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Incentives for Early Arrival in Cost Sharing

Junyu Zhang, Yao Zhang, Yaoxin Ge, Dengji Zhao, Hu Fu, Zhihao Gavin Tang, Pinyan Lu

TL;DR

This work solves 0-1 valued cost sharing games with a novel mechanism called Shapley-fair shuffle cost sharing mechanism (SFS-CS), and extends SFS-CS to a family called generalized Shapley-fair shuffle cost sharing mechanisms (GSFS-CS).

Abstract

In cooperative games, we study how values created or costs incurred by a coalition are shared among the members within it, and the players may join the coalition in a online manner such as investors invest a startup. Recently, Ge et al. [10] proposed a new property called incentives for early arrival (I4EA) in such games, which says that the online allocation of values or costs should incentivize agents to join early in order to prevent mutual strategic waiting. Ideally, the allocation should also be fair, so that agents arriving in an order uniformly at random should expect to get/pay their Shapley values. Ge et al. [10] showed that not all monotone value functions admit such mechanisms in online value sharing games. In this work, we show a sharp contrast in online cost sharing games. We construct a mechanism with all the properties mentioned above, for every monotone cost function. To achieve this, we first solve 0-1 valued cost sharing games with a novel mechanism called Shapley-fair shuffle cost sharing mechanism (SFS-CS), and then extend SFS-CS to a family called generalized Shapley-fair shuffle cost sharing mechanisms (GSFS-CS). The critical technique we invented here is a mapping from one arrival order to another order so that we can directly apply marginal cost allocation on the shuffled orders to satisfy the properties. Finally, we solve general valued cost functions, by decomposing them into 0-1 valued functions in an online fashion.

Incentives for Early Arrival in Cost Sharing

TL;DR

This work solves 0-1 valued cost sharing games with a novel mechanism called Shapley-fair shuffle cost sharing mechanism (SFS-CS), and extends SFS-CS to a family called generalized Shapley-fair shuffle cost sharing mechanisms (GSFS-CS).

Abstract

In cooperative games, we study how values created or costs incurred by a coalition are shared among the members within it, and the players may join the coalition in a online manner such as investors invest a startup. Recently, Ge et al. [10] proposed a new property called incentives for early arrival (I4EA) in such games, which says that the online allocation of values or costs should incentivize agents to join early in order to prevent mutual strategic waiting. Ideally, the allocation should also be fair, so that agents arriving in an order uniformly at random should expect to get/pay their Shapley values. Ge et al. [10] showed that not all monotone value functions admit such mechanisms in online value sharing games. In this work, we show a sharp contrast in online cost sharing games. We construct a mechanism with all the properties mentioned above, for every monotone cost function. To achieve this, we first solve 0-1 valued cost sharing games with a novel mechanism called Shapley-fair shuffle cost sharing mechanism (SFS-CS), and then extend SFS-CS to a family called generalized Shapley-fair shuffle cost sharing mechanisms (GSFS-CS). The critical technique we invented here is a mapping from one arrival order to another order so that we can directly apply marginal cost allocation on the shuffled orders to satisfy the properties. Finally, we solve general valued cost functions, by decomposing them into 0-1 valued functions in an online fashion.

Paper Structure

This paper contains 13 sections, 11 theorems, 4 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. The Model
  4. 0-1 Valued Cost Functions
  5. Shuffle-Based Cost Sharing Mechanisms
  6. Shapley-fair Shuffle Cost Sharing Mechanism
  7. Proofs of Properties of SFS-CS
  8. OIR
  9. SF
  10. I4EA
  11. A Class of Shuffle-based Cost Sharing Mechanisms
  12. Extension to General Cost Functions
  13. Discussion

Key Result

proposition 1

An online cost sharing mechanism $\phi$ is I4EA if and only if for any player $i$, $\phi_i(N, c, \pi_1)\leq \phi_i(N, c, \pi_2)$ for any two arrival orders $\pi_1$ and $\pi_2$ with only player $i$ being delayed one position, i.e., $\pi_1=[\dots,i,j,\dots]$ and $\pi_2=[\dots,j,i,\dots]$.

Figures (8)

  • Figure 1: The player in red is the marginal player.
  • Figure 2: An attempt on constructing a $\mathtt{shuf}$ for cost sharing game mentioned in Example \ref{['eg:naive shuf']}.
  • Figure 3: An example of $\mathtt{sfs}\text{-}\mathtt{shuf}$. The left side is the original order $\pi$ and for each joining player (colored by blue), the right side shows the construction process of the image ordering $\pi'$. Note that the players colored by red are the marginal players, and the players colored by green are the corresponding related players (see Definition \ref{['Def:related_player']} in Section \ref{['sec:proofs']}).
  • Figure 4: The reconstruction process of $\pi$ from $\pi'$. The left side marks the marginal players (red) and corresponding related players (green) through this process, and the right side shows the identified last players (blue) in each iteration.
  • Figure 5: $\pi$ is the original order. $\pi'$ is the image ordering. $i$ is the marginal player of $\pi'$. Lemma \ref{['lem: I4EA1']} states $S_1\subseteq S_2$.
  • ...and 3 more figures

Theorems & Definitions (27)

  • definition 1: Marginal Cost
  • definition 2: Shapley Value
  • definition 3
  • definition 4
  • definition 5
  • proposition 1
  • definition 6
  • definition 7
  • definition 8
  • proposition 2
  • ...and 17 more