Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

When Effort May Fail: Equilibria of Shared Effort with a Threshold

Gleb Polevoy, Stojan Trajanovski, Mathijs de Weerdt

TL;DR

We investigate shared effort games with a relative contribution threshold $\theta$ and linear project values, formalizing how budgets, projects, and threshold-based sharing shape strategic contributions. The paper delivers a constructive characterization of pure Nash equilibria for $\theta\in\{0,1\}$ and for two-player cases with general $\theta$, derives exact price of anarchy and price of stability values, and extends the analysis to more players, supported by simulations using infinite-strategy fictitious play. A novel concept, cyclically strong equilibrium, narrows Nash equilibria by ruling out cyclical budget deviations, and the authors prove the existence of mixed Nash equilibria in the thresholded, linear setting via Dasgupta-Maskin-type results, with corresponding efficiency bounds extending to mixed and correlated equilibria. The work contributes practical insights for co-authorship, multi-project collaboration, and other shared-effort contexts, and introduces a general methodological tool for relating solution concepts through narrowing or broadening on broad game classes.

Abstract

People, robots, and companies mostly divide time and effort between projects, and \defined{shared effort games} model people investing resources in public endeavors and sharing the generated values. In linear $θ$ sharing (effort) games, a project's value is linear in the total contribution, thus modelling predictable, uniform, and scalable activities. The threshold $θ$ for effort defines which contributors win and receive their share, equal share modelling standard salaries, equity-minded projects, etc. Thresholds between 0 and 1 model games such as paper co-authorship and shared assignments, where a minimum positive contribution is required for sharing in the value. We constructively characterise the conditions for the existence of a pure equilibrium for $θ\in\{0,1\}$, and for two-player games with a general threshold, and find the prices of anarchy and stability. We also provide existence and efficiency results for more than two players, and use generalised fictitious play simulations to show when a pure equilibrium exists and what its efficiency is. We propose a novel method for studying solution concepts by defining a new concept and proving its equivalence to a previously known on a large subclass of games. This means that the original concept narrows down to a more demanding concepts on certain games, providing new insights and opening a path to study both concepts conveniently. We also prove mixed equilibria always exist and bound their efficiency.

When Effort May Fail: Equilibria of Shared Effort with a Threshold

TL;DR

We investigate shared effort games with a relative contribution threshold and linear project values, formalizing how budgets, projects, and threshold-based sharing shape strategic contributions. The paper delivers a constructive characterization of pure Nash equilibria for and for two-player cases with general , derives exact price of anarchy and price of stability values, and extends the analysis to more players, supported by simulations using infinite-strategy fictitious play. A novel concept, cyclically strong equilibrium, narrows Nash equilibria by ruling out cyclical budget deviations, and the authors prove the existence of mixed Nash equilibria in the thresholded, linear setting via Dasgupta-Maskin-type results, with corresponding efficiency bounds extending to mixed and correlated equilibria. The work contributes practical insights for co-authorship, multi-project collaboration, and other shared-effort contexts, and introduces a general methodological tool for relating solution concepts through narrowing or broadening on broad game classes.

Abstract

People, robots, and companies mostly divide time and effort between projects, and \defined{shared effort games} model people investing resources in public endeavors and sharing the generated values. In linear sharing (effort) games, a project's value is linear in the total contribution, thus modelling predictable, uniform, and scalable activities. The threshold for effort defines which contributors win and receive their share, equal share modelling standard salaries, equity-minded projects, etc. Thresholds between 0 and 1 model games such as paper co-authorship and shared assignments, where a minimum positive contribution is required for sharing in the value. We constructively characterise the conditions for the existence of a pure equilibrium for , and for two-player games with a general threshold, and find the prices of anarchy and stability. We also provide existence and efficiency results for more than two players, and use generalised fictitious play simulations to show when a pure equilibrium exists and what its efficiency is. We propose a novel method for studying solution concepts by defining a new concept and proving its equivalence to a previously known on a large subclass of games. This means that the original concept narrows down to a more demanding concepts on certain games, providing new insights and opening a path to study both concepts conveniently. We also prove mixed equilibria always exist and bound their efficiency.
Paper Structure (21 sections, 20 theorems, 18 equations, 6 figures)

This paper contains 21 sections, 20 theorems, 18 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Model
  4. Pure Nash Equilibrium
  5. Characterization for Two Players with Close Budgets
  6. Existence of Nash Equilibrium
  7. Efficiency
  8. Narrowing down Nash to Cyclically Strong Equilibrium
  9. Mixed Nash Equilibrium
  10. Related Work
  11. Contribution to Projects
  12. Background: Equilibria and Efficiency
  13. Background: Mixed Equilibria and their Existence
  14. Extant Models of Sharing Effort
  15. Conclusions and Future Work
  16. ...and 6 more sections

Key Result

Theorem 1

A game with $M_{\text{eq}}^{0}$ admits a potential function, a pure NE exists and $\mathop{\mathrm{PoA}}\nolimits = \mathop{\mathrm{PoS}}\nolimits = 1$.

Figures (6)

  • Figure 1: The co-authors invest what is shown in the arrows that go up, every project's value is defined as the $P$ function of the total contribution, and it is equally shared among the contributors who contribute above the relative threshold of $0.2$. The obtained shares are denoted by the arrows that go down.
  • Figure 2: The existence and efficiency of NE for $2$ players as a function of the ratio of the project functions coefficients and the ratio of the two largest budgets. The first row plots the results of the simulations, and the second row shows the theoretical predictions. Black means that an equilibrium has not been found.
  • Figure 3: The existence and efficiency of NE as function of project functions for $2, 3, 4, 5$ and $6$ players. Black means that Nash Equilibrium has not been found.
  • Figure 4: The efficiency as a function of the ratio of the project value functions for $2, 3, 4, 5$ and $6$ players. Efficiency of $0$ means that Nash Equilibrium has not been found.
  • Figure 5: The existence and efficiency of NE as function of the largest and the second largest budgets. Black means that Nash Equilibrium has not been found, and gray hatching indicates the non-defined area, since the second highest budget may not be larger than the highest one.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Example 1
  • Definition 1
  • Example 1: Continued
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • ...and 21 more