The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games

Ismael Tito Freire; Konstantinos E. Psannis; Marti Sanchez-Fibla; Nikolaos Al. Papadopoulos

REVIEW 5 major objections 4 minor 1 cited by

The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games

T0 review · 5 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read In a repeated multi-agent coordination game, Q-learning policies that look fair by conventional metrics actually alternate access to a shared reward worse than random action selection, by 34–74% on the paper's primary alternation metric.

desk verdict Plausible and worth testing, but the current write-up doesn't pin down what the reported 'worse than random' numbers actually measure. read the letter →

arxiv 2603.05789 v5 pith:XXJOVYPA submitted 2026-03-06 cs.MA cs.GTcs.LG

Nikolaos Al. Papadopoulos , Ismael Tito Freire , Marti Sanchez-Fibla , Konstantinos E. Psannis This is my paper · ORCID

classification cs.MAcs.GTcs.LG

keywords temporalfairnessturn-takingalternationmetricsmulti-agentreinforcementlearningrepeatedgamesBattleoftheExescoordinationQ-learning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Repeated interactions among self-interested agents can produce high aggregate payoffs while the agents are failing to actually take turns. This paper argues that standard fairness and efficiency metrics are temporally blind: they can score coordinated rotation, random access, and monopolization almost identically. To expose the difference, the authors define Perfect Alternation (PA) as a reference turn-taking regime and introduce six Alternation (ALT) metrics, then compare simple Q-learning agents against analytically derived random-policy baselines in a multi-agent Battle of the Exes game. The central finding is that the learned policies score worse than random on every primary ALT metric, even as reward fairness often exceeds 0.9, so conventional metrics can conceal severe coordination failure as agent count grows. The paper thereby makes a measurement point: temporal coordination must be evaluated relative to an explicit chance-level null process.

What carries the argument

The load-bearing object is Perfect Alternation (PA): the n-periodic round-robin sequence in which each of n agents wins exactly once per block of n episodes. In the MBoE setting this is the unique allocation achieving exact Temporal Envy-Freeness at every cycle boundary. Against this reference, six ALT metrics (CALT, EALT, AALT, and secondary FALT, qFALT, qEALT) score overlapping batches of n consecutive episodes, penalizing ties and rewarding exclusive wins and winner diversity. A regression-based AltRatio framework converts raw ALT scores into a PA-equivalent count of perfectly alternating agents, and uniformly random action policies are run as explicit null processes so that reported coor

What would settle it

Compute the ALT metrics only on the final block of episodes after epsilon has reached its minimum, and average over many independent seeds; if post-convergence Q-learning then meets or exceeds the random baselines on CALT, the paper's central claim that learned policies are actively worse than random is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that in the Multi-Agent Battle of the Exes (MBoE)—a Markov game where n agents compete for exclusive access to a single high-reward state—independent tabular Q-learning agents systematically under-coordinate relative to chance. Across 20 configurations of state representation, reward scheme, and agent count, Q-learning achieved reward fairness values between 0.49 and 0.993 and efficiency up to 0.677, yet all primary Alternation metrics (CALT, EALT, AALT, FALT) were below random-policy baselines, with CALT 34–74% lower and PA-equivalent coordination falling to roughly one-fifth of the population at n=10. The paper interprets this as a measurement fai

Load-bearing premise

The central comparison assumes the Q-learning runs are evaluated fairly and represent converged learners; because ALT scores are computed over full training histories, early-exploration and epsilon-decay artifacts may contribute to the below-random result.

Editorial extensions

If this is right

If the claim holds, any evaluation of emergent coordination that reports only efficiency, reward fairness, or turn-taking fairness is incomplete and can certify failure as success.
Reported coordination results in repeated games should include a random-policy null baseline; without one, high traditional metrics are uninterpretable.
Temporal fairness measures such as the ALT family should accompany outcome-based fairness in multi-agent reinforcement learning evaluation, especially as agent counts grow.
The sharp decline from 56.8% to roughly 22% PA-equivalent coordination between 2 and 8–10 agents suggests that coordination difficulty in this class of game scales non-linearly with population size.
Independent tabular Q-learning, in this setting, does not spontaneously converge to turn-taking behavior, meaning decentralized learners need additional structure (memory, signaling, or centralized training) to achieve temporal fairness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same temporal blindness likely affects other aggregate metrics in multi-agent systems beyond this game—any setting where fairness of cumulative shares is reported without checking the sequence of allocations.
A testable extension: adding an alternation-based reward bonus or using ALT scores as an early-stopping signal would likely push independent learners above the random baseline, which would confirm that the deficit is a credit-assignment failure rather than an intrinsic impossibility.
The result suggests that comparing learned policies to an analytic random null is a cheap and general diagnostic that could be standardized across coordination benchmarks, not just turn-taking games.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the takeaway: the paper's central claim — independent Q-learners in the MBoE game do worse than random on turn-taking metrics even when traditional fairness looks high — is plausible and worth testing, but as written it is not demonstrated. The main reason is that the evaluation protocol is under-specified. I cannot tell from the text or tables whether the Q-learning ALT values come from full training histories or post-convergence rollouts. Figure 5 shows CALT decreasing as epsilon decays, so this matters. If the numbers average over training, then "worse than random" is about learning trajectories, not learned policies.

What is genuinely new: the MBoE Markov-game formalization, the PA reference regime, the six ALT metrics, and the systematic random-baseline comparison. Prior BoE work has mostly stayed two-agent and rarely reports chance baselines. The reported pattern — high reward fairness alongside below-random alternation — is a real observation if it survives scrutiny. The authors also ship code and data, which is more than many papers in the area.

Now the soft spots, in rough order of importance. First, the abstract and Table 4 disagree on magnitudes: the abstract says 'up to 92% on EALT' but Table 4 shows -75.9% as the worst EALT relative change. That suggests the headline numbers came from a different pipeline than the main tables. Second, no multiple seeds. The only spread cited is across the four configuration modes, not stochastic replicates, so the 'systematically worse than random' gap has no confidence interval. Third, the PA-equivalent values come from a regression in Supplementary S1 that is not described in the main text; a headline number should not depend on an invisible fit. Fourth, the 'PA is uniquely TEF' claim is imported from Choi & Li (2026) and asserted rather than proven for MBoE; smaller issue, since the metrics don't stand or fall on it.

These are fixable. The core idea is good, the code exists, and the direction is probably right. What is missing is a clear evaluation protocol, seed variance, and consistent numbers between abstract and tables.

Who should read it: people working on MARL evaluation, temporal fair division, or coordination metrics. It deserves a serious referee — an editor should send it out — but the referee should ask for the protocol to be pinned down and the numbers regenerated with seeds and post-convergence evaluation.

Referee Report

5 major / 4 minor

Summary. This paper formalizes a multi-agent variant of the Battle of the Exes as an episodic Markov game and proposes Perfect Alternation (PA) as a reference regime, together with six ALT metrics (CALT, EALT, AALT, FALT, qFALT, qEALT) intended to measure temporal coordination. It also proposes an AltRatio regression mapping to PA-equivalent coordination, and random-policy baselines as null processes. In Q-learning experiments with n ∈ {2,3,5,8,10} and four state/reward configurations, the paper reports that Q-learning policies fall below random baselines on all primary ALT metrics while traditional fairness/efficiency metrics remain high, concluding that conventional metrics mask temporal coordination failure. The central empirical claim is that 'Q-learning agents have learned policies that are actively worse than random action selection for achieving turn-taking coordination' (Section 5.5.2).

Significance. If the central empirical claim were cleanly established, the paper would make a useful contribution: it provides a concrete diagnostic family for temporal coordination, an explicit null baseline for turn-taking evaluation, and a striking demonstration that outcome-based fairness metrics can be high while temporal alternation is below chance. The availability of code and archived data is a strength, as is the attempt to ground PA in the temporal fair division literature. However, the current manuscript does not yet support the headline quantitative claims because the evaluation protocol for the Q-learning ALT values is not specified and no seed-level variance is reported.

major comments (5)

[§5.2, Table 4, Figure 5] The manuscript never states whether the Q-learning ALT values in Table 4 are computed over the full training history, over a post-convergence evaluation, or over final-policy rollouts. Figure 5 shows CALT decreasing during training as epsilon decays, so the metric is phase-dependent. If Table 4 uses full-history averages, the comparison is not 'learned policies vs random' but 'learning-trajectory average vs random', while the random baseline uses 10,000 fresh uniform-random episodes. This is load-bearing for the Section 5.5.2 claim. The authors must specify the protocol and report post-convergence evaluation of final greedy policies over a matched number of fresh episodes.
[§5.5.2, Figure 1] No multiple independent seeds are reported. The only spread shown in Figure 1 is across the four Type-A/B × ILF/IQF configurations, not across stochastic replicates. Consequently, there is no confidence interval or standard error for any Q-learning ALT value, and the statement that Q-learning is 'systematically' below random is unsupported statistically. Multiple seeds (at least 10, with typical dispersion reported) and matched episode counts are needed for the central empirical claim.
[Abstract vs Table 4] The reported deficit magnitudes are inconsistent. The abstract states 'up to 92% on EALT', but Table 4 gives the worst EALT Relative Change as -75.9%; the full-text abstract says 'up to 81% below random' and Section 7.1 says '-81% (qEALT)', a secondary metric not in Table 4. The reader cannot tell which numbers are the headline result or which evaluation protocol generated them. These inconsistencies must be reconciled, with all headline numbers traceable to a named table entry.
[§4.1, Contribution 2] The claim that Perfect Alternation is the unique allocation achieving exact Temporal Envy-Freeness 'at every n-episode cycle boundary' is asserted via Theorem 17 of Choi and Li (2026), but the mapping from their scheduling model to MBoE is not proved, and uniqueness is not established. Since this is presented as independent theoretical grounding for PA, the authors should either provide a self-contained proof of the mapping and uniqueness or explicitly soften the claim to a correspondence that holds under stated assumptions.
[§5.3, Table 5] The AltRatio/PA-equivalent mapping is central to the abstract claim that 10-agent coordination is 'roughly one-fifth of the population', but the main text only says regression equations are in Supplementary Section S1. Table 5 states 'ALT Ratio = sqrt(CALT − 1.879e−10)' with no derivation or explanation of the constant. The PA-equivalent percentages in Section 5.5.3 depend on this unexplained mapping. The regression construction should be summarized in the main text, or the PA-equivalent claims should be clearly labeled as dependent on a supplementary calibration.

minor comments (4)

[§4.3, Eq. (9)] The CALT definition uses β_j^{qFALT} before qFALT is defined. Reorder the metric presentations or add a forward reference so Eq. (9) is self-contained.
[Figure 1 caption] The caption says 'Error bars represent standard deviation across Type-A/Type-B and ILF/IQF configurations.' This is a spread across configurations, not a statistical uncertainty estimate. Clarify this in the caption to avoid misleading readers into inferring seed-level variance.
[§7.1] The claim 'Relative Change reaches as low as -81% (qEALT)' and the single positive '+5.4% for 10-agent qEALT' are not visible in any main-text table; they are deferred to supplementary material. Since these are used as summary statistics, include the secondary-metric table or at least the relevant rows in the main text.
[§5.2, Eq. (15)] The episode-count schedule is described as heuristic, and the paper states that 'pilot runs with alternative scalings produced qualitatively similar trends' without showing these runs. A brief description of the pilot sensitivity analysis would strengthen confidence that the reported patterns are not an artifact of the chosen schedule.

Circularity Check

1 steps flagged · score 4.0 of 10

PA-equivalent analysis is a monotone rescaling of CALT rather than an independent benchmark; central 'worse than random' claim remains self-contained.

renaming known result [Section 5.3, Table 5 caption, Section 5.5.3]
"To interpret ALT values, we define AltRatio=x/n∈[0,1], where x represents the equivalent number of perfectly alternating agents... Multiplying by n yields the PA-equivalent... ALT Ratio=√(CALT−1.879×10−10); PA Equiv.=n×ALT Ratio... The finding that 10 Q-learning agents coordinate as effectively as only 2.19 perfectly alternating agents constitutes a notable finding."

The PA-equivalent analysis is not an independent behavioral benchmark: it is a direct algebraic transform of the paper's own CALT metric. Table 5 defines PA Equiv. = n·√(CALT), so the headline '10 agents ≈ 2.19 perfectly alternating agents' is simply 10·√(0.0482) ≈ 2.19. Similarly, the random-baseline PA-equivalents in Figure 2 (69.7% at n=2, 33.3% at n=10) equal √(0.486) and √(0.111), i.e., the same transformation applied to random CALT. Thus 'coordination as well as x perfectly alternating agents' is a relabeling of CALT, not an empirical finding established by simulating x alternating agents. The central worse-than-random comparison does not depend on this renaming, so the circularity is partial.

full rationale

The core empirical claim that Q-learning agents fall below random-policy baselines on ALT metrics is self-contained and not circular by construction: Relative Change and Coordination Score (Eq. 16) compare independently simulated random baselines (10,000 episodes, Table 3) against Q-learning runs, and the ALT metric definitions (Eqs. 8-14) are given in the paper rather than imported as black boxes. The main caveat is that the paper never states whether Table 4 Q-learning ALT values are computed over the full training history or over post-convergence policies, and Figure 5 shows CALT decreases during training; this is a protocol/robustness limitation, not a circular derivation. The PA reference regime is grounded partly in an external theorem (Choi and Li, 2026), not in the authors' own prior work, and the self-citations for ALT metrics (Papadopoulos and Sanchez-Fibla, 2021; Papadopoulos et al., 2025) are accompanied by in-paper definitions, so they are not load-bearing in a circular sense. The one concrete reduction is the AltRatio/PA-equivalent mapping, which is a monotone rescaling of CALT presented as an interpretable benchmark finding. Because the central 'worse than random' result is independent of that rescaling, the overall circularity is limited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central empirical claim depends on the ALT metrics being meaningful measures of temporal coordination, on the random-policy null being appropriate, and on the Q-learning runs being representative. The PA-as-TEF grounding is an asserted mapping to the temporal fair division setting, and the PA-equivalent headline values are obtained from a fitted regression. These are assumptions or calibrations introduced by the paper rather than independently evidenced facts.

free parameters (3)

AltRatio regression coefficients = Exact coefficients not reported in main text; formula ALT Ratio = sqrt(CALT - 1.879e-10) given in Table 5 caption
PA-equivalent performance percentages in Table 5/Figure 2 are obtained from a regression fitted to extreme-case simulations (n in [2,40]); the mapping determines the 'one-fifth of the population' headline and is a fitted quantity, not derived from first principles.
Q-learning hyperparameters and episode schedule = gamma=0.999, alpha=0.3, epsilon 0.9 to 0.004, BASE=1000; Episodes(n)=BASE*(n choose 2)^2*(1+ln(n!/2!))
Chosen by hand; the episode-count schedule is described as heuristic. These choices shape the ALT trajectories and no sensitivity analysis or seed averaging is reported.
Reward scaling r_low = r_high=100; r_low = r_high/n (ILF) or r_high/n^2 (IQF)
The two reward schemes are design choices that affect efficiency values; the ALT metrics are claimed to be reward-magnitude independent, so this is secondary but still a hand-chosen input.

assumptions (4)

domain assumption MBoE's one-good-per-episode structure 'corresponds precisely' to the identical-days house-allocation setting in temporal fair division.
Section 4.1 uses this asserted correspondence to claim via Choi and Li (2026) Theorem 17 that PA is the unique exact-TEF allocation. The mapping is not proven for MBoE dynamics where agents choose move/stay and can tie.
domain assumption Uniformly random action selection is a valid null process for 'no temporal coordination'.
The Relative Change and Coordination Score statistics define all deficits relative to this null. If the random baseline is not a meaningful comparator, the 'worse than random' framing weakens.
ad hoc to paper ALT metrics satisfy their stated properties: boundedness, ALT=1 under PA, monotonic degradation under monopolization or ties.
Section 4.2 asserts these properties; no formal proof or validation against ground-truth alternation is provided. Even FALT is admitted to be insensitive to monopolization, yet it is included among the six metrics.
ad hoc to paper The Q-learning agents have converged sufficiently for their ALT values to be representative of the learned policy.
The paper acknowledges synchronized epsilon decay at 75% of episodes may lock in suboptimal regimes. Without a post-convergence evaluation protocol, training-average ALT may not represent the final policy.

invented entities (2)

Perfect Alternation (PA) reference regime
purpose: Reference turn-taking regime used as the benchmark for all ALT metrics and PA-equivalent statements.
Defined as any order in which each agent wins once per n-episode block. It is a normative benchmark, not an independently observed entity; its status rests on an asserted equivalence to the identical-days TEF setting.
ALT metric family (CALT, EALT, AALT, FALT, qFALT, qEALT)
purpose: Six scoring functions intended to measure temporal alternation quality in repeated games.
Newly proposed by the authors (building on prior self-cited metrics). No external benchmark or axiom set validates that they measure what they claim, so they carry no independent evidential handle.

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Cite this review

Pith. "Pith review of The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games." pith.science (2026). https://pith.science/paper/XXJOVYPA

@misc{pith2026260305789,
  author       = {Pith},
  title        = {Pith review of: The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXJOVYPA}},
  note         = {Machine review of arXiv:2603.05789}
}

read the original abstract

Repeated multi-agent interactions require evaluation metrics that capture not only payoff distributions but also their temporal organization. Conventional outcome-based fairness measures can assign similar aggregate scores to temporally distinct coordination patterns, obscuring whether access to a shared resource is genuinely rotating or persistently monopolized. We study this problem in the Honey-Jar Game (HJG), a minimally dynamic repeated threshold-congestion Markov game in which n agents compete for exclusive access to a single high-reward resource. We introduce Perfect Alternation (PA), a reference turn-taking regime corresponding to the n-periodic round-robin picking sequence, together with six novel Alternation (ALT) metrics and a benchmarking methodology mapping ALT values to interpretable PA-equivalent performance. Using Q-learning agents as a minimal adaptive baseline against analytically derived random-policy baselines, we uncover a clear measurement failure: despite deceptively high traditional metrics (e.g., reward fairness often exceeding 0.9), learned policies perform worse than random on every ALT metric, by 34-74% on CALT and up to 92% on EALT, with PA-equivalent coordination falling to roughly one-fifth of the population at n=10. EALT further reveals two distinct failure patterns: without episodic memory (Type-A), the deficit grows from -20% at n=2 to -92% at n=10; with episodic memory (Type-B), it reaches a trough of -76% at n=5 before partially recovering to -10% at n=10. Conventional efficiency and fairness metrics do not reveal these differences. The ALT framework complements the temporal fair division and picking-sequence literature by diagnosing whether temporal fairness emerges spontaneously in decentralized adaptive systems rather than how to enforce it.

Figures

Figures reproduced from arXiv: 2603.05789 by the authors.

**Figure 2.** Perfect Alternation Equivalent Performance Degradation. Q-learning agents (blue [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 3.** Traditional Metrics Fail to Detect Coordination Failure. Each column shows one [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Symmetric Coordination Failure in 3-Agent Systems. Despite nearly identical high [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

**Figure 5.** Figure 5: CALT Progression During Q-Learning Training. CALT values (solid lines) decrease [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Temporal Fair Division in Multi-Agent Systems: From Precise Alternation Metrics to Scalable Coordination Proxies
cs.MA 2026-05 unverdicted novelty 6.0 of 10

RP detects turn-taking failures invisible to reward fairness and reproduces ALT-metric rankings with Spearman ρ ≥ 0.95 while running 12–25× faster in n=2..10 experiments.

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