The paper introduces a compact sixteen‑box benchmark (M=16, N=8, clue accuracy p=0.20) to isolate a mechanism the author calls the Shared Discovery Paradox. Pooling noisy private clues into a single posterior and then applying a one‑answer (consensus) rule improves the average quality of the recommended action (from 0.20 to 0.3835) but dramatically reduces the probability that the group discovers the hidden jackpot (from 0.8322 under decentralized clue‑following to 0.3835 under consensus). The author shows that the loss is a protocol failure, not an information failure, by comparing four protocols: private clue‑following, pooled consensus, a coordinated planner that assigns the top‑L posterior boxes, and an anonymous equal‑split market where self‑interested searchers split the prize. Exact enumeration yields discovery probabilities Gconsensus=0.3835, Gprivate=0.8322, Gplanner=0.8594, Gmarket^sym=0.5991, and a blind coordinated portfolio benchmark of 0.50. The paper defines protocol loss, attainable frontier, and recovery budget (L = 7 coordinated actions replace eight private searchers). It then studies incentives: the equal‑split game is an exact potential game with a unique anonymous symmetric equilibrium that follows a water‑filling rule; discovery equals N times the common payoff, and the worst‑case price of anarchy is bounded by 2‑1/N (tight). A sole‑rescue reward (pay only the unique finder) implements the planner’s optimal portfolio in every pure Nash equilibrium. A latent common‑cue copying model introduces correlation among clues; copying probability c collapses effective channels, reduces distinct proposals, and reverses the ordering between private search and the market at a crossover c≈0.788. The author also derives a proportional sparse‑evidence limit (M,N→∞ with N/M→α and fixed likelihood ratio) showing that consensus discovery vanishes while diversified protocols converge to positive limits (e.g., Gmarket^sym→0.547, Gprivate→0.847, G_portfolio→0.874). The paper situates the benchmark within team theory, optimal search, information cascades, congestion games, and organizational learning, and discusses practical implications for R&D, venture capital, science, and multi‑agent AI. Limitations and extensions are outlined, emphasizing the benchmark’s role as a reusable quantitative reference.
Topics
Defining the Shared Discovery Paradox
Formal definition of the paradox where a shared‑information protocol raises average action quality (Q) but lowers group discovery (G). Includes the mathematical condition Q(Π^S)>Q(Π^P) and G(Π^S)<G(Π^P).
Sixteen‑Box Benchmark Architecture
Exact specification of the toy environment: M=16 boxes, one jackpot, N=8 searchers, private clue accuracy p=0.20, independent signals, and the enumeration method for report‑count classes.
Protocol Comparison: Private, Consensus, Planner, and Market
Derivation of discovery probabilities for four protocols (private clue‑following, pooled consensus, coordinated planner portfolio, anonymous equal‑split market) and the blind coordinated portfolio control. Highlights the paradoxical ordering of Q and G across protocols.
Protocol Loss, Attainable Frontier, and Recovery Budget
Introduction of V_L(F) (attainable frontier), L_L(Π;F) (protocol loss), and the recovery budget L* = 7 that shows how many coordinated actions replace decentralized search. Provides the accounting identity LQ−G = E[(K−1)^+].
Incentive Design: Anonymous Symmetric Market and Water‑Filling Equilibrium
Analysis of the equal‑split game as an exact potential game, derivation of the unique anonymous symmetric equilibrium via a water‑filling rule, and the welfare identity G_market^sym = N·λ. Shows that market discovery exceeds consensus but remains below private and planner benchmarks.
Exact Price‑of‑Anarchy Bound (2 − 1/N)
Proof that the worst mixed Nash equilibrium of the equal‑split game loses at most a factor 2 − 1/N relative to the optimal top‑N portfolio, with a tight example. Connects to smoothness arguments in algorithmic game theory.
Sole‑Rescue Reward as Full Implementation of the Planner
Definition of the sole‑rescue rule (pay only the unique finder), proof that the game becomes an exact potential game whose pure Nash equilibria coincide with the planner’s optimal portfolio, achieving first‑best discovery.
Latent Common‑Cue Copying Model and Channel Collapse
Model where each searcher copies a common cue with probability c, leading to effective channel count N_eff = N−K+1{K≥1}. Derives closed‑form private discovery G_private(c) and shows how increasing c reduces distinct proposals and eventually makes the market outperform private search (c*≈0.788).