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August 6th, 2017, 04:45 AM  #1 
Member Joined: Jul 2017 From: europe Posts: 51 Thanks: 0  Paper, scissor, rock
Let's consider a game of paper, scissor, rock. We have two players: player A and player B. Player A always do paper; player B always uses a mixed strategy, where he tosses a die, and if it comes out 1 or 2 he goes rock, if it comes out 3 or 4 he goes paper, and if it comes out 5 or 6 he goes scissor. The game is repeated many, many, many, many times. Who is most likely to get more wins? If the question is not concrete enough and can't be answered: let's say the game is played 100 times, who is most likely to get more wins? P.S.: we are considering only wins, so lets ignore the draws in the context of my question. 
August 6th, 2017, 05:43 AM  #2 
Senior Member Joined: Oct 2009 Posts: 428 Thanks: 144 
Both are expected to get equal wins.

August 6th, 2017, 08:17 AM  #3 
Member Joined: Jul 2017 From: europe Posts: 51 Thanks: 0  So, if we are considering one single "shot" of paper, scissor, rock.. mixed strategy is not better than pure strategy? Mixed strategy is better than pure strategy only on the long run... i.e. if we are considering a series of plays. Last edited by DesertFox; August 6th, 2017 at 08:49 AM. 
August 6th, 2017, 08:55 AM  #4 
Senior Member Joined: Oct 2009 Posts: 428 Thanks: 144  Well, these are notions from statistics, and they only show up in the long run anyway. But yes, you are correct. However, in other similar games, one might as well be better than some other. It depends on the given implementation of the game.

August 6th, 2017, 09:28 AM  #5 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
The notion of long run or short run is irrelevant here. Besides the long run or short run may be subjective. What matters is whether your opponent has identified your strategy or not. For example, even after a "long run" series of play, if your opponent absolutely have no clue about the pure strategy you are sticking with, you can be better off still playing that strategy. In the other case, after three plays (just to make it look like there s a trend) if your opponent is aware that you r adopting a pure strategy, you would be better off switching to a mixed strategy. Here you are in the very short run 
August 6th, 2017, 09:28 AM  #6  
Member Joined: Jul 2017 From: europe Posts: 51 Thanks: 0  Quote:
Once I think everything is clear... the next moment I am clouded with new problem. Could you give me an example of mixed strategy in tic, tac, toe? I know mixed strategies doesn't work in that game... but just give me an example.  
August 6th, 2017, 09:57 AM  #7  
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Quote:
 

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