Mixed Strategies (Game Theory)
Pure vs Mixed Strategy
A Pure Strategy is when a player picks one specific move and sticks to it.
For example, in the Prisoner's Dilemma game, the smartest move is a pure strategy: both players usually choose to "tell" on each other every single time.
But there are games where a mix of actions is best, and that's called a Mixed Strategy.
Mixed Strategy
A Mixed Strategy lets a player be unpredictable and can increase their success.
Example: Goal!
It's a penalty shootout, and you are about to kick. Which way should you kick?
If you always kick right, the goalie will learn this and always dive right. Same problem if you always kick left.
So is 50-50 best? Maybe flip a coin each time?
But a player is often better at one side. Let's say you typically score 40% of the time when kicking left, and 70% when kicking right:
(Explanation: if both go left, there's no score. Same for both going right. But, for example, for "Kick Right, Block Left" you score 0.7 of the time.)
Now let's have some fun and try to outsmart the Goalie.
- If we always kick left, the goalie blocks left: result 0
- If we always kick right, the goalie blocks right: result 0
- How about we kick 50% randomly? Then they are best always blocking right because that's our best side: result is half the time we get 0.4, so 0.2
- So if we think they will always block right we could add in a few extra lefts as they won't be there. How about 60% left: result 0.6 x 0.4 = 0.24
- Maybe we could go 70% left: result is 0.7 x 0.4 = 0.28, but they will wise up and start always blocking left, so instead we get 0.3 x 0.7 = 0.21
OK, instead of guessing, let's calculate it exactly. This illustrates it (check the points above and see if it makes sense):
Where k stands for how often we kick left (our bad side):
To find where the 0.4k line meets the 0.7(1–k) line set them equal:
0.4k = 0.7(1–k)
0.4k = 0.7 – 0.7k
1.1k = 0.7
k = 0.7/1.1 = 0.636...
Result is 0.636 x 0.4 = 0.254 (or 0.364 x 0.7 = 0.255)
So this kicker's best strategy is to kick left randomly 64% of the time
Cool, hey?
In fact a study (https://www.jstor.org/stable/3083302) found that professional players do randomize, and that:
- kickers kick to their best side only 45% of the time (55% to their bad side)
- goalies go to that side 57% of the time
What about the Goalie? Take their point of view and see what you come up with.
When all players are using their best mixed strategies at the same time, the result is called a Mixed Strategy Nash Equilibrium.