Thursday, March 13, 2008

More Fun and Games from Econ 307

Here are some other fun games from Econ 307 (officially entitled Managerial Economics, although it is really a course in Industrial Organization).

Rock, Paper, Scissors as a Simultaneous Game:
Find the equilibria of the following game.


Player 2

Player 1


Rock

Paper

Scissors

Rock

0 , 0

-1 , 1

1 , -1

Paper

1 ,-1

0 , 0

-1 , 1

Scissors

-1 , 1

1 , -1

0 , 0

Answer: There is no pure-strategy equilibrium in this game. There is a mixed-strategy Nash equilibrium, however, in which each player plays each strategy with probability 1/3. If you model this as an extended form (or sequential) game, the second player always loses, for obvious reasons. (I'd post that model, too, but it would involve working up some images, and I don't have time for that at the moment).

Warfare as a Collective Action Problem:

A soldier considers marching into a hail of gunfire with his comrades, or running away. All his comrades face the same decision. Suppose we model this decision by considering a representative pair of soldiers:


Soldier 2

Soldier 1


Fight

Run Away

Fight

10,000 , 10,000

-10,000,000 , 10,000,000

Run Away

10,000,000 , -10,000,000

10,000,000 , 10,000,000

What is the equilibrium outcome? Can you tell a story that makes sense of the payoffs in the table? (That is, why does the fellow who fights when his buddies run earn -10,000,000? Why do they get smaller but positive payoffs if they all fight? Why do they get 10,000,000 if they all run away?)

Answer:
(Run Away, Run Away) is the dominant strategy equilibrium. Here is one story that makes sense of the payoffs:

If all the soldiers fight, they stand a fair chance of surviving, as they might win. Therefore the payoff is positive. But if only one soldier tries to fight, he is certain to die, while those who run away are happy to be alive. Finally, if all the soldiers run away, they’ll all be happy to survive, although the war will not go well if this persists.

Suppose we alter the story so that an officer stands behind the soldiers with a machine gun. The machine gun is pointed at these (i.e., his own) soldiers, not at the enemy. Why might the officer do this, and how does it alter the payoffs?

Answer:

The officer exists to change the positive payoffs from running away into negative payoffs. By doing so he makes (Fight, Fight) the dominant strategy equilibrium. This example is not so far-fetched; the Soviet Army did this during the battle for Stalingrad during World War II (and probably on many other occasions). It has long been the case that soldiers who desert their post can be court-martialed (The excellent Kirk Douglas movie Paths of Glory deals with the case of soldiers in WWI who refused to march to certain death in the no-mans-land between the trenches, and were court-martialed and executed as a result). This should serve as a reminder that there can be ways around the Prisoner’s Dilemma. Using a third party as an enforcer is a powerful option.

Helmetless Hockey Players

(This problem was inspired by Robert Frank's The Economic Naturalist.) Consider the following game, in which hockey players choose whether or not to wear a helmet. Consider players 1 and 2 to be representative of any two players in a hockey game.


Player 2

Player 1


Wear a helmet

No helmet

Wear a helmet

0 , 0

-2 , 3

No helmet

3 , -2

-1 , -1

What is the equilibrium of this game? Is this the socially optimal outcome? Of what famous game does this remind you?

Answer: (No helmet, No helmet) is a dominant strategy equilibrium. This is similar to the prisoner’s dilemma.

Can you tell a story that makes sense of these payoffs and this equilibrium? To put it another way, why would hockey players support a rule that all players have to wear helmets, and why would such a rule be necessary?

Answer: If one player goes without a helmet, he has an advantage over the other players, in terms of visibility, mobility, and reduced weight. But each player reasons this way; “if other players are going without helmets, I, too, must forego my helmet, so that I will not be at a disadvantage.” As a result, they all end up going without helmets, eliminating the advantage of not having them while simultaneously putting themselves at greater risk of being injured. They would all favor a rule because it would prevent coordination failure--they all end up safer, and no one has an unfair advantage.