Co-Investment under Revenue Uncertainty Based on Stochastic Coalitional Game Theory

TL;DR

This paper tackles co-investment for Edge Computing infrastructure under revenue uncertainty by formulating a stochastic coalitional game among an Infrastructure Provider and multiple Service Providers. It introduces a nominal (expectation-based) game to establish convexity and core stability, then derives a probabilistic lower bound, $\nu^{\text{LB}}$, on grand-coalition stability under bounded/unpredictable revenues, using Hoeffding’s inequality and independence assumptions. When revenue fluctuations are highly correlated, it complements stability analysis with profitability guarantees, showing that profitability probabilities are at least as large as stability probabilities and improving with longer investment horizons. The framework is instantiated for MEC/EC deployment, with concrete cost and utility models, and two traffic uncertainty settings (bounded and fractional Brownian motion with Hurst parameter $H$), demonstrating practical viability of fair, dynamic resource sharing via Shapley-value-based payoff distribution. Overall, the approach enables robust, fair, and scalable coordination among InPs and SPs to deploy edge resources under uncertainty, with clear guidance on coalition stability, profitability, and payoff sharing.

Abstract

The introduction of new services, such as Mobile Edge Computing (MEC), requires a massive investment that cannot be assumed by a single stakeholder, for instance the Infrastructure Provider (InP). Service Providers (SPs) however also have an interest in the deployment of such services. We hence propose a co-investment scheme in which all stakeholders, i.e., the InP and the SPs, form the so-called grand coalition composed of all the stakeholders with the aim of sharing costs and revenues and maximizing their payoffs. The challenge comes from the fact that future revenues are uncertain. We devise in this case a novel stochastic coalitional game formulation which builds upon robust game theory and derive a lower bound on the probability of the stability of the grand coalition, wherein no player can be better off outside of it. In the presence of some correlated fluctuations of revenues however, stability can be too conservative. In this case, we make use also of profitability, in which payoffs of players are non-negative, as a necessary condition for co-investment. The proposed framework is showcased for MEC deployment, where computational resources need to be deployed in nodes at the edge of a telecommunication network. Numerical results show high lower bound on the probability of stability when the SPs' revenues are of similar magnitude and the investment period is sufficiently long, even with high levels of uncertainty. In the case where revenues are highly variable however, the lower bound on stability can be trivially low whereas co-investment is still profitable.

Figures (9)

• Figure 1: Traffic load with bounded fluctuations over five days within the investment period
• Figure 2: Expected traffic load and the corresponding resource allocation for a typical day during the investment period
• Figure 3: Lower bound on the probability that the the grand coalition is stable under different levels of uncertainty $\sigma$ and varying coalition sizes
• Figure 4: Stochastic realizations used to model traffic load variations over five days within the investment period
• Figure 5: Violin plots of revenue, payment, and payoff redistributions for InP, SP1, and SP2 under varying levels of uncertainty $\alpha$, using 100 generated realizations

Theorems & Definitions (24)

• Proposition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Proposition 11
• Proposition 12
• Theorem 13