Economic inefficiency in resource allocation games

In this paper we consider the economic efficiency of multi-tiered resource allocation bidding systems where allocations are based on monetary bids leading to a competitive congestion game model. We consider resources that are priced and proportionally divided among the users. This paper focuses on two aspects: (i) the impact of wealth and (ii) the inefficiency of Nash equilibrium. Motivated by the recent debate on Net-Neutrality we consider the impact of two distinct categories of players, one with higher endowment than the other. We define Wealth impact factor (WIF) as the measure of disparity of pay-offs between the rich and the poor when the game is at NE. Surprisingly, improving WIF requires quadratic effort by the poor players. which shows the disparity between the rich and the poor when considering multiple tiers of service. We also consider the inefficiency of Nash equilibrium that arises in resource allocation. The inefficiency of utilities achieved in Nash equilibrium, measured by the price of anarchy, has been shown to be at least 3/4 by Johari and Tsitsiklis. Since the effective utilities of the players depends on the payments, we define the social objective as a function of pay-offs and express the price of anarchy in terms of a measure that we term as the economic efficiency factor (ECF). We show show that this inefficiency can be as large as n, the number of players for linear utilities. Interestingly, for strictly concave utilities the ECF is shown to be bounded, based on the behavior of the derivatives of the utility functions.