## Game Theory and the Iditarod

The Iditarod Trail Sled Dog Race began in 1973 and consists of a 1,150 mile race through Alaska by mushers and their dog team. The trail, which starts in Anchorage and ends in Nome, had originally been used as a route for transporting supplies and was crucial during an epidemic of diphtheria in Nome, in which drugs were delivered by mushers who used the trail. The founders of the race set out to preserve the sled dog culture as well as the trail itself. Mushers must make it through several checkpoints along the trail and are required to make one 24 hour stop and two 8 hour stops.  Each musher starts out with 16 dogs that have gone through extensive training to run over long distances.

In the 2008 race, Lance Mackey, who won the race the previous year, and Jeff King, a four time champion, were the two leaders up until the end. Both stopped in Elim, a few checkpoints away from Nome. King had expected Mackey to rest in Elim for a bit, so King did as well. However, after King closed his eyes, Mackey took off, gaining the lead and he was able to hold it until the end.

In the 2008 race, Lance Mackey, who won the race the previous year, and Jeff King, a four time champion, were the two leaders up until the end. Both stopped in Elim, a few checkpoints away from Nome. King had expected Mackey to rest in Elim for a bit, so King did as well. However, after King closed his eyes, Mackey took off, gaining the lead and he was able to hold it until the end.

This situation can be simplified and modeled by game theory. Two mushers happen to be at the same checkpoint. They have two choices: they can leave now or rest for bit. These mushers have gone without rest for a while so they would really like to take a nap. We can look at the payoffs for what they can do once they get to the checkpoint. Since they would get a larger payoff from resting, the Nash equilibrium for this set up would be if they both rest now.

 A/B Leave Now Rest Leave Now A:   1     B:   1 A:   1   B: 3 Rest A:   3     B:   1 A:   3   B: 3

However, this matrix doesn’t take into account whether they would win depending on their decision. It only takes into account the fact they have gone through an exhausting journey. King must’ve seen this matrix and believed that the only thing that matters at that point was to get some rest. However, Mackey, who had lost a couple dogs and believed that his team was slower than King’s, saw the following matrix.

 A/B Leave Now Rest Leave Now A:   0    B:   3 A:   3   B: 2 Rest A:   2    B:   3 A:   2   B: 5

One was subtracted from the payoff of the player who lost in each scenario and two was added to the payoff of the player that won. If both had left at the same time, Mackey would have lost. The only scenario in which he could win would be if he had left before King, which is exactly what he did.

In this game, the two players don’t know each other’s payoffs and so these matrices are made up based on what the each player think the other player’s payoffs are. In the games we’ve looked at, both players know the playoffs of the other players but in cases such as these, when different things are taken into consideration for each individual player, the matrices become different and players can’t predict what the other person will do. This is exactly why King couldn’t predict that Mackey would leave.

References:

http://iditarod.com/