IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 2, APRIL 2012 117 Congestion, Information, and Secret Information in Flow Networks Khoa Tran Phan, Mihaela van der Schaar, Fellow, IEEE, and William R. Zame Abstract—Some users of a communications network may have more information about traffic on the network than others do—and this fact may be secret. Such secret information would allow the possessor to tailor its own traffic to the traffic of others; this would help the secret information possessor or informed user and (might) harm other uninformed users. To quantitatively study the impact of secret information, we formulate a flow control game with incomplete information where users choose their flows in order to maximize their (expected) utilities given the distribution of the actions of others. In this environment, the natural baseline notion is Bayesian Nash Equilibrium (BNE); we establish the existence of BNE. Next, we assume that there is a user who knows the realized congestion created by other users, but that the presence of this informed user is not known by other uninformed users; thus, the informed user has secret information. For this environment, we define a new equilibrium concept: the Bayesian Nash Equilibrium with Secret Information (BNE-SI) and establish its existence. We establish rigorous estimates for the benefit (to the informed user) and harm (to the uninformed users) that result from secret information; both the benefit and the harm become smaller for large networks. Interestingly, simulations demonstrate that secret information may in fact benefit all users. Secret information may also harm uninformed users in particular scenarios. This analysis can be used as a starting point for securing communications networks, both from the network manager and the user’s perspectives. Index Terms—Bayesian game, communication games, secret information. I. INTRODUCTION T HE smooth functioning of many communication net- works depends on the information users have about the network and about other users. In such settings, it is of concern that some users (whom we call informed users) might acquire “illicit” information, and that this illicit information might aid informed users and—perhaps more importantly—harm others (whom we call uninformed users). This potential for harm might reduce the willingness of uninformed users to pay for network services; if the potential for harm is great enough, Manuscript received March 30, 2011; revised July 25, 2011; accepted De- cember 25, 2011. Date of publication January 02, 2012; date of current version March 09, 2012. The work of K. T. Phan and M. van der Schaar was supported in part by the National Science Foundation (NSF) under Grant 0830556 and the work of W. R. Zame was supported by in part by the NSF under Grant SES-0518936. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Amir Leshem. K. T. Phan and M. van der Schaar are with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095 USA (e-mail: kphan@ee.ucla.edu; mihaela@ee.ucla.edu). W. R. Zame is with the Department of Economics, University of California, Los Angeles, CA 90095 USA (e-mail: zame@econ.ucla.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTSP.2011.2182496 it might deter uninformed users from joining the network at all. Thus the designer/manager of a network has an important incentive to secure the network so that no users can obtain illicit information. Because such security might be costly, it is important to know the extent to which illicit information would be useful to an informed user and harmful to uninformed users. Many kinds of illicit information might be relevant; in this paper we focus our attention on illicit information about the be- havior of other users. We set our study in the context of flow control. We consider a network of users of which only some may be online at a given moment. Users are distinguished by their utility functions, which we think of as their types. Each of the users chooses a flow to send to the network and derives a utility that depends on its own flow and on network conges- tion, which we proxy by the ratio of total flow to the capacity of the network. In our baseline scenario, the distribution of char- acteristics of users is commonly known but the realization of characteristics of users who are online at a given moment is not. Hence users can work out—or learn—the distribution of con- gestion, but not the realized congestion. For this scenario, an ap- propriate solution notion is Bayesian Nash Equilibrium (BNE). Under appropriate assumptions, we show that BNE exist. To ex- plore the impact of secret information, we depart from the base- line scenario by assuming that some (informed) user knows, not only its own type or utility function and the distribution of types of potential users, but also the realized congestion created by other users. Furthermore, other users do not know this; thus, the informed user has secret information. 1 For this scenario, an appropriate solution notion is what we call Bayesian Nash Equi- librium with Secret Information (BNE-SI); under the same as- sumptions as before, we show that BNE-SI exist. To see the distinction in a simple setting, suppose there are two players whose utilities depend on the actions of both players. Player 1 chooses an action first, player 2 chooses an action second. It is clear that the “correct” action for player 1 depends not only on whether or not player 2 sees the action of player 1 before choosing his own action, but also on whether or not player 1 knows whether player 2 sees his action. The standard Harsanyi framework [3], [21] would incorporate a probability that player 2 sees this action and assume that is common knowledge. Hence, player 1 believes with probability that player 2 sees player 1’s action, and player 2 believes that player 1 believes with probability that player 2 sees player 1’s action, and so forth. In our framework, player 1 believes (with probability 1) that player 2 does not see, and player 2 knows this. In the Harsanyi framework, the equilibrium behavior of 1 We assume here that the informed user learns all the relevant informa- tion—in this case, the realized congestion—at no cost. A more elaborate model ought to take account of the amount of information that might be acquired and the cost of acquiring it, and the steps that a network manager might undertake to prevent users from acquiring such information. 1932-4553/$31.00 © 2012 IEEE