# Distributed Task Management in Cyber-Physical Systems: How to Cooperate under Uncertainty?

## TL;DR

This paper addresses task allocation in cyber-physical systems under uncertainty, modeling it as a stochastic cooperative game and demonstrating the existence of stable, optimal solutions that can be implemented via Walrasian equilibrium.

## Contribution

It introduces a novel multi-state stochastic cooperative game framework for task allocation in CPSs, proving the existence and properties of the strong sequential core.

## Key findings

- Strong sequential core is non-empty and unique under certain conditions.
- The core is equivalent to Walrasian equilibrium under state uncertainty.
- Task allocation solutions can be implemented using Walras' tatonnement process.

## Abstract

We consider the problem of task allocation in a network of cyber-physical systems (CPSs). The network can have different states, and the tasks are of different types. The task arrival is stochastic and state-dependent. Every CPS is capable of performing each type of task with some specific state-dependent efficiency. The CPSs have to agree on task allocation prior to knowing about the realized network's state and/or the arrived tasks. We model the problem as a multi-state stochastic cooperative game with state uncertainty. We then use the concept of deterministic equivalence and sequential core to solve the problem. We establish the non-emptiness of the strong sequential core in our designed task allocation game and investigate its characteristics including uniqueness and optimality. Moreover, we prove that in the task allocation game, the strong sequential core is equivalent to Walrasian equilibrium under state uncertainty; consequently, it can be implemented by using the Walras' tatonnement process.