## Saturday, August 25, 2012

### How to Win Games Part 4: Counter Prudential Strategies

Last week, we talked about how to counter someone who plays spitefully by using your prudential strategy. However, what if you know someone is playing prudentially?

This week, we will conclude the four game theory posts by learning how to determine your counter prudential strategy, which does exactly what it says: counters the prudential strategy.

First, you will have to figure out their prudential strategy though. Let's go back to the police-criminal game from last week.

Crime Lay Low
Patrol
3, -5
0, 1
Donuts
-2, 3
2, 0

We had determined that the police's prudential strategy is 4/7 patrol and 3/7 donuts. So, what do the criminals do?

Well, this isn't their mixed-strategy equilibrium. Remember that the mixed-strategy equilibrium is a way to make the other player's payoffs equal so that they don't have an advantage by playing one way or another.

Since this is the prudential strategy, and therefore not the mixed-strategy equilibrium, the criminals do have an advantage by playing one way or another. So, let's determine the expected payoff of playing either strategy.

For committing crime, they will get -5 4/7 of the time and 3 3/7 of the time. So, we do:

-5(4/7) +3(3/7) = -11/7

For laying low, we do the same thing:

1(4/7) + 0(3/7) = 4/7

Since 4/7 is greater than -11/7, the criminals should lay low every single time for their counter prudential strategy.

Though this type of thing isn't a proof or pattern like I normally post about, I find it really cool that you can analyze games just like you analyze math. If you have a lot more time, you can analyze games like chess, poker, black jack, and even sports.

Answer: Here is the answer to July's problem of the week. Make sure you do last week's as well!

Easy:
e = 24
h = 30
m = 12
n = 18
A = 81

Hard:
t = 45
z = 4.5
s = 50
a = 1
b = -1250
c = 390625
x = 625
P = 50π

For the rope problem from the hard problem of Monday, you must first set both ends of one rope on fire and set only one end of the other rope. Half an hour later, your first rope has burned completely leaving your second rope with 30 minutes left. Now, set the other end, and put your plant over the flame for its fifteen minute cooking.