Desperate help needed from a Math person

[joesomebody]

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?

[joesomebody]

I have finished A, B, and D.

[joesomebody]

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

[Pushead2]

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

[joesomebody]

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.

[joesomebody]

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.

[joesomebody]

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)?

[joesomebody]

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.

[joesomebody]

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.

[joesomebody]

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

[Pushead2]

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.

[joesomebody]

D: Tables

[joesomebody]

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.

[joesomebody]

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

[CrazyPhuD]

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)?

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...

[dtebbe]

Come on Joe.. you don't have Dr. Don on speed dial? He'd be all over it and love to hear from a former student! :)

DT

[cdcox]

Joe --

In the interest of maintaining academic integrity I'll kind of outline the procedure. It really just falls out of your work for A and B.

1. Write a general form of B(i,n+1) - S(i,n). You are trying to show that this is greater than zero for all possible values of i and n.
2. Under what conditions will this expression be a minimum? You are going to make use of your proofs in steps A and B here.
3. Show that under the conditions listed in 2, the minimum value of B(i,n+1) - S(i,n) >0

Hope this helps.

[alnorth]

A, B, and C are easy. There's a much easier explanation (as in, understandable for the average person) than what was given in the first few posts.

I wont flat-out give the answer, but I'll show the path. For A, what does dominant mean? To me, it means that in is always better than out in all situations for that country. OK, lets prove that. Examine the worst-case scenario where only the first country chooses to join the union, and everyone else stays out. Is the first country still better off choosing in, even though they are alone in the union vs being out with the other 11? (If the instructor is picky about proofs, then show every scenario where in is always better, not just the worst one)

B, solve similar to A. Assuming you prove in is the dominant strategy for the first country, then for the second country, the worst-case scenario if they choose in is that ONLY country 1 and 2 choose in and the other 10 stay out. Which option is better for country #2, in or out?

C, basically the same as B, repeated 10 more times.

Your on your own for D, this is what I came up with after a couple minutes and I have to go to work.

edit: actually, D is very easy too. You need to compare "everyone choosing in" (B(i,n) where n=11 for all i)) and "everyone choosing out" (S(i,n) where n=0 for all i)) for all countries, and see how many are worse off when all choose out.

[Sofa King]

i would drop that class,effective immediately...

[cdcox]

Also, you have a small arithmetic error in part B.

[Dayze]

i would drop that class,effective immediately...

this.

[Hog's Gone Fishin]

When I can't figure out a mathematical equation, I usually just plug in the number (42).

Sometimes it's right but not very often.

[CoMoChief]

http://i134.photobucket.com/albums/q120/sohrabthebad/batman.jpg

[CoMoChief]

i would drop that class,effective immediately...

Eh.....that's a tad bit too late for that, considering finals week is either now, or just around the corner for most schools.

[Sofa King]

Eh.....that's a tad bit too late for that, considering finals week is either now, or just around the corner for most schools.

ok, Plan B.

Sit next to someone smart and cheat.

[joesomebody]

Come on Joe.. you don't have Dr. Don on speed dial? He'd be all over it and love to hear from a former student! :)

DTI really considered calling him. I'm good friends with 3 of his son's. I know he is semi-retired now, and I didn't want to bug him at home. Wasn't sure if I could catch him at school. Also, I know when I was in school, most of the teachers weren't all that into email on a regular basis.

Oh well, I think I got it worked out. Dr. Don could have provided proofs, a background on said proofs, useful tips for the proofs in other situations, as well as an update on the St. Louis Cardinals in about 6 minutes had I got a hold of him.

Great guy.

[joesomebody]

Thanks to all who tried to help. I am pretty sure I got it worked out. I don't think I'll get FULL credit on part C, as I didn't quite master all of the necessary proofs. I provided tables for each instance of i, and all possible outcomes of B and S for each n. This provides an answer; but in these types of classes, teachers want you to provide proofs that would work for any situation of n and i usually, not just this specific 12 nation problem.

I bet I get 80% of the points for that sub question.

Thanks again.

[Chief Chief]

Blame this all on the origination of the euro!

[keg in kc]

My last math class was in 1992. I can't even add and subtract without a calculator these days.

[dtebbe]

I really considered calling him. I'm good friends with 3 of his son's. I know he is semi-retired now, and I didn't want to bug him at home. Wasn't sure if I could catch him at school. Also, I know when I was in school, most of the teachers weren't all that into email on a regular basis.

Oh well, I think I got it worked out. Dr. Don could have provided proofs, a background on said proofs, useful tips for the proofs in other situations, as well as an update on the St. Louis Cardinals in about 6 minutes had I got a hold of him.

Great guy.

One of my favorite teachers ever. Pretty awesome having a college professor for your high school math & computer science. I guarantee you would have made his day!

DT

[cdcox]

OK, since you turned it in....

Show that B(i,n+1) - S(i,n) > 0

B(i,n+1) - S(i,n) = 2.2 + (n +1) - i - (i - n) = 3.2 + 2n - 2i

For a given value of i, B(i,n+1) - S(i,n) will be a minimum at the smallest possible value of n.

Based on parts A and B, the least value of n = i-1 (all the previous countries will join).

B(i,n+1) - S(i,n) = 3.2 + 2 (i - 1) - 2i = 1.2

[Over-Head]

Funny how I managed to graduate and NEVER honestly passed a math course from grade 9 to 12.
If my Gr12 math teacher hadn't done a pole dance at the bar I was house band in (and I managed to get pic's) I probably NEVER would have gotten that 50 final mark.
Now 20 odd years later, math which I hated, and copuld not fathom, turns out to be one of the biggest parts to me doing a custom ceiling.
If the math don't work out the ceiling probably wont look right





Oh btw:
i have NO fuggen idea what this thread is about other than math :)