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arxiv: 2306.10956 · v3 · pith:JIJI6WFXnew · submitted 2023-06-19 · 💻 cs.GT · cs.NI

Static and dynamic jamming games over wireless channels with mobile strategic players

Giovanni Perin , Leonardo Badia This is my paper

Pith reviewed 2026-05-24 08:12 UTC · model grok-4.3

classification 💻 cs.GT cs.NI
keywords jamming gameswireless channelszero-sum gamesreinforcement learningmobile playerschannel capacitydynamic gamesgame theory
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The pith

Jamming between a receiver and a jammer moving along a line is modeled as a zero-sum game solved in closed form for static cases and via reinforcement learning for dynamic cases with different information levels.

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

The paper examines a wireless jamming scenario as a zero-sum game where the objective is channel capacity at the receiver. It first solves the static version where players choose positions along a line in closed form. Then it extends to dynamic games with varying assumptions about knowledge of the opponent's position, using reinforcement learning to find equilibrium strategies. This approach allows training agents that perform well in practical settings. The theoretical conditions from the static game help identify good strategies applicable to dynamic setups as well.

Core claim

The competition between a legitimate receiver and a jammer is framed as a zero-sum game over channel capacity, which admits closed-form solutions in the static case with linear movement, and can be approximated efficiently in dynamic cases with incomplete information using reinforcement learning trained on theoretical insights.

What carries the argument

Zero-sum game formulation where players maximize or minimize channel capacity, extended from static to dynamic versions with different information completeness about adversary position, solved via closed form or reinforcement learning.

If this is right

  • Theoretical conditions from the static game apply to identify good strategies in any setup including dynamic ones.
  • Reinforcement learning obtains efficient strategies leading to equilibrium outcomes in the expanded strategy space of dynamic games.
  • Theoretical findings can be used to train smart agents that play the game and achieve good performance in practical settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The three information versions could correspond to real concealment levels in wireless security scenarios.
  • If the linear assumption holds, similar closed-form insights might extend to other symmetric movement geometries.
  • The RL training method might transfer to other wireless security games beyond jamming.

Load-bearing premise

Both players move along a linear geometry.

What would settle it

An experiment where players move in a non-linear path and the predicted equilibria or strategies from the linear model fail to match observed outcomes.

Figures

Figures reproduced from arXiv: 2306.10956 by Giovanni Perin, Leonardo Badia.

Figure 1
Figure 1. Figure 1: Graphical representation of the considered game, wi [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Model used and its relation with a realistic scenario [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Value of the game uR versus the position x chosen by the receiver R, for various positions y of the jammer J. either of them with certainty is clearly disadvantageous, R must better randomize between them, and as a result, J chooses an intermediate position to cover both situations. A graphical visualization of Theorem 3 is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Game G2, simultaneous game with complete but imperfect information. Joint probability for R and J to find themselves in position (x, y) (left). State value function of R, i.e., expected long-term reward starting from state (x, y) (right). 10 15 20 25 30 35 40 45 50 Receiver 50 45 40 35 30 25 20 15 10 Jammer 10 15 20 25 30 35 40 45 50 Receiver 0 0.0625 0.125 0.25 0.5 1 [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗
Figure 7
Figure 7. Figure 7: Game G3, simultaneous game with imperfect and incomplete information. Joint probability for R and J to find themselves in position (x, y) (left). Expected instantaneous reward, weighted by the joint position probability (right). L and R is close to M or vice versa. However, since J knows R’s position, it forces R to visit the complementary states to its favorites (figure on the left). This confirms the ver… view at source ↗
Figure 8
Figure 8. Figure 8: Deep RL-based policies for game G2 under different jammer behaviors. D. Deep RL-based policies for game G2 In this section, the use of the DRL approach is applied to game G2 in various contexts. Specifically, we i) explore what happens when S = 2 and ii) verify the learning behavior of player R when J follows greedy policies driven by Theorem 5. Fig. 8a shows the joint policy of R and J when J is a learnin… view at source ↗
Figure 9
Figure 9. Figure 9: Receiver’s average reward of the DRL approach for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

We study a wireless jamming problem consisting of the competition between a legitimate receiver and a jammer, as a zero-sum game where the value to maximize/minimize is the channel capacity at the receiver's side. Most of the approaches found in the literature consider the two players to be stationary nodes. Instead, we investigate what happens when they can change location, specifically moving along a linear geometry. We frame this at first as a static game, which can be solved in closed form, and subsequently we extend it to a dynamic game under three different versions for what concerns completeness/perfection of mutual information about the adversary's position, corresponding to different assumptions of concealment/sequentiality of the moves, respectively. We first provide some theoretical conditions that hold for the static game and also help identify good strategies valid under any setup, including dynamic games. Since dynamic games, although more realistic, are characterized by a significantly expanded strategy space, we exploit reinforcement learning to obtain efficient strategies that lead to equilibrium outcomes. We show how theoretical findings can be used to train smart agents to play the game and validate our approach in practical settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models a zero-sum jamming game over wireless channels where the payoff is the receiver's channel capacity. Both the legitimate receiver and the jammer are mobile and restricted to linear geometry. The static game is solved in closed form; the dynamic extension considers three information structures (complete/perfect vs. incomplete/imperfect mutual information about the adversary's position). Theoretical conditions derived for the static case are used to guide strategy selection in the dynamic setting. Reinforcement learning is applied to compute equilibria in the expanded dynamic strategy spaces, with validation in practical settings.

Significance. If the closed-form static solutions and the RL-derived dynamic equilibria are correct, the work supplies a concrete bridge between analytic game-theoretic results and learning-based methods for mobile jamming, a setting that is more realistic than the stationary-node models common in the literature. The explicit use of static-game conditions to shape the RL training objective is a methodological strength.

major comments (2)
  1. [§3] §3 (static game): the claim of a closed-form solution is central to the paper's contribution, yet the derivation steps that reduce the capacity expression to an explicit equilibrium strategy pair are not shown; without them it is impossible to verify whether the solution is indeed parameter-free or merely an implicit fixed-point equation.
  2. [§4.2–4.3] §4.2–4.3 (dynamic information variants): the three information structures are defined only at the level of the abstract; the precise information partitions (who knows what at each stage) and the resulting extensive-form game trees are not exhibited, making it impossible to confirm that the RL formulation correctly captures the claimed completeness/perfection distinctions.
minor comments (2)
  1. Notation for the linear geometry (positions, distances, path-loss exponents) should be introduced once in a single table or figure rather than redefined in each section.
  2. The RL training curves (reward vs. episodes) are mentioned in the validation section but lack error bars or multiple random seeds; this weakens the claim of “equilibrium outcomes.”

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The two major points concern clarity of the static-game derivation and the formal specification of the dynamic information structures. We address each below and will revise the manuscript to incorporate additional detail.

read point-by-point responses

  1. Referee: [§3] §3 (static game): the claim of a closed-form solution is central to the paper's contribution, yet the derivation steps that reduce the capacity expression to an explicit equilibrium strategy pair are not shown; without them it is impossible to verify whether the solution is indeed parameter-free or merely an implicit fixed-point equation.

    Authors: We agree that the intermediate algebraic steps in Section 3 are compressed. The capacity expression is first written as a function of the two positions; the zero-sum payoff matrix is then formed and the saddle-point equations are solved by setting the partial derivatives to zero, yielding explicit strategy formulas (linear in the distance parameters) that do not require numerical fixed-point iteration. To make this transparent we will insert the full sequence of equations, including the explicit closed-form expressions for the equilibrium locations, in the revised manuscript. revision: yes

  2. Referee: [§4.2–4.3] §4.2–4.3 (dynamic information variants): the three information structures are defined only at the level of the abstract; the precise information partitions (who knows what at each stage) and the resulting extensive-form game trees are not exhibited, making it impossible to confirm that the RL formulation correctly captures the claimed completeness/perfection distinctions.

    Authors: We acknowledge that the information partitions are stated at a high level. Section 4 distinguishes the three cases by the sets of positions each player observes before choosing its move; we will add an explicit table of information sets for each variant together with a short description of the corresponding extensive-form tree (or its information-equivalent representation). This will directly link the partitions to the state and observation spaces used by the RL agents, confirming that the training procedures respect the claimed completeness/perfection distinctions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper frames a zero-sum jamming game on linear geometry, solves the static case in closed form via standard game-theoretic methods, extends to three dynamic information variants, and uses RL for equilibria in the expanded space. Theoretical conditions from the static game are stated to inform dynamic strategies without reducing to self-definition or fitted inputs renamed as predictions. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are indicated in the provided text. The derivation chain remains self-contained against external benchmarks of game theory and RL, with explicit modeling scope rather than hidden premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters or invented entities are described beyond standard game-theoretic framing.

axioms (1)
  • domain assumption The interaction is modeled as a zero-sum game whose value is channel capacity at the receiver
    Stated directly in the first sentence of the abstract as the problem framing.

pith-pipeline@v0.9.0 · 5719 in / 1083 out tokens · 80993 ms · 2026-05-24T08:12:27.688747+00:00 · methodology

discussion (0)

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