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 August 4th, 2017, 10:33 AM #1 Member   Joined: Jul 2017 From: europe Posts: 51 Thanks: 0 Game theory.. Please help... What is the difference between pure strategy and mixed strategy? I need (1) clear definitions and ( 2) a simple and illustrative example.. Information around the web is not really helpful (at least for me).
 August 4th, 2017, 12:52 PM #2 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 In game theory, there are usually several possible strategies. If you use only one that is a pure strategy. If you use different ones in some mix, they are mixed strategies. Simple example - coin toss: pure strategy, always call heads; mixed strategy, sometimes call heads and sometimes call tails. Thanks from DesertFox
 August 4th, 2017, 01:30 PM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Wikipedia has a pretty good explanation: https://en.wikipedia.org/wiki/Strategy_(game_theory) Thanks from DesertFox
 August 4th, 2017, 01:41 PM #4 Senior Member   Joined: Oct 2009 Posts: 841 Thanks: 323 Let's do a game of paper, scissor, rock. Now, I have the following strategy: always do paper. Or: alternate paper and rock. These are pure strategies. The problem is that if you are sufficiently smart, you will see through my pure strategy and make me lose, unless my strategy is sufficiently complicated. However, it is better to do a mixed strategy, where I toss a die, and if it comes out 1 or 2, I go rock, if it comes out 3 or 4, I go paper, and otherwise I go scissors. Here I let chance determine the outcome, and this is a really optimal strategy against a smart opponent. Thanks from DesertFox
August 4th, 2017, 10:50 PM   #5
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 Originally Posted by Micrm@ss Let's do a game of paper, scissor, rock. Now, I have the following strategy: always do paper. Or: alternate paper and rock. These are pure strategies. The problem is that if you are sufficiently smart, you will see through my pure strategy and make me lose, unless my strategy is sufficiently complicated. However, it is better to do a mixed strategy, where I toss a die, and if it comes out 1 or 2, I go rock, if it comes out 3 or 4, I go paper, and otherwise I go scissors. Here I let chance determine the outcome, and this is a really optimal strategy against a smart opponent.
Thank you so much! That was very clear answer to my question, even for poor mathematician like me!! Really I appreciate the efforts .....
mathman, thank you too, buddy!

Now I grasp the basic idea.
But immediately a new question arises. Let's do a game of tic, tac, toe. In this case: both of the players uses (if he is smart enough) a strategy, which guarantees at least draw (parity); both of them see through the opponent's pure strategy, but nobody can do it better. If we are smart enough, we will reach some kind of optimum: no matter how many times we play the game, there will be no winner.
I can't think of mixed strategy in this case.

Let's consider one more example of very simple game, where mixed strategy is impossible.
A guard is hired to protect two safes in separate locations: S1 contains 10 000 dollars and S2 contains 100 000 dollars. The guard can protect only one safe at a time from a safecracker. The safecracker and the guard must decide in advance, without knowing what the other party will do, which safe to try to rob and which safe to protect. When they go to the same safe, the safecracker gets nothing; when they go to different safes, the safecracker gets the contents of the unprotected safe.
In this case: no matter how many times this game is re-played, the smart guard always will choose S2 and the smart safecracker will always hit S1 ... mixed strategy doesn't exist here.

As I already understand, in some games there is mixed strategy, in other games- there is not mixed strategy (this is the case, when the player has the only one "right" strategy).

So, my question goes: what is the factor, which determines whether in a specific game mixed strategy is possible or not?

Tic, tac, toe: no mixed strategies;
Paper, scissor, rock: there are mixed strategies;
Guard vs. safecracker: no mixed strategies;
Coin toss: there are mixed strategies.

As a layman, I have the following conclusion: a player uses mixed strategies, only when there is not single "right" strategy; and when there is not single "right" strategy: it is better to use mixed strategy, because pure strategy is easier to reveal.
Am I right???

Thank you buddies,
Have a nice day full with energy!

August 5th, 2017, 03:03 AM   #6
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Quote:
 Originally Posted by Micrm@ss Let's do a game of paper, scissor, rock. Now, I have the following strategy: always do paper. Or: alternate paper and rock. These are pure strategies. The problem is that if you are sufficiently smart, you will see through my pure strategy and make me lose, unless my strategy is sufficiently complicated. However, it is better to do a mixed strategy, where I toss a die, and if it comes out 1 or 2, I go rock, if it comes out 3 or 4, I go paper, and otherwise I go scissors. Here I let chance determine the outcome, and this is a really optimal strategy against a smart opponent.
My last post in this topic is really messy! Please, don't try to answer it.

I managed to re-arrange my thoughts a little bit.
Please, give me a simple example where it is better to use pure strategy in comparison to mixed strategy?

I really can't think of such an example..
It seems that mixed strategy is always better than pure strategy...

August 5th, 2017, 04:54 AM   #7
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Quote:
 Originally Posted by Micrm@ss Let's do a game of paper, scissor, rock. Now, I have the following strategy: always do paper. Or: alternate paper and rock. These are pure strategies. The problem is that if you are sufficiently smart, you will see through my pure strategy and make me lose, unless my strategy is sufficiently complicated. However, it is better to do a mixed strategy, where I toss a die, and if it comes out 1 or 2, I go rock, if it comes out 3 or 4, I go paper, and otherwise I go scissors. Here I let chance determine the outcome, and this is a really optimal strategy against a smart opponent.
Let's consider coin toss. In this case, I think that pure strategy and mixed strategy work equally good.

In conclusion: (1) there are games, where mixed strategy is better than pure strategy and (2) there are games, where mixed strategy and pure strategy are equally good.... but (3) there are no games, where pure strategy is better than mixed strategy.

Are my conclusions right?

 August 5th, 2017, 05:26 AM #8 Member   Joined: Jan 2017 From: California Posts: 80 Thanks: 8 am still wondering if a mixed strategy would be adequate for the guard or safe cracker since with a pure strategy the safe cracker is guaranteed 10,000 everytime. Probabilities must come to play . If we assume equal probabilities then the expected loss/gain is 55,000. But does it still warrant adopting a pure strategy Last edited by dthiaw; August 5th, 2017 at 05:44 AM.
August 5th, 2017, 08:30 AM   #9
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 Originally Posted by dthiaw am still wondering if a mixed strategy would be adequate for the guard or safe cracker since with a pure strategy the safe cracker is guaranteed 10,000 everytime. Probabilities must come to play . If we assume equal probabilities then the expected loss/gain is 55,000. But does it still warrant adopting a pure strategy
I think what you are missing is the basic definitions for pure strategy and mixed strategy. I am not competent enough, that's why I created this thread. Now I think I have better understanding....

Here are my thoughts:
There are two kinds of mixed strategy: 1) when you let chance to determine the outcome (Micrm@ss gave very clear example) 2) when you take into account the opponent's strategy... in other guess, when you guess his move.

Pure strategy doesn't care about these two conditions.... that's why it is called PURE, you don't mix it with other factors, you don't take into account nothing more.... it's your PURE strategy....

In the case of the guard and the safecracker: if you are the safecracker and if you guess that the guard would prefer to save the bigger sum... you're already using MIXED strategy. Or you can toss a coin: for example, head- for the safe with 10 000 dollars, and tail- for the safe with 100 000 dollars. This is mixed strategy again... but in this case it will be really unadequate. And finally, you use pure strategy if you go only for the first safe for example.... you go for the first safe without taking account of other factors, it's just your pure strategy......

I hope my comment is not misleading....
Let's wait for the mathematicians to say more about the topic...

Wish all the best and thank you for the replies....!!!

August 5th, 2017, 02:49 PM   #10
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ok if you say so....

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