Game Theory Mixed Strategy
Pure vs Mixed Strategy
The Prisoner's Dilemma is an example of a Pure Strategy, where a specific course of action can be taken by a player:
| Dana | |||
| Stay Quiet | Blame Casey | ||
| Casey | Stay Quiet | -1, -1 | -10, 0 |
| Blame Dana | 0, -10 | -6, -6 | |
In this case the result is for them to blame each other (-6, -6).
But there are games where a mix of actions is best, and that is called a Mixed Strategy.
Mixed Strategy
A Mixed Strategy let's a player be unpredictable and can increase their success.
Example: Goal!
It is a shootout and you get to kick for your team. Which way should you kick?
If you always kick right the Goalie can always go right and block you.
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:
| Goalie | |||
| Block Left | Block Right | ||
| Kicker | Kick Left | 0, 0 | +0.4, –0.4 |
| Kick Right | +0.7, –0.7 | 0, 0 | |
(Explanation: if both go left, there is 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 they will always block left: result 0
- If we always kick right they will always block right: result 0
- How about we kick 50% randomly? Then they are best always blocking right because that is 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.