The Tour de France, public goods, and the chicken game

I finally finished watching this year's Tour de France on Sunday. Yes, I was a week behind. That's because I was overseas when it started, and it took me that long to catch up (with big thanks to Sky On Demand!). Jonas Vingegaard well deserved his win. The individual time trial he rode on Stage 16 was amazing to watch (even if his team Jumbo Visma says so themselves).

Anyway, this is a blog about economics. Sports provide lots of great examples of economics in action, because economics is ultimately about choices, and so are sports. One striking example of economics in action in cycling road races occurs when there is a breakaway, and it is getting close to the finish line. The riders in the breakaway face a difficult choice. They can ride hard at the front of the breakaway, ensuring that the breakaway won't be caught by the peloton, and one of the breakaway riders will surely win the race. Or they can hold back, riding in the slipstream of the rider who is riding at the front, which lets them conserve energy for a sprint finish, but at the risk that the peloton catches them.

This exact scenario played out in Stage 18 of the Tour de France this year, with three riders approaching the finish. Victor Campenaerts rode hard towards the finish, ensuring the breakaway would succeed. However, it was Kasper Asgreen who won the stage, having conserved his energy for the final sprint among the breakaway riders.

Let's think about the incentives for a breakaway rider. Riding hard is a public good. It is non-rival (one cyclist benefiting from a rider riding hard at the front of the breakaway doesn't reduce the amount of the benefit available for the other riders in the breakaway) and non-excludable (if a rider is riding hard at the front of the breakaway, they can't easily prevent the other breakaway riders from sitting in their slipstream and conserving their energy).

Public goods, like riding hard at the front of the breakaway group, suffer from a free rider problem (pun intended!). Other riders can benefit from the front rider's hard work, without paying any of the cost themselves. It is difficult for a rider to justify riding hard at the front if other riders are unwilling to contribute, since they face all of the cost of riding hard, but the benefit (in terms of a better chance of winning the race) goes to the other riders (the free riders).

Ordinarily, the provision of public goods breaks down. They cannot be privately provided, because of the free rider problem. In this case though, cycling has developed norms that ensure some cooperation within the breakaway group. The riders tend to take turns at the front of the breakaway group, helping to increase the chances of success. However, the closer the race gets to the finish, the greater the incentives to free ride become. Regular cycling fans will no doubt remember many instances where a breakaway group has been caught, within sight of the finish line, because they failed to work together.

Another way of thinking about the incentives within a breakaway group is to use game theory. To make the problem simpler, let's say that the breakaway group only consists of two riders, and there are two strategies: (1) to ride hard; or (2) to hold back. We'll assume each rider makes their decision just once, and they make their decisions at the same time (a simultaneous game).  The payoffs for this scenario are shown in the table below. If both riders ride hard, they have a 50% chance of winning the race (since they will both be equally tired). If one rider rides hard and the other holds back, the rider that holds back wins the race for sure. If both riders hold back, then they are caught by the peloton, and neither of them wins (and they don't even finish in the top two in the race). What will happen?

To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:

1. If Rider B chooses to ride hard, Rider A's best response is to hold back (since winning for sure is better than a 50/50 chance of winning) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If Rider B chooses to hold back, Rider A's best response is to ride hard (since losing and finishing in the top two is better being caught by the peloton and finishing much lower in the order);
3. If Rider A chooses to ride hard, Rider B's best response is to hold back (since winning for sure is better than a 50/50 chance of winning); and
4. If Rider A chooses to hold back, Rider B's best response is to ride hard (since losing and finishing in the top two is better being caught by the peloton and finishing much lower in the order).

In this scenario, there are no dominant strategies. Neither rider has a strategy that is always better for them, no matter what the other rider chooses to do. However, there are two Nash equilibriums (outcomes where both players are playing their best response), which occur when one rider rides hard, and the other holds back. Neither rider will want to be the rider that rides hard, so both may be holding out hoping that the other rider will ride hard. This is the free rider problem described earlier. This game is an example of the chicken game (which I have discussed here). If both riders hold back, hoping that the other rider will ride hard, both riders will be caught by the peloton.

The chicken game is an example of a coordination game. To end up at one of the equilibriums (or another), the players need to coordinate their actions. However, in this case neither rider really wants to coordinate on the other rider's preferred equilibrium. Both really want to hold back, especially closer to the finish line, which is why the breakaway can often be caught.

Riders are motivated by the chance to win the race. That is why breakaway groups form in the first place. However, the incentives outlined above work against the breakaway succeeding. And riders are aware of these issues. One thing that often happens is that, towards the end of a race, one rider will ride especially hard, breaking away from the breakaway group. There is no free rider problem when a rider is riding by themselves. Sadly, solo breakaways are seldom successful (except on mountain stages), because the effort required to remain clear from a group of breakaway riders who suddenly become more motivated to work together and catch the solo breakaway rider is very high. The solo breakaway rider is often caught, after which the chicken game and free riding begins again.

One thing that can increase the success of a breakaway is to have multiple teammates in the breakaway group. Teammates are more likely (but not certain) to be able to coordinate their strategies, and work together, reducing the free riding problem. That's why riders in the peloton are more vigilant and energetic in chasing down an early breakaway group that has multiple riders from the same team. Most of the time, a breakaway group will only go clear if every rider in the group is from a different team. Riders in the peloton don't want the breakaway to succeed, and having all breakaway riders from different teams decreases the chance that a rider from the breakaway wins the race.

There is a lot of strategy in sports, and cycling is no exception. There are also a lot of choices for athletes to make, and choices involves trade-offs. That, along with the transparent rules and the obvious goals of the athletes involved (they want to win), is why sports can provide a lot of useful illustrations of economic concepts.