The Game Theory of 21

On Sunday a great basketball rivalry was renewed. It was not the Kings and Timberwolves fighting to stay out of last place, it was Andy, Jimmy and I playing 21 (a basketball game). In general when we play I win probably two thirds of the time, Andy will win one third, and Jimmy will never win (Burn!*). Recently though, Jimmy has started winning, highlighted by a game Sunday when Jimmy had 19, I came back to 17, and then Andy let Jimmy get a free layup for 21! With two victories in the last nine games (and zero for Andy) it appears that my secret has been uncovered: don't guard Jimmy.

Turns out, this is a classic Prisoner's dilemma. In all situations it is better for both Andy and me to not guard Jimmy (example odds of winning in each case are illustrated in the table). As a result though, both of us have significantly lowered our odds of winning the game.

This sort of idea comes up often in life, and a fair bit in recreational sports too. The same phenomenon applies to capture the flag: Offense is fun, but defense wins, so both teams play defense and have no fun, but still each have 1/2 chance of winning.

Could this phenomenon happen in national sports leagues? Absolutely. Off the top of my head here are a couple examples.

Helmet hits in NFL: Knocking the other team's star player out helps you win, but both teams are better off if there are no helmet to helmet hits.

College Recruiting Violations: Paying to get the best players can help you win (SMU), but if everybody does it then everybody loses.

Steroids in Baseball: Again, it's an advantage for you if you're the only one on the juice, but once everyone does it then the playing field is level again and your health sucks.

How do you get out of a Prisoner's Dilemma? Put a negative incentive on the negative behavior, including fines, sanctions, and suspensions. The leagues are all actually controlling for it fairly well. That doesn't mean I'll guard Jimmy though.

*Jimmy actually played really well when we played today. Somewhere over the past couple years he has learned to shoot and take advantage of his athleticism on the boards. He should probably get better than 0% odds on winning. That wouldn't amuse me as much though.