Research Spotlights

Abstract

A new section with a new direction. Research Spotlights replaced Expository Research Papers on January 31, 2012. Papers in Research Spotlights should cover topics in applied and computational mathematics of particularly wide interest and importance. Contributions must also be accessible to the broad and diverse SIAM Review readership. The principal theme of Research Spotlights is author flexibility with the intent that an expanded format promotes creativity and spurs innovative articles. Authors now have latitude in terms of considering a standard research paper or else a nontraditional article such as a mini-survey, a timely communication, a software description, or a new mathematical perspective within an application area. Prospective authors are encouraged to first consult with the section editor Ray Tuminaro (rstumin@sandia.gov) about potential contributions—especially those that lie outside the scope of traditional SIAM articles. While they can have a more relaxed format, ultimately articles must be of broad interest, must be accessible to the community, and must meet the technical review standards of SIAM journals. Focus groups provide feedback on potential new products: Which juice is more appealing to you, the slimy green or the yucky yellow one? Systems of sensors process signals to decide: Is there an intruder or not? These are examples of “group decision making,” a process where individuals work together to make a collective decision. The problem of figuring out how the collective arrives at a decision occurs in areas as varied as cognitive psychology, economics, political science, and signal processing. Margot Kimura and Jeff Moehlis in their paper “Group Decision-Making Models for Sequential Tasks” consider the “two-alternative forced-choice test,” where one must choose between two hypotheses: slimy green or yucky yellow; intruder or no intruder. Decisions must be made quickly and can only tolerate certain error rates. This means that there are limits on how often the sensors are allowed to miss an intruder, or signal a false alarm. The model in this paper assumes $N$ independent decision makers, each of whom receives observations, sequentially, one at a time. Each decision maker continues to process the observations until s/he is able to make a decision. The incoming observations are represented by independent random variables, with known prior probabilities for each decision. The processing consists of applying the “sequential probability ratio test” to each new observation. Based on the prespecified error rates for the number of misses and false alarms, this test either reports a decision or continues to process the next observation. The authors also consider a continuous version of this test, which becomes a drift-diffusion model as the time between observations goes to zero. Once a decision maker has come up with a decision, s/he reports it to the “fusion center,” which is responsible for arriving at a collective decision. The fusion center can operate in one of three modes: race (report only the very first decision that arrives), majority-total (wait until all $N$ decisions have arrived and then report the majority), and majority-first (wait until $N/2$ identical decisions have arrived, and then report this smallest possible majority). For each such mode, the authors derive probability distribution functions for the collective error rates and decision times, from the error rates and decision times of the individual decision makers. Simulations are presented to compare the different modes. Which mode turns out to be the most efficient is not at all obvious and depends on the scenario at hand, whether decision makers can have different error rates or can malfunction. Finally, the authors extend their analysis to more general modes, where the fusion center makes a collective decision based on the first $\eta$ decisions that arrive. The approach presented here has many advantages. It is general and applies to many situations that require collective decision making based on sequential observations, including even “cybernetic groups” with human observers and nonhuman detectors. Furthermore it is systematic and elegant, because it provides a clear path for deriving the efficiency of collective decisions from those of individual decision makers. Especially appreciated is a list of acronyms thoughtfully included by the authors at the beginning of the paper, which makes it easy to decipher the many acronyms in this area.