60 CHAPTER 2 Playing with Equilibria in Wireless Non-Cooperative Games This concept has been applied by Altman et al. (2009a) to study decentralized MACs with constraints. The impact of these constraints is to correlate the players' actions. As mentioned at the beginning of this section, there can be multiple NE in this type of game. This is precisely what happens in decentralized MACs. One of the problems that arises in such contexts is to know how a player values the fact that the constraints on another player are satisfied or violated. Some extreme cases are as follows: i. A player is indifferent to satisfaction of constraints of other players. ii. Common constraints: if a constraint is violated for one player then it is violated for all players. The concept of normalized equilibrium, applicable to concave games, is one possible way of predicting the outcome of such a game and/or selecting one of the possible equilibria. Specifically, the authors of Altman et al. (2009a) have shown that, in the context of multiple access game with multiuser detection, the normalized equilibrium achieves maxmin fairness and is also proportionally fair (for these notions see for example Mo and Walrand (1998)). 2.4.2 The Role of Dynamics in Equilibrium Selection As pointed out in Section 2.1, Nash equilibria of a certain game can be observed in a degraded version of this game where players have less information; or more generally, a different knowledge. Typically, a certain static game with complete infor- mation can have multiple equilibria, but if it is played several times under the assump- tion of partial information, it might happen that only one equilibrium is effectively observed. For example, if the players play sequentially such that each player observes the actions played by the others, and reacts to them by playing his best response, and then the others update their strategy accordingly, and so on, it can happen that this game converges to the NE that would be obtained if the players knew the game com- pletely and played it in one shot. Figure 2.5 shows the possible NE in the power allocation game of Belmega et al. (2009a) in two-band two-user interference relay channels. In this figure, i represents the power fraction user i allocates to a frequency band, 1 - 1 being the fraction allocated to the other band. It can be shown that the (2) (3) (0) (2) (1) (0) (1) sequence i ; -i = BR -i i ; i = BR i -i ; -i = BR -i i , . . . will converge to one of the three possible NE, depending on the game starting point, i.e., (0) on the value of i {1, 2} and the value of i [0, 1]. As a more general conclusion, we see that the initial operating state of a network can determine the equilibrium state in decentralized networks with certain convergence properties. We will not expand on this issue here, but it is important to know that games with standard best-responses, potential games, and S-modular games have attractive convergence properties. For example, the authors of Sastry et al. (1994b) have shown how simple learning proce- dures, based on mild information assumptions, converge to the NE predicted in the associated game with complete information.