Prospect Theoretic Analysis on Weighted Vertex Cover of Networks

Weighted vertex cover (WVC) is a classical combinatorial optimization problem, which has rich practice significance. Indeed, the existing WVC studies based on game theory has introduced the traditional expected utility theory (EUT), where each vertex is treated as a rational player. However, the uncertainty caused by the weight factor is neglected. So how does this uncertainty affect players' behaviour? In this paper, the complete theoretical analysis is shown. Specifically, we model the WVC problem as an non-cooperative game on weighted networks and respectively establish EUT-based and prospect theory-based utility functions. Then, we prove the existence of mixed NE on this weighted game. Further, by designing a state function, we find that the vertices' state under mixed NE satisfies weighted vertex cover state of the network. Finally, a fictitious play algorithm is proposed to get a NE of the game. Our findings will pave the way for the future numerical simulation, which can demonstrate the impact of the prospect theory on weighted vertex cover.