Encyclopedia of Systems and Control

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Network Games

  • R. SrikantEmail author
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DOI: https://doi.org/10.1007/978-1-4471-5102-9_35-2
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Abstract

Game theory plays a central role in studying systems with a number of interacting players competing for a common resource. A communication network serves as a prototypical example of such a system, where the common resource is the network, consisting of nodes and links with limited capacities, and the players are the computers, web servers, and other end hosts who want to transfer information over the shared network. In this entry, we present several examples of game-theoretic interaction in communication networks and a simple mathematical model to study one such instance, namely, resource allocation in the Internet.

Keywords

Congestion games Network economics Price-taking users Routing games Strategic users 

Introduction

A communication network can be viewed as a collection of resources shared by a set of competing users. If the network were totally unregulated, then each user would attempt to grab as many resources in the network as possible, resulting in poor network performance, a situation commonly referred to as the tragedy of the commons (Hardin, 1968). In reality, there is a carefully designed set of network protocols and pricing mechanisms which provide incentives to users to act in a socially responsible manner. Since game theory is the mathematical discipline which studies the interactions between selfish users, it is a natural tool to use to design these network control mechanisms. We now provide a few examples of network problems which naturally lend themselves to game-theoretic analysis. Later, we will elaborate on the game-theoretic formulation of one of these examples.
  • Resource Allocation: A network such as the Internet is a collection of links, where each link has a limited data-carrying capacity, usually measured in bits per second. The Internet is shared by billions of users, and the actions of these users have to be regulated so that they share the resources in the network in a fair manner. Equivalently, this problem can be viewed as one in which a network designer has to design a collection of protocols so that the users of the network can equitably allocate the available resources among themselves without the intervention of a central authority. Such protocols are built into every computer connected to the Internet today, to allow for seamless operation of the network. The problem of designing such protocols can be posed as a game-theoretic problem in which the players are the network and the traffic sources using the network (Kelly, 1997).

  • Routing Games: Finding appropriate routes for each user’s data traffic is a particular form of resource allocation mentioned above. However, routing has applications beyond communication networks (with the other major application area being transportation networks), so it is useful to discuss routing separately. In communication networks, each user may attempt to find the minimum-delay route for its traffic, with help from the network, to minimize the delay experienced by its packets. In a transportation network, each automobile on the road attempts to take the path of least congestion through the network. An active area of research in game theory is one which tries to understand the impact of individual user decisions on the global performance of the network (Roughgarden, 2005). An interesting result in this regard is the Braess paradox which is an example of a road transportation network in which the addition of a road leads to increased delays when each user selfishly choose a route to minimize its delay. Of course, if routes are chosen to minimize the overall delay experienced in the network, such a paradox will not arise.

  • Peer-to-Peer Applications: Many studies have indicated that file sharing between users (also known as peers) directly, without using a centralized web site such as YouTube, is a dominant source of traffic in the Internet. For such a peer-to-peer service to work, each peer should not only download files from others but should also be willing to sacrifice some of its resources to upload files to others. Naturally, peers would prefer to only download and not upload to minimize their resource usage. The design of incentive schemes to induce users to both download and upload files is another example of a game-theoretic problem in a network (Qiu and Srikant, 2004).

  • Network Economics: In addition to end-user interaction, Internet service providers (ISPs) have to interact with each other to allow their customers access to all the web sites in the world. For example, one ISP may have a customer who wants to access a web site connected to another ISP. In this case, the data traffic must cross ISP boundaries, and thus, one ISP has to transport data destined for a customer of another ISP. Thus, ISPs must be willing to contribute resources to satisfy the needs of customers who do not directly pay them. In such a situation, ISPs must have bilateral agreements (commonly known as peering agreements) to ensure that the selfish interest of each ISP to minimize its resource usage is aligned with the needs of its customers. Again, game theory is the right tool to study such inter-ISP interactions (Courcoubetis and Weber, 2003).

  • Spectrum Sharing: Large portions of the radio spectrum are severely underutilized. Typically, portions of the spectrum are assigned to a primary user, but the primary user does not use it most of the time. There has been a surge of interest recently in the concept of cognitive radio, whereby radios are cognitive of the presence or absence of the primary user, and when the primary user is absent, another radio can use the spectrum to transmit its data. When there are many users and the available spectrum is split into many channels, it is impossible for users to perfectly coordinate their transmissions to achieve maximum network utilization. In these situations, game-theoretic protocols which take into account the noncooperative behavior of the users can be designed to allow secondary users to access the available channels as efficiently as possible (Saad et al., 2009).

In the next section, we will elaborate on one of the applications above, namely, resource allocation in the Internet, and show how game-theoretic modeling can be used to design fair resource sharing.

Resource Allocation and Game Theory

Consider a network consisting of L links, with link l having capacity cl. Suppose that there are R users sharing the network, with each user r being characterized by a set of links which connect the user’s source to its destination. Since each user uses a fixed route in our model, we will use r to denote both the user and the route used by the user. We use the notation l ∈ r to denote that link l is a part of route r. Let xr denote the rate at which user r transmits data. Thus, we have the following natural constraints, which state that the total data rate on any link must be less than or equal to the capacity of the link:
$$\displaystyle \begin{aligned} \sum_{r:l\in r}x_r \leq c_l, \qquad \forall l. \end{aligned} $$
(1)
Associated with each user is a concave utility function Ur(xr) which is the utility that user r derives by transmitting data at rate xr. The network utility maximization problem is to solve
$$\displaystyle \begin{aligned} \max_{x\geq 0}\sum_r U_r(x_r), \end{aligned} $$
(2)
subject to the constraint (1). In (2), x denotes the vector (x1, x2, …, xR), and x ≥ 0 means that each component of x must be greater than or equal to zero. Note that the goal of the network in (2) is to maximize the sum of the utilities of the users in the network.
Let pl be the Lagrange multiplier corresponding to the capacity constraint in (1) for link l. Then the Lagrangian for the problem is given by
$$\displaystyle \begin{aligned} L(x,p)=\sum_r U_r(x_r)-\sum_l p_l (y_l-c_l), \end{aligned} $$
(3)
where we have used the notation yl :=∑r:lrxr to denote the total data rate on link l. If p is known, then the optimal x can be calculated by solving
$$\displaystyle \begin{aligned} \max_{x\geq 0} L(x,p). \end{aligned}$$
Notice that the optimal solution for each xr can be obtained by solving
$$\displaystyle \begin{aligned} \max_{x_r \geq 0} U_r(x_r)-q_r x_r, \end{aligned} $$
(4)
where qr =∑lrpl. Thus, if the Lagrange multipliers are known, then the network utility maximization can be interpreted as a game in the following manner. Suppose that the network charges each user qr dollars for every bit transmitted by user r though the network. Then, qrxr is the dollar per second spent by the user if xr is measured in bits per second. Interpreting Ur(xr) as the dollars per second that the user is willing to pay for transmitting at rate xr, the optimization problem in (4) is the problem faced by user r which wants to maximize its net utility, i.e., utility minus cost. Thus, the individual optimal solution for each user is also the solution to the network utility maximization problem. The above game-theoretic interpretation of the network utility maximization problem is somewhat trivial since, given the pl’s or qr’s, there is no interaction between the users. Of course, this interpretation relies on the ability of the network to compute p. We next present a scheme to compute p, which couples the users closely and thus allows for a richer game-theoretic interpretation.
Suppose that the network wants to compute p but does not have access to the utility functions of the users. The network asks each user r to bid an amount wr which is interpreted as the dollars per second that the user is willing to pay. The network then assumes that user r’s utility function is \(w_r\log x_r\) and solves the network utility maximization. While this choice of utility function may seem arbitrary, the resulting solution x has a number of attractive properties, including a form of fairness called proportional fairness. The proportionally fair resource allocation solution to (4) is given by
$$\displaystyle \begin{aligned} \frac{w_r}{x_r}=q_r. \end{aligned} $$
(5)
The network then allocates rate xr to user r and charges qr dollars per bit. From (5), the amount charged to user r per second is wr, thus satisfying the original interpretation of wr. Knowing that the network charges users in this manner, how might a user chooses its bid wr? Recall that user r’s goal is to solve (4). Substituting from (5), the problem in (4) can be rewritten as
$$\displaystyle \begin{aligned} \max_{w_r \geq 0} U_r\left(\frac{w_r}{q_r}\right)-w_r. \end{aligned} $$
(6)
Thus, the users’ problem of selecting w can be viewed as a game, with each user’s objective given by (6). Note that qr is given by (5) and thus depends on all the wr’s. Depending upon the application, the game can be solved under one of two assumptions:
  • Price-Taking Users: Under this assumption, users are assumed to take the price qr as given, i.e., they do not attempt to infer the impact of their actions on the price. This is a reasonable assumption in a large network such as the Internet, where the impact of a single user on the link prices is negligible, and it is practically impossible for any user to infer the impact of its decisions on the prevailing price of the network resources. When the users are price taking, the socially optimal solution, i.e., the solution to the network utility maximization problem, coincides with the Nash equilibrium of the game. To see this, note that the solution to (6) is given by
    $$\displaystyle \begin{aligned} \frac{1}{q_r}U_r^{\prime}\left(\frac{w_r}{q_r}\right)-1=0, \end{aligned}$$
    under the assumption that the utility function is differentiable and the solution is bounded away from zero. Using (5), this equation reduces to
    $$\displaystyle \begin{aligned} U_r^{\prime}(x_r)=q_r, \end{aligned}$$
    which maximizes the Lagrangian (3). It is not difficult to see that the complementary slackness equations in the Karush-Kuhn-Tucker conditions are satisfied since the constraints for (2) and the proportionally fair solution are the same. Thus, under the price-taking assumption, the equilibrium of the game solution is the same as the socially optimal solution provided the network computes qr using the proportionally fair resource allocation formulation.
  • Strategic Users: In networks where the number of users is small, it may be possible for each user to know the topology of the network, and thus each user may be able to solve for the proportionally fair resource allocation if it has access to other users’ bids. In other words, it may be possible to compute a Nash equilibrium by taking into account the impact of the wr’s on the qr’s. When the users are strategic, the socially optimal solution could be quite different from the Nash equilibrium. The ratio of the network utility under the socially optimal solution to the network utility under a Nash equilibrium is called the price of anarchy.

There is a rich literature associated with both interpretations of the network congestion game. In the case of price-taking users, much of the emphasis in the literature has been on designing distributed algorithms to achieve the socially optimal solution (Shakkottai and Srikant, 2007). In the case of strategic users, the focus has been on characterizing the price of anarchy (Johari and Tsitsiklis, 2004; Yang and Hajek, 2007).

Summary and Future Directions

We have presented a number of applications which involve the interactions of selfish users over a network. For the resource allocation application, we have also described how simple mathematical models can be used to provide incentives for users to act in a socially optimal manner. In particular, we have shown that, under the reasonable price-taking assumption and an appropriate computation of link prices, selfish users automatically maximize network utility. In the case where the users are strategic, the goal is to characterize the price of anarchy.

Moving forward, two areas which require considerable further research are the following: (i) inter-ISP routing and (ii) spectrum sharing. The Internet is a fairly reliably network, and any unreliability often arises due to routing issues among ISPs. As mentioned in the introduction, peering arrangements between ISPs are necessary to make sure that ISPs carry each others’ traffic and are appropriately compensated for it, either through reciprocal traffic-carrying agreements or actual monetary transfer. Thus, the policy that an ISP uses to route traffic may be governed by these peering agreements. The more complicated these policies are, the more chances there are for routing misconfigurations that lead to service interruptions. This interplay between policies and technology in the form of routing algorithms is an interesting topic for further study.

Cognitive radios and spectrum sharing are expected to be significant technological components of future wireless networks. Designing algorithms for selfish radios to share the available spectrum while respecting the rights of the primary user of the spectrum is a challenge that requires considerable further attention. This area of research requires one to combine sensing technologies to sense the presence of other users with game-theoretic models to ensure fair channel access to the secondary users, subject to the constraint that the primary user should not be affected by the presence of the secondary users.

Cross-References

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical and Computer Engineering and the Coordinated Science LabUniversity of Illinois at Urbana-ChampaignChampaignUSA

Section editors and affiliations

  • Tamer Başar
    • 1
  1. 1.Coordinated Science Laboratory and Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA