Monday, 21 April 2014

Stealing the opponents team sheet and mixed strategies

Recently, in the Premiership football, Crystal Palace beat Cardiff City 3-0. Nothing particularly unusual about that. But it subsequently came to light that Crystal Palace had used underhand means to obtain the Cardiff team line up ahead of schedule. Cardiff, obviously, cried foul play. This incident reminded of the 2008 Ryder Cup golf when Nick Faldo was photographed with a set of team pairings. In that case there was no foul play. There was, though, the notion that the US gained from knowing the European pairings.
       To make sense of why it may, or may not, be useful to know an opponents strategy we need to look at mixed strategies. To illustrate, consider the simple matching pennies game below. Europe can choose two possible team line ups, A or B, and the US can also choose two possible line ups, X or Y. If Europe chooses Team A and the US Team X then Europe wins (which is why Europe gets payoff 1 and the US payoff -1). If Europe chooses Team A and the US Team Y then the US wins, and so on.
         In this game it is clearly not good to let the opponent know what you intend to do. For example, if the US knows that Europe will choose Team B then it can choose Team X and win. This is why there are typically somewhat elaborate procedures to make sure team line-ups are disclosed simultaneously. It is also why mixed strategies become appropriate. Mixed strategies can be difficult to interpret, but the basic idea is simple enough - the only way to keep the opponent guessing what you intend to do is to not know what you intend to do yourself! You should, therefore, randomly decide what to do. Europe, for example, could toss a coin to decide whether to use line-up A or B.
         Now let's go back and think through the practical implications of knowing an opponents plan ahead of time. To fix ideas, let's assume that the US 'knows' Europe's team will be A. Here are three basic possibilities to consider:
1. Suppose no one knows that the US knows Europe's team will be A. Then the US does have a clear advantage. It can choose line up Y and win.
2. Suppose everyone knows that the US knows Europe's team will be A. This was the case in the Ryder Cup given that the photo appeared online, in the newspapers etc. Then Europe clearly has chance to change it's team line up. Which means we are effectively back to the US not knowing Europe's team. In other words, the US has not gained anything.
3. Suppose Europe knows the US knows their team will be A, but the US does not know that Europe knows. This was the case in the Crystal Palace - Cardiff match where Cardiff were told the team had been leaked. Given the US thinks Europe will choose A, they should choose Y. But, that means Europe can choose B and win! This time it is Europe that gains from their team being leaked.
          The consequences of knowing what an opponent plans to do depend, therefore, on who knows what about who knows what. So, why do we naturally assume a team gains an advantage from knowing what an opponent plans to do? It could be we have scenario 1 in mind. It could be we just get confused. Or, it could be that there is some other advantage not captured in the simple matching pennies game. One thing not captured in the matching pennies game (but clearly important in sport) is the role of practice. Let's see what difference that makes.
          Suppose it takes a time one week to practice with a particular line up. So, we have the timing: choose the team on Sunday, practice until Friday, make the team line up public on Saturday and then play the match. The game then changes to something like that below. We recognise that Europe may have planned to use line up A, and practiced with that in mind, but changed to line up B at the last minute, and so on. In terms of payoff we see that changing the line up may have costs. For example, if Europe changes from A to B while the US chooses Y as planned then rather than win for certain, either team is equally likely to win (as captured by expected payoffs of 0).
          Consider again scenario 2 where everyone knows that Europe planned line up A. If you look carefully through the numbers you see that the US has still gained no advantage from knowing what Europe planned to do! If the US planned to choose line up X then Europe is in a good position. If the US planned to choose line up Y then the US is in a good position. The crucial thing to take into account, however, is that the US cannot go back in time and re-plan what it was going to do. So, no team gains.
          What about scenario 3? Then Europe is still in a good position despite the US knowing their team. They can no longer be guaranteed to win, because they might have been practising the wrong line-up. But, they still gain an advantage.
          Needless to say, the two simple games considered here do not capture everything that goes on in a sport context. I think the basic message does, however, follow through much more generally: knowing what an opponent plans to do is of very little benefit if the opponent knows that you know what they plan to do. So, Cardiff City do not have too much to complain about.