I commented that accepted answer is not entirely correct regarding original question, but since multiple comments are not encouraged I will post my answer here.
Main difference is in interpretation of what is pirate's goal/priority. Accepted answer assumed that P4 pirate will vote YES for 0 , just to avoid possibility of death if he gets to propose ( even if he stand to gain much more if he proposes ). But original question clearly states:
The other pirates are perfect logicians who always act to maximize
their own expected earnings. If multiple strategies yield the same
expected value, then they randomly select among them. For this puzzle,
maximizing likelihood of survival or bloodthirstiness are not factors.
That means pirates should try to "maximize their own expected earnings", regardless of "likelihood of survival".
To make explanation short, my calculation is similar to accepted answer, up to this point:
- 0/101/101/98 is best for P4 to propose ( or 0 to any one of P123)
- 92.74/92.74/92.74/32.7 is expected gains from viewpoint of each pirate P1234
Difference to accepted solution :
Pirates P1-3 will require at least offer of 93g from P5 to vote yes, and that is correct in accepted answer. Difference is in interpretation what will pirate P4 require. In my opinion, he will require at least 33g to vote yes, otherwise he vote no and expect 32.7g from his own proposal. That 'expect" means he will either get much more (98g) with chance of 1/3, or he will get nothing (0g, and get killed) with chance of 2/3 ... averaging to "expected" 32.7g. Since original question clearly states that he act to maximize expected earnings, he will require at least offer of 33g from P5 to vote YES.
In accepted answer it is assumed that P4 will accept 0g and still vote YES, even if his expected gain is much lower (0g if he vote yes, vs 32.7g if he vote no), in order to survive with 100% chance ( as opposed to 1/3 chance if he gets to propose). While that may be subjective goal of some pirates, it is not what question states.
Edit: it was clarified by JS1 ( who did accepted answer ) that both versions in his answer assume "life over gold, at all costs" ( meaning pirates will avoid any risk of death if they can, regardless of how much gold they stand to earn by risking). His two solutions differ only in "is vote fully or semi secret". So, based on those two elements ( "life or gold" priority and "semi or fully" secret ) we have actually solutions to three problems here:
- Gold+Fully : OP original question and my solution ( 0/0/93/33/174 )
- Life+Semi : accepted answer by JS1 ( 80/80/0/0/140 )
- Life+Fully : answer with "parrot" by JS1 ( 0/93/93/0/114 )
On a side note, I can understand logic that assume "pirates try to avoid death", but "Life" assumption above is much stronger - it assume pirates would avoid ANY chance of death, regardless of how much gold they stand to gain if they take even small risk. My personal opinion is that it would be good assumption if problem was "Five nuns splitting gold", but "pirate" practically has in his job description that he is willing to risk his life to earn some gold.
Solving problem using EV, P5 can offer two things:
- 0/0/93/33/174 ( 93g to any one of P123, 33g to P4 ) : expected 87g due to 50% risk
- 0/93/93/33/81 ( 0 to any one of P123 ) : expected 81g, no risk
Accepted answer suggest only 2nd approach, with different numbers due to offering 0 to P4, but still using "4 votes" principle where P5 makes better offer to 3 additional pirates to ensure majority even if one of them has fever. That way he ensure his proposal will pass, but at a price of expecting less gold.
But first option clearly has better expected value, following same logic as P4 ( and from quoted original question ) that expected value should trump risk of death. So final "optimal" solution should be:
Answer to original question: P5 proposes 0/ 0/ 93/ 33/ 174
About information leakage:
I also tried to see how and if potential fail of P5 proposal could change what others would do, because they would get more info on who may be with fever. For example, if above P5 proposal fail, they will know that either P4 or P3 have fever. So if P4 is not with fever, he will know that P3 has it, and will know that P2 knows it, so he could change his proposal from 0/101/101/98 to eg 101/101/0/98 to ensure vote. But that would change his expected gain to 98g - so should he accept when P5 offers him even 33g, let alone 0g?
My conclusion is that it does not influence answer, since only way information leakage would influence result at the point while they are deciding their vote for P5 proposal is if some pirate without fever decide to vote NO on what appear best deal for him, in hope to use additional information. But that itself would remove additional information, since then it violates "only fevered will vote NO" assumption - in other words, they could not any more assume that pirates offered zero do not have fever.
In short, additional information ( that, if previous vote failed, those offered zero do NOT have fever ) can only be used if question was "what would P4 propose", but have no impact on P5 proposal here.
But even if P4 vote YES, his expected gold is larger than offered 33, due to information leakage: if P5 proposal failed, P4 will know that P3 has fever and will offer 101/101/0/98 and ensure his 98g. Since in this case P4 does not have fever (he voted YES on uneven offer), he also know that chance of P5 offer to pass is not 1/2 (as seen by P5) but 2/3 (fails only if P3 is fevered). Thus his actual expected gold is 2/3(P5 proposal pass)*33+1/3(P5 proposal fails)*98= 54.7
So, should P4 then require 55g in P5 proposal to vote YES?
It may seem so on first look (expected gain is primary decision criteria), and it would force P5 to offer 0/0/93/66/152, BUT ... that hinge on assumption that P4 will vote NO on 0/0/93/33/174 offer from P5 (in order to get more than "54.7g" gain). And this is where my previous explanation why information leakage does not occur is demonstrated: IF P4 vote NO on 55g offer, he can be no longer certain that P3 is one with fever (because P5 proposal failed due to his NO, and not to P3's NO). So his expected gain drops back to 32.7 ...
In short, P4 has expected gain of 54.7g only if he vote YES on P5 proposal, but if he votes NO he has 32.7g expected gain. So P5 can safely tailor his offer according to 32.7 expectation (he needs to turn potential NO vote into YES, not to improve YES vote )
Note that P3, if he is not fevered and voted YES, can apply same logic if P5 proposal fails - he will know that P4 will offer 75/75/75/75 (and fail) and that P2 also know P4 was fevered and P1 is not, so P2 will have to accept 0. Thus P3 offer 0/0/300, and he is certain to get 300g if P4 had fever. So his expected gain while contemplating P5 offer is 2/393+1/3300= 162g . But, again, that is expected gold if he votes YES, since if he votes NO his expected gain drops back to 92.7 ( he cant be certain P4 is fevered in that case) . So, again, P5 can tailor his offer vs expected gain of P3 if P3 votes NO ( no need to win YES vote ).
Same logic apply to P2, if P5 offered 0/93/0/33/174 : if he vote YES and P5 proposal fails, he knows P4 has fever, and knows P3 will offer 0/0/300, so his expected gold is 2/393+1/30=62g. This is interesting twist, since now it appears P5 can offer less that 93g to P2 .. but it cant be done due to same reasons as above: if P2 vote NO, his expected gain is 92.7g, and P5 must turn NO vote into YES, not outbid YES vote. So, again, P5 will have to offer 93g.
In all those cases, key element is that P5 must consider expected gain from viewpoint of pirates if they vote NO on his proposal, and can ignore their potential expected gain if they vote YES. And information leakage (about who is fevered) only happens when they vote YES and other pirate vote NO. Thus, again, information leakage does not influence P5s offer.
Regardless of leakage, P5 still offers : 0/ 0/ 93/ 33/ 174
Few additional comments on expected value:
This question if we must or should use expected value as interpretation of "most gold" raises multiple times. My opinion is that even if it is not clearly stated, if no other method was specified then "expected value" is only logical choice as it is not subjective, and has clear mathematical definition. In this particular question it was clearly stated, but eg in my recent problem here I did not clearly state it, because I though it should be logical choice. I later modified to make it more clear, after suggestion in comment, but I still think that 'expected value" should always be logical approach to choose between two values with different probability to happen.
It follows even broader logic related to any ambiguity while solving problems. Namely, if we can interpret something in multiple ways from problem definition, then :
- if only one interpretation result in unique solution, we should choose that one. If some interpretation allows multiple solutions or no solution at all, that is clearly not right one
- if only one interpretation is objective ( has clear mathematical definition, and does not depend on subjective definitions ), we should prefer that one
- if only one option satisfy above conditions, then such option is clear choice, and problem can still be considered "well defined"
While some people may not agree with above logic, I think that requirements that everything must be spelled up to minute details to avoid any apparent ambiguity ( "apparent" is one that can be resolved with above principles ) would end up with problem creators having to define what "is" is ... in order to be certain that problem is well defined. This especially hold true in logic problems, where we should expect people who solve such problems to manage process of elimination.
In this case "expected value" would beat "subjective value of death" on both account, even if question did not clearly state what should be used ( which it did ). For example, looking at "subjective value of death", what if P4 was about to gain 40g with 99.99% chance? Will he also select to vote for 0g, just to avoid 0.01% chance of death? What if he is even afraid of death, but if he does not return his 20g loan by next week, he is fish bait - which risk of death would he chose ? Answer to that is obviously subjective - while answer on "what is expected value" is always the same.
