Learning Ordinal Response Policies in Rank-Based Stochastic Prize-Collecting Games

Malintha Fernando; Petter \"Ogren; Silun Zhang

REVIEW 4 major objections 6 minor 18 references

Learning Ordinal Response Policies in Rank-Based Stochastic Prize-Collecting Games

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper extends the Team Orienteering Problem to self-interested agents and proves that on complete graphs, rank-based conflict resolution yields a unique pure Nash equilibrium where each agent claims the prize matching its rank.

desk verdict New game formulation and a useful ordinal-rank feature, but Theorem 1's proof ignores travel costs and budget, so the central equilibrium claim is not yet established. read the letter →

arxiv 2510.24515 v2 pith:7E44RK74 submitted 2025-10-28 cs.RO

Malintha Fernando , Petter Ögren , Silun Zhang This is my paper · ORCID

classification cs.RO

keywords StochasticPrize-CollectingGamesTeamOrienteeringProblempureNashequilibriumordinalrankmulti-agentreinforcementlearningfictitiousplaygraphroutingrank-basedconflictresolution

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

The paper builds a bridge between cooperative routing and competitive routing. It introduces Stochastic Prize-Collecting Games (SPCGs), in which self-interested agents on a graph compete for stochastic prizes under travel budgets and a fixed rank-based conflict rule. Its central theoretical claim is that on complete graphs (and star graphs) this game has a unique pure Nash equilibrium: at each stage the $i$-th ranked agent should take the $i$-th largest remaining prize, and the resulting total reward matches the optimal cooperative Team Orienteering solution. On arbitrary graphs a pure equilibrium need not exist, so the paper proposes the ordinal rank, a local rank inside an agent's current neighborhood, as the right information for learning best responses. Experiments on road networks report that ordinal-rank-conditioned policies reach 87–95% of the optimal cooperative total reward and scale to far more agents than global-rank-conditioned policies.

What carries the argument

The argument is carried by two objects. The first is the rank-based conflict-resolution rule: when several agents visit the same node at the same time, the lowest-index (highest-ranked) agent takes the whole prize and everyone else receives zero. The second is the ordinal rank, $I_i = 1 + |\{j \in \lambda(i) : j < i\}|$, where $\lambda(i)$ is the connected component containing $i$ in the separating graph, whose edges join agents with overlapping reachable sets. The ordinal rank collapses global position and opponent locations into a single local number; on complete graphs it coincides with global rank, and Theorem 1 reduces best response to a one-line rule, 'take the $i$-th largest remaining prize'. On incomplete graphs, it is the information that lets a myopic agent best-respond to its immediate opponents, and it is the conditioning variable that makes the shared learned policies generalize and scale.

What would settle it

Run a two-agent complete graph with start $s$, terminal $d$, budget $L=3$, edge costs $w(s,A)=1$, $w(A,d)=1$, $w(s,B)=2$, $w(B,d)=2$, $w(s,C)=0.5$, $w(C,d)=0.5$, and prizes $p_A=10$, $p_B=9$, $p_C=8$, with $p_d$ very large. If agent 1 takes $A$, agent 2's prescribed equilibrium action is $B$ (the second-largest prize), yielding 9 and no terminal reward because $3-2 < w(B,d)=2$; deviating to $C$ yields $8+p_d$, which is larger. This instance would refute Theorem 1 as stated.

Watch

Extended reading notes

Core claim

At its core, the paper claims that rank-based conflict resolution makes competitive prize-collecting routing strategically tractable. On a complete graph, for every prize realization, the $i$-th ranked agent's unique best response at every stage is to move to the $i$-th largest remaining prize; this profile is the unique pure Nash equilibrium, and its total team reward equals the optimal solution of the equivalent Team Orienteering Problem, so competition costs nothing in that setting. On arbitrary graphs the situation changes: the paper exhibits a two-agent game with no pure Nash equilibrium, showing that simultaneous moves destroy the Stackelberg guarantees of earlier leader-follower formulations. To make the equilibrium playable under local information, the paper defines the ordinal rank of an agent inside its connected component of the separating graph, computes it with Ordinal Rank Search (ORS), and trains policies with Fictitious Ordinal Response Learning (FORL), a sequential entropy-regulated fictitious-play scheme. Empirical evaluations on road networks report 87–95% of the optimal cooperative TOP reward, better zero-shot transfer to imbalanced prize distributions, and near-linear scaling to 25 agents for ordinal-rank-conditioned shared policies.

Load-bearing premise

The load-bearing premise is that the $i$-th largest remaining prize is always reachable with the agent's remaining budget and still leaves enough budget to reach the terminal; the proof of Theorem 1 only checks the budget as a terminal condition, so with general edge costs a cheaper nearby prize can dominate the rank-matching prize.

Editorial extensions

If this is right

On complete graphs, self-interested agents do not need to model opponents' intentions: the rank-based conflict rule turns best response into a simple 'take the prize matching your rank' rule, and the equilibrium total reward equals the cooperative TOP optimum for the same budget.
On arbitrary graphs, pure equilibria can fail, so the right design question is what local information lets a learning agent approximate best responses; the paper shows that ordinal rank is one such effective statistic.
Because ordinal-rank conditioning aliases many global ranks into the same local state, a policy trained on a small team can be deployed zero-shot on larger teams with roughly linear reward growth in the tested road-network settings.
The FORL training procedure provides a convergent scheme in independent learning: it terminates at a pure Nash equilibrium when one exists and otherwise at a policy whose entropy lies between prescribed confidence bounds.
In the tested environments, the gap between the competitive equilibrium and the optimal cooperative routing solution is 5–13% of total reward, quantifying the price of anarchy for these games.

Reading between the lines

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

The paper leaves open whether ordinal-rank conditioning helps in other graph-structured multi-agent games, such as congestion games or auction-based task allocation, where an agent's payoff depends mainly on nearby better-ranked rivals.
The complete-graph result suggests a natural next test: SPCGs with random edge failures or time-varying travel costs, where the rank-matching rule may need to be replaced by a cost-aware ordinal response.
Because the Theorem 1 proof treats the budget only as a terminal condition, a stronger and untested extension would be a reachability-aware variant of the ordinal response rule; such a variant could close the gap between the paper's equilibrium claim and settings with highly non-uniform edge costs.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The new piece here is real: a simultaneous-move stochastic prize-collecting game with rank-based conflict resolution, plus the ordinal-rank (OR) representation and the FORL training scheme. The empirical story—OR-conditioned shared policies scaling better than global-rank ones, and beating global-state baselines—is plausible and worth a look. This is not just a rehash of the leader-follower paper [8]; the simultaneous-move setting genuinely changes the problem.

What the paper does well: the experiments are suggestive, the OR idea is practical, and the two-agent counterexample in Theorem 2 (no PNE on a general graph) is a clean sanity check. The prose is mostly clear about what is being claimed, and the connection to TOP/MILP is useful.

The soft spots are in the theory, and they are load-bearing. Theorem 1 asserts a unique pure Nash equilibrium on complete graphs where the i-th agent always takes the i-th largest remaining prize. The proof only uses travel cost and remaining budget as a terminal condition; it never checks feasibility or opportunity cost of choosing a distant large prize versus a nearby smaller one plus terminal reward. The stress-test counterexample is valid and breaks the claim as stated. So the abstract's equivalence with the TOP optimum and Remark 1 do not follow. This is not a minor gap—it is the paper's central theoretical anchor. There is also a circular flavor in Proposition 3: identifying zero-entropy with PNE just restates that pure profiles have zero entropy; it does not prove best-response. The convergence claim therefore overreaches. No code or data are released, which makes the empirical claims harder to verify.

Who this is for: readers working on competitive multi-robot routing, especially people interested in state aliasing via local rank features. They will get a useful model and an interesting training trick, but they should not cite the equilibrium theorem as proven. The ordinal-rank idea deserves a second paper with repaired theory and released artifacts.

For peer review: I would send it out, because the problem and the OR representation are worth referee time, but it needs substantial revision before acceptance. My verdict is skeptical on the current form, not on the idea.

Referee Report

4 major / 6 minor

Summary. The paper proposes Stochastic Prize-Collecting Games (SPCGs), a game-theoretic extension of the Team Orienteering Problem to self-interested agents with energy constraints, stochastic prizes, and rank-based conflict resolution. The central theoretical claim is that on complete graphs there is a unique pure Nash equilibrium in which the i-th ranked agent repeatedly takes the i-th largest remaining prize, and that this equilibrium coincides with the optimal routing solution of an equivalent cooperative TOP. The paper then introduces Ordinal Rank Search (ORS) to compute local ordinal ranks, and Fictitious Ordinal Response Learning (FORL) as an entropy-regulated sequential learning algorithm claimed to converge to a PNE when one exists. Empirical experiments on road-network graphs compare FORL, independent policy learning, and parameter sharing, and report 87%-95% optimality relative to a MILP solution of an equivalent TOP.

Significance. If the theoretical results were correct, the paper would offer a valuable and unusual result: a finite-horizon, simultaneous-move stochastic game on graphs with a unique pure equilibrium that matches a cooperative optimum under a simple rank-based priority rule. The ordinal-rank representation is also a plausible and interesting inductive bias for multi-agent RL on graphs, and the empirical study addresses an important practical setting. However, the paper's load-bearing theoretical claims are not sound as stated. Theorem 1 is falsified by a simple finite-budget example, and Proposition 3's convergence argument conflates determinism with equilibrium. Since these flaws directly support the abstract's uniqueness claim, the TOP-equivalence claim, and the interpretation of the empirical results as Price of Anarchy, the paper cannot be accepted in its current form.

major comments (4)

[Section IV, Theorem 1 and Remark 1] The proof of Theorem 1 is not sound as stated. The argument compares only the immediate prize of the i-th largest remaining node with deviations 'left' and 'right', and treats the budget only in the terminal clause 'If the i-th prize is 0, or l_t is not sufficient to reach another prize, then move to d.' But the total payoff includes the remaining budget and the terminal reward, and on a complete graph with arbitrary travel costs W these determine whether a deviation is profitable. A concrete counterexample is V={s,A,B,C,d}, n=2, L=2, W(s,A)=W(s,B)=W(s,C)=W(s,d)=1, W(A,d)=W(B,d)=10, W(C,d)=1, and p_A=10, p_B=9, p_C=8, p_d=100. The claimed profile is agent1->A, agent2->B. Under the standard orienteering interpretation that the terminal reward is collected only if d is reached within the remaining budget, agent2 receives 9 from B but can receive 8+100=108 by deviating to C, so B is not a best response; agent1 similarly prefers d (100) to A (10). If instead d is reachable at any time regardless of budget, the theorem fails even with zero travel costs because the senior agent should always move to d immediately. Thus Theorem 1 needs either additional feasibility restrictions or a different terminal model; as written it does not support the claimed unique PNE, nor does Remark 1's assertion that the PNE team reward equals the equivalent TOP optimum.
[Section V.B, Proposition 3] Proposition 3's convergence claim is circular and unsupported. The proof argues that if a PNE exists, then it is a pure strategy profile, hence has entropy zero, and the algorithm will not terminate until H_stop=0. This only establishes that a terminating policy may be deterministic; zero entropy is not a certificate of Nash equilibrium, and the algorithm's stopping rule is an entropy threshold, not a best-response check. Consequently the claim that FORL terminates at a PNE when one exists is not established. In the non-PNE case, the asserted chain H_{κ(t+1)} ≤ H(π^{θ_t}_j) ≤ H(π^{θ_t}_i) ≤ H_{κ(t)} for j<i is stated without proof and, even if true, bounds entropies rather than best-response distances. The abstract's statement that FORL is an 'entropy-regulated algorithm to obtain convergent policies' therefore overstates what is proved.
[Section IV, Remark 2 and Section V] The generalization from complete graphs to incomplete graphs via ordinal ranks is not formally established. Remark 2 says an agent only needs its rank, the player count, and the graph state to compute the PNE, but this rests entirely on Theorem 1, which applies at best to complete graphs. For incomplete graphs, Section V asserts that best responding to one's immediate opponents is sufficient, but no theorem or derivation is provided for this claim. Because the subsequent empirical success of OR-conditioning is interpreted through this unproved sufficiency, the paper should either prove a formal statement about localized stage games or clearly present the OR claim as a heuristic rather than as a consequence of the equilibrium analysis.
[Section VI.B] The reported 87%-95% figures are described as Price of Anarchy, but PoA is a worst-case ratio over equilibria, whereas the experiments evaluate one learned policy trajectory. The comparison to the MILP solution additionally presupposes both that the learned profile is at a PNE and that the PNE team reward equals the equivalent TOP optimum, both of which depend on the unsupported Theorem 1 and Proposition 3. These numbers should be presented as empirical optimality gaps of specific trained policies, not as formal PoA bounds.

minor comments (6)

[Abstract] The abstract contains a duplicated phrase: 'we also propose we propose Fictitious Ordinal Response Learning'.
[Section I] The text 'the robotsmustcompetein reward-scarce environments' is missing spacing; please correct the formatting.
[Section III] In the prize re-drawing rule, the condition 'p_t = 0' should presumably be 'p_v^t = 0' or similar; the current notation is ambiguous.
[Algorithm 2] The pseudocode appears to have an ordering issue for n=2: after j=1 satisfies the entropy condition, line 10 increments j to 2 and then line 11 immediately triggers the reset because j≥n, so the second agent is never updated in the fictitious-play stage. Please clarify the intended control flow.
[Section V.A] The sentence 'compute their ordinal ranks... by sorting their global ranks in the ascending order' contains the typo 'λ . ∈Λ'; also the notation around ordinal rank computation could be made more explicit.
[References] The MILP benchmark is attributed to [18], but [18] is a survey on the orienteering problem; a specific MILP formulation reference would be more appropriate.

Circularity Check

1 steps flagged · score 5.0 of 10

FORL convergence proof equates zero entropy with PNE; other theoretical claims are not circular.

self definitional [Section V-B, Proposition 3 and its proof (page 5-6)]
"Proposition 3. Suppose t max is chosen sufficiently large and H_stop = 0. Then algorithm 2 will terminate with a joint policy π* = ×_{i∈N}π^{θ*_i}_i which satisfies (1) if the game has a PNE, policy π* is a PNE... Proof: (Case 1). If the game has a PNE, denoted by π# = ×_{i∈N}π^{θ#_i}_i, then H[π#] = 0 since it is a pure strategy. The algorithm will not terminate until reaching H_stop = 0 for a large enough t_max."

The proof of clause (1) never checks the no-profitable-deviation condition required by Definition 1. It only notes that a pure PNE has zero entropy and that the algorithm stops when entropy reaches H_stop=0, then labels the terminal policy a PNE. This reduces the game-theoretic conclusion to the algorithmic stopping rule: zero entropy, i.e., pure determinism, is treated as synonymous with PNE. The absence of profitable unilateral deviations is assumed rather than derived, so the claimed convergence to PNE is asserted by construction of the stopping criterion, not established by best-response verification.

full rationale

The paper's central equilibrium theorem (Theorem 1) is a direct application of the rank-based reward rule and is not circular; its known weakness is a correctness gap about travel costs and remaining budgets, not a self-referential reduction. The ordinal-rank representation is definitional, but it is not fitted to the results and does not smuggle in the conclusions. Self-citations are limited to reference [12] and are not load-bearing for the main claims. The empirical optimality comparisons against an external MILP benchmark (87%-95%) provide independent, non-circular support for the learning results. The only substantive circularity is in Proposition 3, where the proof infers PNE from zero entropy, effectively equating the stopping condition with the equilibrium property. Because FORL's convergence-to-PNE guarantee is a central algorithmic claim, this is partial circularity rather than a minor blemish, warranting a score of 5 rather than 2.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests less on fitted constants than on unstated modeling assumptions. No scientific constant is fitted, but the algorithm's termination properties depend on hand-set entropy thresholds, and the theoretical results depend on the feasibility of rank-ordered prize choices under budget constraints, on an unproved fictitious-play convergence, and on the sufficiency of one-hop reachability for defining immediate opponents.

free parameters (3)

H_stop = set to 0 in Proposition 3
The early-stopping entropy threshold is hand-chosen, and the convergence proof sets it to 0 so that zero-entropy policies are treated as pure and hence Nash, which is the core unsupported step.
H_0 = 0.6 to 0.8 times H_max in experiments
The initial freezing point is selected by hand and determines which agent updates first in the bootstrapping stage; no sensitivity analysis is provided.
delta_h = not specified
The entropy-decay step controls the termination schedule of FORL, but no value is reported and its effect on convergence is not analyzed.

assumptions (5)

standard math Nash's theorem guarantees at least one mixed-strategy equilibrium in finite SPCGs.
Invoked in Section IV before the pure equilibria analysis to justify that mixed equilibria always exist.
domain assumption Rank-based conflict rule in Eq. (2) assigns each prize wholly to the lowest-index agent visiting the node.
This is the environment model that drives all equilibrium results, not a derived consequence.
ad hoc to paper A stage-wise choice of the i-th largest prize is feasible and optimal under arbitrary travel costs W and finite budget L_max.
Theorem 1's proof uses this without deriving it from the game dynamics; it is false when a distant large prize prevents the agent from reaching the terminal or future prizes.
ad hoc to paper Alternating fictitious play with entropy thresholds converges to a pure Nash equilibrium whenever one exists in SPCGs.
Proposition 3 asserts termination with a PNE from entropy reaching 0, but no convergence theorem for general-sum stochastic games is provided or cited.
domain assumption One-hop reachable-set intersections identify all agents whose strategies can affect i's payoff.
Definitions 2 through 4 rely on this assumption, but multi-step future conflicts between agents whose current reachable sets are disjoint are not captured.

invented entities (1)

Ordinal rank I_i
purpose: Compact local rank feature for policy conditioning, claimed to act as an inductive bias and improve generalization.
Defined internally in Eq. (4); its benefit is only demonstrated in this paper's simulations, with no external falsifiable prediction.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Learning Ordinal Response Policies in Rank-Based Stochastic Prize-Collecting Games." pith.science (2026). https://pith.science/paper/7E44RK74

@misc{pith2026251024515,
  author       = {Pith},
  title        = {Pith review of: Learning Ordinal Response Policies in Rank-Based Stochastic Prize-Collecting Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7E44RK74}},
  note         = {Machine review of arXiv:2510.24515}
}

read the original abstract

The Team Orienteering Problem (TOP) generalizes many real-world multi-agent scheduling and routing tasks that occur in autonomous mobility, aerial logistics, and surveillance applications. While many flavors of the TOP exist for planning in multi-agent systems, they assume that all the agents cooperate toward a single objective; therefore, they do not extend to settings when they compete in reward-scarce environments. We propose Stochastic Prize-Collecting Orienteering Games (SPCOG) as an extension of the TOP to plan in the presence of self-interested agents operating on a graph, under energy constraints and stochastic transitions. A theoretical discussion on complete and star graphs establishes that there is a unique pure Nash equilibrium in SPCOGs that coincides with the optimal routing solution of an equivalent TOP under rank-based conflict resolution. We propose the concept of Ordinal Rank (OR) as a concise representation of an agents' global rank and its location within a topological, well-defined neighborhood. Empirical evaluations conducted on real-world, road-network graphs under both dynamic and stationary prize distributions show that in parameter-sharing settings, the policies that leverage local information can outperform those policies leverage global information when the former is conditioned on the OR rather than the global rank, indicating that the OR acts as a strong inductive bias in multi-agent games on graphs. The OR-conditioned policies also generalize much better to games with large number of agents compared to global-rank conditioned policies. Finally, we also propose we propose Fictitious Ordinal Response Learning (FORL) as an entropy-regulated algorithm to obtain convergent policies in independent-learning settings in prize-collecting games on graphs.

Figures

Figures reproduced from arXiv: 2510.24515 by the authors.

**Figure 1.** Left: A multi-robot prize-collecting problem on a graph. Here, S, D denote the start and terminal nodes. One robot gets priority during conflicts. In the TOP, the optimal policy (shown) routes the robots to maximizes the total reward regardless of their ranks. However, when the robots are self-interested, the senior robot must directly move to prize (2.5) ignoring the prize (1) to prevent being cut-in-front, while t… view at source ↗

**Figure 2.** A two-agent SPCG played on a graph defined by [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 5.** (a) The optimality-gap between the SPCG and an equivalent TOP. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: (a) Convergence plots for a dynamic prize distribution graph with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 6.** Figure 6: (a) Zero-shot transfer to region-specific, normally-distributed prizes. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 15 canonical work pages

[8]

Prize collecting multiagent orienteer- ing: Price of anarchy bounds and solution methods,

T. Murray, J. Garg, and R. Nagi, “Prize collecting multiagent orienteer- ing: Price of anarchy bounds and solution methods,”IEEE Transactions on Automation Science and Engineering, vol. 19, no. 1, pp. 531–544, 2020

work page 2020
[1]

The team orienteering problem,

I.-M. Chao, B. L. Golden, and E. A. Wasil, “The team orienteering problem,”European journal of operational research, vol. 88, no. 3, pp. 464–474, 1996

work page 1996
[2]

Multi- robot scheduling for environmental monitoring as a team orienteering problem,

A. Mansfield, S. Manjanna, D. G. Macharet, and M. Ani Hsieh, “Multi- robot scheduling for environmental monitoring as a team orienteering problem,” in2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6398–6404, 2021

work page 2021
[3]

A spatio-temporal representation for the orienteering problem with time-varying profits,

Z. Ma, K. Yin, L. Liu, and G. S. Sukhatme, “A spatio-temporal representation for the orienteering problem with time-varying profits,” in2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6785–6792, 2017

work page 2017
[4]

Orienteering Problem: A survey of recent variants, solution approaches and applications,

A. Gunawan, H. C. Lau, and P. Vansteenwegen, “Orienteering Problem: A survey of recent variants, solution approaches and applications,” European Journal of Operational Research, vol. 255, pp. 315–332, Dec. 2016

work page 2016
[5]

Learn to solve the min-max multiple traveling salesmen problem with reinforcement learning,

J. Park, C. Kwon, and J. Park, “Learn to solve the min-max multiple traveling salesmen problem with reinforcement learning,”

work page
[6]

Competing for the most profitable tour: The orienteering interdiction game

E. ´Alvarez Miranda, M. Sinnl, and K. Tanınmıs ¸, “Competing for the most profitable tour: The orienteering interdiction game,” July 2024. arXiv:2407.02959 [math]

work page Pith review arXiv 2024
[7]

DIRECT: A scalable approach for route guidance in Selfish Orienteering Problems,

P. Varakantham, H. Mostafa, N. Fu, and H. C. Lau, “DIRECT: A scalable approach for route guidance in Selfish Orienteering Problems,”

work page

Show all 18 references

[9]

Collaborative dynamic scheduling in a self-organizing manufacturing system using multi-agent reinforcement learning,

Y . Gui, Z. Zhang, D. Tang, H. Zhu, and Y . Zhang, “Collaborative dynamic scheduling in a self-organizing manufacturing system using multi-agent reinforcement learning,”Advanced Engineering Informatics, vol. 62, p. 102646, 2024

2024
[10]

Optimizing task scheduling in human- robot collaboration with deep multi-agent reinforcement learning,

T. Yu, J. Huang, and Q. Chang, “Optimizing task scheduling in human- robot collaboration with deep multi-agent reinforcement learning,”Jour- nal of manufacturing systems, vol. 60, pp. 487–499, 2021

2021
[11]

A dynamic task assignment model for aviation emergency rescue based on multi- agent reinforcement learning,

Y . Shen, X. Wang, H. Wang, Y . Guo, X. Chen, and J. Han, “A dynamic task assignment model for aviation emergency rescue based on multi- agent reinforcement learning,”Journal of Safety Science and Resilience, vol. 4, no. 3, pp. 284–293, 2023

2023
[12]

Graph at- tention multi-agent fleet autonomy for advanced air mobility,

M. Fernando, R. Senanayake, H. Choi, and M. Swany, “Graph at- tention multi-agent fleet autonomy for advanced air mobility,”ArXiv, vol. abs/2302.07337, 2023

2023 arXiv
[13]

Consensus-based decentralized auctions for robust task allocation,

H.-L. Choi, L. Brunet, and J. P. How, “Consensus-based decentralized auctions for robust task allocation,”IEEE transactions on robotics, vol. 25, no. 4, pp. 912–926, 2009

2009
[14]

Optimal cost-sharing in general resource selection games,

V . Gkatzelis, K. Kollias, and T. Roughgarden, “Optimal cost-sharing in general resource selection games,”Operations Research, vol. 64, no. 6, pp. 1230–1238, 2016. 9

2016
[15]

Nash q-learning for general-sum stochastic games,

J. Hu and M. P. Wellman, “Nash q-learning for general-sum stochastic games,”Journal of machine learning research, vol. 4, no. Nov, pp. 1039– 1069, 2003

2003
[16]

Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations,

Y . Shoham, “Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations,” p. 532
[17]

Stabilizing Transformers for Reinforcement Learning,

E. Parisotto, H. F. Song, J. W. Rae, R. Pascanu, C. Gulcehre, S. M. Jayakumar, M. Jaderberg, R. L. Kaufman, A. Clark, S. Noury, M. M. Botvinick, N. Heess, and R. Hadsell, “Stabilizing Transformers for Reinforcement Learning,” Oct. 2019. arXiv:1910.06764 [cs, stat]

2019 arXiv
[18]

Orienteering problem: A survey of recent variants, solution approaches and applications,

A. Gunawan, H. C. Lau, and P. Vansteenwegen, “Orienteering problem: A survey of recent variants, solution approaches and applications,” European Journal of Operational Research, vol. 255, no. 2, pp. 315–332, 2016

2016

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [8]

Prize collecting multiagent orienteer- ing: Price of anarchy bounds and solution methods,

T. Murray, J. Garg, and R. Nagi, “Prize collecting multiagent orienteer- ing: Price of anarchy bounds and solution methods,”IEEE Transactions on Automation Science and Engineering, vol. 19, no. 1, pp. 531–544, 2020

work page 2020

[2] [1]

The team orienteering problem,

I.-M. Chao, B. L. Golden, and E. A. Wasil, “The team orienteering problem,”European journal of operational research, vol. 88, no. 3, pp. 464–474, 1996

work page 1996

[3] [2]

Multi- robot scheduling for environmental monitoring as a team orienteering problem,

A. Mansfield, S. Manjanna, D. G. Macharet, and M. Ani Hsieh, “Multi- robot scheduling for environmental monitoring as a team orienteering problem,” in2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6398–6404, 2021

work page 2021

[4] [3]

A spatio-temporal representation for the orienteering problem with time-varying profits,

Z. Ma, K. Yin, L. Liu, and G. S. Sukhatme, “A spatio-temporal representation for the orienteering problem with time-varying profits,” in2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6785–6792, 2017

work page 2017

[5] [4]

Orienteering Problem: A survey of recent variants, solution approaches and applications,

A. Gunawan, H. C. Lau, and P. Vansteenwegen, “Orienteering Problem: A survey of recent variants, solution approaches and applications,” European Journal of Operational Research, vol. 255, pp. 315–332, Dec. 2016

work page 2016

[6] [5]

Learn to solve the min-max multiple traveling salesmen problem with reinforcement learning,

J. Park, C. Kwon, and J. Park, “Learn to solve the min-max multiple traveling salesmen problem with reinforcement learning,”

work page

[7] [6]

Competing for the most profitable tour: The orienteering interdiction game

E. ´Alvarez Miranda, M. Sinnl, and K. Tanınmıs ¸, “Competing for the most profitable tour: The orienteering interdiction game,” July 2024. arXiv:2407.02959 [math]

work page Pith review arXiv 2024

[8] [7]

DIRECT: A scalable approach for route guidance in Selfish Orienteering Problems,

P. Varakantham, H. Mostafa, N. Fu, and H. C. Lau, “DIRECT: A scalable approach for route guidance in Selfish Orienteering Problems,”

work page