I have had a request to follow up on my previous post on the World Cup betting pool that I ran in my department. Specifically, I was asked to address whether the betting satisfied the four desiderata that I outlined in the previous post.

For a brief refresher, I used a model where we treat each of the teams (or a group of teams) in the World Cup as a room in a share house, and each of the participants in the betting pool as a tenant in the house. Then the goal is to match rooms to tenants, and allocate shares of the rent across tenants, so that:

• the outcome should be efficient
• no one should envy anyone else’s room/rent combination
• the sum of the rents should be equal to the total rent payable for the house
• the mechanism should be incentive compatible (i.e. no one should be able to manipulate the outcome by lying about their preferences)

So, did my betting pool satisfy these four criteria? The short answer is that it is impossible to guarantee all four at once.

The longer answer is that, in summary, if we can assume that no one lied about their preferences, then the other three conditions are automatically satisfied. If we think this assumption might be violated, then the first three conditions will still be satisfied if people have misrepresented their preferences in an optimal fashion. If we think that people might have misrepresented their preferences sub-optimally, then condition three will still be satisfied, but there is nothing that we can say about the first two conditions.

For more details, read on!

There are two approaches we can take to answering this question. The first approach is to assume that everyone told the truth about their preferences (note that this assumption is probably false, but it might work as a decent approximation). In this case, then the first three conditions are satisfied automatically by the model. To see why, let’s take a look at how the pool played out.

I had 8 people participating in the pool, and there were 32 teams in the World Cup, so I split the teams into 8 flights of 4 teams. I tried to keep the flights as evenly matched as possible (the algorithm I used will work best when people have differing opinions of which flight of teams is the best). The eight flights were:

The highlighted squares show the allocations that the algorithm made. I will note at this point that Chad spent some time with the betting odds calculating the expected values for each of the flights, and submitted those as his bids. So we can (roughly) think of Chad’s bids as the values that a disinterested outside observer should have had for each of the flights. I made my bids by looking at the betting odds (although I used a somewhat cruder model than Chad did), and then adjusting those numbers up for teams that I liked and down for teams that I didn’t like.

The prices that the algorithm awarded to each of the flights are given next:

Notice that the sum of the prices is equal to $80, so condition number three is satisfied.

To check the other two conditions, we want to look at how much surplus each participant would get from each of the flights. Under the assumption that everyone truthfully represented their preferences, the surplus is calculated as the bid submitted for the flight minus the price of the flight. The table of surpluses is reproduced below.

Condition number two will be satisfied if everyone was allocated the flight that gives them the greatest surplus. It is easy to check that this indeed the case (although there are lots of cases where people are indifferent between two or more flights). Condition number one requires that the sum of the shaded boxes produces the largest number possible, taking into account all possible prices and allocation combinations. It’s not exactly easy to see, but it is the case that this condition is also satisfied.

Now, remember that all of this only holds when we assume that people were telling the truth. When we make that assumption, we can formulate the surplus table as above, and everything is straightforward. But is this assumption reasonable? I suspect that it is probably violated here. I would be very surprised if, for example, Paul actually valued flight 5 $40 more than he valued flight 1.

So what can we say if we don’t think that people truthfully revealed their preferences? Well, the algorithm that I used guarantees that in any equilibrium the first three conditions will be satisfied. What this means is that if people are manipulating their responses in an optimal fashion then the first three conditions will be satisfied. However, this is a pretty strict bar – it’s not at all clear to me what is the optimal way to manipulate responses to improve your outcome. For this reason I truthfully reported my preferences when submitting my bids; one implication of the discussion above is that if you truthfully reveal your preferences then you are guaranteed to receive a non-negative surplus.

If we think that people might have tried to manipulate the mechanism in a sub-optimal way, then their is not much to say. Condition three is trivially satisfied by the algorithm, but we can’t calculate the surplus matrix (because we don’t know people’s true values). And we can’t be sure that the other two conditions are satisfied. In fact, if people really get their manipulation attempts wrong then it becomes quite possible that, in particular, condition two is violated.