# The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

## TL;DR

This paper investigates the computational complexity of subgame perfect equilibria in multiplayer quantitative reachability games, proving the problem is PSPACE-complete and introducing a new iterative algorithm for characterizing SPE outcomes.

## Contribution

It establishes the PSPACE-completeness of the constrained existence problem for SPE in these games and proposes a novel fixpoint-based algorithm for outcome characterization.

## Key findings

- The constrained existence problem for SPE is PSPACE-complete.
- A new iterative algorithm constructs a finite graph representing SPE outcomes.
- The algorithm's complexity aligns with PSPACE, confirming the problem's computational difficulty.

## Abstract

We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership.