Accuracy and Decision Time for Decentralized Implementations of the Sequential Probability Ratio Test Sandra Hala Dandach Ruggero Carli Francesco Bullo Abstract— This paper focuses on decentralized decision mak- ing in a population of individuals each implementing the sequential probability ratio test. The individual decisions are combined by a fusion center into a collective decision via an aggregation rule. For distinct aggregation rules, we study how the population size affects the performance of the collective decision making, i.e., the decision accuracy and time. We analyze two aggregation rules, the fastest rule and the majority rule. In the fastest rule, the group decision is equal to the first decision made by any individual. In the majority rule, the group decision is equal to the majority of the decisions. Under the assumption of measurement independence among individuals, we introduce a novel numerical representation of the performance of decentralized decision making. We study our settings analytically as well as numerically. Our numerical results and simulations characterize the tradeoff between ac- curacy and decision time as a function of the population size. I. I NTRODUCTION This work aims to understand how grouping individual decision makers (DM) affects the speed and accuracy with which a collective decision is made. This class of problems has a rich history and some of its variations are studied in the context of distributed detection in sensor networks [1], [2], [3], [4], [5] and Bayesian learning in social networks [6], [7]. In this paper we consider a group of N individuals each of them individually implementing the standard sequential probability ratio test (SPRT) with the purpose of deciding between two hypothesis H 0 and H 1 . We refer to the individ- uals as decision makers which we denote as DMs hereafter. In our setup no-communication is allowed between the DMs. Once a DM has provided a decision it communicates it to a fusion center. The fusion center collects the various decisions and provides a global decision via an aggregation rule. In this paper we focus on two aggregation rules, the fastest rule and the majority rule. In the fastest rule, the fusion center makes a global decision as soon as one hypothesis gets more votes than the opponent hypothesis, while in the majority rule the fusion rule makes a global decision as soon as one of the two hypothesis gets ⌊N/2⌋ +1 decisions in its favor. For both rules and for distinct sizes of the group of DMs, we study the performance of the collective decision making, i.e., the decision accuracy and expected number of observations required to provide a decision. This work has been supported in part by AFOSR MURI Award F49620- 02-1-0325. S. H. Dandach and R. Carli and F. Bullo are with the Cen- ter for Control, Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106, USA, {sandra|carlirug|bullo}@engr.ucsb.edu. The framework we analyze in this paper is related to the one considered in many papers in the literature, see for instance [8], [1], [9], [2], [10], [3], [11] and references therein. The focuses of these works are mainly two-fold. First, determining which type of information should the DMs send to the fusion center. Second, computing optimal decision rules both for the DMs and the fusion center where optimality refers to maximizing accuracy. One key implicit assumption made in these works, is the assumption that the aggregation rule is applied by the fusion center only after all the DMS have provided their local decisions. The work presented in this paper, relaxes the assumption on the local decisions. Indeed the fusion center might provide the global decision much earlier than the time needed for the local decisions to be made by the whole group. Our main concern is the exact computation of both the accuracy of the final group decision as well as the expected number of observations required by the group, in order to provide the final decision. In this work we accomplish these objectives by proposing exact expressions for the conditional probabilities of the group giving the correct and wrong final decisions, at any time instant. We perform this analysis both for the fastest rule and the majority rule for a varying group sizes. This represents the major contribution of this paper. In the second part of the paper we use the expressions that we provide in the first part of the paper, to numerically characterize the tradeoff between the accuracy and the ex- pected time and the group size. For illustration, we consider a discrete distribution of the Koopman-Darmoi-Pitman form. We find that the majority rule provides, as N increases, a remarkable improvement in terms of of the accuracy while the fastest rule provides a remarkable improvement in terms of the expected number of observations required to provide the final decision. The rest of the paper is organized as follows. In Section II we review the standard SPRT. In Section III we formally introduce the problem studied in this paper. In Section IV we present our novel numerical method useful to analyze the problem of interest. In Section V we provide some numerical results. We conclude in Section VI. II. A BRIEF REVIEW OF SPRT AND OF RELATED ANALYSIS METHODS In this section we discuss the classical sequential proba- bility ratio test (SPRT) for a single decision maker; to do so we follow the treatment in [12].