Apparently, I'm not understanding this simple concept. What are the differences between the two? Can a person have multiple pure strategies that change throughout the game?

  • $\begingroup$ When the definition of the game allows the action being drawn to a distribution both should be equal. In general when the strategy set is finite there is a difference. $\endgroup$ – Listing Nov 7 '13 at 18:51
  • $\begingroup$ "When the definition of the game allows the action being drawn to a distribution both should be equal." Is this true even when the strategy set is finite? $\endgroup$ – larry Nov 7 '13 at 19:23
  • $\begingroup$ The strategy set can't be finite in that case because the amount of strategy distributions is uncountably infinite (I assume there is more than 1 strategy). $\endgroup$ – Listing Nov 7 '13 at 19:25
  • $\begingroup$ I misread something. I removed the stuff about distributions. $\endgroup$ – larry Nov 7 '13 at 19:29

A pure strategy determines all your moves during the game (and should therefore specify your moves for all possible other players' moves).

A mixed strategy is a probability distribution over all possible pure strategies (some of which may get zero weight). After a player has determined a mixed strategy at the beginning of the game, using a randomising device, that player may pick one of those pure strategies and then stick to it.

Also see Wikipedia.

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I want to add an observation that might make mixed strategies more intuitive. This observation is based on [1].

If you are player $1$ and the other players are $\{2,3,4, \dots, n\}$; and the other players are playing strategies $p_2, p_3, \dots$ .

Suppose that you have a choice between strategies $p_1$ or $p_1 \prime,$ such that each one of these actions give you an equal (average) utility, and they are both "Best responses" to your oppoent's strategies. How do you choose between them? One approach is to not choose either, but to choose a randomized mixture of both of them: whereby you select $x \in [0,1]$ : with probability $x$ you play $p_1$, else you play $p_1 \prime$.

By randomly choosing between your two actions, you can confuse your opponents, as they won't know which you'll do. This confusion might give you an advantage.

In these cases, a mixed strategy can be rational.

Of course, other players will also mix their strategies simultaneously.

Note that if all other players are playing pure ($ie$ non-random) strategies, then you should too. This is because if you "know" how other players play, then you should just play your exact optimal response, there is no advantage to randomizing in this case.

[1] http://oyc.yale.edu/sites/default/files/mixed_strategies_handout_0.pdf

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  • $\begingroup$ Actually, informally speaking what [1] states is that if the other players have fixed their strategies, even if they play mixed strategies, then you don't need to mix. $\endgroup$ – Sasha Sep 18 '18 at 9:09

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