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pure & mixed strategies

Hi,

Could anyone tell me, in a 2 payoff matrix, how is it possible to have both pure and mixed strategies? If there is a pure strategy, then you choose that option no matter what. Thus, it has a probability of 1. That leaves 0 for the other option, meaning it is never chosen, hence no possibility of mixed strategies....or am I getting something totally mixed up? :s-smilie:

Reply 1

rjwilliams3

We would vary whether we would play a pure or mixed strategy depending on the situation. If we were playing a repeated game then we may adopt a mixed strategy to add some confusion.

If each player has a finite number of pure strategies then there must exist at least one mixed strategy. If there are no pure strategies then there must exist at least one mixed strategy.

If we think about a game of chicken it is possible to have a pure strategy Nash equilibria and a mixed strategy Nash equilibria. We can see from the game below (I hope it displays okay!) that the pure strategy Nash equilibria are Drive,Swerve and Swerve,Drive. Equally we can construct a mixed strategy Nash equilibria (see below), but this would only be possible if we assume that the players were able to repeat the game (presumably they're not injured too badly and have another car to use).
Drive Swerve
Drive -5,-5 5,-3
Swerve -3,5 0,0

Providing I have dont my calculations correctly, I find the MSNE to be (5/7,5/7).

In short, there are some games with no pure strategy (these will have mixed strategy), and there are games with pure strategies. If these games with pure strategies are paid repeatedly then we may wish to pursue a mixed strategy.

This might make more sense - http://www.econ.ucsb.edu/~garratt/Econ171/Lect07and08_Slides.pdf

Reply 2

Chr0n

Original post by marcsaccount

If there is a pure strategy, then you choose that option no matter what. Thus, it has a probability of 1. That leaves 0 for the other option, meaning it is never chosen.

This is exactly the simplest mixed strategy you can choose. Mixed strategy simply means you randomize across your pure strategies. I.e. you put some probabilities on the strategies.

In your case, you put probability 1 on strategy 1 and probability 0 on strategy 2, making it a mixed strategy.

Every pure strategy can be seen as a mixed strategy with putting all probability on that strategy and 0 on the others.

More generally a mixed strategy is if you put probability p on strategy 1 and probability 1-p on strategy 2 where 0 <= p <= 1.

rasclerhys

In short, there are some games with no pure strategy (these will have mixed strategy), and there are games with pure strategies. If these games with pure strategies are paid repeatedly then we may wish to pursue a mixed strategy.

Did you mean there are some games which have no pure strategy nash equilibrium? I am not quite sure how you could play a mixed strategy (i.e. put some non-negative probability on your pure strategies) without any pure strategies existing in the game? Of course, I might be wrong given that I have only taken one game theory module. Please explain if this is the case.

(edited 9 years ago)

Reply 3

rjwilliams3

Original post by Chr0n

Yes, sorry, that paragraph should be taken to refer to Nash equilibria, and I should have said "played" not "paid".

Reply 4

marcsaccount

Thanks for your help guys.

...but it's not possible to have a pure strategy with probability less than 1? So you could never have a pure strategy (with prob=1) and a mixed strategy with probability greater than 0?

Reply 5

Chr0n

Original post by marcsaccount

...but it's not possible to have a pure strategy with probability less than 1? So you could never have a pure strategy (with prob=1) and a mixed strategy with probability greater than 0?

You can have any combination of probabilities so long as they sum up to 1.
If you have two strategies, A and B, then to play a mixed strategy you would play:

A with probability p and
B with probability 1-p
where p + (1-p) = 1 and p >= 0.

So, as said above the simplest mixed strategy is to to set p = 1 (you will play A with 100% certainty) or p = 0 (you will play B with 100% certainty). However, you can also set it to anything in between. If you put p = 0.5, you play both strategies with 50% certainty. If you set p=0.3, you play A with 30% certainty and B with 70% certainty.