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nowimcalm

Dividing money brain teaser

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Hi everyone, thougt i would post this intersting one:

 

7 Sales executives have 1 million dollars to divide amongst themselves. The most senior sales executive propses a particular split and then everyone votes (each person's vote is equal). If at least 50% of the people accept, then the money is divided the way that was suggested. Otherwise the sales executive who propsed it gets fired...and then we move on to the next senior sales exec and the whole process repeats. The executives are rational (want to keep their jobs first and also get as much money as possible), and they also would prefer fewer executives in the group if given a choice (all else equal).

 

How should the cash be split ?

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well the optimal group is going to be 4, because after 3 people get fired for not winning they will start to get nervous. and split the money evenly between them all.

so imo 1mill/4

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Is wheelin' and dealin' allowed ("Vote against this next proposal; my turn is next and I'll reward you nicely")?


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Without wheelin' and dealin', the top person offers a four-way split amongst himself and execs #3, #5, and #7.

 

Let's suppose it goes all the way down to #6. #6's proposal is easy: Everything goes to #6. #6 will vote for this proposal, garnering the needed 50%. Exec #7 will be screwed if it gets down to #6.

 

Before it gets down to #6, #5's proposal needs to be sunk. Unlike #6, #5 needs 1 cohort to go along with his proposal. #7 will go along with anything #5 proposes so long as it is better than nothing. Exec #6 will be screwed if it gets down to #5.

 

Before it gets down to #5, #4's proposal needs to be sunk. #4 also needs 1 cohort to garner 50%, the obvious target being #6. Execs #5 and #7 will be screwed by #4's proposal.

 

Before it gets down to #4, #3's proposal needs to be sunk. #3 needs two cohorts, the two who would be screwed by #5's proposal. #3 offers a three-way split between himself, #5, and #7. Execs #4 and #6 will be screwed if it gets down to #3.

 

Before it gets down to #3, #2's proposal needs to be sunk. #2 also needs two cohorts to get 50%, and these are execs #4 and #6. Execs #3, #5, and #7 will be screwed if it gets down to #2.

 

Before it gets down to #2, #1's proposal needs to be sunk. #1 needs three cohorts: Execs #3, #5, and #7. Exec #1 should offer a four-way split amongst himself and execs #3, #5, and #7.

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Hi everyone, thougt i would post this intersting one:

 

7 Sales executives have 1 million dollars to divide amongst themselves. The most senior sales executive propses a particular split and then everyone votes (each person's vote is equal). If at least 50% of the people accept, then the money is divided the way that was suggested. Otherwise the sales executive who propsed it gets fired...and then we move on to the next senior sales exec and the whole process repeats. The executives are rational (want to keep their jobs first and also get as much money as possible), and they also would prefer fewer executives in the group if given a choice (all else equal).

 

How should the cash be split ?

 

The second least senior executive should vote no on every proposal, and attempt to get everyone else to vote no. Once it gets down to the second least senior executive and the least senior executive, the second least, would now be in the position to choose how to delegate the money, and his proposal will pass because 1/2 is 50%.

 

The most senior executive and most likely the second most, and third most are going to get sacked, no matter which way they decide to split the money up.

 

The problem is that every person next in line to decide how to split up the money is going to vote no on the current proposal. And the person who ends up with the ultimate position of power in the end is the second least senior executive, and he can choose to keep it all, and his vote will make the 50% needed.

 

Maybe, the third least senior executive has a good shot to counter this though, if the deal comes to him, he can delegate half to him and half to the least senior executive, leaving the second least senior executive out. And two out of three would pass. For this reason, the second least senior executive, is going to want to deal with the fourth senior executive. If the fourth senior executive comes in and offers half to to second least and half to him, he/she could pass that delegation with the 50%.

 

This will cause the least senior, and the third least senior to want to team up with the fifth most senior to delegate the money between them.

 

Then the second, fourth, will include the sixth and will want to team up to delegate the money between them so they don't get left out.

 

In response the least senior, third, and fifth and seventh will team up and delegate the money between them as 1/4ths.

 

So I seem to have come to the same conclusion as DH, which makes me feel pretty confident in my conclusion considering DH seems to be a pretty smart guy.

Edited by toastywombel

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The senior can split the money equally between 4 people (the ones he likes), and nothing (zero) to the three last ones. He would have gathered 4 votes against three, and win. Quite unfair, but 250.000 in his pocket.

At least, no one is fired.

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I think it depends on how much value there is to removing competition, and how much value to losing one's job (although that last bit may not matter if you assume pure rationality).

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The senior can split the money equally between 4 people (the ones he likes), and nothing (zero) to the three last ones. He would have gathered 4 votes against three, and win. Quite unfair, but 250.000 in his pocket.

At least, no one is fired.

There is no reason the second most senior exec would accept such a deal. Suppose he rejects the deal along with those other three. The first deal is sunk, and so is the topmost exec. The new top exec only needs to find two cohorts to attain the requisite 50% vote. That's a bigger slice of the pie for our former #2 (new #1) exec -- and he's the new #1 exec.

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If the order of voting was known before hand, the first exec would split the money evenly between #2, 3 and 4, forgoing his own compensation (or grant himself one penny) but keeping him his job and giving #2, #3 and #4 a split they would not otherwise be entitled to due to their place in line.


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Oops, hang on a minute... scratch that. #4 would be entitled to $500,000.

 

So yeah, it is in #6s best interest to always vote no, so #6 will never be floated any money. So the money would be divided like this:

 

#1 - $0.01

#2 - $333,333.33

#3 - $333,333.33

#4 - $0

#5 - $0

#6 - $0

#7 - $333,333.33

 

This way #2 and #3 get a split greater than or equal to what they would expect anyway, and #7 Would certainly vote for it as they would be eligible for no money otherwise.

 

For #1, passing on the money is the only way to keep their job.

Edited by jryan
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Read post #7, jryan. Why would exec #2 vote for your proposal? If he joins execs #4, #5, and #6 and votes against exec #1's proposal, exec #2 can get the same amount of money as offered by exec #1 and he will get exec #1's job.

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Let's keep score. The fraction indicates a n-way split but not necessarily evenly.

 

Let's suppose it goes all the way down to #6. #6's proposal is easy: Everything goes to #6. #6 will vote for this proposal, garnering the needed 50%. Exec #7 will be screwed if it gets down to #6.

 

Score:

Exec #1: loss of job

Exec #2: loss of job

Exec #3: loss of job

Exec #4: loss of job

Exec #5: loss of job

Exec #6: 1 mill, -5 competitors

Exec #7: -5 competitors

 

 

Before it gets down to #6, #5's proposal needs to be sunk. Unlike #6, #5 needs 1 cohort to go along with his proposal. #7 will go along with anything #5 proposes so long as it is better than nothing. Exec #6 will be screwed if it gets down to #5.

 

Score:

Exec #1: loss of job

Exec #2: loss of job

Exec #3: loss of job

Exec #4: loss of job

Exec #5: 1/2 mil, -4 competitors

Exec #6: -4 competitors

Exec #7: 1/2 mil, -4 competitors

 

 

Before it gets down to #5, #4's proposal needs to be sunk. #4 also needs 1 cohort to garner 50%, the obvious target being #6. Execs #5 and #7 will be screwed by #4's proposal.

 

Score:

Exec #1: loss of job

Exec #2: loss of job

Exec #3: loss of job

Exec #4: 1/2 mil, -3 competitors

Exec #5: -3 competitors

Exec #6: 1/2 mil, -3 competitors

Exec #7: -3 competitors

 

Before it gets down to #4, #3's proposal needs to be sunk. #3 needs two cohorts, the two who would be screwed by #5's proposal. #3 offers a three-way split between himself, #5, and #7. Execs #4 and #6 will be
screwed if it gets down to #3.

 

Score:

Exec #1: loss of job

Exec #2: loss of job

Exec #3: 1/3 mil, -2 competitors

Exec #4: -2 competitors

Exec #5: 1/3 mil, -2 competitors

Exec #6: -2 competitors

Exec #7: 1/3 mil, -2 competitors

 

Before it gets down to #3, #2's proposal needs to be sunk. #2 also needs two cohorts to get 50%, and these are execs #4 and #6. Execs #3, #5, and #7 will be screwed if it gets down to #2.

 

Score:

Exec #1: loss of job

Exec #2: 1/3 mil, -1 competitor

Exec #3: -1 competitor

Exec #4: 1/3 mil, -1 competitor

Exec #5: -1 competitor

Exec #6: 1/3 mil, -1 competitor

Exec #7: -1 competitor

 

Before it gets down to #2, #1's proposal needs to be sunk. #1 needs three cohorts: Execs #3, #5, and #7. Exec #1 should offer a four-way split amongst himself and execs #3, #5, and #7.

 

Exec #1: 1/4 mil

Exec #2:

Exec #3: 1/4 mil

Exec #4:

Exec #5: 1/4 mil

Exec #6:

Exec #7: 1/4 mil

 

If -1 competitor is worth 1/4 mil, then this proposal will be rejected by #7.

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Well shoot. That puts some unknowns into the original question (salary differences, etc.) that would need to be considered.

 

But in your #3,5 and 7 scenario it is not in #5's best interest to vote for #1's solution as you spelled it out as he is better off voting against #1 and holding out for a $500,000 pay day that he would likely get in later votes.

Edited by jryan

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Let's keep score. The fraction indicates a n-way split but not necessarily evenly.

Nice job.

 

That the spit does not have to be even is, I think, crucial.

 

Since the OP did not provide any information regarding the value of each position the question is essentially unanswerable. The OP also did not answer my question regarding wheelin' and dealin', and that can obviously change the outcome immensely.

 

If -1 competitor is worth 1/4 mil, then this proposal will be rejected by #7.

Of course, one way to overcome this is to not offer an even split. That #1 gets to keep his job might well be worth a lot more than 1/4 million. Heck, it might be worth so much that exec #1 will sweeten the pot with some of his own money. He could offer $1 million each to execs #3, 5, and 7, for example.

 

I would expect that jumping up a notch in the hierarchy is worth more to exec #3 than it would be to exec #7. For most people, pay levels out as one progresses. Pay raises can be quite phenomenal for fresh-outs. Some fresh-outs simply aren't qualified to do fresh-out level work, and the pay for them reflects that. Once fresh-outs have proven their worth their pay jumps by quite a bit (percentage wise). After that, pay raises start becoming rather pathetic; eventually they barely keep pace with inflation.

 

This is not the case in the cutthroat executive world. Pay starts going off the charts the higher one climbs. Exec #1 is most likely paid more, a whole lot more, than #2, #2 is paid a lot more than #3. Things probably start to flatten out from there. To forestall a rebellion by any one of his odd numbered cohorts, exec #1 may want to offer more to #3 than #5, and more to #5 than #7.

 

As said earlier, the OP didn't supply enough information needed to truly solve the problem. All we can do is speculate.


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But in your #3,5 and 7 scenario it is not in #5's best interest to vote for #1's solution as you spelled it out as he is better off voting against #1 and holding out for a $500,000 pay day that he would likely get in later votes.

You are forgetting that by rejecting #1's proposal exec #2 gets a shot. Exec #2 only needs two cohorts -- and those two cohorts most likely will not include exec #5. Exec #5 stands to get zero by rejecting #1's proposal.

 

Without any other info, this is an even numbered versus odd numbered execs proposition, and the evens have a distinct advantage each step of the way. Exec #1 needs three cohorts but exec #2 only needs two. Exec #3 needs two cohorts but exec #4 only needs one. Exec #5 needs one cohort but exec #6 does need any.

Edited by D H
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I agree that it's unanswerable without more info. You've got two commodities (money and reduced competition) without an "exchange rate," and a situation where the utility of each might well be different depending on where you are on the list and/or how many remain. It could well be that eliminating competitors is more important than money for everyone, in which case the guaranteed result is #6 getting a million, #7 getting nothing, and everybody else getting fired.

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You are forgetting that by rejecting #1's proposal exec #2 gets a shot. Exec #2 only needs two cohorts -- and those two cohorts most likely will not include exec #5. Exec #5 stands to get zero by rejecting #1's proposal.

 

Without any other info, this is an even numbered versus odd numbered execs proposition, and the evens have a distinct advantage each step of the way. Exec #1 needs three cohorts but exec #2 only needs two. Exec #3 needs two cohorts but exec #4 only needs one. Exec #5 needs one cohort but exec #6 does need any.

 

Well, I think that #2s strategy might have to include #5 as #6 can hold out for a $1 million pay day, #3 will be eyeing the senior spot leaving 4 and 7 I suppose... which would then need to bet a $0-$500,000-$500,000 split to make it worth #4 and #7's consideration as they are already looking at a $500,000 payday if it makes it to #4s choice.

 

So in #2s best interest, answering the question with the data provided, the proposal I stated still seems to be the best option for all 4 yes voters as #1 skips the pay out in favor of their job and #2 and #3 get a pay out that they would have had to skip otherwise, and #7 plays the pivotal vote as they were never going to get more than $500,000 anyway, but the cascading cancer of seniority + Bonus would likely mean that #6 would wind up with all the money and seniority anyway.

Edited by jryan

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Look at it from a bottom-up perspective, jryan. Posts #3, #4, and #10. Without any other info from the OP, this is clearly an evens-versus-odds proposition.


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To illustrate how much harder this is when the odd numbered execs get their shot versus the evens, consider that exec #1 needs 3 cohorts, #2 only needs 2, etc. In tabular form,

 

[math]\begin{array}{ccc}

\text{Turn \#} & \text{\# cohorts} & \text{As percent} \\

1 & 3 & 43 \\

2 & 2 & 33 \\

3 & 2 & 40 \\

4 & 1 & 25 \\

5 & 1 & 30 \\

6 & 0 & \phantom{0}0

\end{array}[/math]

 

There is no turn #7. The last exec will never get a shot. Exec #6 gets the requisite 50% from his/her own vote.

Edited by D H
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True, as such #7 is also always a yes vote when you give them money. But they can never make more than $500,000. But I stand by the decision that the plan creator must accept $0 or $0.01 of the total payout simply to maximize their chance to keep their job. As such, by the second in line viting "No" to any plan they are essentially voting themselves into seniority, but with $0 compensation.

 

With the exception of #6 who will always vote against any plan other than their own because they are the only one with the potential to get everything.

 

Leaving #2, 3, 4 and 5 at play, obviously.

 

Also, I would say that we have to decide based on the info given since it's not a given that various sales executives would even know each others total compensation, so that knowledge should not be required.

 

Actually, given that compensation is not known, we will have to ignore the seniority aspect, I think, and realize that the decision maker will always have their job on the line and therefor try to maximize the votes by taking $0 on $0.01 in compensation.... except that bastard #6. :)


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So, if we assume that #1 is gone, and the vote choice falls to #2, his choice would be:

 

#2 - 0

#4 - $500,000

#? - $500,000

 

In this case #7 is not necessary to win, so it is really a toss up for #5 and 7... as so I think it is in #2, 3 and 7s bets interest to take option 1 rather than risk losing everything (though it is impossible for #7 to lose their job, so their vote is always dependent on bonus money).

Edited by jryan

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I agree that it's unanswerable without more info. You've got two commodities (money and reduced competition) without an "exchange rate," and a situation where the utility of each might well be different depending on where you are on the list and/or how many remain. It could well be that eliminating competitors is more important than money for everyone, in which case the guaranteed result is #6 getting a million, #7 getting nothing, and everybody else getting fired.

 

Not quite. I think that if keeping one's job is the primary motivation, then the top exec will want to bribe the lower execs, since all the top execs are afraid to lose their job. While #2 might like to fire #1, doing so puts him on the chopping block. If you assume superrationality (that everyone reaches the same conclusions), then #2 considers that because a majority would want to get rid of #1, a very similar majority would want to get rid of him as well.

 

In this case, (let's assume that the money is essentially worthless), #6 and #7 are guaranteed their jobs. #5 can't reject #4's proposal because of that, since #6 and #7 will kick him out. So #5 is fairly secure in his job, as are all the others below him. I guess that means that these four will have the majority and outvote the others.

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You're right. If losing even 1 competitor is worth more than $1 million, it would be #4's proposal that is accepted. The votes of 1-3 don't even matter, because 4-7 would vote no on everything until then, when it would be 4 and 5 for yes and 6 and 7 for no. 4 is safe voting no on 1-3 because he knows he can count on 5 when his time comes, because 5 has to avoid his own turn.

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Here is a quick matrix, based on the assumption that the proposer is most interested in keeping their job, and therefor has no real leverage:

 

 

Round 1 - 7, total votes needed=3, pool=5, share=$333,333.33

Round 2 - 6, total votes needed=2, pool=4), share=$500,000.00

Round 3 - 5, total votes needed=2, pool=3), Share=$500,000.00

Round 4 - 4, total votes needed=1, pool=2), share=$500,000.00

Round 5 - 3, total votes needed=1, pool=1), Share=$????????????

Round 6 - 2, total votes needed=0, pool=0), Share=$1,000,000.00

 

By round 5 Exec#5 really has no real control, and may need to hand all $1,000,000 to Exec#7 just to keep their job.

 

So I see no reason to treat #7 any differently than #6. As a matter of fact, #6 now seems completely out of the running for any money as round 5 would almost certainly resolve the issue in #7s favor as #5 would be looking to keep their job.

 

#5 would definitely have leverage over 7 in a "take it or leave it" fashion, but then #7 would have leverage over #5 in a "make me happy or you're fired" fashion.


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After some consideration I think that #7 should be left out of the equation rather than #6. If #6 is a rational person then they have to realize that, while they can't lose their job, they have no chance of ever getting to make their "$1 million to me" proposal. #5 would split the bonus with #7 before that (though the nature of that split would be interesting!).

 

So I would change my group to 2,3 and 6 getting $333,333.33 as 2 and 6 have nothing to lose by voting yes, and 3 won't get a better deal either way.


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Also, I figured you all might be interested in this article:

 

http://euclid.trentu.ca/math/bz/pirates_gold.pdf

 

It's a discussion of a similar application of game theory.. but I think the restrictions and demands are sufficiently different that we can cast out their conclusion.

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Here is a quick matrix, based on the assumption that the proposer is most interested in keeping their job, and therefor has no real leverage:

jryan, you are thrashing here. You are just throwing out solutions with no logic behind then. This is not the way to solve these kinds of problems.

 

Also, I figured you all might be interested in this article:

 

http://euclid.trentu.ca/math/bz/pirates_gold.pdf

 

It's a discussion of a similar application of game theory.. but I think the restrictions and demands are sufficiently different that we can cast out their conclusion.

First off, interesting find. I suspect the OP changed the problem from pirates to executives so we wouldn't be able to find a solution on the 'net. Do note that the author came to the same conclusion that I came to: This is an evens versus odds proposition. The following paragraph in that paper is key (epmphasis mine):

The secret to analyzing all such games of strategy is to work backward from the end.
At the end, you know which decisions are good and which are bad. Having established that, you can transfer that knowledge to the next-to-last decision and so on. Working from the beginning, in the order in which the decisions are actually taken, doesn’t get you very far. The reason is that strategic decisions are all about “What will the next person do if I do this?” so the decisions that follow yours are important. The ones that come before yours aren’t, because you can’t do anything about them anyway.

 

This pirate problem is almost exactly the same problem as the topic of this thread, only it is a bit better specified. The reason that the pirate problem is a bit better stated is that there is no particular advantage accrued by moving up a notch on the fierceness totem pole. This problem is about executives, not pirates. In most companies, a huge financial advantage accrues from moving up the executive seniority totem pole.

 

That isn't true in all companies. Some are much more egalitarian. (Prototypical example: Ben & Jerry's, at least up until 2000 when Ben and Jerry sold out.) Compensation is fairly flat in such companies; what is accrued in advancing up the totem pole is a bit more glory at the expense of a lot more headaches. The division in this kind of company would be simple: Even shares for all, with everyone patting each other on the back for a job well done.

 

No mention of wheelin' and dealin' (the thing that salescritters do for a living). Imagine a more cutthroat corporation than a Ben & Jerry's. Exec #4 pulls aside execs #5, #6, and #7. "We can sink every proposal up to mine. I'll split the pot between us, and we can all move three notches up the ladder. BTW, hit men are cheap; don't think of voting down my proposal."

 

This wheelin' and dealin' makes for a solution that is not a Nash equilibrium. The simplistic Nash equilibrium solution is simple: Exec #1 offers $1 each to execs #3, #5, and #7. The remaining $999,997 dollars goes to exec #1.

 

This simplistic split is not realistic. Even ignoring the dollar value inherent in moving up the executive ladder, one of those three odd numbered executives may reject exec #1's proposal on grounds of fairness. So they lose a buck; exec #1's proposal is not fair. We humans (and other animals) appear to have some kind of built-in fairness mechanism. Google "Ultimatum game" for more.

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This simplistic split is not realistic. Even ignoring the dollar value inherent in moving up the executive ladder, one of those three odd numbered executives may reject exec #1's proposal on grounds of fairness. So they lose a buck; exec #1's proposal is not fair. We humans (and other animals) appear to have some kind of built-in fairness mechanism. Google "Ultimatum game" for more.

 

A good point, and one of the many reasons that we are not rational (or at least sometimes appear not to be). In part this is because the game of life has multiple rounds. If this one game was all there were, rejecting a proposal on the basis of unfairness might not be rational. But in a multi-round game you can make it clear that you reject "unfair" proposals, logic be damned. In the end, this sense of fairness can benefit you, as others fear to offer you an unfair proposal.

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A good point, and one of the many reasons that we are not rational (or at least sometimes appear not to be). In part this is because the game of life has multiple rounds. If this one game was all there were, rejecting a proposal on the basis of unfairness might not be rational. But in a multi-round game you can make it clear that you reject "unfair" proposals, logic be damned. In the end, this sense of fairness can benefit you, as others fear to offer you an unfair proposal.

 

Indeed. In fact, that's probably the whole function of anger. Being spiteful is self-harming pretty much by definition, but the threat of spite warns others not to mess with you.

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But in a multi-round game you can make it clear that you reject "unfair" proposals, logic be damned. In the end, this sense of fairness can benefit you, as others fear to offer you an unfair proposal.

 

This is a good observation. While being "fair" and "irrational" may lose you the $1 you may have otherwise gained, it also forces the others to offer more than the $1 because they do need your participation (or at least the participation of a majority).

 

If you are the top guy and a majority of the others won't settle for $1, wouldn't you offer, say $100k each so that you can get perhaps $400k instead of nothing?

 

If you aren't playing "fair" and are being entirely rational, all you will get is the $1 if you are not the top guy. Why settle for that when by being irrational you can get more?


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Indeed. In fact, that's probably the whole function of anger. Being spiteful is self-harming pretty much by definition, but the threat of spite warns others not to mess with you.

 

And then sometimes you have to play it out so that others know you aren't just bluffing. So what if you, as exec # 7, lose $1? Its not a big deal because on the next time everyone else knows they have to offer more, perhaps considerably more, to get your participation.

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jryan, you are thrashing here. You are just throwing out solutions with no logic behind then. This is not the way to solve these kinds of problems

 

We all are until the obvious solution arises. As you mentioned, the fairness aspect plays into this decision, which is why #5 is a problem. 7 can't happen, and 6 is easy, but #5 places "Job" directly at odds with "Bonus", and without job specifics (salary, turnover, etc.) we are left guessing. If #5 made $350,000 a year salary, and #7 made $150,000 then the fair split would be $400,000 to #5 and $600,000 to #7.... but we don't know that so we can't answer that. If we assume that #5s compentation is $1 million or greater then we can assume that the fair split is to give all the money to #7 and #7 really has all of the leverage in that case.

 

I think that even absent the information there is enough here to answer the question, however, and there is a way to deduce a fair distribution in round 1 that would garner support from the 3 necessary voters due to the variability of possibilities after the initial offer is rejected.

 

A bird in the hand is worth two in the bush.

 

In this case, assuming $1 million+ salaries, neither 6 or 3 are guaranteed any money in any permutation and would accept the offer made by #1 because it is the only given that have while #2 would accept it because they are guaranteed money and job which the wouldn't be in round #2.

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I think that even absent the information there is enough here to answer the question, however, and there is a way to deduce a fair distribution in round 1 that would garner support from the 3 necessary voters due to the variability of possibilities after the initial offer is rejected.

No, there isn't. You are making up information about salaries and such precisely because the problem lacks specificity. The only solution that makes any sense given the (limited and incomplete) information at hand is a 999997,0,1,0,1,0,1 split. The even numbered execs get nothing, execs numbers 3, 5, and 7 get a pittance, and exec #1 gets almost everything.

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