## 3 Materials and Methods
### An Enhanced Intelligent Decision Support System That Minimizes Counterfactual Regret  
The Counterfactual Regret Minimization algorithm, a widely utilized iterative technique in game theory and games of incomplete information, aims at calculating equilibrium strategies and approximations to the Nash equilibrium [38]. Unlike traditional regret minimization algorithms [39] that require exhaustive exploration of the entire strategy space, CFR operates by iteratively updating strategies based on the regrets associated with each action taken at different information sets. This research employs CFR as the cornerstone for an intelligent decision support system due to its efficiency in updating strategies. The core concept involves minimizing regret values within each information set to progressively converge towards optimal strategies.  
In the context of an information set \(I\), counterfactual regret for player \(i\) not choosing action \(a\) is defined as the difference between the utility of taking action \(a\) and the actual utility under strategy \(\sigma_{t}\). Mathematically, it is represented as Equation (1):  
\[R_{t}^{i}(a)=\sum_{h\in I}\pi^{-i}(h)[u^{i}(h,a)-u^{i}(h,\sigma_{t})] \tag{1}\]  
where \(\pi^{-i}(h)\) denotes the probability of reaching leaf node \(h\) by players other than \(i\), and \(u^{i}(h,a)\) represents the utility of player \(i\) choosing action \(a\) at history state \(h\). After \(T\) iterations, the cumulative counterfactual regret is given by Equation (2):  
\[R_{T}^{i}(a)=\sum_{t=1}^{T}R_{t}^{i}(a) \tag{2}\]  
Figure 2: Decision tree for the game scenario.  
Subsequently, the current strategy is updated via regret matching, as given by Equation (3):  
\[\sigma_{T}^{i}(a)=\left\{\begin{array}{ll}\frac{\max(R_{T}^{i}(a),0)}{\sum_{b \in A}max(R_{T}^{i}(b),0)}&\text{if }\sum_{b\in A}\max(R_{T}^{i}(b),0)>0\\ \frac{1}{|A(I)|}&\text{otherwise}\end{array}\right. \tag{3}\]  
where \(A(I)\) denotes the set of actions that can be taken under information set \(I\).  
Through continuous iteration, the strategy converges to the Nash equilibrium. This convergence property makes CFR particularly suitable for complex games like Kuhn poker, where the number of possible strategies can be vast. Prior to experimentation, the CFR algorithm was applied to train on Kuhn poker scenarios. Specifically, 100,000 rounds of training were conducted to ensure robustness and accuracy in approximating Nash equilibrium outcomes across various information sets.  
These outcomes provided foundational guidelines for the decision support system's recommendations. For example, in the Kuhn poker game, if we denote \"p\" as pass and \"b\" as bet, the detailed strategies for both the first and second players are shown in Tables 1 and 2, respectively. These tables illustrate how the probabilities of actions, i.e., passing or betting, vary depending on the specific information set encountered during gameplay. The iterative process of the strategies of the two players with different information sets is shown in Figure 3a,b. By comparison, the solution obtained by the CFR algorithm is closer to that obtained by the theory, which verifies the effectiveness of the algorithm.  
These strategies demonstrate the dynamic adjustment of probabilities based on the evolving game states, providing participants with informed decisions throughout the experiment.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Information Set** & **1** & **1 pb** & **2** & **2 pb** & **3** & **3 pb** \\ \hline pass & 0.73 & 1.00 & 1.00 & 0.39 & 0.18 & 0.00 \\ \hline bet & 0.27 & 0.00 & 0.00 & 0.61 & 0.82 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 1: First player strategy.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Information Set** & **1b** & **1p** & **2b** & **2p** & **3b** & **3p** \\ \hline pass & 1.00 & 0.65 & 0.66 & 1.00 & 0.00 & 0.00 \\ \hline bet & 0.00 & 0.35 & 0.34 & 0.00 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Second player strategy.