equivalent mixed and behavior strategies

Definition (equivalent mixed and behavior strategies)

A mixed strategy $\sigma_i$ and behavior strategy $b_i$ of player $i$ in an extensive-form game are equivalent to each other if for every mixed/behavior strategy vector $\sigma_{-i}$ of the players $N \setminus \{i\}$ and every vertex $x$ in the game tree,

i.e. the mixed strategy $\sigma_i$ and behavior strategy $b_i$ are equivalent if for every mixed/behavior strategy vector $\sigma_{-i}$ the two strategy vectors $(\sigma_i, \sigma_{-i})$ and $(b_i, \sigma_{-i})$ induce the same probability of arriving at each vertex in the game tree, in particular for each leaf $x$.

Theorem (equivalent utility of equivalent mixed and behavior strategies)

If mixed strategy $\sigma_i$ is equivalent to behavior strategy $b_i$, then for every mixed/behavior strategy vector $\sigma_{-i}$ of the other players and every player $j \in N$,

Corollary

Let $\sigma = (\sigma_i)_{i \in N}$ be a vector of mixed strategies. For each player $i$ let $b_i$ be a behavior strategy that is equivalent to $\sigma_i$, and denote $b = (b_i)_{i \in N}$. Then for each player $i$,

Conditions for existence of equivalent mixed strategy to any behavior strategy

Definition (action)

Let $x$ be a vertex in the game tree that is not the root, and $x_1$ a vertex on the path from the root to $x$. The (unique) edge emanating from $x_1$ on the path from the root to $x$ is called the action at $x_1$ leading to $x$.

Corollary

If there exists a path from the root to some vertex x that passes at least twice through the same information set $U_i$ of player $i$, and if the action leading in the direction of $x$ is not the same action at each of these information sets, then player $i$ has a behavior strategy that has no equivalent mixed strategy.

Theorem

Let $\Gamma = (N,V,E,v_0, (V_i)_{i \in N \cup \{0\}}, (p_x)_{x \in V_0}, (\mathcal{U}_{i \in N}), O, u)$ be an extensive form game that satisfies the condition that at every vertex there are at least two actions. Every behavior strategy of player $i$ has an equivalent mixed strategy if an only if each information set of player $i$ intersects every path emanating from the root at most once.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 223, 226.