Accuracy and Decision Time for Sequential Decision Aggregation

This paper studies prototypical strategies to sequentially aggregate independent decisions. We consider a collection of agents, each performing binary hypothesis testing and each obtaining a decision over time. We assume the agents are identical and receive independent information. Individual decisions are sequentially aggregated via a threshold-based rule. In other words, a collective decision is taken as soon as a specified number of agents report a concordant decision (simultaneous discordant decisions and no-decision outcomes are also handled). We obtain the following results. First, we characterize the probabilities of correct and wrong decisions as a function of time, group size and decision threshold. The computational requirements of our approach are linear in the group size. Second, we consider the so-called fastest and majority rules, corresponding to specific decision thresholds. For these rules, we provide a comprehensive scalability analysis of both accuracy and decision time. In the limit of large group sizes, we show that the decision time for the fastest rule converges to the earliest possible individual time, and that the decision accuracy for the majority rule shows an exponential improvement over the individual accuracy. Additionally, via a theoretical and numerical analysis, we characterize various speed/accuracy tradeoffs. Finally, we relate our results to some recent observations reported in the cognitive information processing literature.

💡 Research Summary

The paper investigates how a group of identical, independent agents that each perform binary hypothesis testing can be aggregated sequentially using a simple threshold‑based rule. Each agent observes a stream of data and runs a sequential probability ratio test (SPRT): when its cumulative log‑likelihood ratio crosses an upper or lower bound it instantly declares hypothesis H₁ or H₀, respectively. All agents share the same false‑alarm and miss‑detection probabilities and the same average stopping time, and their observations are mutually independent.

At the collective level the authors introduce an “k‑agreement” rule: as soon as k agents have reported the same decision, that decision becomes the group’s final output. The rule also specifies how to handle simultaneous contradictory decisions or the case where no agent has yet decided. This framework subsumes two well‑known special cases: the “fastest” rule (k = 1) where the first individual decision is adopted, and the “majority” rule (k = ⌈N/2⌉) where a simple majority is required.

The first major contribution is an exact, time‑dependent characterization of the probability that the group makes a correct or an incorrect decision under the k‑agreement rule. By modeling each agent’s decision time as a point process and treating the arrival of k concordant decisions as the first‑k‑arrival time of a combined Poisson‑like process, the authors derive closed‑form expressions for P_correct(t) and P_error(t). Importantly, these calculations scale linearly with the group size N, in contrast to naïve state‑space enumeration which would be exponential.

The paper then focuses on the two extreme thresholds. For the fastest rule (k = 1) the group decision time converges, as N → ∞, to the minimum of the individual stopping times. Using extreme‑value theory the authors show that the expected group decision time shrinks roughly as 1/(N λ), where λ is the individual decision rate, so a large swarm can decide almost instantaneously. However, the error probability remains identical to that of a single agent because no additional evidence is accumulated.

For the majority rule (k = ⌈N/2⌉) the analysis relies on the central limit theorem. Each agent’s binary outcome is a Bernoulli trial with success probability p > 0.5 (the individual accuracy). The sum of N such trials is approximately normal with mean Np and variance Np(1‑p). The probability that at least ⌈N/2⌉ agents are correct therefore behaves like

 P_correct(N) ≈ 1 − exp(‑c N)

for some constant c that depends on p and the SPRT thresholds. Consequently, the group’s accuracy improves exponentially with N, while the average decision time grows only modestly (still O(E