I hesitate to bring cricket to Soccernomics but the debate that occurred in the match between England and Australia that finished on Sunday raised an issue which appears in all sports. The issue surrounded the concept of “walking”. If a batsman in cricket hits the ball and it is caught by a fielder before hitting the ground then he is out (a bad thing). Sometimes the ball only just touches the bat, and so it is difficult to tell if the batsman is really out. However, the batsman, being closest to the action, often knows if the ball really did touch the bat; the umpire who has to decide the issue is 22 yards away. As can be expected, and as is confirmed by technology, umpires occasionally get it wrong.
In cricket there is a longstanding tradition, going back at least to the 19th century, that a batsman should admit the truth if he is out and “walk”, even if the umpire is unsure. At one point in last week’s game the replay technology showed that an English batsman hit the ball and was caught out, but the umpire somehow failed to see or hear what everyone else did (and replays confirmed). It was impossible to imagine that the batsman did not know the truth, but he did not “walk”.
This triggered a lot of debate in the press both in England and Australia about declining moral standards. In fact the Australian captain and vice-captain have said they saw nothing wrong with the actions of the England player, and generally the practice of “walking” has declined dramatically over the last three decades- most players now wait for the umpire to decide. However, when it is so obvious that the player hit it, most players would walk.
The simple point I want to make here is that this is a very good example of the prisoner’s dilemma. We talk about the prisoner’s dilemma in Soccernomics, and almost any pop book on economics explains it. But good examples are always worth noting. To show how it works, you have to simplify this to binary choices for each team. Suppose that before the game each time decides (independently) on a policy “walk” or “don’t walk”. Then you have to define the payoffs to these actions. Clearly if one side chooses “don’t walk” while its opponent chooses “walk” then it has a slight advantage because it benefits from umpire error while the other team does not.
Let’s measure the benefit in terms of extra runs scored in the game (this is the currency of cricket). Let’s say it’s worth +25 runs to the “don’t walk” team, which logically means it’s worth -25 to the other team. What if they both choose “walk”? Then the net gain to each must be 0. If they both choose “don’t walk” then one team might gain because randomly the umpire makes more errors on their side, but in expectation the net gain should be zero. So having said all this we can write down a payoff matrix that looks like this
Australia |
|||
Walk |
Don’t walk |
||
England |
Walk |
(0, 0) |
(-25,+25) |
Don’t walk |
(+25,-25) |
(0, 0) |
In each bracket, the first payoff refers to England, the second to Australia.
To figure what happens you have to focus on the “best response” of each team. If you’re England, what would you do if Australia chose “don’t walk”? You get -25 if you choose “walk”, but 0 if you choose “Don’t walk” which is obviously better for England. If Australia chooses “walk”, England gets 0 if it also chooses “walk”, and +25 choosing “don’t walk”, so again the better choice is “don’t walk”. So we conclude that whatever Australia chooses, England’s best choice is “don’t walk”. But it should also be fairly obvious that the problem looks the same to Australia, so clearly their best decision is to choose “don’t walk”, whatever England chooses. Thus we conclude that the equilibrium of this game is that both choose “don’t walk” (the bottom right hand corner of the matrix).
However, thus far we have only considered the decision to walk in terms of its contribution to winning the game. For many people, cricket is a better game if the players choose to walk- you’re not just watching skilled people perform, you’re watching “good” (i.e. honest) people. As the comments of the Australian players demonstrates, this is not something that most players seem to feel, but much (though certainly not all) of the viewing public does.
How to incorporate this into our matrix? We have to identify the exchange rate between moral sentiment and runs. There is clearly no objective standard, but we can speculate a little. First note that the players are the only ones who can decide to walk, so this is all about how indirect social pressure might weigh upon their decision. Suppose each player thought that the social pressure (e.g. criticism in the press, ostracism at home) arising from the decision “don’t walk” was equivalent to losing 25 runs in the game. Then the matrix would look like this
Australia |
|||
Walk |
Don’t walk |
||
England |
Walk |
(0, 0) |
(-25,0) |
Don’t walk |
(0, -25) |
(-25, -25) |
If you go through the same exercise as before, you should see that now each side is exactly indifferent between choosing “ walk” and “don’t walk” – the social pressure exactly offsets the potential advantage in the game. However, if we end up in the “don’t walk” scenario, then both sets of players are worse off than if they chose the “walk” scenario (this is the famous prisoner’s dilemma- pursuit of your own self- interest can lead to outcomes which are disastrous for everyone, including you).
So here’s the conclusion. If players feel the social pressure, but it’s not that great (say valued at only 10 runs) then we end up in a prisoner’s dilemma- the equilibrium is “don’t walk” when both sides would be better off choosing “walk”. If, on the other hand, if the value the social pressure is greater than 25 runs, then both sides will choose “walk” anyway, which is the best outcome.
One account of cricket’s development in the last 30 years is that the value of social pressure has fallen to move us from a “walk” equilibrium into a “don’t walk” equilibrium, which is a prisoner’s dilemma, because the value of the social pressure is not zero (as the media outcry demonstrates). Much the same can be said, I think, about other sports, including football. While many people like to complain about the moral standards of the players, one inference that you might draw from this analysis is that it’s more to do with the moral standards of the fans.
Unless of course the game was repeated, over a series perhaps, which itself is played regularly, with an indefinite end, in which case we might expect a cooperate play of the game. So has Broad just deviated from the best response?
I wasn’t going to go into this, but those not versed in game theory should know that it is possible to show that if a game of the type described above is repeated indefinitely then it is possible to avoid the prisoner’s dilemma – so long as both sides recognise that deviations from good behaviour (e.g. “walk”) will be “punished” by matching behaviour in the future. This could explain why “walk” was sustained as an equilibrium for a very long time (i.e. from the 1870s to the 1970s) but not why the practice has since almost disappeared, at least at professional level. I thinking changes in outside social pressure are a better explanation.
Apart from the tacit collusion resolution, it’s also worth noting the outside intervention possibility (albeit impractical in this case) that the ICC imposes heavy penalties for ‘not walking’ and a ‘walking’ equilibrium results (like Rod Fort’s ‘Beer Wars’ example). Actually Stef, as opposed to the social pressure angle, perhaps an alternative PD framework here would have been to assume that both teams put a low weight on draws relative to wins, and to note that the ‘no walking’ eqilibrium extends the conclusion of the match and increases the probability of a draw – a jointly less favourable outcome.