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Stability, Optimality and Complexity of Network Games with Pricing and Player Dropouts

We study basic properties of a class of noncooperative games whose players are 
 selfish, distributed users of a network and the game's broad objective 
 is to optimize Quality of Service (QoS) provision. 
 This class of games was previously introduced by the authors 
 and is a generalization of 
 well-studied network congestion games. The overall goal is to determine a minimal set of 
 static game rules based on pricing that result in stable and near optimal 
 QoS provision. We show the following. (i) 
 Standard techniques for exhibiting stability or existence of Nash 
 equilibria fail for these games—specifically, neither are 
 the utility functions convex, nor does a generalized potential 
 function exist. (ii) The problem of finding whether a 
 specific game instance in this class has a Nash equilibrium is NP-complete. To offset the apparent instability of these games, we show 
 positive results. 
 (iii) For natural subclasses of these games, although 
 generalized potential functions do not exist, approximate Nash equilibria 
 do exist and are easy to compute. (iv) 
 These games perform well in terms of price of stability and price of anarchy. I.e., 
 all of these approximate Nash equilibria nearly optimize 
 a communal (or social) welfare function, and there 
 is atleast one Nash equilibrium that is optimal. Finally, we give computer experiments illustrating the 
 basic dynamics of these games which indicate that 
 price thresholds could speed up convergence to Nash equilibria.