Desperate help needed from a Math person
 

Okay, I have homework due tomorrow in Game Theory (Econ 4340). I've been at it for about 6 hours. I have finished all but one of the 8 problems. This one I've been working on for about 3 hours.

Here is the problem:

A group of 12 countries is considering whether to form a monetary union. They differ in their assessments of the costs and benefits of this move, but each stands to gain more from joining, and lose more form staying out, when more of the other countries choose to join. The countries are ranked in order of their liking for joining, 1 having the highest preference for joining and 12 the least. Each country has two actions, IN and OUT. Let

B(i,n) = 2.2 + n - i

be the payoff to country with ranking i when it chooses IN and n others have chosen IN. Let

S(i,n) = i - n

be the payoff to country with ranking i when it chooses OUT and n others have chosen IN.

(a) Show that for country 1, IN is the dominant strategy.
(b) Having eliminated OUT for country 1, show that IN becomes the dominant strategy for country 2.
(c) Continuing in this way, show that all countries will choose IN.
(d) Contrast the payoffs in this outcome with those where all choose OUT. How many countries are made worse off by the formation of the union?

I have finished A, B, and D.

i = any whole number (1 to 12)
n = Any whole number (0 to 11)
N = 12
B(i,n) = 2.2 + n - i
S(i,n) = i - n

****, at 6:51am you're asking for the answer?! Chiefsplanet can hardly read english or numbers...

A:

i = 1

Country (i = 1) will choose option IN as long as

B(1,0) > S(1,0) and [B(1, n + 1) - S(1,n)] > 0

B(1,0)= 2.2 + 0 - 1 = 1.2 > S(1,0) = 1 - 0 = 1
1.2 > 1

and

B(1,n+1) - S(1,n) = 2.2 + 1 - 1 - (1-0) = 1.2
1.2 > 0

Marginal gain of choosing IN with respect to n is B(1,n') which is 1.
Marginal gain of choosing OUT with respect to n is S(1,n') which is -1.

For all n: (0 through 11), country (i = 1) is better off choosing IN.

B is really similar, it's just that you've proved n must be >= 1 since we have proven that country (i=1) will choose IN, so N now must = at least 1.

B:
i=2
Country (i = 2) will choose option IN if:
B(2,n+1)-S(2,n) > 0
2.2 + 2 - 2 - (2 - 1) = 0.2
0.2 > 0
Therefore, Country (i = 2) will choose IN for all n.

C is what I can't figure out.

Do they want me to do the same steps from B for all i: (3-12) manually? Or is there some mathematical proof I'm missing that would prove that the minimum n will always be i -1, and then using that I can prove that all countries must choose IN (Payoff B)?

D is pretty easy, I just need a payoff table for all i where n = 11, then compare that payoff table to when n = 0.

Any and all help will be much appreciated it. I'm going to nap for a few hours before class and hopefully figure this out then.

There are a lot of really smart planeteers. I'm hoping one or more of them that are really bored at work this morning will take a few minutes to ponder this for me. I've been up all day and night working on papers, homework, and studying for finals. I hate this time of year.

:( I'm sorry I am not smart, I have failed.

D: Tables

D:
From the tables, looks to me that 6 countries lose out by the formation of the union.

That finishes the problem other than C. I just don't think manually working through each i is what the problem is asking for there. Seems to menial and basic. Please let me know if any of you come up with a proof.

I swear they call up everything from 7th grade through College level Calc 2 and I simply cannot recall everything on demand like they want. The math isn't crazy hard by any means, it's just too much for me at this ungodly hour. Night.

Theres probably some proof/formula..to tired/rusty to look at it now, but if you do it all manually then you'll likely be safe, and if there is a formula odds are good it will pop out at you as you are doing them all...