The Stackelberg Equilibria of the Kelly Mechanism
The Kelly mechanism dictates that players share a resource proportionally to their bids. The corresponding game is known to have a unique Nash equilibrium. A related question arises, which is the nature of the behavior of the players for different prices imposed by the resource owner, who may be viewed as the leader in a Stackelberg game where the other players are followers. In this work, we describe the dynamics of the Nash equilibrium as a function of the price. Toward that goal, we characterize analytical properties of the Nash equilibrium by means of the implicit function theorem. With regard to the revenue generated by the resource owner, we provide a counterexample which shows that the Stackelberg equilibrium of the Kelly mechanism may not be unique. We obtain sufficient conditions which guarantee the set of Stackelberg equilibria to be finite and unique in the symmetric case. Finally, we describe the dependency between the resource’s signal and the maximum revenue that the resource owner can generate.
KeywordsKelly mechanism Nash equilibrium Stackelberg equilibrium
- 3.Başar, T., Olsder, G.: Dynamic Noncooperative Game Theory. Society for Industrial and Applied Mathematics, 2nd edn. (1998)Google Scholar
- 4.De Pellegrini, F., Massaro, A., Goratti, L., El-Azouzi, R.: Bounded generalized Kelly mechanism for multi-tenant caching in mobile edge clouds. In: Network Games, Control, and Optimization. pp. 89–99. Springer International Publishing, Cham (2017)Google Scholar
- 5.Johari, R.: Efficiency Loss in Market Mechanisms for Resource Allocation. Ph.D. thesis, Department of Electrical Engineering and Computer Science (2004)Google Scholar