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For a situation with $n$ team members having to share a prize of $x$ dollars, is there a mathematical formulation of how a voting system can be used to split the money in a way that would satisfy the majority?

For instance, each member could provide their opinion on what share of the money all the other members of the team he/she believes they deserve. In the end, the votes are combined and a final breakdown of the money is made after accounting for parameters on how far the members’ votes can sway the division of the money from the trivial $\frac{x}{n}$ for each member.

Example

Assume a band of $n$ pirates just looted some treasure and has to find a system to distribute it justly among the members depending on their contribution. It is important that

  • everyone gets a say in the vote.
  • the voting is confidential (in order not to lead to conflict).

What I’m looking for is a method where members who are judged by the others to be worthy of a larger share of the prize end up receiving that, and vice versa.

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  • $\begingroup$ So in a Quidditch team, the three chasers might vote for chaser-shares to be larger than the share of the single seeker, even though we know that the game is always won by Harry catching the golden snitch? $\endgroup$ Feb 11, 2017 at 12:28
  • $\begingroup$ Can there only be one vote? Or could there be multiple votes to fine tune the distribution? Also, you seem to be asking for a voting system which guarantees a unanimous decision. This is almost self-contradictory, especially if only one vote is allowed. Can you expand on your requirements for the vote to be considered unanimous? $\endgroup$
    – Jens
    Feb 14, 2017 at 1:23
  • $\begingroup$ @Jens I realized that unanimous was not the right word to use. I have rephrased that part of the question. $\endgroup$
    – hb20007
    Feb 14, 2017 at 12:58
  • $\begingroup$ How many people is the money being distributed between? Some systems that work well for large groups may not work well for smaller groups. $\endgroup$ Feb 14, 2017 at 21:45
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    $\begingroup$ @BenjiAltman It would be interesting to hear about what difference group size makes, but I am more concerned with smaller groups $\endgroup$
    – hb20007
    Feb 14, 2017 at 21:49

2 Answers 2

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This is a classical problem with no single approach. One can approach the problem as a bargaining problem. For that just google 'Rubinstein bargaining' (two people) or 'Baron Ferejohn bargaining in legislatures' (multiple people).

When it comes to voting. Any division is Pareto efficient (there does not exist other division that all would weakly prefer and some would strictly prefer) and hence any division is a Condorcet winner (there does not exist other division that would beat a division in a binary vote).

Can you be more specific with your question?

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A simple system to divide the wealth would be the following:

a. Ask each person to rank the others (no person can rank themself) according to how much of a contribution they feel each person had in the success. Those they feel contributed the most they give a $1$, those they feel contributed slightly less they give a $2$, etc. More than $1$ person can therefore be allocated a given rank. Thus, if a person feels everyone contributed equally, that person would give everyone a rank of $1$.

b. Sum the voted ranks of everyone. Give the smallest summed rank the value $1$, the second smallest summed rank the value $2$, etc. We now have the overall ranking of each persons's contribution as the team sees it.

c. Now to distribute the wealth. This can be done in infinitely many ways. We could decide that each person of rank $k$ gets $20$% (or p%) more than a person of rank $k+1$. Or we could decide that those of rank $1$ get a full share, those of rank $2$ get half a share, etc, where a share would be $x$ divided by the sum of the inverse of the ranks. There are, as said, infinitely many ways to do this once you have an ordering of the persons in the team.

Whether the above system is "just" I leave for others to decide. Whether it is "unanimous" I leave for the OP author.

I was going to add an example here, but it is past my bedtime. Will edit within 24 hours.

EDIT

As the first example, let's see what happens when $n=2$:

enter image description here

In this case, there can be only one outcome, namely that both team members are ranked equally. And whatever wealth distribution system we decide to use, they all have one thing in common, namely that they are solely based on differences in rank. As there is no difference here, each member gets an equal amount.

As a second example, let's try the one suggested by @Hagen von Eitzen in the comments. Here $n=4$ and $3$ of the members (the "chasers") are in collusion to make sure they get more than the fourth member, Harry. How well can they do? Well, their best bet would be to rank each other as $1$ and Harry as a $2$. Assuming Harry ($a4$ in the table below) suspects the collusion, his best bet is to rank the chasers equally, i.e. all with rank $1$. This gives the following result:

enter image description here

We see that Harry cannot completely avoid the consequences of a collusion between a majority of his team, but then I doubt any voting system could. And he is only $1$ rank lower than the "collusionists", so with a fairly mild wealth distribution system he would do okay.

No more examples for now, but I look forward to seeing other, perhaps fairer, voting systems.

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