The tactics of big cycle races, such as the Tour de France, provide a fascinating area for game theory to analyze.
Each rider has a couple of basic options:
He can either wait in the pack and conserve his energy for a sprint finish, but there will be a lot of riders so he will have a small chance of winning; or
He can try to be in a small breakaway group which will give him a much better chance of winning
The interesting dynamic is that as the size of the breakaway increases it has a greater chance of successfully staying away from the main group, but each individual rider in the breakaway has a smaller chance of winning. How do these factors work together and is there an optimum size of breakaway.
As the size of the breakaway group increases they have more chance of successfully staying away from the main group, but only if they work together. A rider at the front of the group has to work harder as they are the one that is cutting through the wind, the rest can enjoy an easier time in the first rider’s slipstream. The breakaway has more chance of working if they take it in turns to ride at the front.
As the group gets larger there is more chance that one or more members of the group will try to get a ‘free ride’ and not take their fair share of time at the front of the group. This is the first element of game theory, the riders have to co-operate to be successful but, when the group is large, it is easier for a rider not to co-operate which will be better for them but not the group as a whole. We will assume the optimum group size is six to eight before the free rider effect starts to become a problem.
This chart shows how this might look. The chance of success increases until a group size of seven and then starts to fall away again:
The second factor to look at is the chance of winning if the group breaks away successfully. Clearly if a single rider breaks away, and manages to stay away from the main pack, then he will win. If two breakaway then they will each have a 50:50 chance in the final sprint (assuming they are equally good sprinters). You can see as the breakaway group gets bigger each rider’s chance of winning the final sprint reduces.
This is shown in the chart below:
If we combine the two factors then we get a chart like this:
This shows the best chance of success for an individual rider is at a group size of between five and seven riders. Of course the reality is more complicated because some races are on the flat and some are in mountains and different riders have different abilities. Nonetheless it is still interesting to see that large groups are not good because the free rider effect will stop them working as a team and even if they do work an individual rider has less chance of winning from a larger group.
The optimum size sized group for a rider to have the best chance of winning is smaller than the best group size to successfully breakaway.