Category Archives: Game Theory

Identifying high-order rationality

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Identifying high-order rationality is a forthcoming paper in Econometrica from my favourite behavioural game theorist, Terri Kneeland. For those of you who are not familiar with the inside baseball, Econometrica is one of the top two journals in economics and good economists can go their entire careers without publishing a paper in Econometrica. Getting your first paper published in Econometrica, as Kneeland has, is a pretty awesome achievement.

So what makes Kneeland’s paper so good? Well, it is a simple (yet novel) idea that was implemented in a very clean fashion.[1] Economic models of strategic interactions rely on the notion of rationality, where an agent is defined to be rational if they make optimal decisions given their available information.[3] This definition is actually rather weak and, as Kneeland demonstrates, most people satisfy rationality in simple environments. In some cases, however, economic models require a much stricter notion known as common knowledge of rationality. An agent is called first-order rational if they are rational. They are second-order rational if they are first-order rational and they believe that others are first-order rational. They are third-order rational if they are second-order rational and believe that others are also second-order rational. Common knowledge of rationality is the state where all agents are infinite-order rational.

The innovation in Kneeland’s paper is that she implemented a new experimental design that, with minimal assumptions, can identify a subject’s order of rationality. The process that Kneeland developed is as follows. Suppose that you are standing in a circle with a large number of other people. Everyone in the circle must choose a between A, B and C. You will receive an amount of money that depends on your choice and the choice of the person standing to your left. You are given a table outlining how much money you will receive in each situation. Consider the following table as an example. In this example, you will choose a row, the person to your left will choose a column, and you will receive the amount shown in the corresponding box. Which option would you choose?

A B C
A 12 16 14
B 8 12 10
C 6 10 8

You might have noticed that the option A will give you your best outcome, irrespective of what the person to your left chooses. An economist would call option A a dominant strategy. A first-order rational person will always choose their dominant strategy, if they have one.

 

Next, consider the following table. (Again, you choose a row, and the person to your left chooses a column).

 

A B C
A 20 14 8
B 16 2 18
C 0 16 16

In this case your choice is harder, because there are no dominant strategies for you. If you think that the person to your left will choose A then you should also choose A. But if you think the person to your left will choose C then you should choose B. How should you decide what to do?

In Kneeland’s experiment, you could also see the payoff tables for everyone else in the circle. So to decide what you should do, you should probably look at the payoff table for the person standing to your left. If they have a dominant strategy, then perhaps you think they will choose their dominant strategy and you may then respond accordingly. If you were to behave in this fashion then you would be displaying second-order rationality.

In other, more complicated, situations perhaps the person to your left doesn’t have a dominant strategy either but the person to their left does (that is, they person two spots to your left has a dominant strategy). Perhaps you think that the person to your left will notice that the person to their left has a dominant strategy, and that you could use this to predict the behaviour of the person directly to your left. Then, you can make your choice taking all of this information into account. If you do this then you are third-order rational.

This logical reasoning process can be traced back further to identify people of even higher orders of rationality[2]. This special circular structure (called a ring game) is exactly what Kneeland used to identify the order of rationality of her experimental subjects.

So what were the results? Kneeland finds that 93 of her subjects are first-order rational, 71 percent are second-order rational, 44 percent are third-order rational and only 22 percent are fourth-order rational. Kneeland is very cautious about over-interpreting her results, but in a blog post I need not be so careful. The results indicate, to me, that the standard assumption of rationality is not too far off being correct in simple environments.  However, the results also indicate that at most 22 percent of subjects could possibly satisfy the common knowledge of rationality assumption even in a relatively simple environment, indicating that common knowledge of rationality is not the most realistic of assumptions in many environments.

 

 

 

[1] Econometrica tends to publish two types of papers. The first is simple and clean papers that are good fun to read. The second is technically complicated, but brilliant, papers that can take days or weeks for experts to understand, and are pretty much incomprehensible to everyone else.

[2] An interesting question is to ask how you should reason when no one in the circle has a dominant strategy. The standard (but not entirely uncontroversial) recommendation would be that you should choose a Nash equilibrium strategy, named after John Nash who was played by Russel Crowe in the movie A Beautiful Mind.

[3] Note that this definition of rationality doesn’t mention anything about selfishness, nor make any assumptions about preferences, nor include any other of the crazy stuff that the anti-economics crowd seem to think economists mean by the word “rational”. At the most basic level you are rational if you make “good” decisions, given your information at the time of the decision, for whatever your personal evaluation of “good” is.

World Cup betting pool: the outcomes

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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!

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Quantity precommitment and Bertrand competition yield Cournot outcomes

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This classic paper by David Kreps and Jose Scheinkman is quite possibly my favourite economics paper of all time. It’s easy to explain and understand, but still makes a very deep point that connects two of the most famous economic models in a simple way.

Way back in 1838, Antoine Augustin Cournot wrote what is, I believe, the first mathematical model of competition between firms. The model makes reasonable predictions that are, in a broad sense, supported by empirical observations. One of these predictions is that as more firms enter a market it should become more competitive and prices should decrease.

However, the Cournot model has one deeply unsatisfactory dimension: firms set the quantity that they wish to sell, and then the market determines the price at which that quantity can be sold. This is most certainly not how firms actually make decisions; when a customer goes shopping, the store posts a price and the customer decides how much to buy.

In 1883, Joseph Bertrand came along and wrote a model where firms set prices, and then the market determines the quantity that will be sold at that price. This is a much more satisfactory foundation for a model of firm behaviour. Unfortunately, the Bertrand model generates very poor predictions. One of the implications of the Bertrand model is that two firms are enough to generate extremely intense competition and low prices. Another implication is that adding more firms to the market doesn’t change the outcome. Neither of these implications are compatible with empirical observations.

So we have one model with good assumptions but inaccurate implications, and one model with poor assumptions but reasonable implications. Is there a way that we can resolve this tension?

It took 100 years, but in 1983 Kreps and Scheinkman found the resolution: they key is to use a two stage model. In the first stage, firms install production capacity. Then, in the second stage, the firms compete over prices ala Bertrand. But here’s the kicker: the outcomes that are produced by this model are exactly the same as the Cournot model.

So we now have a model with both good assumptions and reasonable implications. Of course, this model is still highly stylised and leaves a lot of potentially important features unmodelled, but it does provide an extremely compact way of reconciling two very important economic models. Pretty neat.

Additional notes for economists

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An economist’s betting pool

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I am running a betting pool amongst some members of the department for the upcoming World Cup. Now, I suppose that the normal thing to do would be to assign teams randomly amongst the participants sweepstakes style. But for an economist, this seems like a rather inefficient thing to do. What if one person really wants Brazil but draws Germany, and someone else really wants Germany but draws Brazil?

Fortunately, there is an entire literature on what is known as the “rental harmony problem.” In the rental harmony problem, there are a group of house mates who are renting a new house and they wish to work out who should have each room and what share of the rent each should pay. We want the outcome to satisfy a few properties:

  • 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 manipulate the outcome by lying about their preferences)

Unfortunately, it is impossible to satisfy all four of these conditions at once (this can be demonstrated as an application of the Vickrey-Clarke-Groves mechanism; if we enforce strict budget balance no mechanism can be found). Therefore, we must relax one of the conditions, and the usual thing to do is to relax the last condition.

At this point, mathematicians and economists differ in their approach. Mathematicians tend to simply assume that people will tell the truth; this is the approach taken in this extremely elegant paper by Su [1]. Economists, on the other hand, tend to assume that people will lie if it is profitable, and attempt to minimize the damage caused by the lying; this is the approach taken in this paper by Abdulkadiroglu, Sonmez and Unver.

Therefore, for my World Cup betting pool I will be using the mechanism described in the Abdulkadiroglu et al paper. Each team (or group of teams) will be treated as a room in the house, and the total prize money available will be treated as the total rent for the house. Each participant will need to submit a vector of bids (one bid for each team or group of teams), and the algorithm will allocate teams and entry costs to the participants. In a follow up post I will outline the bids that were submitted and the outcome of the algorithm.

 

Additional notes for economists:

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Can we use epsilon equilibrium as a refinement tool?

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The short answer is: no.

That is a watered down version of the conclusions in the linked paper by Matthew O. Jackson, Tomas Rodriguez-Barraquer and Xu Tan.

The link is to an ungated version, but the final version of the paper can be found in Games and Economic Behaviour. The title of this post will probably require a bit of background for some readers, so here goes.

For non-economists:

One of the most fun parts of economics is game theory (it is totally as fun as it sounds!). Game theorists use mathematical tools to analyse behaviour and find equilibria in strategic situations. We can somewhat informally think of these equilibria as predictions; in an equilibrium everyone is happy with their actions (given everything else that is happening in the game) so no one wants to change their behaviour.

A key problem that economists have been working on since at least the 1970s is the fact that most games have multiple equilibria. This is a real problem when you are trying to make predictions – which of the equilibria should you choose as your prediction for the game? Despite working on this for 40 years, economists are yet to come up with an answer that can be applied universally to all games. In fact, economists now think that this question is pretty much impossible to answer – if you give me an algorithm for picking an equilibrium to use as your prediction, I will be able to find a game where your prediction is either absurd or not unique.

The other thing we need to know about is epsilon equilibrium. An epsilon equilibrium is a type of equilibrium that comes about when people don’t care about very small changes in their payoffs. Ask yourself: if someone gave you 5 cents would you feel any better off, even a tiny bit, than you were before? What about 1 cent? What about half a cent? What about one hundredth of a cent? If you answered ‘no’ to any of these questions, then you are a candidate for epsilon equilibrium.

Jackson, Rodriguez-Barraquer and Tan ask the question: can we use the ideas behind epsilon equilibrium to help us choose an equilibrium to use as our prediction? And their answer is no, it is completely useless. Not only can it not make a unique prediction it can’t even rule out any equilibrium, ever.

But now comes the really depressing part. Over the years, economists have come up with a lot equilibrium refinements that work in a reasonably broad range of cases. J, R-B and T use their results to show that a whole lot of these refinements (even some pretty popular ones) can only work if we are absolutely sure that we know everyone’s payoffs in the game absolutely exactly.

If we are even a little bit uncertain about the payoffs in the game (i.e. we are not entirely sure that you like outcome A exactly 2.06 times more than outcome B) then we are in a situation where we can use epsilon equilibrium to approximate the game in question. And when we use epsilon equilibrium we can’t rule out any equilibrium, ever. So none of the equilibrium refinements that rely on approximations to the true game can ever work, unless we are absolutely sure that we have the payoffs correct!

In a sense, you can think of this paper as being a big middle finger to the thousands of papers on equilibrium refinements that have been published over the last 40 years. But in a more positive light, we did learn something from Jackson, Rodriguez-Barraquer and Tan: if you think that you might have misspecified the payoffs in your game, it is important to make sure that you use techniques are robust to the potential misspecification. J, R-B and T have shown that a bunch of techniques that we thought were robust are actually not. This is progress.

For economists:

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