Rugby, dementia, and the prisoners' dilemma

The prisoners' dilemma is probably the most famous game in game theory (I previously discussed it here). The outcome of the game is that both players, acting in their own self-interest, choose actions (strategies) that result in both of them being worse off. In the classic prisoners' dilemma, both prisoners confess to the crime, and both go to jail for a longer time than if they had both stayed silent.

There are many applications of the prisoners' dilemma in real life. Consider exiting a building in the event of a fire. Everyone would be more likely to get out safely if everyone walked to the fire exits in an orderly fashion. However, every person has an incentive to get to the exit as fast as possible to make sure they get out before being engulfed in flames. So, the end result is lots of running, with everyone competing for the exit, and a greater likelihood of people being hurt in the ensuing chaos.

Which brings me to dementia and rugby. The impact of repeated concussions on former rugby players has been in the news quite a bit this year (and follows similar news about NFL players over the last few years). See for example this front page story from the New Zealand Herald:

The first paper from the study into the health of retired rugby players was published in online journal Sports Medicine yesterday. "A Comparison of Cognitive Function in Former Rugby Union Players Compared with Former Non-Contact-Sport Players and the Impact of Concussion History" investigated the difference in brain function between rugby players who experienced concussion and those who didn't.

It found that players who experienced one or more concussions in their career performed worse in tests that measure cognitive flexibility, complex attention, executive function and processing speed. To put it in layman's terms, that is the ability to understand and process information quickly, to make rapid decisions, to switch attention between tasks and to track and respond to information over long periods of time.

How does this relate to the prisoners' dilemma? Consider two rugby players (Player One and Player Two), who have two strategies: (1) to play hard and risk concussion; and (2) to play softer and avoid head injuries. Both players would be better off if they both played softer, since both would avoid head injuries. However, knowing that the other player is playing softer, each player would be better off playing hard since that gives them a better chance of winning. However, if either player plays hard, then both players risk a head injury. The game (in normal form) is laid out in the payoff table below.

What happens in this game? It actually depends on how players evaluate the trade-offs in this game. Let's start by assuming that winning is paramount.

Consider Player One first. They have a dominant strategy to play hard. A dominant strategy is a strategy that is always better for a player, no matter what the other players do. Playing hard is a dominant strategy because the payoff is always better than playing softer. If Player Two plays hard, Player One is better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player Two plays softer, Player One is better off playing hard (because being unsafe with a better chance of winning is better than being safe, because winning is paramount). So Player One would always choose to play hard, because playing hard is a dominant strategy.

Player Two faces the same decisions. They also have a dominant strategy to play hard. If Player One plays hard, Player Two is better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player One plays softer, Player Two is better off playing hard (because being unsafe with a better chance of winning is better than being safe, because winning is paramount). So Player Two would always choose to play hard, because playing hard is a dominant strategy.

Both players will choose their dominant strategy (to play hard), and both will be unsafe. But what if being safe was more important than winning? That changes things considerably. Now, playing hard is no longer a dominant strategy.

Go back to Player One. If Player Two plays hard, Player One is still better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player Two plays softer, Player One is now better off also playing softer (because being safe is better than being unsafe with a better chance of winning, since safety is now paramount). Note that neither strategy is always better for Player One - Player One has no dominant strategy.

Now consider Player Two. If Player One plays hard, Player Two is still better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player One plays softer, Player Two is now better off also playing softer (because being safe is better than being unsafe with a better chance of winning, since safety is now paramount). Note that neither strategy is always better for Player Two - Player Two also has no dominant strategy.

What is the solution to this game? We can find it using the best response method (which we've already described in the last two paragraphs). Any combination of strategies where both players are choosing their best response to the other player's strategy is a Nash equilibrium (named after John Nash). This occurs where both players play hard, and where both players play softer. We have a coordination game (a game with multiple equilibriums).

In a coordination game, if one of the equilibriums is clearly better than the other, we call that a Schelling Point (named after Thomas Schelling), and that equilibrium is the more likely equilibrium to obtain. In this case, both players playing softer is a Schelling Point. So, we could expect that both players would play softer (given our assumption that safety is paramount).

So what? The implication of this rough analysis is pretty clear. When rugby players view winning as paramount, then playing hard is a dominant strategy and all players will continually put their bodies (and minds) on the line in the pursuit of glory. If instead, safety becomes paramount then the outcome of the game changes, and it becomes likely that players will play softer and fewer injuries will result. If reducing rugby players' long-term concussion-related injuries is important, then the challenge for rugby authorities will be to change players' incentives to ensure that safety becomes a higher priority. Unfortunately, that is likely to be easier said than done.