If you’re never seen the show, here is how it works. Each of two contestants independently chooses to split or steal the final prize. If both choose split, then the prize is divided evenly. If one chooses split and the other steal, the person who steals gets the entire prize. If both choose steal, however, then both walk away with nothing.

Here’s the normal form representation of the game:

How should you play this game?

One contestant had an amazingly brilliant strategy that I will discuss below.

The wrong way to play the game

Contestants are allowed to discuss strategy before picking split or steal.

Both realize that split gives a fair 50 percent share to each side, but each also sees the advantage of back-stabbing and stealing the prize.

The discussion usually involves the following strategy. Each person tries to convince the other person to split, and they promise to do the same.

I discussed an example of this in a previous post: strategy in Golden Balls.

In that episode, both were promising they would split the prize, but then one person decided at the last minute to steal all the money. She said she was not proud of the decision, but she herself did not want to be cheated.

So trying to split the money in a conventional way doesn’t work. Is there a better strategy?

Why it’s bad to promise you will split

First, I want to explain why the strategy of splitting does not work. When you promise the other person you will split the prize, you are trying to change the game.

You are telling them that instead of looking at the original payoffs, they should only consider the game under the assumption that you are going to split the prize. So you are telling them to consider the following game:

Do you see what’s wrong with your strategy? If you promise them that you will split the prize, they are faced with a very tempting option to steal. If they split, they will only get 50 percent. But if they steal, they will get the entire 100 percent.

And therein lies the problem: if you promise you’ll SPLIT the prize, then you pretty much are telling them not to worry about the mutual steal option. This makes it a very good idea for them to STEAL the prize.

Clearly it’s a bad idea to promise that you’ll split the prize. Is there another way out?

How to beat the Prisoner’s Dilemma

There’s a remarkably devious way to get cooperation: you must tell them that you will STEAL the prize!

How does this strategy play out? You should watch the following clip to see the negotiation:

Brilliant strategy in Golden Balls

The action proceeds as follows. One contestant, Nick, immediately announces

I want you to trust me. 100 percent I am going to pick the steal ball. I want you to choose split, and I promise you that I will split the money with you [after the show].

The other contestant is completely stunned by this strategy, and the audience finds it amusing too.

The next 2 minutes is a funny exchange between the two. Nick keeps explaining why he is going to steal, and the other is dumbfounded by this terrorist-like action. He wonders, why can’t they both split?

The host reminds them the plan is risky, as there is no legal requirement for the money to be split after the show is over.

Nick is called an idiot and the other contestant just can’t believe he expects cooperation. Nick has taken control of the game, and he has not acted nice. Why should the other person cooperate?

Nick promises over and over that he is an honest person and that he will definitely split the money after the show.

At the 5:21 mark in the video, they reveal their choices. It turns out that both of them choose to SPLIT after all! Thus both end up with a 50 percent share of the money.

Here’s why Nick’s strategy was so brilliant. Nick was credibly explaining that he was going to steal. This changed the game into the following payoffs:

On the one hand, the other contestant could steal and destroy the prize money. On the other, he could split and hope that Nick kept to his word. In other words, Nick has transformed the game so that the weakly dominant best response is to split!

The other contestant is happy at the outcome, but he shouts “Why did you put me through that?” as he really had to struggle over his decision, only to learn Nick would cooperate after all.

I think Nick has shown a brilliant way to beat the Prisoner’s Dilemma in a one-shot game. His statement that he would steal is a credible threat, and it changed the game so the other contestant found split to be the appealing option.

This might not work in a repeated game, as Nick would have burned his reputation by not keeping to his word.

But in a one-shot game, it was a smart way to assure that both go away with a split of the prize.

The father was known to be quirky, and in his will, he specified a rather unusual way for the land to be divided.

On a map of the land, Bob is to mark a location of his choosing. Then Alice gets to mark a location.

Bob will be given all the land that is closer to his marker than Alice’s, and Alice will own the rest of the land.

Assume the land is an equilateral triangle. Which child is favored in this game, if the goal is to get the most land?

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The answer

Alice has a second-mover advantage in this game. Whereever Bob picks, Alice can always pick the same spot (or near his spot) and get a 50 percent share.

It turns out Alice can even do better.

Bob’s best strategy is to locate at the centroid of the triangle. The logic is fairly simple: if Bob does not locate in the center of the triangle, then Alice can mark a spot that will cover an even larger chunk. (The full proof is in this document.)

Alice will locate (infinitesimally) below Bob’s centroid. The perpendicular bisector connecting Alice’s and Bob’s points will be a line that divides the area into two regions. The region in the trapezoid is the locus of points that belong to Alice, and the smaller triangle above is land belonging to Bob.

Using geometry we can figure out the triangle on top is an equilateral triangle with sides 2/3 the length of the original triangle. This means the area of the triangle on top is (2/3)^2 = 4/9 the area of the total triangle. The area of the trapezoid, and Alice’s share, is thus 5/9 of the triangle.

It looks like the father had a soft spot for his daughter and set up rules so that she could get just a bit more land.

One of the useful concepts in voting theory is having an index that determines how powerful particular voters are. One of the voting power indices used is called the Shapley-Shubik index, which I have talked about before.

I came across a great video on Youtube that explains the Shapley-Shubik power index. The video comes from James Hamblin who is a Mathematics Professor at Shippenburg University.

Here is the video on the Shapley-Shubik power index.

Video: Shapley-Shubik Power index

On a concluding note, Professor Hambin has an impressive series of videos and articles Math for Liberal Studies, which are a companion to his book of the same title. These look very interesting and I am slowly making my way through the material.

Here is a fun video that shows an experiment about monkeys and cooperation.

Video: monkey cooperation and fairness

Some of the game theory concepts are:

–reciprocity: once Virgil has all of the nuts, he doesn’t have to offer any back to Vulcan. In fact, in a one-shot game he might be tempted to steal all of the nuts. But in human society and in monkeys, there is a sense you should repay your debts to build good karma.

–jealousy over rewards: in the second game, you’ll notice how Vulcan is originally happy with the biscuit until he sees Virgil getting a juicy grape. He has a jealousy over payouts, so, as the narrator points out, “he would soon have nothing than be short-changed.” Games of jealousy can often be very destructive as I wrote about in this article.

Hi Presh~

Could you write a blog post on game theory tips for fairly dividing household chores/tasks (for roommates, partners, and perhaps coworkers)?

Bonus points for providing easy division methods that people will actually use. In any case, keep up the good work. Thanks.

I thought this was a great question and I wanted to take stab at it. Here are a couple of ideas I came across the following two ideas.

Method 1: “I cut, you choose”

This is one of the most famous examples from fair division. Let’s imagine that two people are trying to split up a triangular slice of cake, but they might have different preferences for the filling or whipped creme topping. What’s the fair way to divide the cake?

The answer is astonishingly simple: one person gets to cut the cake, but the other person gets to choose his piece first. The idea here is that the person cutting will be encouraged to make the pieces as equal as possible so that he ends up with a good piece. If he makes one piece too large, then the other person will just take it, leaving him with a small piece.

This cake-cutting problem is generalizable to all sorts of areas, as I have written about before in a peanut butter commercial and in terms of splitting a can of Coca Cola.

In cake-cutting, the goal is to get as large a piece as possible. For chores, the problem is exactly the opposite: you want to pick the smallest “piece” that corresponds to the least amount of effort.

The problems are analogous, and the “I cut, you choose” method works very well here. Here is how you can divide up chores between 2 people:

Let one person divide the chorses into two separate lists. Let the other person choose the which list to do.

The idea will work out in a very similar way. The person who makes the list of chores has to make the two lists as equal as possible. If not, the other person will choose the easier set of chores.

The “I cut, you choose” method is great for 2 people. It can be extended to 3 and 4 people, but those scenarios can prove to be a lot harder. For details, see this paper about fairly dividing up chores for 4 people.

Method 2: Tug-of-war method

The “I cut, you choose” method is great for infrequent chores like taking out the trash or cleaning the gutters. But it is not so good about dividing up daily chores like cooking.

If you’re in a house where you wish to split the job of cooking, what is the best way to do that?

The idea is you want to split up the amount of work and have some way that both people can agree to the commitment.

I came across a very interesting idea from a webpage posted in 1993. The author turned out to be the economist Robin Hanson who writes the interesting blog Overcoming Bias.

Here is Hanson’s advice found on this page:

When both my wife and I were employed, we split the task of cooking (or picking up) the evening meal with a “Tug-O-War” board, like:

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      |h           /#\   |                 h|
      |i  0    0   ###   0    0    0    0  e|
      |m           \#/   |                 r|
      ---------------------------------------

This has a row of seven holes, with a peg sitting in one of them. The rule is that when you cook, you get to move the peg one step in your direction. If the peg gets all the way to your end, the other person *has* to cook. Now you can cook a few nights in a row if you feel energetic, or wait a few nights if you’re sick. If neither of you wants to cook, the person farther down is expected to cook.

The tug-of-war method is interesting to me for a couple of reasons.

First, with a daily chore it is very hard to remember who has done the work. Each person has a bias towards thinking they have done more of the cooking than they actually have. The board is an unbiased way to keep track of the work, and each person can see the results day by day.

Second, the board has a commitment rule so encourage the work split to stay near equilibrium. If one person has done too much of the work for the past few days, the other person knows it too, and will have to make up for the cooking.

The same technique can be generalized for more people. Instead of a single board, you make several columns with each person’s names. Everyone starts out in the middle, and each time you do a chore, you move your marker up and everyone else’s marker down. Anyone who is at the bottom has to do the chore, and anyone at the top does not have to.

I’m sure there are other interesting ideas for splitting up chores. How do you divide up work in your house?

Classroom discussion questions

1. Let’s say two roommates agree to a division of chores. Suppose one roommate decides to hire help rather than doing the work. The other roommate feels this is unfair because he cannot afford to hire help. Is this a valid complaint?

2. How might you penalize someone for not doing their chore?

3. One roommate feels the kitchen floor should be mopped every two weeks, while the other thinks a month is fine. What is a fair way to decide how often to mop, and who does the mopping?