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Extensive Game and Basics for CF

Extensive Game and Basics for CF

作者: LoveMoM | 来源:发表于2020-11-17 12:25 被阅读0次

Important Definitions

u

​ function mapping each action profile to a vector of utilities for each player

\sigma^t_i

​ a strategy profile which mapping information set I_i and actions' probabilities for player i

\pi^{\sigma}(h)

​ reach probability of game history h with strategy profile \sigma

\pi^{\sigma}(I)=\sum_{h\in I}\pi^{\sigma}(h)

​ probability of reaching information set I through all possible game histories in I

After all things above, we can get these:

We can define the counterfactual\ value at nonterminal history h as:
v_i(\sigma,h)=\sum_{z\in Z,h\sqsubset z}\pi^\sigma_{-i}(h)\pi^\sigma(h,z)u_i(z)
The counterfactual\ regret of not taking action a at history h is defined as:
r(h,a)=v_i(\sigma_{I\rightarrow a,h})-v_i(\sigma,h)
The counterfactual\ regret of not taking action a at information set I is then:
r(I,a)=\sum_{h\in I}r(h,a)
Let r_i^t(I,a) refer to the regret whe players use \sigma^t of not taking action a at information set I belonging to player i. The cumulative\ counterfactual\ regret is defined as:
R_i^t(I,a)=\sum_{t=1}^Tr_i^t(I,a)
Then we can use regret matching to get new strategy:
\sigma^{T+1}_i(I,a)=\left\{ \begin{array}{**lr**} \frac{R_i^{T,+}(I,a)}{\sum_{a\in A(I)}R_i^{T,+}(I,a)}\ if\ \sum_{a\in A(I)}R_i^{T,+}(I,a)>0,\\ \frac{1}{|A(I)|}\ otherwise \end{array} \right.

Derivatives

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