Interview Questions: 5 Bankers

[davidg]

Went to an interview the other day, my first in a decade, kick-off question is (I'm stating it a bit more clearly than the interviewer);

There are five bankers. They have 100 euros. They want to divide it up. The rule is the oldest banker can propose how to divide up the money. If more than half the bankers disagree they kill the oldest banker and start a new round. How will the money be divided up?

For example, the oldest banker could propose a 60,20,10,8,2 split based on seniority. If three of the bankers disagree they kill him and a new round starts with the four remaining bankers, the oldest proposing first.

I've never done this kind of problem explicitly before but recognized it as part of a class of problems known as game theory.

Just thought I'd post it for anyone who enjoys this kind of thing.

[Goat]

If I were the eldest banker (and assuming I didn't want to be killed) I'd propose a 50/50 split between any 2 others.

If that fails you end up with 100% of the money in one bankers hand, i.e. the oldest of the 4 remaining buys one other bankers vote, added to his own then a tie is enough to carry the proposal.

[Deft]

I feel like our interviews are quite dull. We ask people pretty standard questions relating to their CV and the job.

[Bloo Loo]

Went to an interview the other day, my first in a decade, kick-off question is (I'm stating it a bit more clearly than the interviewer);

There are five bankers. They have 100 euros. They want to divide it up. The rule is the oldest banker can propose how to divide up the money. If more than half the bankers disagree they kill the oldest banker and start a new round. How will the money be divided up?

For example, the oldest banker could propose a 60,20,10,8,2 split based on seniority. If three of the bankers disagree they kill him and a new round starts with the four remaining bankers, the oldest proposing first.

I've never done this kind of problem explicitly before but recognized it as part of a class of problems known as game theory.

Just thought I'd post it for anyone who enjoys this kind of thing.

game theory?..more like hypothetical bullshine

[Frank Hovis]

Um, surely he divides it 34:33:33:0:0 the two who get nothing will vote against but lose.

[Gigantic Purple Slug]

Gives it 50:50 to the youngest two.

[sigma]

Assuming they cannot confer

1 2 3 4 5

------------------------------

100,

0, 100

1, 0, 99

0, 1, 0, 99

1, 0, 1, 0, 98

...?

[whocares]

Um, surely he divides it 34:33:33:0:0 the two who get nothing will vote against but lose.

This is sort of what I would have gone for too. But maybe 30:35:35 to make sure 2 of the other 4 are happy.

Or I might have gone with 50:50 (to any two) if I was really worried (about getting killed), and/or not too desperate for the money.

Is a trick question ... as in "no bankers would ever be bothered re splitting a mere €100"? Maybe they should have said "5 waiters in a cheap restaurant" rather than "5 bankers", just to make the question more realistic re the liklihood of serious violence ensueing over €100?!

Where did this notional €100 come from anyway? (Did they obtain it dishonestly?!) I might have wanted more background on the origins of the money and the bankers characters before giving my well thought-through answer!

Hope you will give us the correct answer, once everyone has had a chance to think/reply. (And also what answer you actually gave in the interview, if different!)

PS I am obviously someone who enjoys quizzes and puzzles!

[blobloblob]

Bankers are greedy. Each one who chooses wants to keep it all for him/herself regardless of the consequences. First three get killed. When there are two left, the older of the two wants to keep it all but this time the younger one, unable to form a blocking majority of "more than half" alone, cannot stop it.

So the 4th oldest keeps all the money in the end.

[Goat]

Assuming they cannot confer

1 2 3 4 5

------------------------------

100,

0, 100

1, 0, 99

0, 1, 0, 99

1, 0, 1, 0, 98

...?

Spot on.

The trick is to think of the smallest group possible then work back from there.

[Mikhail Liebenstein]

Trouble with this game is the stakes aren't high enough in cash terms to make it interesting. I don't think any banker is going to risk their life for £100.

Though economists will spoil the real world by pretending people would do anything for £1 more.

So chances are he gives it to the youngest 3 guys, in some risk rated split The reward to the 2nd oldest banker is that he isn't next in line.

The risk of death in terms of possible outcomes is as follows (not sure if I need to also analyse vote outcomes)

Oldest 50% (he either lives or dies on the first round)

2nd Oldest 33% (either the game ends in round 1 and he lives, or it goes to round 2 where he lives or dies)

3rd older 25% (lives if game ends in rounds 1 or 2, and might live or die in round 3)

4th oldest 0% (lives if game ends in round 1,2, or 3 and importantly in round 4, as now there are just 2 bankers left).

5th oldest 0%

You can also do a similar analysis on financial outcome, for the youngest guy, the risk is gets nothing and the 4th oldest gets it all if we got to the third oldest deciding.

For the older guys, 1 , 2 and 3, if they live they'll probably get nothing, as they'll either be next in line or they'll be the oldest.

On that basis I'd say all the proceeds go to the two youngest, but more to 4th oldest as he can never die, but needs to be rewarded for not doing over the youngest in the last round.

[tinker]

Is this how we run business and the kind of managerial ethics we want these days? Greed and self-preservation?

[davidg]

Spot on.

The trick is to think of the smallest group possible then work back from there.

Yes, it is quite an interesting result. I guess it illustrates how the 1% are so rich, they only have to give crumbs to everyone else to stay in the game.

[The Generation Game]

In this scenario, from a greed perspective, is the favoured position the second oldest? He knows that he only needs to convince one other after having killed the eldest.

If it were down to the last 2 then the second youngest could give himself a 100% as the youngest cannot gain a majority vote.

From that perspective, down to the last three, the third oldest could probably get away with a 99,0,1 split, knowing that the youngest can do no better (unless he was very petty).

With 4 left, would 98,0,0,2 be sufficient to gain two votes?

So, with 5, could a 96,0,0,1,3 vote be carried?

[JustYield]

Assuming they cannot confer

1 2 3 4 5

------------------------------

100,

0, 100

1, 0, 99

0, 1, 0, 99

1, 0, 1, 0, 98

...?

What do you mean by "confer"? They must all know the split that each is getting, right?

I don't understand this result. What makes you think those who receive 1 will accept the eldest getting 98 (and so not vote against the proposed split)?

Please explain it to me as if I was a pet labrador.

[erat_forte]

What do you mean by "confer"? They must all know the split that each is getting, right?

I don't understand this result. What makes you think those who receive 1 will accept the eldest getting 98 (and so not vote against the proposed split)?

Please explain it to me as if I was a pet labrador.

The game doesn't say they know what the original stake is. The game is stupid really, there is no correct answer, it is just a tool for discussion.

I would say they would not care as its only €100.

[sigma]

What do you mean by "confer"? They must all know the split that each is getting, right?

I don't understand this result. What makes you think those who receive 1 will accept the eldest getting 98 (and so not vote against the proposed split)?

Please explain it to me as if I was a pet labrador.

I mean that they cannot make agreements with each other and/or do not know anything about each other other than the proposed splits. If either of these are not true then the split will be based on who is the best negotiator/are related/went to school together/etc etc.The outcome is easier to determine if the group is smaller so let's work backwards and start with a single banker:

1 B: 100 (Obviously, he cannot lose the vote)

2 B: 0, 100. (Obviously, the eldest banker cannot lose the vote)

3 B: Now for each banker you need to consider whether their split will increase or decrease if the offer is declined. At two bankers, the youngest got nothing - so he can easily be bribed to vote "yes" with a single dollar. Conversely the second oldest will get 100 if the vote is declined so we can assume he always votes "no". You end up with:

1, 0, 99

4 B: As before, consider whether each banker's split will increase or decrease if the vote is declined. It is more efficient for the eldest to bribe those who will potentially get nothing so in this case we get

0, 1, 0, 99

5 B: As before, the zeros become 1 (and vice versa). The eldest takes the remainder.

1, 0, 1, 0, 98

Hope that makes sense? The younger bankers will accept 1$ because the alternative is getting nothing. I would imagine there's an analogy about divide/conquer tactics waiting to be made here?

[rentbug]

The older banker is a Golman Sachs executive of many years standing. Clearly there is no "right" answer so another methodology must be found to solve the problem. He (there are no women in banking only eye candy) invests the money in a hedge fund and issues derivatives to all of the bankers including himself. The derivatives give each person the option to cash out, but only at 5% of the original stake, equally divided between the 5 participants. So the cash out value is 5% of £20. After one month the cash out value is 5% of the current value of the fund shared equally between the 5 participants. Thus - if the fund is now £105 - the value is 5% of £105. In all instances the balance remains in the fund and is now split amongst the remaining participants. For each month that follows the individuals are entitled to take a further 1% of the equally distributed fund. Thus after 6 months the cash out value is 9% of 20% of the fund at it's real time value. This process stops when all the participants are entitled to 80% of their equal share of the fund (75 months).

When this point is reached provided all participants are still in the fund a dividend of 50% of the fund is paid out equally to all 5 bankers. The cash out value is now reset to 5% for each individual and the monthy incremental counter reinstated until, once again, it reached 80%. This process is repeated until the end of time or until such time as the value of the fund is zero due to bad investment by the hedge fund manager. If 4 of the 5 cash out the remaining participant gets the balance of the fund. Individual bankers can sell their forward option - but only to the other bankers in the scheme.

This is how banking works.

I don't want the job thank you. I am already rich.

[happy_renting]

Just wondering... was the interview for a job in a bank?

[JustYield]

I mean that they cannot make agreements with each other and/or do not know anything about each other other than the proposed splits. If either of these are not true then the split will be based on who is the best negotiator/are related/went to school together/etc etc.The outcome is easier to determine if the group is smaller so let's work backwards and start with a single banker:

1 B: 100 (Obviously, he cannot lose the vote)

2 B: 0, 100. (Obviously, the eldest banker cannot lose the vote)

3 B: Now for each banker you need to consider whether their split will increase or decrease if the offer is declined. At two bankers, the youngest got nothing - so he can easily be bribed to vote "yes" with a single dollar. Conversely the second oldest will get 100 if the vote is declined so we can assume he always votes "no". You end up with:

1, 0, 99

4 B: As before, consider whether each banker's split will increase or decrease if the vote is declined. It is more efficient for the eldest to bribe those who will potentially get nothing so in this case we get

0, 1, 0, 99

5 B: As before, the zeros become 1 (and vice versa). The eldest takes the remainder.

1, 0, 1, 0, 98

Hope that makes sense? The younger bankers will accept 1$ because the alternative is getting nothing. I would imagine there's an analogy about divide/conquer tactics waiting to be made here?

This is very interesting, but plain wrong. You are assuming that the eldest banker is willing to accept a risk, however small, that he may be killed if he makes a proposal unacceptable to the majority. And the assertion that the juniors can be bribed for a marginal dollar is just that - your assertion. They know the splits and they know if it is at all fair or not.

[Dorkins]

Any outcome of money distribution is possible as it doesn't matter how the money is divided up. Even bankers wouldn't have somebody killed over €100, either out of a sense of morality or because they wouldn't want to go to prison.

It will be entirely down to the senior banker's preferences. If he's selfish he might take the money knowing that nobody will kill him. Out of gratitude that he isn't going to die he might give the others €25 each. Without knowing what the senior banker is like as a person we cannot predict his actions.

It is insane to start modelling human beings as empathy-free creatures with time horizons so short that they wouldn't think about the consequences of being arrested and who are willing to murder other people over a few euros. If human beings like that do exist they must be extremely rare.

[erat_forte]

I mean that they cannot make agreements with each other and/or do not know anything about each other other than the proposed splits. If either of these are not true then the split will be based on who is the best negotiator/are related/went to school together/etc etc.The outcome is easier to determine if the group is smaller so let's work backwards and start with a single banker:

1 B: 100 (Obviously, he cannot lose the vote)

2 B: 0, 100. (Obviously, the eldest banker cannot lose the vote)

3 B: Now for each banker you need to consider whether their split will increase or decrease if the offer is declined. At two bankers, the youngest got nothing - so he can easily be bribed to vote "yes" with a single dollar. Conversely the second oldest will get 100 if the vote is declined so we can assume he always votes "no". You end up with:

1, 0, 99

4 B: As before, consider whether each banker's split will increase or decrease if the vote is declined. It is more efficient for the eldest to bribe those who will potentially get nothing so in this case we get

0, 1, 0, 99

5 B: As before, the zeros become 1 (and vice versa). The eldest takes the remainder.

1, 0, 1, 0, 98

Hope that makes sense? The younger bankers will accept 1$ because the alternative is getting nothing. I would imagine there's an analogy about divide/conquer tactics waiting to be made here?

Even this answer is wrong. There are plenty of psychological experiments that show people are unwilling to accept a net benefit to themselves if they percieve that the distribution is "unfair". So even if you suppose that the people are totally ignorant of the original stake or the amount being given to others, then you are basically supposing to go out into the street and saying to random strangers "here's €1, do you want to accept it or refuse it?". Many people will refuse it because the offer looks somehow dodgy.

[Goat]

This is very interesting, but plain wrong. You are assuming that the eldest banker is willing to accept a risk, however small, that he may be killed if he makes a proposal unacceptable to the majority. And the assertion that the juniors can be bribed for a marginal dollar is just that - your assertion. They know the splits and they know if it is at all fair or not.

It's correct from a game theory point of view, of course in the real world no banker is going to risk death for €100 but if that troubles you assume it's £100,000,000 and the player simply drops out of the game.

[JustYield]

It's correct from a game theory point of view, of course in the real world no banker is going to risk death for €100 but if that troubles you assume it's £100,000,000 and the player simply drops out of the game.

Yes, I think the question has been modified somewhere along the line and some important subtleties (like the difference between taking a chance you'll get 98% of the pot or nothing, vs being killed) completely change the "game theory" and the practice.

Obviously €100 is a nonsense amount, but even with more meaningful, life changing amounts the eldest banker will think long and hard after he'd discovered he could take 98% of the pot... he wouldn't do it. As pointed out above, people are incredibly sensitive to perceived unfairness and unlike in real world situations, in this scenario they have the chance to vote to get the man killed for his greed!

There's a great line in Barbarians at the Gate where the CEO (James Garner) is told, "You were just being too greedy." He thought he had the deal wrapped up.

[MrPin]

The old guy should keep it all and then "put out a contract" on those that disagree loudly!