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 Weizsäcker* 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 Weizsäcker “Do we follow others when we should? A simple test of rational expectations”, 2010, American Economic Review 100, 2340-2360.