Playing to Win

Consider the following game: A player starts with 0 points. For 100 rounds, a player picks one of the following three actions in each round.

1. Gets 7 points with probability 1/2, 3 points with probability 1/2.
2. Gets 4 points with probability 1.
3. Gets 10 points with probability 2/5, 0 points with probability 3/5.

The player wins by having at least 500 points at the end of the game. An alternative is to have two or more players with the winner as the one who has the highest score at the end.

What is the right strategy? Initially play action 1 and towards the end possibly switch to action 2 if you are ahead and action 3 if you are behind.

Many sports have these kinds of actions to keep the game exciting even if one player has a lead. Action 2 corresponds to using a closing pitcher, or a prevent defense. Action 3 is using a pinch hitter, pulling the goalie or the "Hail Mary" pass.

Quidditch doesn't have these options rather having a final move that usually dominates the rest of the scoring. The scoring rules of Quidditch is J.K. Rowling's biggest blunder in the Harry Potter universe.

Sometimes you do see action 2 moves earlier in a game. For example in football, after a touchdown a team can either kick for an extra point or run a short play to try for two. Kicks are are rarely missed and the plays are successful slightly more than half the time. Yet most coaches just kick unless there is a significant advantage to go for two.

The choices above apply to many more arenas than just sports. Obama and Clinton have been following actions 2 and 3 respectively over the last few weeks. Which approach will work? We'll find out tomorrow.