When Does Information Sharing Improve Decentralized Discovery? Aggregation, Independent Rescue, and Equilibrium Selection

Organizations often pool dispersed information into one ranking and then allow many agents to act on that shared view. In a discovery problem, this can improve beliefs while reducing coverage. We develop an exactly solvable benchmark with sixteen boxes, one target, eight searchers, and noisy private clues. Pooling raises the accuracy of the best single recommendation from 0.20 to 0.3835, but repeating that recommendation lowers group discovery from 0.8322 under decentralized clue-following to 0.3835. A coordinated eight-action portfolio using the same pooled reports reaches 0.8594, and seven coordinated actions recover the decentralized benchmark. The paradox is a protocol failure, not an information failure: a one-answer rule compresses a portfolio of available actions into one repeated choice. We then replace the planner with self-interested searchers who split a prize. The equal-split game is a potential game. Its anonymous symmetric equilibrium obeys a water-filling rule. In the canonical instance it achieves 0.5991: strictly above consensus, but below both private search and the planner. The exact mixed price of anarchy is 2 - 1/N. A sole-rescue reward, which pays only an agent who covers the target alone, makes every pure Nash equilibrium first-best. Finally, a latent common-cue model shows how correlated reports collapse effective discovery channels. The centralized planner gain rises strictly with copying, and in the canonical environment the symmetric market overtakes decentralized report-following at copying probability c = 0.788462. In a proportional large-market limit the five-protocol ordering survives exactly: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874. The contribution is a compact benchmark that separates information, allocation, incentives, and dependence into exact, reusable quantities.

In decentralized decision-making, several agents connected according to a network graph aim at solving a classification problem by collecting streaming observations. Due to decentralization, they run an iterative algorithm where, at each iteration, they can only exchange information locally with their neighbors. While decentralized estimation solutions have been shown to match the performance of optimal centralized systems, we show here that surprisingly this conclusion does not hold for decentralized decision-making. Specifically, we prove that the error probability for the best decentralized decision strategy exhibits an irreducible loss with respect to the optimal centralized classifier. This result establishes a fundamental limit for the performance of any decentralized decision strategy. We obtain an analytical relation showing that this limit is related to the interplay between decentralization and classification. The first aspect appears through the distances between the nodes in the graph, while the second aspect plays through the moment generating functions of the likelihood ratios that describe the decision problem. By applying the derived closed-form relation to different network topologies and inference problems, we observe some interesting and perhaps unexpected behavior emerging. In particular, we characterize the scaling law (with the network size) for the loss over popular network topologies, showing that the error probabilities might differ by orders of magnitude; and we examine how performance is affected by the relative distance between informative and uninformative agents over the graph.

Social Learning is a decentralized decision-making paradigm in which spatially dispersed agents collect streaming observations regulated by one of a finite number of models (the hypotheses). The agents are interested in assigning probability scores (the beliefs) to the possible hypotheses. To this end, the agents exchange their beliefs according to a certain communication graph. It has been shown that, under reasonable conditions on the identifiability of the decision model and the network connectivity, each agent ultimately places all the belief mass on the true hypothesis governing the data. However, several questions remain unanswered regarding the evaluation of the social learning performance. One recently adopted performance metric is the rejection rate, i.e., the rate at which the beliefs about the erroneous hypotheses vanish. One contribution of this work is to establish that the rejection rate leads to several paradoxes, which make it unsuitable as a valid performance measure. We then focus on studying the error probability measure. For a binary Gaussian problem, we derive an analytical formula characterizing the ratio between the individual agents' probabilities and the optimal Bayesian probability. The formula shows that this ratio is expressed by the product of two terms quantifying the effect of the network connectivity and the role of the prior information. As a result, an irreducible gap emerges between the decentralized and the centralized error probabilities, which is agent-dependent and does not disappear asymptotically.