Reinforcement Learning for Game-Theoretic Resource Allocation on Graphs

Lifeng Zhou; Zijian An

REVIEW 3 major objections 6 minor 39 references

Reinforcement Learning for Game-Theoretic Resource Allocation on Graphs

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Standard deep reinforcement learning, with graph-constrained action masking, learns winning strategies in multi-step Colonel Blotto games on graphs.

desk verdict The action-mask construction and asymmetric-graph experiments earn a look, but the MDP formulation is broken: the state s=d1-d2 discards the information needed to define legal actions, so the theoretical claim does not hold as written. read the letter →

arxiv 2505.06319 v1 pith:GHUK42AR submitted 2025-05-08 cs.LG cs.GT

Zijian An , Lifeng Zhou This is my paper · ORCID

classification cs.LGcs.GT

keywords game-theoreticresourceallocationmulti-stepColonelBlottogameMarkovdecisionprocessaction-displacementadjacencymatrixdeepQ-networkproximalpolicyoptimizationasymmetricgraphsgraph-constrainedactionmasking

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 tries to establish that the multi-step Colonel Blotto game on a graph, where two players move indistinguishable resource units along edges over multiple turns to control a majority of nodes, can be treated as a Markov decision process and solved by off-the-shelf deep reinforcement learning. The authors introduce the action-displacement adjacency matrix, which derives the set of legal moves from the current resource distribution and the graph's adjacency matrix, so a fixed neural-network output layer can be masked to respect graph constraints. Experiments on five small graphs with DQN and PPO show win rates above random and greedy baselines, convergence to a balanced 50% win rate when the two sides use the same learned policy, and exploitation of asymmetric structures, including by the structurally disadvantaged player.

What carries the argument

The action-displacement adjacency matrix $J$ is the load-bearing object. For a distribution $d$ with $M$ units, the paper assigns each resource a one-hot row for its node, forms $J = \hat{d} H'$, where $H'$ is the adjacency matrix $H$ with row $i$ cyclically left-shifted by $i$ positions, and reads $J_{j,k}=1$ as "resource $j$ may move $k$ steps clockwise." This matrix converts the graph constraint into a mask over a fixed $M$-by-$N$ action table, so DQN's Q-values and PPO's policy logits can be filtered to legal actions; Theorem 1 proves the mask is exactly the set of legal moves, and Theorem 2 gives the deterministic update $d' = \sum_j \hat{d}_{j,:} P^{a_j}$ for the next distribution. Together they turn a moving-boundary action space into a fixed-size, distribution-dependent mask.

What would settle it

Compare two positions that give the same state $s$ but different distributions: on a two-node path with $M=3$, $d_1=(3,0), d_2=(2,1)$ and $d_1'=(2,1), d_2'=(1,2)$ both give $s=(1,-1)$, yet the first player's legal moves from $d_1$ differ from those from $d_1'$. If a DQN trained on $s$ alone assigns a different Q-value to the same $(s,a)$ pair depending on which hidden distribution produced it, or if replacing $s$ by the full pair $(d_1,d_2)$ changes the reported win rates, the MDP formulation's sufficiency claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that an MDP formulation with state $s = d_1 - d_2$, reward $+1/0/-1$ by node majority, and the action-displacement matrix $J$ (built by cyclically shifting each row of the graph adjacency matrix by the node index and lifting the resource distribution to one-hot rows) makes the graph-constrained multi-step Blotto game learnable by DQN and PPO. Theorem 1 states that for a resource at node $n_j$, a displacement $k$ is legal exactly when $J_{j,k}=1$, i.e. $H_{n_j,(n_j+k) \bmod N}=1$. Empirically the trained policies beat random and greedy opponents, approach 50-50 play against another learned policy, and on asymmetric graphs exploit permanent-control and forced-movement structures enough to raise win rates far above random-policy baselines (DQN and PPO reach 100% on $G_2$, and the disadvantaged side's win rate improves from 20% to 25-33%).

Load-bearing premise

The load-bearing premise is that the difference state $s = d_1 - d_2$ contains enough information to choose an optimal action, although the legal moves at each step depend on each player's full distribution and two different distributions can yield the same difference.

Editorial extensions

If this is right

On the five tested graphs, both DQN and PPO beat random and greedy opponents, so graph-constrained Blotto does not require bespoke game solvers; standard RL with legal-action masking suffices for the studied sizes.
When both players are trained RL agents, win rates settle near 50%, meaning mutual learning reaches a balanced outcome under symmetric initialization on $G_0$ and $G_1$.
Asymmetric graph structure is exploitable: DQN and PPO reach 100% win rate on $G_2$ where one node is unreachable, and the disadvantaged player still improves from 20% to 25-33% win rate by learning.
Randomness in initial distributions trades off against generalization: agents trained on fixed deterministic initializations beat the trained opponent 100% but drop to about 50% against a random opponent, while random initialization improves robustness.
With unequal resources (7 vs 8), RL raises the weaker player's win rate from 17% under random play to 23-24%, showing the learned policy partially compensates for numeric disadvantage.

Reading between the lines

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

The state $s = d_1 - d_2$ may be information-deficient: two positions with the same difference can have different legal action sets, so an $s$-only MDP has transitions that are not well-defined; a testable fix is to include each player's own distribution (or a factored encoding) and check whether win rates against adaptive opponents improve.
Because $J$ is built from cyclic shifts of the adjacency matrix, the masking operation is differentiable and batched, so the same construction should extend to larger graphs and to graph neural network policies, not just MLP DQN and PPO.
The asymmetric-graph results suggest learned policies discover structural 'fortresses' (e.g., holding an isolated node with one unit); formalizing this as a graph-theoretic condition on winning regions could predict when RL will succeed before running experiments.
Self-play convergence near 50% on symmetric graphs is consistent with either a mixed equilibrium or policy cycling; distinguishing the two by tracking the policies' entropy or by evaluating the final policy against a best-response oracle would clarify what the agents actually learned.

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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

Quick take: the action-displacement adjacency matrix is a nice piece of engineering, and the asymmetric-graph experiments show RL can exploit topology. But the paper's central formal claim — that the game is an MDP with state s = d1 − d2 — does not hold. Legal moves for each player depend on their own distribution d, and s does not determine d. So the transition kernel P(s'|s,a) is not well-defined, and the Bellman target max_{a'} Q(s',a') is not a function of s' alone. The paper even uses d1 and d2 to generate action masks in Figure 2, so the actual observation is richer than s. This is not a minor notational slip; it invalidates the "we formulate MCBG as an MDP" claim in Section III-A. The learned policies are heuristics for a partially observable problem, which is fine if presented that way, but the paper oversells the theory.

What is genuinely new: applying DQN/PPO to the multi-step Blotto game on graphs, with a graph-derived action mask. The J-matrix construction in Section III-B is a compact way to encode adjacency and resource counts into per-resource valid actions; Theorems 1 and 2 are simple linear algebra and the intended statements are correct. Theorem 1's proof, however, has a sign error: Eq. (4) defines P_{ij}=1 when j ≡ i−1 (mod N), which is a right shift, not the left shift claimed by e_kP = e_{k+1}. The statement is still true if you flip the convention, but as written it is internally inconsistent. Eq. (10) also looks misindexed in the unequal-resource case.

Empirically, the scope is small — 4–5 nodes, 7–8 resources — and the tables lack error bars, seeds, and hyperparameters. The greedy baseline is built from a pretrained DQN, which is unusual but not circular. The 50% self-play claim is plausible but not surprising: in a symmetric game with identical state representations, near-50% win rates are what you expect. The experiments on asymmetric graphs (G2–G4) are the most interesting: the RL agent learns to hold an isolated node or exploit forced moves. Those results are worth reporting as heuristics.

Bottom line: this is a solid workshop/short-paper candidate once the state-sufficiency issue is addressed head-on. The authors can either expand the state to (d1, d2) or (d1, s), which fixes the math but changes their dimensionality argument, or explicitly reframe as a POMDP and drop claims of optimality. They also need to release code and seeds, fix the notational errors, and temper the 'MDP' language. The action-mask construction is citable; the MDP claim is not. I would send it to peer review, but the referee should insist on these revisions.

Referee Report

3 major / 6 minor

Summary. The paper proposes a reinforcement learning framework for a multi-step Colonel Blotto game on graphs (GRAG). The authors formulate the game as an MDP with state s = d1 - d2 (the difference of the two players' resource distributions), introduce an action-displacement adjacency matrix that generates the set of valid graph-constrained actions at each step, and train DQN and PPO agents against random, greedy, and self-play opponents on small symmetric and asymmetric graphs. They report win rates on five graphs and conclude that both algorithms outperform baselines, exploit structural asymmetries, and converge to a balanced 50% win rate in self-play.

Significance. If the MDP formulation were sound, the paper would offer a useful engineering contribution: a concrete action-masking mechanism for graph-constrained Blotto games and an empirical comparison of two standard RL algorithms in that setting. The action-displacement adjacency matrix construction in Section III-B is concrete, and the linear-algebraic characterization in Theorems 1 and 2 is a helpful formalization. A strength is that the reported win rates are obtained against external baselines (random and greedy opponents) rather than fitted targets, so there is no circular fitting of the headline numbers. However, the central state-compression claim is not justified: the state s = d1 - d2 is not a sufficient statistic for the players' decision problem, which undermines the claim that the game is modeled as an MDP and the theoretical status of the learned policies. The experiments are on very small graphs and are reported without error bars or seed information, so the broad generalization claims should be tempered.

major comments (3)

[Section III-A, definition of S; Section III-B, Eq. (3)] The state s = d1 - d2 is not a sufficient statistic for the MDP. The valid action set in Eq. (3) is generated from the player's own full distribution d_t, and the next state s' = T(d1,a1) - T(d2,a2) depends on the absolute distributions through the resource locations. Two different underlying states can give the same s but different legal moves and different transitions. For example, with N=2, M=3 and an adjacency matrix H = [[1,1],[1,0]], s=(1,-1) is consistent with d1=(2,1), d2=(1,2) and also with d1=(1,2), d2=(0,3); in the first case Player 1 has two resources on node 0 and one on node 1, while in the second case the counts are reversed, so the valid action sets differ. Consequently P(s'|s,a) is not well-defined as a function of s alone, and the Bellman target max_{a'}Q(s',a') is not a function of s'. This invalidates the claim in Section II-A that the game satisfies the Markov property for the chosen state representation. In the implementation, the agent additionally conditions on its own distribution through the action mask, so the learned object is not actually a policy π(a|s) as described. To make the MDP formulation valid, the state must include the full distributions (d1,d2) or a sufficient statistic; otherwise the learning problem is a POMDP with an insufficient observation and the convergence claims do not follow.
[Section III-B, Theorem 1 proof, Eq. (4)-(6); Theorem 2, Eq. (8)] The permutation matrix P defined by P_{ij}=1 iff j ≡ i-1 (mod N) satisfies e_k P = e_{k-1}, not e_{k+1} as claimed in the text; it implements a right shift, not a left shift. The same sign error propagates into Theorem 2, where e_{n_j} P^{a_j} would move a resource to node n_j - a_j rather than the intended n_j + a_j. The intended left-shift construction requires P_{ij}=1 iff j ≡ i+1 (mod N), or equivalently the use of P^T. The final formulas in Eq. (7) and Eq. (8) are correct under the intended convention, so the theorem statements are likely correct, but the proofs as written are internally inconsistent and need correction.
[Abstract and Tables I-IV] The abstract claims that both DQN and PPO 'consistently outperform baseline strategies,' but several PPO test results are at or below chance against a random policy. For example, Table II (C3, Greedy π1, Random column) reports 43%, and Table IV (C4, Greedy π1, Random column) reports 43%; the RL-vs-RL rows in Tables II and IV also contain values such as 37%, 44%, and 40%. These entries contradict the unqualified 'consistently' claim. The performance statement should be qualified to distinguish training performance from generalization performance and to acknowledge that PPO often fails to beat a random opponent after training against a fixed greedy policy.

minor comments (6)

[Section III-A] The acronym 'CRAG' appears in the first paragraph; it should be 'GRAG'.
[Section III-C, Eq. (9)] The definition of J'_1 for virtual resources is unclear. Virtual resources should be constrained to action 0, so the corresponding rows should have a single 1 in the first column, but the notation '1T_{M-M1,0}' is ambiguous and appears to suggest an all-ones block. Please clarify the construction.
[Figure 7 caption] The caption refers to graphs G1 and G3, while the text describes the episodes as being on G2 and G4. These references should be made consistent.
[Section IV-E and Table V] The text states that PPO achieves a 71% win rate on G3, but Table V reports 72%. The numbers should be reconciled.
[Tables I-IV] The header 'Train/Test' and the notation '100%∼50%' are not defined. Please clarify which entries are training win rates and which are test win rates against random and greedy opponents, and define the '∼' notation.
[Section IV] No number of random seeds, standard deviations, or hyperparameter settings are reported for the win rates in Tables I-V. Adding this information is necessary to assess the reliability of the empirical claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RL win-rate results are empirical comparisons, and the valid-action encoding is definitional rather than a fitted prediction.

full rationale

The paper's central reported results—win rates against random, greedy, and self-play opponents—are obtained by training DQN and PPO in the described environment, not by fitting parameters to those outcomes. The action-displacement adjacency matrix J is constructed directly from the adjacency matrix H and current distribution d_t (Eq. 2), and Theorem 1 proves the equivalence J_{j,k}=H_{n_j,(n_j+k) mod N}; this is a formal encoding of the graph constraint, not a prediction derived from its own conclusion. No fitted constant is renamed as a discovery, and the 50% self-play convergence is an emergent empirical observation rather than an input. The only self-citations ([17], used for background on the on-graph CBG and in future work on heterogeneous resources) are not load-bearing: the paper redefines the game and MDP in Sections II-A and III-A/B, so those citations are background rather than the justification for the reported results. The potential insufficiency of the state s=d1-d2, where valid actions depend on absolute distributions, is a modeling-soundness concern, not a circularity: even if the MDP formulation were flawed, the experimental win rates would remain empirical outputs rather than identities with the model's inputs. Therefore no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the game being a zero-sum simultaneous-move MDP, on the sufficiency of the difference state s=d1-d2, on the availability of a self-loop for staying still, and on the episodic termination rule. The hyperparameters listed are unstated training choices that affect the reported win rates. No new physical or domain entities are introduced; the action-displacement adjacency matrix is a mathematical construction rather than a postulated entity.

free parameters (3)

Discount factor gamma = not reported
Standard DQN/PPO hyperparameter selected by hand; the value is not stated and affects the learned policy through reward discounting.
Exploration schedule and training iterations = not reported
Exploration probability, its decay, and total training steps are not stated; these choices affect convergence and final win rates.
Neural network architecture for DQN and PPO = not reported
The action space is N^M, e.g., 4^8=65536 for G0, and the paper does not describe how the network outputs Q-values or action probabilities over this space, nor the layer sizes used.

assumptions (5)

domain assumption The game is zero-sum with simultaneous moves and rewards of opposite sign (R2_t = -R1_t).
Stated in Section II-B and Section III-D; it justifies treating the two-player game as a symmetric RL problem and underlies the self-play 50% expectation. It is a standard Colonel Blotto property.
domain assumption The state s = d1 - d2 is a sufficient representation of the game for both players.
Section III-A defines S = {d1 - d2}. The paper motivates the compression by shared information and lower dimensionality but provides no proof of sufficiency. The valid action set (Eq. 3) depends on each player's own distribution d, so s alone does not determine the legal actions or the transition kernel. This is a load-bearing modeling assumption for the MDP formulation.
domain assumption A resource may always stay in place: the adjacency matrix H has H_ii = 1 for every node i.
Action 0 is defined as staying still, and Theorem 1 counts it as valid only when H_{n_j,n_j}=1. The example H in Eq. 2 has a diagonal of ones, but Figure 4 draws no self-loops, so the assumption is implicit rather than stated explicitly.
domain assumption The episode terminates as soon as one player controls strictly more nodes, with reward +1, -1, or 0 according to Eq. (1).
Section III-A defines the reward and Section III-D terminates episodes at R_t != 0. This turns the multi-step game into an episodic MDP. It is the authors' game rule and is not validated against alternative stopping rules.
standard math Standard linear algebra for cyclic shifts and one-hot lifting.
Theorems 1 and 2 use cyclic permutation matrices and lifted one-hot position matrices. These are standard; the only issue is the sign inconsistency in the definition of P (Eq. 4), which is a proof defect, not a mathematical input.

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Pith. "Pith review of Reinforcement Learning for Game-Theoretic Resource Allocation on Graphs." pith.science (2026). https://pith.science/paper/GHUK42AR

@misc{pith2026250506319,
  author       = {Pith},
  title        = {Pith review of: Reinforcement Learning for Game-Theoretic Resource Allocation on Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHUK42AR}},
  note         = {Machine review of arXiv:2505.06319}
}

abstract

Game-theoretic resource allocation on graphs (GRAG) involves two players competing over multiple steps to control nodes of interest on a graph, a problem modeled as a multi-step Colonel Blotto Game (MCBG). Finding optimal strategies is challenging due to the dynamic action space and structural constraints imposed by the graph. To address this, we formulate the MCBG as a Markov Decision Process (MDP) and apply Reinforcement Learning (RL) methods, specifically Deep Q-Network (DQN) and Proximal Policy Optimization (PPO). To enforce graph constraints, we introduce an action-displacement adjacency matrix that dynamically generates valid action sets at each step. We evaluate RL performance across a variety of graph structures and initial resource distributions, comparing against random, greedy, and learned RL policies. Experimental results show that both DQN and PPO consistently outperform baseline strategies and converge to a balanced $50\%$ win rate when competing against the learned RL policy. Particularly, on asymmetric graphs, RL agents successfully exploit structural advantages and adapt their allocation strategies, even under disadvantageous initial resource distributions.

Figures

Figures reproduced from arXiv: 2505.06319 by the authors.

**Figure 2.** Game flow and DQN process with Player 1 as the RL agent. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Game flow and PPO network with Player 1 as the RL agent. The game flow structure is the same as in Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Five graphs G0, G1, G2, G3, G4, where a node is indexed numerically and represents a discrete location for resource allocation. Directed arrows between nodes indicate one-way connectivity, specifying the allowable transitions of resources between locations. TABLE I WIN…

**Figure 5.** Figure 5: Illustration of four distinct initial resource distributions [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Examples of gameplay on G0 where Player 1 (red) has seven resources and Player 2 (blue) has eight resources. In our experiments, the RL agent (Player 1) is assigned 7 resources, while Player 2 has 8 resources. We use the case where both players follow a random policy a…

**Figure 7.** Figure 7: Examples of gameplay on G1 (7a) and G3 (7b) where the two players have the same units of resources. also run simulations with Player 2 as the RL agent. If both players follow random policies, their win rate is completely determined by the graph structure. Table V shows…

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S. Deng, Z. Xiang, P. Zhao, J. Taheri, H. Gao, J. Yin, and A. Y . Zomaya, “Dynamical resource allocation in edge for trustable internet-of-things systems: A reinforcement learning method,”IEEE Transactions on Industrial Informatics, vol. 16, no. 9, pp. 6103–6113, 2020

2020
[28]

Reinforcement learning agents in colonel blotto,

J. C. G. Noel, “Reinforcement learning agents in colonel blotto,”arXiv preprint arXiv:2204.02785, 2022

2022 arXiv
[29]

Human-level control through deep reinforcement learning,

V . Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski et al., “Human-level control through deep reinforcement learning,” nature, vol. 518, no. 7540, pp. 529–533, 2015

2015
[30]

Prox- imal policy optimization algorithms,

J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, “Prox- imal policy optimization algorithms,”arXiv preprint arXiv:1707.06347, 2017

2017 arXiv
[31]

Adaptiveε-greedy exploration in reinforcement learning based on value differences,

M. Tokic, “Adaptiveε-greedy exploration in reinforcement learning based on value differences,” inAnnual conference on artificial intel- ligence. Springer, 2010, pp. 203–210

2010
[32]

Pettingzoo: Gym for multi-agent reinforcement learning,

J. Terry, B. Black, N. Grammel, M. Jayakumar, A. Hari, R. Sullivan, L. S. Santos, C. Dieffendahl, C. Horsch, R. Perez-Vicenteet al., “Pettingzoo: Gym for multi-agent reinforcement learning,”Advances in Neural Information Processing Systems, vol. 34, pp. 15 032–15 043, 2021

2021
[33]

Tianshou: A highly modularized deep reinforcement learning library,

J. Weng, H. Chen, D. Yan, K. You, A. Duburcq, M. Zhang, Y . Su, H. Su, and J. Zhu, “Tianshou: A highly modularized deep reinforcement learning library,”Journal of Machine Learning Research, vol. 23, no. 267, pp. 1–6, 2022. [Online]. Available: http://jmlr.org/papers/v23/21-1127.html

2022
[34]

Deep reinforcement learning with double q-learning,

H. Van Hasselt, A. Guez, and D. Silver, “Deep reinforcement learning with double q-learning,” inProceedings of the AAAI conference on artificial intelligence, vol. 30, no. 1, 2016

2016
[35]

Dueling network architectures for deep reinforcement learning,

Z. Wang, T. Schaul, M. Hessel, H. Hasselt, M. Lanctot, and N. Freitas, “Dueling network architectures for deep reinforcement learning,” in International conference on machine learning. PMLR, 2016, pp. 1995– 2003

2016
[36]

Prioritized experience replay,

T. Schaul, J. Quan, I. Antonoglou, and D. Silver, “Prioritized experience replay,”arXiv preprint arXiv:1511.05952, 2015

2015 arXiv
[37]

A novel ddpg method with prioritized experience replay,

Y . Hou, L. Liu, Q. Wei, X. Xu, and C. Chen, “A novel ddpg method with prioritized experience replay,” in2017 IEEE international conference on systems, man, and cybernetics (SMC). IEEE, 2017, pp. 316–321

2017
[2022]

Available: https://api.semanticscholar.org/CorpusId: 250089045

[Online]. Available: https://api.semanticscholar.org/CorpusId: 250089045
[2023]

Available: https://api.semanticscholar.org/CorpusId: 259539732

[Online]. Available: https://api.semanticscholar.org/CorpusId: 259539732

Pith tools

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

[1] [1]

To compute or not to compute? adaptive smart sensing in resource-constrained edge computing,

L. Ballotta, G. Peserico, F. Zanini, and P. Dini, “To compute or not to compute? adaptive smart sensing in resource-constrained edge computing,”IEEE Transactions on Network Science and Engineering, vol. 11, pp. 736–749, 2022. [Online]. Available: https://api.semanticscholar.org/CorpusID:252090339

work page 2022

[2] [2]

Energy-efficient industrial internet of things in green 6g networks,

X. N. Fernando and G. L ˘az˘aroiu, “Energy-efficient industrial internet of things in green 6g networks,”Applied Sciences, 2024. [Online]. Available: https://api.semanticscholar.org/CorpusID:272909251

work page 2024

[3] [3]

Cooperative air and ground surveillance,

B. Grocholsky, J. Keller, V . Kumar, and G. Pappas, “Cooperative air and ground surveillance,”IEEE Robotics & Automation Magazine, vol. 13, no. 3, pp. 16–25, 2006

work page 2006

[4] [4]

Planning for opportunistic surveil- lance with multiple robots,

D. Thakur, M. Likhachev, J. Keller, V . Kumar, V . Dobrokhodov, K. Jones, J. Wurz, and I. Kaminer, “Planning for opportunistic surveil- lance with multiple robots,” in2013 IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE, 2013, pp. 5750–5757

work page 2013

[5] [5]

Failure-Resilient Coverage Maximization with Multiple Robots

I. E. Rabban and P. Tokekar, “Improved resilient coverage maximization with multiple robots,”ArXiv, vol. abs/2007.02204, 2020. [Online]. Available: https://api.semanticscholar.org/CorpusId:220364130

work page Pith review arXiv 2007

[6] [6]

Game tree search for minimizing detectability and maximizing visibility,

Z. Zhang, J. Smereka, J. Lee, L. Zhou, Y . Sung, and P. Tokekar, “Game tree search for minimizing detectability and maximizing visibility,” Autonomous Robots, vol. 45, pp. 283 – 297, 2021. [Online]. Available: https://api.semanticscholar.org/CorpusId:231668346

work page 2021

[7] [7]

Stackelberg game approaches for anti-jamming defence in wireless networks,

L. Jia, Y . Xu, Y . Sun, S. Feng, and A. Anpalagan, “Stackelberg game approaches for anti-jamming defence in wireless networks,” IEEE Wireless Communications, vol. 25, pp. 120–128, 2018. [Online]. Available: https://api.semanticscholar.org/CorpusId:44149661

work page 2018

[8] [8]

Jamming Intrusions in Extreme Bandwidth Communication: A Comprehensive Overview

M.-S. Alouini, Y . Ata, R. Priyadarshani, and K. Park, “Jamming intrusions in extreme bandwidth communication: A comprehensive overview,”ArXiv, vol. abs/2403.19868, 2024. [Online]. Available: https://api.semanticscholar.org/CorpusId:268793863

work page Pith review arXiv 2024