A Short Proof of Convexity of Step-out Step-in Sequencing Games

TL;DR

This paper provides concise proofs demonstrating the convexity of the step-out step-in sequencing game and the correctness of a greedy algorithm for coalition valuation, simplifying previous complex case analyses.

Contribution

It offers simplified, short proofs of the convexity and coalition valuation correctness in the sequencing game, improving understanding and accessibility.

Findings

The game is convex.

A greedy algorithm correctly computes coalition valuations.

Simplified proofs replace complex case analyses.

Abstract

The Step out-Step in sequencing game is a particular example of a game from the sequencing game framework of Curiel, Perderzoli, and Tijs, where coalitions of players in a queue may reorder themselves to improve the their overall cost, under some restrictions. Musegaas, Borm and Quant proved, in two papers, that a simple greedy algorithm correctly computes the valuation of a coalition, and that the game is convex. These proofs entail rather involved case analyses; in this note, we give short proofs of both results.