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A pure strategy provides a complete definition for a way a player can play a game. In particular, it defines, for every possible choice a player might have to make, which option the player picks. A player's strategy space is the set of pure strategies available to that player.
A mixed strategy is an assignment of a probability to each pure strategy. It defines a probability over the strategies, and reflect that, rather than choosing a particular pure strategy, the player will randomly select a pure strategy based on the distribution given by their mixed strategy. Of course, every pure strategy is a mixed strategy (the function which takes that strategy to  and every other one to 0).
The following notation is often used:
 for the strategy space of the  -th player for a particular element of  ; that is, a particular pure strategy for a mixed strategy. Note that
![$ \sigma_i\in S_i\rightarrow [0,1]$](http://images.planetmath.org:8080/cache/objects/3204/l2h/img7.png "$ \sigma_i\in S_i\rightarrow [0,1]$") and
 . for the set of all possible mixed strategies for the  -th player for
 , the set of all possible combinations of pure strategies (essentially the possible outcomes of the game) for
  for a strategy profile, a single element of   for
 and
 for
 , the sets of possible pure and mixed strategies for all players other than  . for an element of  and
 for an element of
 .
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"strategy" is owned by Henry.
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(view preamble)
See Also: game
| Also defines: |
strategy, pure strategy, mixed strategy, strategy space |
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Cross-references: outcomes, function, distribution, reflect, game, player
There are 36 references to this entry.
This is version 4 of strategy, born on 2002-07-24, modified 2002-07-28.
Object id is 3204, canonical name is Strategy2.
Accessed 11614 times total.
Classification:
| AMS MSC: | 91A99 (Game theory, economics, social and behavioral sciences :: Game theory :: Miscellaneous) |
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Pending Errata and Addenda
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