DD-008B
Common-Source Analysis
General-N source-choice threshold theorem with a bounded independent audit and exact interior counterexample.
Completed finite studyChecked independently
The question
For arbitrary finite `N`, what thresholds characterize unilateral independent- source acquisition and planner acquisition, when does an all-common equilibrium coexist with a planner preference for independence, and must private teams always acquire weakly fewer independent sources than the planner?
What we found
- DD-C-0057: For every finite N at least 2 and accuracy p strictly between zero and one in the frozen common-source game, the common-to-independent gross payoff thresholds A_k=p(1-p)E[1/(1+X_k)-1/(N-k+X_k)] are strictly decreasing from A_0=p(1-p)(N-1)/N to A_(N-1)=0 and exactly characterize all weak pure source-count equilibria; planner thresholds are B_k=p(1-p)^(k+1) through k=N-2 and B_(N-1)=0, so the all-common equilibrium/planner-independence trap holds exactly on [p(1-p)(N-1)/N,p(1-p)), an interval of width p(1-p)/N.
- DD-C-0058: Universal private under-acquisition is false even inside the frozen common-source model; at N=3, p=4/5, and c=13/375, the private threshold at k=1 is 14/375 while the planner threshold is 4/125=12/375, yielding the unique source-count equilibrium k=2 and unique planner optimum k=1.
What this result covers
Checked independently. The formal evidence wording and registry state remain available below.
Technical details
- Study phase
complete-bounded-study- Registry status
complete-bounded-study- Evidence status
general-n-theorem-with-independent-finite-audit-and-exact-negative-result
Reproducible evidence
20260721T192412Z_DD-008B_649deb08_29dbeaf3a9
Files and data
What comes next
preserve the theorem and counterexample regressions; use them with explicit scope in the Common-Source Trap paper