DD-001
Private Information Teams
Exact private-team role and alignment results under zero communication.
Completed finite studyChecked evidence
The question
Is clue-following team-optimal? When can role-dependent signal policies or territories improve discovery? How do best-known and certified policies vary with \(M,N,p\), and what can be bounded at the canonical \((16,8,0.20)\) instance?
What we found
- DD-C-0018: In the finite static common-payoff DD-001 model, some deterministic policy profile weakly dominates every ex-ante randomized team strategy.
- DD-C-0019: Direct clue-following is weakly below the private-team frontier, which is weakly below the pooled planner frontier when the planner observes all private signals and can emulate every private-team action profile.
- DD-C-0020: The DD-001 primary run certifies exact deterministic private-team optima for all 21 configured tiny cases; direct clue-following is optimal at 5 points and strictly suboptimal at 16.
- DD-C-0021: At M=3, N=2, and p=2/5, the exact private-team optimum is 7/10, strictly above direct clue-following 16/25 by 3/50; a constant-plus-signal hybrid policy attains the optimum.
- DD-C-0022: In the configured canonical coordinate-ascent search, direct clue-following was an exact coordinate fixed point and all 17 non-direct starts converged to its value 325089/390625; no configured start improved the lower bound.
- DD-C-0023: In the finite DD-001 model, a policy's targetwise incoming counts and fixed-point indicators determine its conditional success vector and hence every fixed-profile discovery value; such a signature is realizable exactly when the local count conditions and singleton residual Hall bounds hold.
- DD-C-0024: The DD-001A primary signature run independently reproduces all 21 previously certified tiny optima and their raw-policy multiset tie counts, including the M=3, N=2, p=2/5 value 7/10, while exact feasibility and reconstruction agree with every raw policy signature through M=5.
- DD-C-0025: For M=16 there are exactly 148348284928 feasible labeled policy signatures and 5806 individual-signature target-label orbits; without a proved cross-agent target quotient, the eight-agent signature-multiset space has 5817544508742203415773519264794208893444139238056614203268215932897118628684812510720 states.
- DD-C-0026: For M at least 3 and N=2, among two direct policies, two distinct constant territories, and the one-reroute constant-plus-signal hybrid, territorial is best below p=1/M, the hybrid is best between 1/M and 1/(M-1), and direct is best above 1/(M-1), with exact adjacent ties at the boundaries.
- DD-C-0027: For M in {3,4,5}, N=2, and every p in the continuous informative interval [1/M,1], unrestricted deterministic private-team optimality follows the same hybrid-then-direct envelope, with the switch at p=1/(M-1).
- DD-C-0028: At p=0 and N=2, the unrestricted exact optima are 11/12 for M=3 and 2/3 for M=4, strictly above the corresponding best direct/territorial/one-reroute values 2/3 and 1/2.
- DD-C-0036: The canonical deterministic zero-communication private-team frontier T_8(16,1/5) lies in the exact interval 325089/390625 <= T_8(16,1/5) <= 860391662035297/1001129150390625, whose gap is 27224111644672/1001129150390625, approximately 0.0271934061994395.
- DD-C-0037: For every M >= 2, N >= 1, and symmetric-report accuracy p in [0,1], the joint-target-column global-count-budget relaxation in ADR-0009 is an admissible upper bound on discovery by deterministic zero-communication private teams.
- DD-C-0038: In the canonical M=16, N=8, p=1/5 fixture, the deterministic and ex-ante randomized zero-communication private-team optima both equal 325089/390625 = 0.83222784, attained by direct clue-following.
What this result covers
Checked evidence. The formal evidence wording and registry state remain available below.
Technical details
- Study phase
complete-bounded-study- Registry status
active- Evidence status
exact-tiny-results-signature-reduction-thresholds-and-exact-canonical-optimum
Reproducible evidence
20260720T200447Z_DD-001_6eb12861_ba766d1eba20260720T221139Z_DD-001_b1d8d431_40bf5b06a520260720T223829Z_DD-001_b2cc23f4_5e16a90ad120260721T012208Z_DD-000_8e4b55e2_e8321d104820260721T022739Z_DD-001_358cb1eb_cd16846ba5
Files and data
What comes next
characterize when the certified count-budget relaxation is tight outside the canonical fixture