By playing the way we learned last week, we are playing less to make ourselves win and more to make the other player lose. We make their payoff at the bare minimum of what we can get away with so we have a better chance of winning. This is called playing spitefully.
What if you think the other player is playing spitefully? What do you do?
You could play spitefully as well. If you use the same logic, you can actually determine who will win the game. You determine both expected payoffs, and in some cases, you will be guaranteeing yourself a loss by playing spitefully.
Rather than playing spitefully back, you can use your prudential strategy, which is basically your best counter for spiteful play. And you solve for it almost the same way as solving for mixed strategy equilibria: what we learned last week.
Let's go back to the police and criminal game from last week.
Let's say you are the police, and you know that the criminal is playing spitefully. This means he is committing crime 2/7 of the time and laying low 5/7 of the time.
For the police's counter, you solve it by first realizing that the criminals are only using your payoffs to create their strategy. So, you can take that into consideration by pretending their payoffs are not what they are and instead the negative of your payoff (they want your payoff to be low, so they want the negative of yours to be high). The game then looks like this:
This is called a zero sum game because if you add the two payoffs in each box, it equals zero.
Now, we solve for our mixed strategy equilibrium in this new game.
-3x + 2(1 - x) = 2 - 5x
0x - 2(1 - x) = 2x - 2
2x - 2 = 2 - 5x
7x = 4
x = 4/7
So, the police should patrol 4/7 of the time and eat donuts 3/7 of the time, which seems to make a little more sense. They are doing their job more often, and they will catch a lot more criminals.
But how does the criminal counter that? Shouldn't they shy away from committing crime then? Next week, we will conclude my game theory posts for now, and we will learn how to counter the prudential strategy.