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.
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