Risk measures on networks and expected utility

• We discuss risk acceptance and theoretically develop this theme on the basis of network theory.
• We propose an original framework for describing the algebraic structure of the set of the networks, when they are viewed as risks.
• We introduce the risk measures on networks, which induce total orders on the set of networks.
• We state conditions on the risk measures on networks to let the induced risk-acceptance criterion be consistent with a new formulation of the expected utility theory.

In reliability theory projects are usually evaluated in terms of their riskiness, and often decision under risk is intended as the one-shot-type binary choice of accepting or not accepting the risk. In this paper we elaborate on the concept of risk acceptance, and propose a theoretical framework based on network theory. In doing this, we deal with system reliability, where the interconnections among the random quantities involved in the decision process are explicitly taken into account. Furthermore, we explore the conditions to be satisfied for risk-acceptance criteria to be consistent with the axiomatization of standard expected utility theory within the network framework. In accordance with existing literature, we show that a risk evaluation criterion can be meaningful even if it is not consistent with the standard axiomatization of expected utility, once this is suitably reinterpreted in the light of networks. Finally, we provide some illustrative examples.