In this video there is a game with perfect recall, see the snippet below. They claim that they have an example of a mixed strategy that is not a behavior strategy. However the following holds and is stated in the book by Maschler: Game Theory:
Theorem 6.15 (Kuhn [1957]) In every game in extensive form, if player i has perfect
recall, then for every mixed strategy of player i there exists an equivalent behavior strategy.
This seems a contradiction to me.
EDIT What is the equivalent behavior strategy for this mixed one: $$(.6(A,G),.4(B,H))$$
The answer will shed some light on my OQ.
EDIT2 I've tried to solve this: $$p\cdot s=0.6$$ and $$(1-p)\cdot (1-s)=0.4$$
but this system has only imaginary solutions.
Here, $p$ is probability taking action $A$ and $s$ taking $G$.
I guess the confusion here arises because of the meaning of "equivalent strategies". Two strategies of Player 1 (they can be mixed strategies, behavior strategies, or one mixed and the other behavior) are equivalent if for every strategy of the other player (be it mixed or behavior), the two pairs induce the same probability distribution over the leaves of the tree. For example, when Player 1 selects A in her first decision node, the play cannot reach her second decision node, and hence the pure strategies (A,G) and (A,H) of Player 1 are equivalent under this definition.
Now, Kuhn's Theorem states that in games with perfect recall every mixed strategy has an equivalent behavior strategy, and vice versa. Since the game discussed here has perfect recall, Kuhn's Theorem holds in this case. The mixed strategy ( 0.6(A,G), 0.4(B,H) ) selects with probability 0.6 the pure strategy (A,G), and with probability 0.4 the pure strategy (B,H). The behavior strategy that is equivalent to it is: in the first decision node, select the mixed action (0.6 A, 0.4 B), and in the second decision node, select H with probability 1.
Note that mixed strategies and behavior strategies are mathematically different objects. A mixed strategy is a probability distribution over pure strategies (where a pure strategy is a function from the set of all information sets of the player to actions), while a behavior strategy is a function from the set of all information sets of the player to mixed actions. So a mixed strategy cannot be a behavior strategy - they lie in different spaces. Yet, as Kuhn's Theorem states, there is a natural equivalence between these two classes of strategies.