On Performance Gauge of Average Multi-Cue Multi-Choice Decision Making: A Converse Lyapunov Approach Mehdi Firouznia and Qing Hui, Member, IEEE Abstract—Motivated by the converse Lyapunov technique for investigating converse results of semistable switched systems in control theory, this paper utilizes a constructive induction method to identify a cost function for performance gauge of an average, multi-cue multi-choice (MCMC), cognitive decision making model over a switching time interval. It shows that such a constructive cost function can be evaluated through an abstract energy called Lyapunov function at initial conditions. Hence, the performance gauge problem for the average MCMC model becomes the issue of finding such a Lyapunov function, leading to a possible way for designing corresponding computational algorithms via iterative methods such as adaptive dynamic programming. In order to reach this goal, a series of technical results are presented for the construction of such a Lyapunov function and its mathematical properties are discussed in details. Finally, a major result of guaranteeing the existence of such a Lyapunov function is rigorously proved. Index Terms—Cognitive modeling, decision making, Lyapunov function, multi-cue multi-choice tasks, performance gauge.   I. Introduction F OR many critical infrastructure systems, such as power transmission networks, water distribution systems, and gas pipeline networks, human operators are naturally part of the system decision making process in which they have absolute authority to hold off the decision made by automation or control systems. The question of how the human operators’ decision will affect the performance of such systems is the main challenge in designing functional, human-intelligent control systems [1]. Also the idea of exploring the possibility of optimizing human-in-the-loop decision making by mainly controlling the parts that may affect human decision performance sounds plausible with the ever increasing presence of artificial intelligence in daily life. For this purpose, the first logical step is to develop appropriate models for human decision process in the control system by integrating cognitive perspectives on human decision making. The concept of temporal integration of evidence through sequential probability ratio test (SPRT) has been widely employed in decision-making modeling studies [2]. According to the drift diffusion model (DDM) to which SPRT converges in its continuum limit, decisions are made by accumulating noisy stimulus information until the decision variable reaches either positive or negative threshold in two-alternative forced choice (2AFC) tasks. The DDM has been proven to be successful in emulating the process of decision, and due to its connection to SPRT, it is optimal in a sense of maximizing any reward criterion that is monotonically decreasing with respect to decision time [3]. In other words, DDM renders the shortest possible decision time, given a specific accuracy. However, as pointed out by [4], this optimality description does not consider any cost associated with behavior, or a cost function for gathering information dynamically. By increasing the number of decision choices and attributes in multi-choice multi-cue tasks (MCMC), race models with mutual inhibition were adopted to depict the decision process [3]. Although these models are intuitively plausible in describing the decision process, the notion of asymptotic optimality of SPRT [5] cannot be applied. Note that for the specific case of two-choice tasks, inhibition models can be reduced to DDM, and hence, renders the optimal solution under specific circumstances [6]. This leads to the question of how to address optimality for general MCMC tasks. Even before we talk about optimality, a cost associated with optimal performance needs to be defined and its evaluation needs to be tackled. In this work we take a control-theoretic approach to tackle the performance evaluation of MCMC tasks modeled by mutual inhibition race pools, as discussed later in Section II. Our focus is to construct a performance gauge, which accounts for the performance of average MCMC and at the same time deals with the potential sources of deviation from optimality at the psychological level [7]. To do so, we first briefly review the mathematical abstract models in decision making in Section II and then formulate our proposed problem in Section III. In Section IV, some mathematical preliminaries of a novel method, called converse Lyapunov approach, are developed to prepare for construction of such a performance gauge later, which is based on converse Lyapunov results for switched and nonlinear semistable systems. As a major contribution of the paper, a performance gauge function is constructed in Section V. Finally, some conclusion about this Manuscript received March 17, 2020; revised April 23, 2020; accepted June 13, 2020. Recommended by Associate Editor Zhen Song. (Corresponding author: Qing Hui.) Citation: M. Firouznia and Q. Hui, “On performance gauge of average multi-cue multi-choice decision making: A converse Lyapunov approach,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 1, pp. 136–147, Jan. 2021. M. Firouznia is with the Department of Electrical and Biomedical Engineering, University of Vermont, Burlington, VT 05405 USA (e-mail: mfirouzn@uvm.edu). Q. Hui is with the Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588 USA (e-mail: qing.hui@unl.edu). Digital Object Identifier 10.1109/JAS.2020.1003471 136 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 8, NO. 1, JANUARY 2021