# A strategic model of job arrivals to a single machine with earliness and tardiness penalties

## TL;DR

This paper models decentralized job arrivals to a single machine with penalties for deviations from due dates, showing the existence of pure symmetric Nash equilibria and analyzing their properties under various conditions.

## Contribution

It characterizes the structure of Nash equilibria in a decentralized scheduling game with earliness and tardiness penalties, including conditions for social optimality and robustness of equilibria.

## Key findings

- Pure symmetric equilibria exist for weighted absolute deviation costs.
- Multiple equilibrium arrival times form a continuum.
- Equilibrium properties are robust to various model relaxations.

## Abstract

We consider a game of decentralized timing of jobs to a single server (machine) with a penalty for deviation from a due date, and no delay costs. The jobs' sizes are homogeneous and deterministic. Each job belongs to a single decision maker, a customer, who aims to arrive at a time that minimizes his deviation penalty. If multiple customers arrive at the same time then their order of service is determined by a uniform random draw. We show that if the cost function has a weighted absolute deviation form then any Nash equilibrium is pure and symmetric, that is, all customers arrive together. Furthermore, we show that there exist multiple, in fact a continuum, of equilibrium arrival times, and provide necessary and sufficient conditions for the socially optimal arrival time to be an equilibrium. The base model is solved explicitly, but the prevalence of a pure symmetric equilibrium is shown to be robust to several relaxations of the assumptions: restricted server availability, inclusion of small waiting costs, stochastic job sizes, randomly sized population, heterogeneous due dates, and non-linear deviation penalties.