## 6 Comparison of Two Chatbots: The \"Paradox\"  
The goal of this section was to uncover a counterintuitive property of the general AI chatbot information extraction problem. When using optimal search policy, an AI based chatbot must find and extract target information in a multi-round iterative process. We supposed that since the search chatbot was unreliable, the human user was never satisfied with the chatbot's first answer and asked the latter to improve it with a refined query; in other words, a successful round was always preceded by at least one unsuccessful round, i.e., we assumed that the parameter \(u(i)\) exceeded \(1\) for any \(i\). Suppose we have two search chatbots with the same time/efficiency characteristics and the same discount but with different reliabilities. Let us denote chatbots 1 and 2 and assume that both are sufficiently reliable in the sense that their overlook probabilities are quite small: \(0<a_{i}^{s}(s)<0.5\), \(s=1\), 2, and in all locations, \(i=1\), \(\ldots\), \(N\). For definiteness, let us assume that the first chatbot is more reliable, that is, \(0<a_{i}^{(1)}<a_{i}^{(2)}\), \(i=1\), \(\ldots\), \(N\). Both chatbots operate in the same environment, but due to different reliabilities, their optimal search policies may be different. Let \(\mathbf{\pi}^{\star(1)}\) be the optimal policy for chatbot \(1\) and \(\mathbf{\pi}^{\star(2)}\) be the optimal policy for chatbot \(2\). Accordingly, let \(F_{1}(\mathbf{\pi}^{\star(1)})\) and \(F_{2}(\mathbf{\pi}^{\star(2)})\) denote the satisfaction functions obtained by the first and second chatbot, respectively. The main theoretical result of this work is as follows:  
Proposition 2: _If the chatbots are sufficiently reliable, that is, \(0<a_{i}^{(1)}<a_{i}^{(2)}\), \(i=1\), \(\ldots\), \(N\), then, when using the optimal search policy, a less reliable chatbot is more efficient, in the sense that its search efficiency is greater than the efficiency of a more reliable chatbot, i.e., \(F_{1}(\mathbf{\pi}^{\star(1)})<F_{2}(\mathbf{\pi}^{\star(2)})\)._  
To prove Proposition 2, we first need to prove the following auxiliary claim:  
Proposition 3: _Let the function f(x) be \(x^{k-1}\)(\(1-x\)), where \(k\) is a positive integer greater than \(1\). Then, f(x) is increasing in x when \(0<x<0.5\)._  
Proof: We have: \(f(x)=x^{k-1}(1-x)=x^{k-1}-x^{k}\), \(k>1\). Then, \(f^{\prime}(x)=(x^{k-1}-x^{k})^{\prime}=x^{k-2}(k-1-kx)\). Since \(x>0\), \(f^{\prime}(x)>0\) iff \(k-1-kx>0\), or, equivalently, \(f^{\prime}(x)>0\) iff \(x<(k-1)/k\). Note that \((k-1)/k\geq 0.5\) for any integer \(k>1\). Then, for \(k>1\), we have: \(0<x<0.5\leq(k-1)/k\). Therefore, \(f^{\prime}(x)>0\), and, hence, \(f(x)\) is increasing. \(\square\)  
Now, we return to the proof of Proposition 2. Let us denote the optimal expected discounted satisfactions for chatbots 1 and 2 by \(F_{1}(\mathbf{\pi}^{\star(1)})=E(R_{disc\cdot 1}(\mathbf{\pi}^{\star(1)}))\) and \(F_{2}(\mathbf{\pi}^{\star(2)})=E(R_{disc\cdot 2}(\mathbf{\pi}^{\star(2)}))\), respectively. Since the reliabilities of the two chatbots are different, the optimal search strategies of these smart robots, denoted as \(\mathbf{\pi}^{\star(1)}=(\mathbf{\pi}_{1}^{\star(1)},\mathbf{\pi}_{2}^{\star(1)},\ldots,\ \mathbf{\pi}_{m}^{\star(1)},\ldots)\) and \(\mathbf{\pi}^{\star(2)}=(\mathbf{\pi}_{1}^{\star(2)},\mathbf{\pi}_{2}^{\star(2)},\ldots,\ \mathbf{\pi}_{m}^{\star(2)},\ldots)\), respectively, in the general case can be different, despite all other characteristics of them being pairwise identical. The optimal values of the corresponding expected discounted satisfaction, \(F_{1}(\mathbf{\pi}^{\star(1)})\) and \(F_{2}(\mathbf{\pi}^{\star(2)})\), can also be different.