Difference between pure and mixed strategy

Question

In game theory, what is the difference between pure and mixed strategy? I arrived to that mixed strategy related with probability of the other agent do the action. I'm not sure that the result is correct. and even this result is correct, that is not enough.

Answer 1

A player is considered to have a number of options and that the game will be played multiple times. A pure strategy says the player will always select the same option. A mixed strategy will choose different options different times, usually based on a random selection with some probabilities. For example, in rock-paper-scissors, a pure strategy would be to always play rock. A mixed strategy would be to play rock $1/3$ of the time, paper $1/2$ of the time, and scissors $1/6$ of the time based on rolling a die. Some games make a pure strategy better than any mixed strategy, some make a mixed strategy optimal. For rock-paper-scissors you should play all options equally often. I don't understand anything after your question mark.

Answer 2

The set of strategies available to a player is a full list of the choices he can make.

There are two major conventions to interpret this definition. The first one, following natural language, is that there is a set of choices that the player can make. These "native" options are called pure strategies. F.i., in a Prisoners' dilemma, you can stay mum (cooperate) or incriminate your partner (defect).

A more extensive interpretation is that a player can inject randomness in his choice, by selecting a randomising device to pick one of his pure strategies. F.i., he may decide to toss a coin and then cooperate/defect based on getting heads/tails. The options to randomise over pure strategies are called mixed strategies. The set of pure strategies can be associated with the subset of mixed strategies that pick a specific pure strategy with probability one.

The mathematical motivation for mixed strategies is that they are necessary to prove existence of a value in zero-sum two-player finite games or existence of equilibrium in finite games. An intuitive motivation is that sometimes your best option is making yourself unpredictable: f.i., think of playing hide-and-seek --- if you choose your hiding spot randomly, you give the seeker no clue where to look for.