Pure strategies

The pure strategies of a player are maps from all the information sets played by this player to actions available at these information sets. A pure strategy is like a masterplan that a player can design in advance, so that at any possible configuration of the game, the player can read off their strategy what to do next.

Pure strategies are a Nashian concept. In non-Nashian game theory, pure strategies have no meaning as a non-Nashian strategy is the reaction to a particular game outcome (Stackelberg competition against the past) and only assigns actions to those information sets that are actually active.

Let us come back to this example:

We have two players, Alice playing at A and Alfred playing at X, Y, and Z.

Alice's pure strategies are:

Strategy number	A
1	a
2	b
3	c

Alfred's pure strategies are:

Strategy number	X	Y	Z
1	0	0	0
2	0	0	1
3	0	1	0
4	0	1	1
5	1	0	0
6	1	0	1
7	1	1	0
8	1	1	1

Strategy profile

If every player chooses a pure strategy for themselves, the combination of all players' pure strategies is called a (pure) strategy profile. It assigns an action to every information set in the game.

Strategy profiles are also a Nashian concept. Non-Nashian game theory is by essence contextual, and only assigns actions to information sets that are active in the same history.

These are the strategy profiles in our example:

Alice's strategy number	Alfred's strategy number	A	X	Y	Z
1	1	a	0	0	0
1	2	a	0	0	1
1	3	a	0	1	0
1	4	a	0	1	1
1	5	a	1	0	0
1	6	a	1	0	1
1	7	a	1	1	0
1	8	a	1	1	1
2	1	b	0	0	0
2	2	b	0	0	1
2	3	b	0	1	0
2	4	b	0	1	1
2	5	b	1	0	0
2	6	b	1	0	1
2	7	b	1	1	0
2	8	b	1	1	1
3	1	c	0	0	0
3	2	c	0	0	1
3	3	c	0	1	0
3	4	c	0	1	1
3	5	c	1	0	0
3	6	c	1	0	1
3	7	c	1	1	0
3	8	c	1	1	1

Given any (pure) strategy profile, exactly one history is obtained. However, different (pure) strategy profiles can lead to the same history.

It is the same to build the pure strategies, or the (pure) strategy profiles, from the spacetime game or from any of its extensive forms.

Strategic form

Given all pure strategies available to each player in a spacetime game, or in a game in extensive form, it is possible to build an equivalent game in normal form with exactly these players and pure strategies. Thus, the pure strategies of all the player form a static "matrix" and each cell in this matrix corresponds to one history. It is however possible for several cells to correspond to the same history.

Imagine, as a though experiment, if every Chess player had decided in advance what to do for any configuration of the chessboard. Then, at a tournament, every player would bring their pure strategy and one directly reads off the matrix who wins. Of course, the gigantic size of the strategy space makes it unfeasible. And interesting fact: if this were feasible, then there is a theorem of game theory that says that either there is a winning strategy for the white player, or there is a winning strategy for the black player, or there are pat strategies for both players. Tournaments would be more boring than they are today!

Reduced strategic form

Because of the structure of the game, some pure strategies may contain superfluous information. For example, if a player decides in their strategy to play A=a, and B (played by the same player) never gets activated in any history where A=a, then it is superfluous to assign any action to B in the pure strategy as this would not change the outcome of the game no matter what opponents do.

Removing such superfluous assignments is a simplification that is called the reduced strategic form of the game.