is the nature of "irrationality
        Subjective preference orderings
 
 Humans are pretty limited information processors.  Hence what appears irrationality may be either differences in subjective judgement or limited information processing

Expected Utility and Decision Making

Rational Decision making

    Do states act as rational actors
        (what does B. BdM say about states as rational actors?)
 

    E(Decision O1 vs O2) = p_o2(B_O1 - C_O1) + p_O2(B_O2 - C_O2)

    Lottery payoff

    This is simple logic.