Tuesday, 29 January 2013

HS2 and cost-benefit analysis

The UK government has recently announced the route for the second leg of the long awaited high speed rail link between London and the north of England. This prompted the usual appeals to cost-benefit analysis to argue one way or the other. Most ‘experts’ seemed to be arguing that the supposed benefits are largely illusory and so the link does not make economic sense. But what does that really mean?
   Cost-benefit analysis is one of the most basic tools in the economist’s armoury. It’s pretty clear, however, that the general public don’t like it. Many people I heard on the radio talking about the rail link were annoyed by the experts cost-benefit calculations. Similarly, when health economists argue that a particular cancer drug should not be made available on the National Health Service because the costs exceed the benefits, people aren’t too pleased. So, what’s wrong with cost-benefit analysis?
    Given that I’m an economist it will be no surprise to hear that I don’t think anything is wrong with cost-benefit analysis. It’s one of the most basic principles of economics for a good reason –something is worth doing if and only the benefit exceeds the cost. What I have much less sympathy for is the way economists often use cost-benefit analysis.
   Doing a proper cost-benefit analysis is difficult because measuring potential benefits and costs can be a very tricky thing indeed. I’m excited by a high speed rail link because I want to see technological progress and development; how can you measure this benefit? Similarly, a cancer drug might add two years to a person’s life; how can you measure the benefit of that? I know there are ways we can attempt to measure these things, such as, willingness to pay and quality adjusted life years, but I also know how imperfect these measures are!
   So, my two concerns with the way economists often do cost-benefit analysis are as follows: (i) They make life easy for themselves by counting the simple to measure financial costs and benefits while ignoring the more difficult to measure costs and benefits; this often biases against the benefits. (ii) They underplay the huge margin for error in the calculation of costs and benefits; this leads to a sense of decisiveness that is not justified.
   It’s important, therefore, to appreciate the limitations of cost-benefit analysis. It is a useful tool but it cannot be expected to give clear cut answers. On tricky issues it is best used as a tool to inform and open debate rather than as a means to decide. Decisions should come down to questions of individual or democratic judgement. I, for one, would vote for the high speed rail link. Maybe you disagree. Just don't think cost-benefit analysis holds all the answers.   

Saturday, 19 January 2013

Information cascades and standing up at football

The issue of standing up at football grounds has been attracting some attention recently. The basic problem people are talking about is one of externalities and property rights: Post the Hillsborough disaster, standing is banned at the major football grounds in Britain. Clearly, however, that does not stop some fans wanting to stand up in order to enjoy the game that bit more. But, if a fan stands up that creates a negative externality for anyone sitting behind – at best the person behind has to also stand up to see the game, at worst the person cannot see even if they stand up. Some fans are arguing that football is not football without standing – others are annoyed at not being able to see the game. My perspective would be to look who has the property rights. And given that standing at grounds is banned the property rights clearly stand with those who want to sit and see the game. So, ‘sit down’. 
   Thoughts of standing at football remind me of watching football in my childhood years – going around the country to watch Aston Villa. When the football was not good – you never have long to wait with the Villa – I would often watch the crowd. And I find crowds fascinating. Indeed, it doesn’t seem hyperbole to say that I became a game theorist watching football crowds. For example, one thing you can learn about watching a football crowd are information cascades. Information cascades are traditionally applied to analyze consumer choice and the stock market, but a football crowd is just as interesting. Let me try and explain.
   Picture a packed stadium with all the supporters sitting down comfortably. Then the home team starts attacking and it looks as though something exciting might happen. If something exciting does happen then the supporters would rather be standing up in order to let off energy. If nothing exciting happens they would rather have stayed sitting. In real time, as the team attacks, each supporter must decide whether to stand up or remain seated. This scenario has the two key ingredients we need for an information cascade to occur:
(a) Each supporter has their own beliefs about whether something exciting might happen. Some may be optimistic, some pessimistic, some may have a better view, others a worse view, etc. In game theory parlance each supporter has a private signal of whether something exciting may happen. The key word here is ‘private’ – only the supporter knows what his signal and beliefs are.
(b) Choices are made sequentially with the possibility to observe what others a doing. If a supporter stands up then all the supporters behind can clearly see that he has stood up. Note, however, that only the action is observable. The reasons behind the action, i.e. the private signal or beliefs remain private.
   Let’s roll forward time a little until a first supporter decides to stand up. Suppose his name is Darius. What does Darius’ action – him standing up – tell us about his signal? Probably a lot. It might be that he is pessimistic anything exciting will happen and just stood up to go and get a cup of coffee. Much more likely, however, is that he stood up because he is really confident something exciting will happen. Suppose that once Darius has stood up there is a cascade of other people standing up. For example, imagine that Sam, whose sitting a few rows behind Darius, stands up. What does Sam’s action tell us about his signal? Probably very little. It could be that Sam is standing up because he was always confident something exciting might happen. Equally, however, Sam could be standing up because the actions of Darius and others have caused him to update his beliefs – initially he was pessimistic something exciting would happen but has changed his mind. Once we have reached the point where a Sam’s action tell you nothing about his private signal then we have an information cascade.     
    Information cascades have lots of interesting properties. For example, they mean that mass action can convey very little information. That Darius stands up tells us something. That the 2,000 supporters around him stand up tells us very little. The main consequence of this is that information cascades can be very misleading. If we combined the private signal of every supporter we might get a good prediction of the chances of something exciting happening. But, that’s not how it works. Darius triggered the whole thing and that’s just one private signal which could easily have been wrong. So, supporters can expect to be up and down like yo-yo’s. Another interesting property is fragility. For example, suppose that while Darius and the other 2,000 supporters are standing up, Brian stays firmly in his seat. What does that tell us about Brian’s signal? Potentially quite a lot. It suggests he has a strong signal that nothing exciting is going to happen – he might, for instance, have seen that the linesman has flagged to stop play. Given that we know little about the signal of the 2,000 supporters who stood up we are just left with a good idea of Darius and Brian’s signals which ‘cancel each other out’. Brian staying seated can easily, therefore, stop the cascade.  
   Well that’s the theory. What about the practice. We know from the fascinating work of Georg Weizs├Ącker* that people are biased in situations where information cascades may occur. They tend to underestimate the information conveyed by Darius or Brian’s action while overestimating the information conveyed by the 2,000 others who stand up. Such bias is probably not too surprising. What’s interesting is to know how big the bias is – how easily are people misled by mass action. My experience of watching football crowds suggests we are not too easily misled. Indeed, my impression is that in real situations people have a fairly good intuition of how information cascades work. For example, my anecdotal evidence, is that you often get supporters playing the part of Brian by remaining seated while everyone in front stands, and others reacting to that. This means we get the fragility that is predicted in theory but unlikely if people are strongly misled by mass action.
   The problem we have is that our theoretical understanding of information cascades remains largely foccussed on nice textbook cases that are far removed from real world settings. For example, we know very little about what should happen in the real time setting that we find in a football ground, or the stock exchange. This is one area, therefore, of game theory that needs a lot more work before we can be too confident what is going on. So, football crowds can teach us something.
     *G Weizs├Ącker “Do we follow others when we should? A simple test of rational expectations”, 2010, American Economic Review 100, 2340-2360.

Friday, 4 January 2013

Prisoners dilemma or stag hunt

Over Christmas I had chance to read The Stag Hunt and the Evolution of Social Structure by Brian Skyrms. A nice read, very interesting and thought provoking. There’s a couple of things in the book that prompt further discussion. The one I want to focus on in this post is the distinction between the stag hunt game and the prisoners dilemma game.
   To be sure what we are talking about, here is a specific version of both type of game. Adam and Eve independently need to decide whether to cooperate or defect. The payoff matrix details their payoff for any combination of choices, where the first number is the payoff of Adam and the second number the payoff of Eve. For example, in the Prisoners Dilemma, if Adam cooperates and Eve defects then Adam gets 65 and Eve gets 165.

Prisoners Dilemma


140, 140
65, 165

165, 65
90, 90

Stag Hunt


140, 140
10, 70

70, 10
70, 70

The key thing about the prisoners dilemma is that cooperating is a dominated strategy. It doesn’t matter what Eve does, it is in Adam’s interest to defect. Similarly, it doesn’t matter what Adam does, it is in Eve’s interest to defect. So, we have a clear game theoretic prediction that both Adam and Eve should defect. Simple enough. This result, however, is a bit depressing given that both Adam and Eve would get much higher payoffs if they were to cooperate. It’s this trade-off between individual rationality and collective rationality that has resulted in the prisoners dilemma, despite its seeming simplicity, being easily the most analyzed game in game theory. The key questions asked are: (i) whether people cooperate in the prisoners dilemma, (ii) if they do (many do) then why, and (iii) if they do not (many do not) then how can we get them to cooperate.
   The main thing I liked about Skyrms’ book is his suggestion that we should focus a little less on the prisoners dilemma and a little more on the stag hunt game. There are, at least, two reasons to focus more on the stag hunt game. The reason emphasized by Skyrms is that this game is often a better description of the applied context we’re interested in than the prisoners dilemma. A more subtle reason, not explicitly mentioned by Skyrms but a theme throughout the book nonetheless, is that an understanding of the stag hunt game can possibly tell us more about the prisoners dilemma than an analysis of the prisoners dilemma can do. So, what’s different about the stag hunt game?
   In this game cooperate is not a dominated strategy. If Eve cooperates then it is in Adam’s interest to also cooperate. Which suggests that it should be a lot easier to get cooperation? That, however, is where things get interesting. If you ask people to play the stag hunt game then the outcome is remarkably similar to what you get if you ask people to play the prisoners dilemma. This is the case in the two player versions given above, or in the more general many player versions (which correspond to a linear public good game and minimum effort game) where defection quickly becomes the norm. This empirical finding potentially tells us a lot. The standard story is that people defect in the prisoners dilemma because that is the rational thing to do. That story, however, sounds a little suspect if people defect to a similar extent in the stag hunt game. In the stag hunt game defection cannot be explained as the ‘rational thing to do’ and is almost certainly a consequence of people avoiding a risky option. Something similar may be going on in the prisoners dilemma. If so, it would be a mistake to put a lack of cooperation in the prisoners dilemma down to defection being the rational thing to do.
   I’m not saying that different things may not be happening in the prisoners dilemma and the stag hunt game. Clearly, the problems of obtaining cooperation in the prisoners dilemma appear greater than in the stag hunt game. My point is more of a ‘let’s walk before we can run’ nature. It seems ambitious to try and get people to cooperate in the prisoners dilemma when we don’t know how to get them to cooperate in the stag hunt game (and we don’t). My hope would be that ways of obtaining cooperation in the stag hunt game would work pretty well for the prisoners dilemma as well. And to get cooperation in the stag hunt game the emphasis must surely be on making people more confident that the person they are playing with will cooperate. This line of reasoning is quite different to that found in most of the research on the prisoners dilemma. But, it still leaves open the question of how to get cooperation in the stag hunt game. Skyrms had a lot to say on that question, which gives me a nice topic for a future post.