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Coordination Mechanisms with Rank-Based Utilities

Gilad Lavie, Tami Tamir

TL;DR

The paper studies coordination mechanisms with rank-based utilities in job-scheduling games, focusing on equilibrium existence, computational complexity, and efficiency under various machine models and competition structures. It proves NP-completeness for deciding NE existence even with restricted job lengths, and identifies several tractable classes where NE exists or can be computed efficiently, including two identical machines via Inversed-Policies and two related machines under linear-time procedures. It analyzes how competition affects makespan-related efficiency via PoA, PoS, and sink equilibria, revealing that competition can both improve and deteriorate outcomes depending on the class and parameters. It also extends the model to competition classes, proving NE existence and BRD convergence under seniority-based and partition-based priority schemes, thereby providing design principles for stable, competitive scheduling environments with rank-based incentives.

Abstract

In classical job-scheduling games, each job behaves as a selfish player, choosing a machine to minimize its own completion time. To reduce the equilibria inefficiency, coordination mechanisms are employed, allowing each machine to follow its own scheduling policy. In this paper we study the effects of incorporating rank-based utilities within coordination mechanisms across environments with either identical or unrelated machines. With rank-based utilities, players aim to perform well relative to their competitors, rather than solely minimizing their completion time. We first demonstrate that even in basic setups, such as two identical machines with unit-length jobs, a pure Nash equilibrium (NE) assignment may not exist. This observation motivates our inquiry into the complexity of determining whether a given game instance admits a NE. We prove that this problem is NP-complete, even in highly restricted cases. In contrast, we identify specific classes of games where a NE is guaranteed to exist, or where the decision problem can be resolved in polynomial time. Additionally, we examine how competition impacts the efficiency of Nash equilibria, or sink equilibria if a NE does not exist. We derive tight bounds on the price of anarchy, and show that competition may either enhance or degrade overall performance.

Coordination Mechanisms with Rank-Based Utilities

TL;DR

The paper studies coordination mechanisms with rank-based utilities in job-scheduling games, focusing on equilibrium existence, computational complexity, and efficiency under various machine models and competition structures. It proves NP-completeness for deciding NE existence even with restricted job lengths, and identifies several tractable classes where NE exists or can be computed efficiently, including two identical machines via Inversed-Policies and two related machines under linear-time procedures. It analyzes how competition affects makespan-related efficiency via PoA, PoS, and sink equilibria, revealing that competition can both improve and deteriorate outcomes depending on the class and parameters. It also extends the model to competition classes, proving NE existence and BRD convergence under seniority-based and partition-based priority schemes, thereby providing design principles for stable, competitive scheduling environments with rank-based incentives.

Abstract

In classical job-scheduling games, each job behaves as a selfish player, choosing a machine to minimize its own completion time. To reduce the equilibria inefficiency, coordination mechanisms are employed, allowing each machine to follow its own scheduling policy. In this paper we study the effects of incorporating rank-based utilities within coordination mechanisms across environments with either identical or unrelated machines. With rank-based utilities, players aim to perform well relative to their competitors, rather than solely minimizing their completion time. We first demonstrate that even in basic setups, such as two identical machines with unit-length jobs, a pure Nash equilibrium (NE) assignment may not exist. This observation motivates our inquiry into the complexity of determining whether a given game instance admits a NE. We prove that this problem is NP-complete, even in highly restricted cases. In contrast, we identify specific classes of games where a NE is guaranteed to exist, or where the decision problem can be resolved in polynomial time. Additionally, we examine how competition impacts the efficiency of Nash equilibria, or sink equilibria if a NE does not exist. We derive tight bounds on the price of anarchy, and show that competition may either enhance or degrade overall performance.

Paper Structure

This paper contains 25 sections, 16 theorems, 5 equations, 19 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Problem Statements
  3. Rank-Based vs. Cost-Based Utilities
  4. Related Work
  5. Our Results
  6. Computational Complexity
  7. Greedy Algorithms and Stability Conditions
  8. Identical Machines
  9. Related Machines
  10. Identical Machines - Equilibrium Existence and Computation
  11. Machine-Dependent Priority Lists and Arbitrary Job Lengths
  12. Global Priority List and Unit-Length Jobs
  13. Machine-Dependent Priority Lists and Unit-Length Jobs
  14. Related Machines - Equilibrium Existence and Computation
  15. Global Priority List
  16. ...and 10 more sections

Key Result

Theorem 2.1

Given a coordination mechanism with rank-based utilities, the problem of deciding whether the game has a NE is NP-complete, even when the machines are identical and for all jobs $p_j \in \{1, 2\}$.

Figures (19)

  • Figure 1: A BR-sequence in a game with two unit-length players and no NE.
  • Figure 2: A cost-increasing rank-reducing deviation.
  • Figure 3: Let $T=\{t_1=\{x_1,y_1,z_1\}, t_2=\{x_1,y_2,z_3\}, t_3=\{x_2,y_2,z_2\}, t_4=\{x_2,y_1,z_2\}, t_5=\{x_2,y_3,z_1\}, t_6=\{x_3,y_3,z_3\}, t_7=\{x_3,y_1,z_1\}\}$. Note that $\{t_1,t_3,t_6\}$ is a perfect matching. (a) a NE profile. $(b)$ a non-beneficial deviation of $d_3$.
  • Figure 4: Let $T=\{t_1=\{x_1,y_1,z_1\}, t_2=\{x_1,y_2,z_3\}, t_3=\{x_2,y_2,z_1\}, t_4=\{x_2,y_1,z_2\}, t_5=\{x_2,y_3,z_1\}, t_6=\{x_3,y_3,z_3\}, t_7=\{x_3,y_1,z_1\}\}$. Note that a perfect matching does not exist. (a) a possible profile. $(b)$ a beneficial deviation of $v_4$.
  • Figure 5: (a) The sets $P,\ P_1$ and $P_2$ in $s$. (b) A beneficial migration of job $j \in P_1$ from $M_1$ to $M_{m - c +1}$, assuming condition $2$ is not satisfied and $j \prec_{m - c + 1} j'$.
  • ...and 14 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Claim 2.2
  • Claim 2.3
  • Claim 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 4.2
  • ...and 32 more