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Adversarial Knapsack for Sequential Competitive Resource Allocation

Omkar Thakoor, Rajgopal Kannan, Victor Prasanna

TL;DR

The paper addresses sequential competitive resource allocation via an adversarial knapsack framework, contrasting with traditional simultaneous Colonel Blotto. It formulates a bilevel leader-follower game that allows deterministic pure strategies and fractional payoffs, deriving a value-proportional solution in the fractional case and developing efficient heuristics for discrete resources. It also analyzes the 0-1 (all-or-nothing) variant, proving NP-hardness and containment in $\Sigma_2^P$, and proposes an MILP heuristic to mitigate local maxima. Numerical results show the proposed BB and GD-based heuristics perform well across budget regimes, offering scalable methods with practical impact for sequential allocation problems in domains like security, scheduling, and competitive budgeting.

Abstract

This work addresses competitive resource allocation in a sequential setting, where two players allocate resources across objects or locations of shared interest. Departing from the simultaneous Colonel Blotto game, our framework introduces a sequential decision-making dynamic, where players act with partial or complete knowledge of previous moves. Unlike traditional approaches that rely on complex mixed strategies, we focus on deterministic pure strategies, streamlining computation while preserving strategic depth. Additionally, we extend the payoff structure to accommodate fractional allocations and payoffs, moving beyond the binary, all-or-nothing paradigm to allow more granular outcomes. We model this problem as an adversarial knapsack game, formulating it as a bilevel optimization problem that integrates the leader's objective with the follower's best-response. This knapsack-based approach is novel in the context of competitive resource allocation, with prior work only partially leveraging it for follower analysis. Our contributions include: (1) proposing an adversarial knapsack formulation for the sequential resource allocation problem, (2) developing efficient heuristics for fractional allocation scenarios, and (3) analyzing the 0-1 knapsack case, providing a computational hardness result alongside a heuristic solution.

Adversarial Knapsack for Sequential Competitive Resource Allocation

TL;DR

The paper addresses sequential competitive resource allocation via an adversarial knapsack framework, contrasting with traditional simultaneous Colonel Blotto. It formulates a bilevel leader-follower game that allows deterministic pure strategies and fractional payoffs, deriving a value-proportional solution in the fractional case and developing efficient heuristics for discrete resources. It also analyzes the 0-1 (all-or-nothing) variant, proving NP-hardness and containment in , and proposes an MILP heuristic to mitigate local maxima. Numerical results show the proposed BB and GD-based heuristics perform well across budget regimes, offering scalable methods with practical impact for sequential allocation problems in domains like security, scheduling, and competitive budgeting.

Abstract

This work addresses competitive resource allocation in a sequential setting, where two players allocate resources across objects or locations of shared interest. Departing from the simultaneous Colonel Blotto game, our framework introduces a sequential decision-making dynamic, where players act with partial or complete knowledge of previous moves. Unlike traditional approaches that rely on complex mixed strategies, we focus on deterministic pure strategies, streamlining computation while preserving strategic depth. Additionally, we extend the payoff structure to accommodate fractional allocations and payoffs, moving beyond the binary, all-or-nothing paradigm to allow more granular outcomes. We model this problem as an adversarial knapsack game, formulating it as a bilevel optimization problem that integrates the leader's objective with the follower's best-response. This knapsack-based approach is novel in the context of competitive resource allocation, with prior work only partially leveraging it for follower analysis. Our contributions include: (1) proposing an adversarial knapsack formulation for the sequential resource allocation problem, (2) developing efficient heuristics for fractional allocation scenarios, and (3) analyzing the 0-1 knapsack case, providing a computational hardness result alongside a heuristic solution.

Paper Structure

This paper contains 8 sections, 6 theorems, 6 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Adversarial Knapsack
  4. Fractional Formulation
  5. Discrete resources
  6. 0-1 Formulation
  7. Numerical results
  8. Conclusions

Key Result

Proposition 1

The leader's optimal solution solution is to set each $w_i$ proportional to its $v_i$, i.e., $w_i = \frac{v_i}{\sum_i v_i} W^{\rm l} \ \forall i$.

Figures (2)

  • Figure 1: Example for discrete fractional setting showing follower payoff varies with follower budget $W^{\rm f}$ for different weights $\bm{w}$. Item 1 ($v_1 = 1.4$) and item 2 ($v_2 = 2.6$) in red, blue resp. Weights (1,3), (2,2), (3,1) shown via dashed, solid and dotted lines resp. with continuous allocation baseline in black.
  • Figure 2: Comparison of our heuristics against the baseline of RR. Each plot shows the %instances where our heuristic performs better than RR (in green), same (in yellow) and worse (in red) as budget ratio varies.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Lemma 1
  • proof : Proof sketch
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • Theorem 1
  • proof