Logic/Game Theory Question
 

A top-tier math school challenges, individually, each of five, math school applicants, who have parallel credentials for admission to the math school, to determine the self-allocation of a $100,000 fellowship (divisible in allocations of $1,000 only). The allocation process is governed by the following conditions:

First, the five applicants draw lots to determine their respective, sequential turns to submit their proposed self-allocations of the $100,000 fellowship. Second, applicant #1 submits his/her allocation that is either accepted or rejected by a majority vote of the five applicants.

Each math-school applicant must take his chronological turn; there can be no abstentions. A tie vote is to be considered a majority vote.

If #1 submits an allocation that does not win a majority of the five votes, then #1's application to the top-tier maths school is rejected, and the challenge is passed to applicant #2. If applicant #2 submits an allocation that does not win a tie or a majority vote, then #2's application to the top-tier maths school is rejected, and the challenge is passed to applicant #3. This process is iterated until either a tie or a majority is reached, or only the remaining applicant #5 is admitted to the top-tier maths school with a $100,000 fellowship. There must be no collusion whatsoever, and the only motivation(s) of each applicant is to win the $100,000 fellowship and/or avoid rejection by the top-tier maths school.

What will #1 offer, assuming everyone acts completely rational?


Notes:

Eveyone votes on a proposed allocation, apart from people who have already had theres turned down. For example, if #2 is making a proposition, then #2-#5 can all vote on it. If its accepted, then #2-#5 all get in, and the money is split according to #2's proposition.

#2 would reject.

Assuming that they are all sadistic, whenever their choice won't make any difference to themselves, and they all know that,

0k (but still rejected)
0k (rejected)
0k (rejected)
100k (and accepted)
0k (and accepted)

First 3 could offer anything, actually, as it wouldn't make any difference.

99k to #2, 1k to #5

k i'm working backwards

if it gets to #4 then #5 will get all the money cos #4 couldn't get a majority

if it gets to #3 then #4 would have to say yes to anything above 0k otherwise he'd be screwed and so #3 has an automatic majority

if it gets to #2 he would have to keep #3 happy so he'd give him 99k and #5 1k for a yes vote. #5 would accept cos if he declined #4 would say yes to anything above 0k on #3's vote and #5 would be shafted.

Ok so this means #2 can't get didly if he doesn't play ball with #1
so #1 would offer #2 1k and #4 2k and himself 97k. #3 and #5 would say no to pretty much anything because of #2's situation if he has to propose a split.

What I think:

#1: 97k
#2: 1k
#3: 0k
#4: 2k
#5: 0k

is that right?

#4 can get himslef 100k

why would he say yes to that ?

how?

"a tie or a majority vote"

well that ****s me up, it says majority twice, then switches to tie or majority.

"A tie vote is to be considered a majority vote."

It refers to a majority of 5 since there can't be a tie, but otherwise it's quite clear.

Comedy 'Nothing to himself, $25k to everyone else' option.

For want of actually understanding the question, I'm going to try configuring bits of linux I have no experience with.

heh

Thats was my initial answer too, but I misunderstood the question in a different way from you (I didnt think the person who proposed an offer got to vote on it).

Assuming they all act completely rationally.

20k each ?

I dunno if I understood the question though.

Or some combination of $0, $50k, $50k maybe.

Assume our guy just wants into the school, he'd be happy taking no money, the other two take their $50k because it's the best offer they're going to get assuming no ties, giving three votes, majority, hurrah?

Edit:
So I'd guess

$0
$50k
$50k
$0
$0

Rationale:
Work backwards as DrNick did, four can't get a majority on his own and so we don't need to buy him off, three will have a majority automatically because four will have to play along if he wants any money, so we need his vote, as we do number 2... And screw five.

i second this.

20/20/20/40 maybe?

Everyone under that situation knows that they can get more though and as there can be no collusion they can't all just agree to split the money and get the place surely?

"Working backwards" is a tactic, not a lifestyle.

Work backwards... screw five

Damn.

Prisoners Dilemma.

Edited out for same reason as above

PS I have a game theory exam on Saturday which I will fail miserably.

correct.

Ah crap, didn't notice a tie constituted a majority

w00t!!!

Edited solution out for other people to try.

i give up. someone PM me the answer.


unless its

50
0
49
1
0


I give up

a tie is 2 against and 2 for right.

so
a third of £100k for the first 3 and nothing for the last two ?

Indeed =/

no

#5 will happily vote yes for only 1k so if it's #2's turn he offers #5 1k and then gives the rest to himself, same for #3

depending on wether #2,3,4,5 are sure that the others are gonna offer the right thing the 1k could also be enough to get #4's vote. then again if they actually know what the others are gonna do what's in it for #5 to vote yes when #1 offers him 1k since he knows #2 and #3 are gonna do the same.

there are a few plausible options:

1) 99k for #2 or #3 and 1k for #5
2) 1k for #4 and 1k for #5 and the rest for #1
3) same as above but 1k to two out of #3,4,5 and rest to #1

it all depends one wether #2-5 assume that the others are gonna make the right offer before they have actually done so and wether #5 votes yes for 1k despite knowing that he will be offered 1k by some1 else later.

I assume the "rational" implies the safe option; #2 would offer the $1K to #4 instead.

Where did you get it Nod? Are there any more like that?

i'm not sure if were are supposed to assume that the 5 student are gonna act rationally or if the 5 students assume that they are gonna act rationally ?!?

You assume "best play", ie every other player will try to get the best outcome, which is predictable.

mornings and readings through carefully dont mix

98k #1
1k #3
1k #5

#5 would say yes to 1k because it's more than he's going to get if it gets down to #4

#4 wouldn't say yes to anything but 100k, unless he thought that he would be outvoted and would therefore end up with nothing. However, if he is outvoted and has something allocated to him even if he is outvoted he gets the money so there is no risk for him to vote against everything except 100k for himself, so he should always recieve 0?

#3 Can't allocate himself 100k because he would be outvoted by #4 and #5, however, if he allocated himself 99k and 1k to #5 they would outnumber #4 so the most he can expect is 99k.

#2 Also cannot allocate himself 100k, because he would be outvoted, he could allocate himself 99k and #5 1k as this would force a tie, so he is also unlikey to accept below 99k

#1 assuming he can vote for himself, shouldn't care how much he gets, so he would get 0k

I'd guess something along the lines of:

#1: 0k
#2: 0k
#3: 99k
#4: 0k
#5: 1k

Assuming he can vote for himself, this would give him a majority, as #3 can never expect to get more than 99k, and #5 cannot risk it getting down to #4, as he would end up getting 0k. 1/3/5 = 3/5 = majority?

#4 is sure that when it's #2's turn he will offer 1k to #5 who will vote yes as u said above and #4 gets nothing - so if #1 offers 1k to #4 he'll vote yes because it's more then he will get from #2

You'd assume #1 to be going for the best deal for himself, which can be achieved 'another way', though I can't see I flaw with that one it just doesn't seem like the choice #1 would make (which apparently was my mistake =/)

1# 98
2# 0
3# 1
4# 1
5# 0

3# and 4# will accept 1#:s offer because otherwise 2# will split it 98-2 with #5.

i am assuming #1 doesn't care about the money and simply wants to get into the school.

this would make more sence, however.

Why would #2 give $2K to #5, when he could give $1K to #4, and keep the rest?

Any student will point out the error here.

he wouldn't but it is irrelevent as it doesn't change his answer...

You are right :(

#97
#0
#1
#2
#0

How about that then?

/edit: but with this rationality thing 4 should realize that 1k is the best offer he is gonna get from #2, which means there is no reason for him not to accept the original :

1# 98
2# 0
3# 1
4# 1
5# 0

which would therefore still be valid.

#1 would get his way, but there's a better solution (as has been mentioned).

PM it to me plz?

I got it then whee!

It makes a difference whether the players accept "don't matter" votes, and whether the proposers can rely on it. Would someone accept $1 from #1, if they know they're gonna get $1 from someone else? Would #1 rely on someone accepting $1 from him instead of one of the later players? If yes to both:

If it gets down to #4, he gets all, so #5 can't let that happen. He will accept ANY offer #3 makes, and as such #3 will always propose 100-0-0 and get it. Thus #4 and #5 know that if it gets down to #3, they get nothing, and as such will accept any offer #2 makes. #2 will thus always propose 100-0-0-0 and get it. So #3 #4 and #5 all know that they'll get nothing no matter what. #1 can safely propose 100-0-0-0-0 and be sure to get it.