Help!!!!! Game Theory.... urgent!!!!?

Question

suppose firm 1 & firm 2 are engaged in a simultaneous move, one shot game in which each firm is setting their advertising levels. there are three options: high advertising, moderate advertising, and no advertising. the chart is a payoff matrix for the two firms.(the payoffs are in millions and should be read firm 1's payoff, firm 2 payoff)

http://i15.tinypic.com/4bgi1ix.jpg

I found that the Nash Equilibrium is (10,10) but im stuck with the following question

1. if the two firms were to collude, what level of advertising do you think they would agree on?

2. suppose that this game were infinitely repeated. what would the interest rate have to be in order to sustain collusion between the firms? (it might be easiest to try and determine what interest rate would be necessary to stop firm B from "cheating")

thanks in advance

Answer 1

Its actually not that hard.

You use what is called called reversion strategy, and the One Shot Deviation Principle.

If they were to collude, they would agree on the pure strategy of (none, none). This has the highest payoff available. Thus they will agree to do this. Now, as you've already noted, this is not Nash. Assuming A plays None, B will want to deviate to Moderate, and vice versa, as the game is symetric.

Here is what their strategy will be for the infinitely repeated game: Play (none,none) if nobody has deviated. If they have, play (moderate, moderate), If they deviate again, play (High, High) forever.

The idea is that if somebody were to deviate (to Moderate), you're going to punish them by playing High forever. This is a Grim Trigger strategy.

Now, the question asks for an interest rate. Here is how you get it. Following the above strategy, we have that if somebody were not to deviate, and to play (none, none) forever, their payoff is:

C = 20+ d*20 + (d^2)*20 + (d^3)*20 +......

If they were to deviate, their payoff would be:

D = 25 + d*11+(d^2)*11 + (d^3)*11 +......

forever and ever. Thus, if deviating were best, C-D would be negative. The question asks what interest rate we would need to keep B from cheating. So we solve for d so that C-D is positive.

C>=D
C-D>=0

Now note that these are geometric series, so we have

C-D = -5 + 9d/(1-d), so

-5 + 9d/(1-d)>=0

or expanding and re-writing, we have

9d>=5-5d, or

d>=5/14


Now, we have the second strategy to look at, so assume there is a deviation, then they get

E = 11 + d*11+d^2*11+....

if they play (moderate, moderate) forever. Now, again, not that there may be a profitabel deviation to low, so,

F =12+ d*10+d^2*10+....

E>F, or
E-F>0

E- F = -1-2*d, and so we see that we must have

d<= 1/2

Now, the question is what keeps him from cheating. Note that the inequalities go different ways on d. B wont deviate in the first stage if d>=5/14, and if d<=1/2. that is, we must have some d such that 5/14<=d<=1/2. That's all you need!

Answer 2

You are going to have a really tough time having someone answer that level of a homework problem for "free" for you.


It is a great problem, but it would take 2 hrs to properly address.