1/17/06

v Strategic Behavior: The Idea

Ø A lot of what we do involves optimizing against nature

§ Should I take an umbrella?

§ What crops should I plant?

§ How do we treat this disease or injury?

§ How do I fix this car?

Ø We sometimes imagine it as a game against a malevolent opponents

§ Finagle's Law: If Something Can Go Wrong, It Will

§ "The perversity of inanimate objects"

§ Yet we know it isn't

Ø But consider a two person zero sum game, where what I win you lose.

§ From my standpoint, your perversity is a fact not an illusion

§ Because you are acting to maximize your winnings, hence minimize mine

Ø Consider a non-fixed sum game--such as bilateral monopoly

§ My apple is worth nothing to me (I'm allergic), one dollar to you (the only customer)

§ If I sell it to you, the sum of our gains is … ?

§ If bargaining breaks down and I don't sell it, the sum of our gains is … ?

§ So we have both cooperation--to get a deal--and conflict over the terms.

§ Giving us the paradox that

· If I will not accept less than $.90, you should pay that, but …

· If you will not offer more than $.10, I should accept that.

§ Bringing in the possibility of bluffs, commitment strategies, and the like.

Ø Consider a many player game

§ We now add to all the above a new element

§ Coalitions

§ Even if the game is fixed sum for all of us put together

§ It can be positive sum for a group of players

§ At the cost of those outside the group

v Ways of representing a game

Ø Like a decision theory problem

§ A sequence of choices, except that now some are made by player 1, some by player 2 (and perhaps 3, 4, …)

§ May still be some random elements as well

§ Can rapidly become unmanageably complicated, but …

§ Useful for one purpose: Subgame Perfect Equilibrium

§ Back to our basketball player--this time a two person game

§ But … Tantrum/No Tantrum game

§ So Subgame Perfect works only if commitment strategies are not available

Ø As a strategy matrix

§ Works for all two player games

§ A strategy is a complete description of what the player will do under any circumstances

§ Think of it as a computer program to play the game

§ Given two strategies, plug them both in, players sit back and watch.

§ There may still be random factors, but …

§ One can define the value of the game to each player as the average outcome for him.

Ø Dominant Solution: Prisoner's Dilemma as a matrix

§ There is a dominant pair of strategies--confess/confess

· Meaning that whatever Player 1 does, Player 2 is better off confessing, and

· Whatever Player 2, does Player 1 is better off confessing

· Even though both would be better off if neither confessed

		Baxter
		Confess	Deny
Chester	Confess	10,0	0,15
	Deny	15,0	1,1

§ How to get out of this?

· Enforceable contract

¨ I won't confess if you won't

¨ In that case, using nonlegal mechanisms to enforce

· Commitment strategy--you peach on me and when I get out …

Ø Von Neumann Solution

§ Von Neumann proved that for any 2 player zero sum game

§ There was a pair of strategies, one for player A, one for B,

§ And a payoff P for A (-P for B)

§ Such that if A played his strategy, he would (on average) get at least P whatever B did.

§ And if B played his, A would get at most P whatever he did

Ø Nash Equilibrium

§ Called that because it was invented by Cournot, in accordance with Stigler's Law

· Which holds that scientific laws are never named after their real inventors

· Puzzle: Who invented Stigler's Law?

§ Consider a many player game.

· Each player chooses a strategy

· Given the choices of the other players, my strategy is best for me

· And similarly for everyone else

· Nash Equilibrium

§ Driving on the right side of the road is a Nash Equilibrium

· If everyone else drives on the right, I would be wise to do the same

· Similarly if everyone else drives on the left

· Multiple equilibria

§ One problem: It assumes no coordinated changes

· A crowd of prisoners are escaping from Death Row

· Faced by a guard with one bullet in his gun

· Guard will shoot the first one to charge him

· Standing still until they are captured is a Nash Equilibrium

¨ If everyone else does it, I had better do it too.

¨ Are there any others?

· But if I and my buddy jointly charge him, we are both better off.

§ Second problem: Definition of Strategy is ambiguous. If you are really curious, see the game theory chapter in my webbed Price Theory

v Solution Concepts

Ø Subgame Perfect equilibrium--if it exists and no commitment is possible

Ø Strict dominance--"whatever he does …" Prisoner's Dilemma

Ø Von Neumann solution to 2 player game

Ø Nash Equilibrium

Ø And there are more