[Paper Review] Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective

Elena Kleiman|arXiv (Cornell University)|Oct 28, 2011

Game Theory and Applications61 references3 citations

TL;DR

This thesis investigates packing, scheduling, and covering problems through a game-theoretic lens, analyzing the efficiency and fairness of equilibria in systems with self-interested agents. It introduces and characterizes Pareto optimal, Strong Nash, and fair equilibria, deriving tight bounds on the Price of Anarchy and Stability for job scheduling, bin packing, and machine covering on identical and related machines.

ABSTRACT

Many packing, scheduling and covering problems that were previously considered by computer science literature in the context of various transportation and production problems, appear also suitable for describing and modeling various fundamental aspects in networks optimization such as routing, resource allocation, congestion control, etc. Various combinatorial problems were already studied from the game theoretic standpoint, and we attempt to complement to this body of research. Specifically, we consider the bin packing problem both in the classic and parametric versions, the job scheduling problem and the machine covering problem in various machine models. We suggest new interpretations of such problems in the context of modern networks and study these problems from a game theoretic perspective by modeling them as games, and then concerning various game theoretic concepts in these games by combining tools from game theory and the traditional combinatorial optimization. In the framework of this research we introduce and study models that were not considered before, and also improve upon previously known results.

Motivation & Objective

• To analyze the efficiency of equilibria in distributed systems where agents act selfishly, focusing on performance degradation due to lack of coordination.
• To characterize and compute Pareto optimal, strictly and weakly Pareto optimal, and Strong Nash equilibria in scheduling and packing games.
• To evaluate fairness and efficiency trade-offs using metrics like Price of Anarchy, Price of Stability, and the maximum envy ratio.
• To extend classical combinatorial optimization problems (bin packing, scheduling, machine covering) to settings with strategic agents and non-cooperative behavior.
• To explore the impact of different cost-sharing mechanisms and structural constraints (e.g., cardinality, variable bin sizes) on equilibrium efficiency and stability.

Proposed method

• Models job scheduling, bin packing, and machine covering as non-cooperative games with self-interested agents minimizing individual cost or maximizing individual utility.
• Applies game-theoretic concepts such as Nash equilibria, Strong Nash equilibria, and Pareto optimality to analyze system performance.
• Uses the Price of Anarchy (PoA) and Price of Stability (PoS) to measure worst-case and best-case efficiency loss due to selfish behavior.
• Derives analytical bounds on PoA and PoS for various machine models (identical, related, uniformly related) and problem variants.
• Introduces and analyzes fairness criteria such as min-max fairness and maximum envy ratio in equilibrium configurations.
• Employs constructive lower bound examples and upper bound proofs via mathematical analysis and structural decomposition of optimal and equilibrium solutions.

Experimental results

Research questions

• RQ1 What is the worst-case efficiency loss (Price of Anarchy) in job scheduling games on identical, related, and unrelated machines?
• RQ2 How do the efficiency and fairness of equilibria differ between strictly and weakly Pareto optimal equilibria in scheduling games?
• RQ3 What are the tight bounds on the Price of Anarchy and Stability in bin packing and parametric bin packing games with selfish items?
• RQ4 How does the structure of cost-sharing (e.g., proportional to item size) affect the existence and efficiency of Strong Nash equilibria?
• RQ5 What is the impact of fairness criteria like min-max fairness and maximum envy ratio on equilibrium performance in machine covering games?

Key findings

• For job scheduling on identical machines, the Price of Anarchy is 2, and the Price of Stability is 1.5, with tight bounds established for specific machine ratios.
• On related machines with speed ratio s ∈ [1, 4/3] ∪ [1.78, 2], the Price of Anarchy is tight at 2 and 1.5 respectively, while the exact value in s ∈ (4/3, 1.78) remains open.
• In the bin packing game with proportional cost sharing, the Price of Anarchy is at most 2, and the Strong Price of Anarchy is at most 2.5, with tight bounds in certain cases.
• For parametric bin packing, the Price of Anarchy is bounded between 1.5 and 2.5, with tight bounds derived for specific parameter ranges.
• The Price of Stability for machine covering on m identical machines is at most 1.5, and the maximum envy ratio is bounded by 2 in equilibrium.
• Min-max fair equilibria exist in the job scheduling game and are strictly Pareto optimal; however, for covering objectives, the min-max-fair Price of Anarchy is not necessarily 1 and requires further study.