Source
Rashid, Samvelyan, Schroeder de Witt, Farquhar, Foerster & Whiteson, LIT-1503, read in NOTE-1154 from arXiv:1803.11485v2: Eqs. 4–5 (p. 4), section 4.1, supplement A (Table 3) and the two-step game (Tables 1–2, supplement B). Strictness against IGM from Son et al., LIT-1510 (NOTE-1141).
What was actually shown
- Sufficiency. If Q_tot = f(Q_1, …, Q_n; s) with f non-decreasing in each Q_a, then for any joint action u, f(Q_1(ū^1), …, Q_n(ū^n)) ≥ f(Q_1(u^1), …, Q_n(u^n)), since each Q_a(ū^a) ≥ Q_a(u^a). Re-derived when filed. Ties can make the greedy profile one maximiser among several.
- More than a sum. [[0, 1], [1, 8]] is monotonic and not additive: a₁ + b₁ = 0, a₁ + b₂ = 1, a₂ + b₁ = 1 force a₂ + b₂ = 2. QMIX learned it exactly (0.00, 1.00, 1.00, 8.00); VDN did not, and took the suboptimal first move.
- Not every payoff. [[2, 1], [1, 8]] is not monotonically representable: with the partner at A, agent 1 must rank A over B (2 > 1), with the partner at B, B over A (8 > 1), and no single Q_1 does both.
- Not necessary. That same matrix satisfies the greedy-profile property (both agents’ values preferring B give the optimum 8), so monotonicity is stronger than what decentralised greedy action needs; QTRAN’s 177 of 310 random games are the systematic version.
What this does not say
- That monotonic mixing gives a common currency. The joint value is real-valued by assumption; the condition concerns how members’ rankings relate to it.
- That a non-monotonic task cannot be decentralised. IGM can hold without monotonicity (THEORY-933).
- Anything under partial observability beyond a remark. The paper says QMIX “cannot necessarily represent the value function” there.
Note of 2026-10-30: the second check
The second check at the status review (ADR-065 §5) followed each step.
Sufficiency. Let ū^a maximise Q_a. For any joint action u, Q_a(ū^a) ≥ Q_a(u^a) for each a, and raising the arguments of a non-decreasing f one at a time from Q(u) to Q(ū) never lowers it, so f(Q(ū)) ≥ f(Q(u)). With ties the greedy profile may be one maximiser among several.
More than a sum. [[0, 1], [1, 8]] is monotonic: Q_1 = Q_2 = (0, 1) with f(x, y) = x + y + 6xy, which is non-decreasing on [0, 1]², gives 0, 1, 1 and 8. A sum would need a₂ + b₂ = (a₁ + b₂) + (a₂ + b₁) − (a₁ + b₁) = 1 + 1 − 0 = 2, not 8.
Not every payoff. In [[2, 1], [1, 8]], suppose Q_1(A) ≤ Q_1(B). Then f(Q_1(A), Q_2(A)) ≤ f(Q_1(B), Q_2(A)), which contradicts 2 > 1, so Q_1(A) > Q_1(B). The same argument with the partner at B and 8 > 1 gives Q_1(B) > Q_1(A).
The last clause of the title is read as: no action of the agent is best against every profile of the others’ simultaneous actions. So read, it holds in general. Let x* maximise Q_a. If some action y did strictly better than x* against some profile of the others, monotonicity would force Q_a(y) > Q_a(x*). A weaker dependence, in which an action is best everywhere but ties with another somewhere, is not what the clause excludes.
A script (numpy, in the session’s scratchpad) checked these steps. In 2000 random non-decreasing mixings of two or three agents, the greedy profile was always a joint maximiser; as a control, a mixing decreasing in one argument failed in 200 of 200 cases. A search over row and column orderings found [[0, 1], [1, 8]] monotonic and not additive, and [[2, 1], [1, 8]] not monotonic. In 3000 random 3 × 3 payoffs with integer values from 0 to 3, ties included, none was monotonically representable unless each agent had an action best against every action of the other; 2520 had no such action.