The Price of Anarchy in Parallel - Serial Competition with Elastic Demand1
John Musacchio Technology and Information Management, University of California, Santa Cruz, CA, USA [email protected] Technical Report No. UCSC-SOE-09-20 Baskin School of Engineering, University of California, Santa Cruz, CA, USA
Abstract We study a game theoretic model of competing network service providers that strate- gically price their service in the presence of elastic user demand. Demand is elastic in that it diminishes both with higher prices and latency, which in turn grows linearly with a network’s usage. The model we study is based on a model first proposed and studied by Acemoglu and Ozdaglar and later extended by Hayrapetyan, Tardos, and Wexler to consider elastic user demand. The model we study supposes a topology of service providers connected in parallel and serial combinations. We consider the Price of Anarchy (PoA), which we define as the ratio of the social welfare of the system when a social planner chooses link prices versus the social welfare attained when link owners choose the link prices selfishly. We make an analogy between the game and an electric circuit, and in this analogy the slope of each provider’s latency function is taken to be the resistance of a circuit branch. Using the analogy, we find a bound on the PoA that depends on the ratio of the conductance of the circuit branch with highest conductance to the combined con- ductance of all the branches. This ratio is a metric of how much of the system’s capability is concentrated in one branch, so our bound measures how efficiency loss increases as the capability of the system becomes more concentrated in a particular serial combination of players.
1 Introduction
We study a a network pricing model based on a model first proposed and studied by Ace- moglu and Ozdaglar [1] and later extended by Hayrapetyan, Tardos, and Wexler [8]. The model studies the pricing behavior of several providers competing to offer users connectivity between two nodes, and it has the features that i) a provider’s link becomes less attractive as it becomes congested; and ii) that user demand is elastic – users will choose not to use any link if the sum of the price and latency of the available links is too high. In the first version of the model studied by Acemoglu and Ozdaglar, the user elasticity is modeled by assuming that all users have a single reservation utility and that if the best available price plus latency exceeds this level, users do not use any service. In this setting, the authors find that the price of anarchy – the worst case ratio of social welfare achieved by a social
1This research was supported in part by NSF Grant CNS-0626397
1 planner choosing prices to the social welfare arising when providers strategically choose their prices – is (√2+1)/2 [1]. (Or expressed the other way, as the ratio of welfare in Nash equilibrium to social optimum, the ratio is 2√2 2.) In [6], Acemoglu and Ozdaglar extend − the model to consider providers in a parallel-serial combination. Traffic chooses between several branches connected in parallel, and then for each branch, the traffic traverses the links of several providers connected serially. Hayrapetyan, Tardos, and Wexler consider a model in which user demand is elastic, which is the form of the demand model we study in the present paper [8]. The topology they consider is just the simple situation of competing links between two nodes. They derived the first loose bounds on the price of anarchy for this model. Later Ozdaglar showed that the bound is actually 1.5, and furthermore that this bound is tight [2]. Ozdaglar’s derivation uses techniques of mathematical programming, and is similar to the techniques used in [1]. In [9], Musacchio and Wu provide an independent derivation of the same result using an analogy to an electrical circuit where each branch represents a provider link and the current represents flow. The present paper generalizes the work of [9] by considering a more general topology of providers connected in parallel and serial combinations. We derive a bound on the price of anarchy using a circuit analogy similar to the argument used in [9]. In addition, our bounds depend on a metric that we suggest is a measure of how concentrated the market power is in the network. That metric is the reciprocal of the slope of the latency function of the network branch with the least slope divided by the harmonic mean of the slopes of all the branches. In terms of the circuit analogy, this is simply the ratio between the conductance of the most conductive branch to the conductance of the whole system. The models of [1], [8], and the present work are part of a stream of recent literature studying the price of anarchy for games of selfish routing. Several works, see for example [10, 11, 12], study games where users are selfish, but the network is passive, the owners of the edges are not strategically choosing their prices. Other work such as [13] consider the problem of having the network choosing prices to induce optimal routing among selfish users, rather than having parts of the network selfishly choose prices to maximize revenue. Another related work is by Johari and Tsitsiklis [14]. In [14] the authors study games where the users request a bit rate from the network, and in turn the network returns to users a price that depends on the sum of the requested rates on each resource. The model of [14] and the model of this paper study very different situations. In particular in [14] the main strategic agents are the users and in the model we study the strategic agents are link operators seeking to maximize profit. However, there are many similarities between the two games in terms of the structure of the payofffunctions of the players. As in this paper, the authors of [14] derive bounds on the efficiency loss.
2 Model
We consider a network consisting of a single source and destination node, connected by an arbitrary topology of providers connected both in parallel and series as depicted in Figure 1. In order to describe the topology, we create hierarchically nested groups. Groups at an odd numbered level of the hierarchy contain groups connected in parallel, and groups at an even level of the hierarchy contain groups connected in parallel. We make this precise as follows. There are n level 1 groups, each denoted as i, i 1, ..., n that are connected G ∈{ } in parallel. We also refer to the level 1 groups as “branches.” Each branch i consists of gi level 2 groups of providers connected in series, and the set of providers in each of these level 2 groups j 1, ..., gi is denoted as ij. Each level 2 group ij is either a single ∈{ } G G provider, or consists of gij level 3 groups connected in parallel, and furthermore, each of these groups is denoted ijk where k 1, ..., gij . In general each nth level grouping has G ∈{ } indices in,in− ,in− , ..., i that indicate membership in an n 1 level grouping, an n 2 1 2 1 − − level grouping, and so on. Each nth level grouping for n 1, 3, 5, ... , consists of gi1,i2...,in ∈{ }
2 V
I
p G n1 112 a 1121 an1
G a a p 1122 111 11221 11222 a111 G 11
p121 p122 a121 a122 G 12
p13
a13 G G 1 n b b1 n
Figure 1: An analogous circuit for analyzing the network. groupings in parallel. Likewise each nth level grouping for n 2, 4, 6, ... consists of ∈{ } gi1,i2...,in groupings in series. Each provider i =(i1,i2, ..., in) charges a price pi per unit flow and has latency li(fi)=aifi where fi is the flow through provider i. Users choose the path to take on the basis of the sum of all the prices and latencies encountered a long a path, in a way that we make precise below. We refer to the combination of price plus latency as “disutility,” and there fore the disutility encountered on a single provider i’s link is li(fi)+pi. In addition to the latency of each provider which scales with flow, we allow each branch i to have a fixed latency bi that does not scale with flow. For each loop free path in P branch i the disutility incurred by flow taking that path is i∈P li(fi)+pi + bi. (Loop free means that the path never passes through the same interconnection point more than ! once.) Users are nonatomic and are free to choose the path that has the lowest disutility. Therefore, in equilibrium, all used paths have a common disutility value, otherwise users would switch paths. Furthermore, the disutility across all groupings connected in parallel that are being used also have the same disutility, because otherwise users would switch to a route via the grouping with lower disutility. This equilibrium of user routing is known as a Wardrop equilibrium [3]. Each player’s i profit or payofffunction is the product of their price pi and the Wardrop equilibrium flow fi, which in-turn is effected by the entire profile of prices from the other players. Users have a limit to how much disutility they will tolerate – if all the paths have a disutility higher than a user’s tolerance or “willingness to suffer”, that user does not use any path. Users are distributed in their willingness to suffer, so we may define a function U(x) to be the disutility that would induce a total flow of x across all paths. To describe it another way, suppose each user has a willingness to suffer that is independent and identically distributed like the random variable W . Let S be the total population of users and let R(d)=SP(W > d) be S times the complementary cumulative distribution
3 function of willingness to suffer W . Then U(x) is the inverse function of R(d). Clearly U(x) is decreasing. We make the additional assumption that U(x) is concave. The assumption that U(x) is concave is a strong assumption, but is necessary to derive the bound in this work. The authors of [2] and [8] make this assumption as well. We also assume U(x) is continuous and almost everywhere differentiable. In summary, the game G is characterized by the following: a disutility function U( ), · latency functions li( ) and fixed branch latencies bi . We are interested in studying the { · } { } Nash equilibrium that results in game G when providers non-cooperatively adjust their prices until they reach a strategy profile (a set of price choices for all providers) such that no provider can improve his profit by unilaterally changing price. We will compare this to the social optimum “configuration” of the game G in which a hypothetical central planner chooses all prices to maximize the social welfare of the whole system – the sum of provider profits plus the welfare of users. Our measure of user welfare is consumer surplus. The classic definition of consumer surplus considers the difference between each user’s willingness to pay and what she actually pay, and then sums this difference across all the users. In this model the surplus is found by integrating the difference between each unit of flow’s willingness to tolerate disutility, and the disutility the flow actually bears. Therefore the surplus, and hence user welfare, is given by
f (U(x) d)dx − "0 where f is the total flow carried by the system, and d is the equilibrium disutility found on all used branches.
2.1 Circuit Analogy The Wardrop equilibrium conditions have an exact analogy in the electric circuit pictured in Figure 1 where voltage and current are analogous to disutility and flow. The diodes in the circuit ensure that a provider’s flow is never negative. The disutility curve is represented by a nonlinear power source whose voltage-current characteristic is given by the disutility curve. In order to make use of the circuit analogy, we define notation for the equivalent resistance of each level of groupings in the circuit.
If i...j is a singleton, then ai...j is the slope of that player’s latency function. • G If i...j is a group of elements connected serially, then • G
ai...j = ai...jk. #k
If i...j is a group of elements connected in parallel, then • G −1 ! −1 ai...j = ai....jk ai...jk . % & $k #k For a given total flow f, the power source has a voltage-current characteristic with slope s(f). Around this operating point, the power source can be approximated as a voltage source of size V with “internal resistance” s. When the power source is connected to a load, that draws f +! flow, the voltage from the power source would be V s(f +!) o(!2). − − Because of concavity, the voltage from the nonlinear source is less than that of its linear approximation. In order to evaluate the conditions for Nash equilibrium, we need to understand how a unilateral price change of a provider will change the flow, and in turn profits, of that provider. In the circuit analogy, this is analogous to the question: how much will the
4 current through a provider increase if the voltage source changes? To answer this question, we can use a basic result from circuit analysis known as Th´evenin’s theorem to reduce a complex resistive circuit to a simple one. Th´evenin’s theorem says that any resistive circuit seen from a “port” or pair of nodes can be reduced to a an open-circuit voltage and a Th´evenin equivalent resistance [4]. The circuit of Figure 1 is not completely resistive in that it has diodes. For the time being, we will suppose that the circuit is in an operating point for which only the diodes at the bottom of each branch are blocking flow. This assumption is equivalent to assuming that providers are not charging a price higher than the disutility of competing paths. We define the set ON to be the set of branches for which the flow is not being blocked by the diodes. We define the following notation for the Th´evenin equivalent resistance seen at different levels of the hierarchy of groups. The quantity δi...j(s, ON ) is the Th´evenin equivalent resistance seen after removing i...j and measuring from the nodes where the group was G connected. At level 1 of the hierarchy, the Th´evenin equivalent resistance is found by the relation
δi(s, ON )= aj s. ON ⊕ j=# i,j$∈ Deeper into the hierarchy, the Th´evenin resistance can be found recursively. If i...jk is a G group of elements connected serially, then
δi...jk(s, ON )= ai...jl δi...j(s, ON ). ⊕ l# k $= If i...jk(s, ON ) is a group of elements connected in parallel, then G
δi...jk = δi...j(s, ON )+ ai...jl. #l=# k 3 Analysis 3.1 Price-Flow Relationships In this subsection we find relations between the price of each link and the flow on each link in both Nash equilibrium and the social optimum configuration. By using these relations we are able to draw an analogy between the pricing game and a network of resistors. In the analogous resistor network, providers choose the resistance on their branch of the circuit. In Nash equilibrium, providers pick a resistance value that is higher than the social optimum value, causing current (flow) to be less than the socially optimal value.
3.1.1 Nash Equilibrium Providers connected serially can have very inefficient equilibria. Consider just two providers connected serially, each of which charges a price larger than the highest point on the disutility curve. The flow through these providers will be zero, and furthermore it is a Nash equilibrium for these providers to charge these prices. This is because one player cannot cause the flow to become positive by lowering his price, so each player is playing a best response to the other player’s prices. In examples for which there are more than one provider connected serially on each branch, one can construct a Nash equilibrium in which no flow is carried, and therefore the price of anarchy across all Nash equilibria is infinite. However these infinite price of anarchy Nash equilibria seem unlikely to be played in a real situation. Intuitively, a provider would want to charge a low enough price so that it would be at least possible that he could carry some flow if the other providers on the branch also had a low enough price. With that in mind, we would like to identify a more restricted
5 set of Nash equilibria that seen more “reasonable” and that have a price of anarchy that can be bounded. In the examples where serial players charge very high prices, the player’s best response is not unique, and we can use that fact to chose a class of equilibria that excludes these examples. For instance, we can consider the notion of strict Nash equilibrium. Like a conventional Nash equilibrium, a strict Nash equilibrium requires that each player’s strategy be a best response to the other players. However a strict Nash equilibrium requires that each player be playing a unique best response. In this context, a strict equilibrium requires that every player carry a positive flow, because if otherwise their profit would be zero and there are many prices that lead to a profit of zero. Ozdaglar and Acemoglu [6] use the notion of strict equilibrium in order to have a class of equilibria with bounded price of anarchy for parallel-serial competition without elastic demand. In our model, for some cases it is not possible to find a Nash equilibrium in which all players carry a positive flow, which is required in strict equilibrium. For example, if the fixed latency bi on a branch i is large enough, it will not carry flow regardless of what the provider profits are. For our model, we would like to define a class of equilibria that has the property that it always exists, and that it is possible to bound the price of anarchy. To that end, we define the notion of a zero-flow zero-price equilibrium. A zero-flow zero-price equilibrium is a Nash equilibrium for which players who are carrying zero-flow must charge a zero-price. The intuition motivating this definition is that a player who is not attracting any flow will at least try lowering his price as far as possible in order to attract some flow. We will later show that there always exists a zero-flow zero-price equilibrium, and that the price of anarchy is bounded. At this point, we need to identify the condition for a player to be playing a best response in order to construct Nash equilibria. In the following Theorem we show that a player playing a strict best response must have a price to flow ratio equal to the Th´evenin equivalent resistance he sees from the rest of the circuit, plus the slope of his latency function.
Theorem 1. Consider a set of prices pi and flows fi that satisfy the Wardrop equilib- { } { } rium conditions. Let ON + be the set of branches that are on following a small increase in price by player i, and let ON − be the set of on branches following a small decrease. Like- wise let be s+ and s− be the magnitude of the right-hand and left-hand derivatives of the + disutility function at the operating point. More precisely s = lim!↓0(U(f + !) U(f))/! − − − and s = lim!↑ (U(f + !) U(f))/!. − 0 − A necessary and sufficient condition for a player i to be playing a strict best response is that
pi + + − − [δi(ON ,s )+ai,δi(ON ,s )+ai]. (1) fi ∈
If ON + = ON − = ON and s+ = s− = s, the condition reduces to
pi = δi(ON ,s)+ai. fi
Proof. Consider a player i charging price pi and carrying flow fi. A unilateral change in + price by an amount +! will change the flow the provider carries by !/(δi(ON ,s)h + − ai) o(!2). The term o(!2) is negative because the actual flow coming from the power − − source is never more than the estimate given buy the source’s linear approximation. The profit resulting from this price change is
+ 2 (pi + !) fi !/(δi(ON ,s)+ai) o(! ) = − − + −1 2 pifi +[ pi(δi(ON ,s)+ai) + fi]! o(! ) + , − −
6 The above expression is less than the original profit pifi for all nonzero ! if and only if
pi + δi(ON ,s)+ai. fi ≥ A similar analysis for a unilateral price decrease yields
pi − δi(ON ,s)+ai. fi ≤
At this point we introduce the concept of the Nash circuit. The Nash circuit with resistances ai + δi(ON ,s) is the circuit that results when the voltage source representing { } each provider i’s price (see Figure 1) is replaced with a resistor of size ai + δi(ON ,s). The Nash circuit construction is useful, because a solution to it is a Nash equilibrium. The following corollary makes this idea precise
Corollary 1. Suppose that there exists a set of branches ON and slope s such that for flows fi that satisfy the KVL and KCL conditions of the Nash circuit with resistances { } ai + δi(ON ,s) : { }
fi > 0 i ON i, ⇔ ∈ ∀ U &(f)= s, − then the prices fi(ai + δi(ON ,s)) constitute a Nash equilibrium. { } The corollary requires that there be an ON set such that when the resistances δi(ON ,s) , { } the resulting circuit has flows where the branches that are actually “on” (carrying flow) correspond to the ON set. Furthermore, the corollary requires the existence a “fixed point” slope s that when used to determine the resistances δi(ON ,s) , the resulting total flow f { } occurs at a point on the disutility curve with slope s. We address the problem of finding the set ON and slope s in section 3.2, and we will find that there are cases where we cannot find a pair ON and slope s. But in such cases, we will show that a Nash equilibrium can be constructed by using the condition that the price to flow ratio be in a range when the sets ON + and ON − are different. At this point in the development it suffices to merely recognize that the Nash circuit can be used to construct Nash equilibria.
3.1.2 Social Optimum
Simple Parallel Serial Case
Now consider the socially optimal pricing. We first give an intuitive argument why the ∗ ∗ socially optimal price should be aifi where fi is the flow of link i in social optimum. The theory of Pigovian taxes says that one way to align the objectives of individuals with that of society as a whole is to make market individuals pay the cost of their externalities on others [7]. Following this notion the socially optimal pricing should price the flow so that each user bears the marginal cost to society of each additional unit of new flow. ∗ The latency on each branch is aifi + bi. However, the social cost is the latency times ∗2 ∗ the amount of flow bearing that latency. Therefore the cost is aifi + bifi . Thus the ∗ ∗ marginal cost is 2aifi + bi. The latency borne by users is aifi + bi. Therefore to make the ∗ disutility borne by users reflect marginal social cost, the price should be set to aifi . We call the circuit of Figure 1 where each provider i’s pricing voltage source is replaced by a resistor of size ai as the optimal circuit. In the lemma that follows, we show that the solution to the optimal flow problem can be found by solving Kirchoff’s current and voltage laws for the optimal circuit. To express the optimization problem, we define be T the set of group indices that refer to groups consisting of a single provider.
7 Lemma 1. Consider the game G. A social optimum assignment of flows to routes satisfies Kirchoff’s current and voltage laws (KCL and KVL) for the optimal circuit. Therefore, pi prices satisfying the relation ∗ = ai for each provider i induce optimal flows. fi Proof. A socially optimal assignment of flows to routes solves the following constrained optimization problem:
f 2 max U(x)dx aifi bifi − − 0 i∈T i -" . # # s.t. fi...jk fi...j = 0 for each parallel group i...j, k − G fi...jk fi...jl = 0 for each serial group i...j, ! − G (2) and (i...jk), (i...jl) i...j, ∈G fi 0 i. ≥ ∀ Note that the object is concave and Slater’s constraint qualification condition holds so there is no duality gap if we use Lagrangian techniques to find the optimal solution [15]. We therefore may find optimality conditions for the solution to the above problem by writing the Lagrangian, evaluating the first order conditions as well as the complementary slackness conditions for the Lagrange multipliers associated with the inequality constraints and simplifying. For groups i...j consisting of parallel singleton elements this procedure reveals that G ∗ 2ai...jkf = V ( i...j) (i...jk) i...j i...jk G ∀ ∈G where V ( i...j) is the Lagrange multiplier on the flow conservation constraint. This implies G that ∗ V ( i...j) fi...j = G . 2ai...j This relation is analogous to the circuit relation that current through resistors connected in parallel is found by dividing the voltage frop by the equivalent resistance of the parallel combination of resistors. For each serial group i...j of singletons or parallel groups of singletons G ∗ 2ai...jkf = V ( i...jk). i...j G # #k
For convenience, we can define V ( i...j) to be the right side of the above expression. V ( i...j) G G is analogous to a voltage drop across resistors connected serially. For each parallel grouping i...j of arbitrary sized subgroups, we find that G
V ( i...j)=V ( i...jk) i...jk i...j. G G ∀G ∈G In other words, the voltage drops across all branches connected in parallel are the same. This relationship, along with the flow conservation constraint leads to
∗ V ( i...j) fi...j = G . 2ai...j
For serial groupings of i...j of arbitrary sized subgroups, we can define G
V ( i...j) ! V ( i...jk). G G #k so that ∗ V ( i...j) fi...j = G . 2ai...j
8 For the flow across the N main branches, we have that
∗ ∗ ∗ D 2aifi bi =0 fi > 0 ∗ − − ∗ /D bi < 0 fi =0. − where D∗ = U(f ∗) is the disutility of the used branches in the optimal solution. Thus we have verified the exact parallel between the optimality conditions of the flow assignment problem, and the KCL and KVL equations of the optimal circuit.
Lemma 2. Consider the optimal circuit with all diodes removed. (This allows the flow in some branches to possibly be negative.) Then in a solution to circuit equations, the quantity
f 2 U(x)dx aifi bifi − − 0 i i -" . #∈T # is at least as large as the social welfare in social optimum of game G.
Proof. Consider the following optimization problem:
f 2 max U(x)dx aifi bifi − − 0 i i -" . #∈T # s.t. fi...jk fi...j = 0 for each parallel group i...j, k − G fi...jk fi...jl = 0 for each serial group i...j, ! − G and (i...jk), (i...jl) i...j. ∈G This can be seen to be the same problem as (2) except that the constraint that flows be positive is relaxed. Relaxing the constraints cannot decrease the solution to the problem. It is easy to verify that the first order conditions of the Lagrangian of this problem correspond to the KVL and KCL equations of the optimal circuit with the diodes removed.
3.2 Nash Equilibrium Existence In Theorem 1 we have found conditions for a provider to be playing a strict best response. In this subsection we address whether a Nash equilibrium exists. To start with, we will suppose that the set ON (the set of branches that are “on.”) for purposes of computing Th´evenin equivalents is fixed to a particular set even if that S assignment leads to a different set of branches actually being on in the circuit. Later, we will resolve this by finding a fixed point so that = ON . We start with the following S Lemma.
Lemma 3. Consider either the optimal or Nash circuits. Fix a constant s and set so S that δi(s, ) are taken to be constant resistances for all i. The circuit has a unique solution. S Proof. It is straightforward to verify that all but the diodes blocking backwards flow from each voltage source bi maybe deleted without effecting the KCL and KVL conditions fora solution to the circuit. With these deletions, each of the highest level groups i consist of G strictly passive circuit elements and thus have a unique solution for each voltage across each group’s connection point, and furthermore each group can be by replaced by a Th´evenin equivalent resistancea ˜i without effecting the solution to the rest of the circuit. ( See [4]) It remains to show that there is a unique solution of currents traversing the highest level groups. A solution can be constructed by the following procedure. If mini bi >U(0) then { } having zero flow on all branches, and all diodes blocking is a valid solution. Let OFF = bi ∗ ∗ { } and ON = ∅ and f = 0. Pick i = mini OFF. If U(f) >bi∗ let OFF OFF i and ← − ON ON + i∗.Then find a flow assignment assuming that the branches in ON have on ← diodes and the branches in OFF are replaced with open circuits. Such a solution should
9 be unique because the circuit elements are strictly passive and monotone [4]2. Let f be the sum of flows across all on branches. Repeat until OFF is exhausted or until U(f) >bi∗. Flow f is monotone increasing at each step because we are always adding branches for more current to flow, and each added branch has a bi less than the voltage of the terminal the branch is connected to. Because the bi’s of the already on branches are lower, the flow through them never becomes negative. Uniqueness can be shown as follows. Suppose that there are two solutions to the circuit. Let f and f & represent the flow vectors in both solutions. If the same diodes are on and off, the two solutions would represent non unique solutions to the same passive circuit (one where on diodes are replaced with closed circuits and the offones open.) which is not possible. Therefore the two solutions must have different sets of on and offdiodes. Consider two cases: i) Suppose that the set of ON diodes in each case are not subsets & & & of the other ON " ON and ON " ON . Then there exists bi Lemma 4. Consider the Nash circuit where the quantity s used to determine the resistances & δi(s, ) is allowed to vary with the slope of the disutility function U (f), however we fix a S set . There is a unique solution to this circuit. S Proof. For each fixed s Lemma 3 says that there is a unique solution to the Nash circuit. An increase in s increases each δi(s, ) which therefore must decrease the total flow in S the unique circuit solution. This relation describes a monotone decreasing function s → f(s). The range is restricted to [0,f ] where f = U −1(0) and the domain is [0, ) max max ∞ The slope of U(f) is monotone increasing by concavity therefore U &(f) is a monotone − increasing function in f mapping [0,fmax] to [0,smax]. The two functions must have a unique intersection point, and this point is the unique solution. Theorem 2. There exists at least one Nash equilibrium. Furthermore, a Nash equilibrium of one of the following two types exists: 1. There is a set of Nash equilibria that satisfies the following two properties: i) In the reduced player game with only players carrying positive flow, the players are in strict Nash equilibrium, ii) The players carrying no flow are in branches such that even if all players in the branch reduced their price to zero, the flow in the branch would still be zero, thus the prices of players in these branches are arbitrary. 2. There exists a set of Nash equilibria that satisfies the following properties: i) The players carrying positive flow are playing strict best responses, ii) There is at least one branch i with bi = U(f) and all players in that branch charge 0 price. iii) Branches with bi >U(f) carry no flow, and the prices of the players can be chosen arbitrarily. Proof. Without loss of generality, number the branches in increasing order of bi. Let D(k) be the value of U(f) found by analyzing the Nash circuit with resistances ai +δi(s(f), k) { S } where k = 1, ..., k . The solution is well defined and unique by Lemma 4. The function S { } D(k) is monotone non-increasing by the following reasoning. The larger k, the smaller the value of δi(s, k) for each i, and thus the larger f. Since U(f) is monotone non-increasing, S it follows that D(k) is monotone increasing. A branch i is not on if bi D(k). Therefore ≤ as k increases, the number of on branches will decrease. Let J(k) be the number of on branches as a function of k. If there is a fixed point J(k)=k that fixed point corresponds to a Nash equilibrium with k branches are turned on, and the rest undercut. 2 A strictly passive, monotone circuit element is such that for any different voltages v1, v2, the corresponding flows f1 and f2 satisfy the relation (v1 v2)(f1 f2) > 0 [4]. − − 10 If there is no fixed point, then there is a crossing with J(k) >kand J(k + 1) F (δi(s(f), k)) F (δi(s(f), k)) < f < F (δi(s(f), k )). S S +1 By continuity, there exists an assignment of resistances δi [δi(s(f), k ),δi(s(f), k)] ∈ S +1 S for which the flow is f = F (δi), and thus U(f)=bk+1. Suppose now that all providers i in branches k charge prices satisfying the price to flow ratio auli + δi and providers in S branch k + 1 charge 0. Branch k + 1 carries no flow because U(f)=bk+1. However, if any provider in branches 1...k increase price, they will induce flow in branch k + 1, so the Th´evenin equivalent they see for price increases is δi(s(f), k ) and for price decreases it S +1 is δi(s(f), k). Thus having the price-flow ratio in the above range satisfies the necessary S and sufficient condition for a best response given in Theorem 1. Corollary 2. There exists a zero-flow zero-price equilibrium. Proof. From Theorem 2 one can construct a set of Nash equilibrium of either type 1 or 2 from the statement of Theorem 2. For type 1, the prices of the players carrying no flow are arbitrary, so they can be set to zero. For type 2, each branch i with bi >U(f) have arbitrary prices, so they can be set to zero. In either case, the resulting Nash equilibrium satisfies the zero-flow zero-price property. Corollary 3. In any zero-flow zero-price equilibrium, or in any Nash equilibrium of the type described in the statement of Theorem 2, there exist constants duli , with duli { } ∈ [0, 1] such that the Nash equilibrium prices and flows satisfy the KVL and KCL equations of a Nash circuit with resistances ai + diδi(f,ON ) where f is the total flow in Nash { } equilibrium, and ON is the set of branches carrying positive flow. Proof. This result follows directly from the proof of Theorem 2. 3.3 Equivalent Resistances in the Nash Circuit Corollary 3 says that in Nash equilibrium, the prices and flows must solve the Nash circuit with resistances ai + diδi(s, ON ) for some set of non-negative constants di that are { } { } each less than 1. In this subsections we will find expressions for the equivalent resistances of branches of the Nash circuit by composing the resistances corresponding to individual providers. This process of composing resistances will allow us to treat arbitrary parallel- serial topologies by finding an “equivalent” simple parallel serial topology. Knowing the overall total resistance in each branch of the circuit will allow us to write expressions for the total flow in each branch, which is critical to being able to express the total social welfare achieved in Nash equilibrium. To write expressions for welfare we will also need to express the cost of latency – which is the latency of each provider times the 11 flow that has to bear that latency. In the Nash circuit, there is a resistor associated with each provider’s price and a separate resistor associated with the provider’s latency. & The flow across each provider’s i s resistor ai which is associated with latency incurs 2 a latency aifi and a latency loss of aifi , which of course is the power dissipated by the resistor in the circuit. Similarly the power dissipated by the resistor of size ai + diδi in the Nash circuit corresponds to the provider’s profit. As we compute the effective resistance of groupings, we need to keep track of what fraction of the effective resistance is due to latency cost. In other words if r is the equivalent resistance of a group, when flow across the group is f, the power dissipated is rf2 some fraction of that will be latency loss. If the power dissipated as latency loss is lf2, we call l the equivalent latency resistance of the group. Each group i...j, whether it contains elements (subgroups) connected in parallel or G serial, is connected with the rest of the circuit at two nodes. Let ηi...j be the maximum number of providers that can be crossed in a loop free path across group i...j connecting G these two nodes. (Loop free means that the path never traverses the same circuit node twice.) We will pursuer the following line of reasoning. Lemma 5 finds the equivalent resistances for a serial combination of singleton providers. Similarly, Lemma 6 finds the equivalent resistances for a serial combination of parallel providers. Finally, Lemma 7 finds the equivalent resistances of arbitrarily complex groups using an induction argument. Lemma 5. Consider the Nash circuit with resistances ai + diδi in place for the price { } for all the providers, where di [0, 1] for each i. The equivalent resistance ri...j of a group ∈ i...j of single providers connected serially is G ri...j = (1 + mi...j)ai...j + mi...jdi...jδi...j for some mi...j [1,ηi...j] and di...j [0, 1]. Furthermore, the equivalent latency resistance ∈ ∈ is ai...j. Proof. The resistance of the group i...j is just the sum across the resistances of each G provider (i...jk), which included both the resistance ai...jk + di...jkδi...jk representing price and the resistance ai...jk representing latency. Thus the total resistance is given by ri...j = 2ai...jk + di...jkδi...jk k # Substuting the definition of δi...jk we have ri...j = 2ai...jk + di...jkδi...j + di...jkai...jk". " #k k#=# k Since di...jk 1 and the sumation has ηi...j 1 terms, there exists an mi...j [1,ηi...j] and ≤ − ∈ di...j [0, 1] satisfying ∈ ri...j = (1 + mi...j)ai...j + mi...jdi...jδi...j. Since the group is connected serially, all the flow passes through each resistor. Therefore the equivalent resistance is the sum of the latency resistors ai...jk which by definition is ai...j. ! Lemma 6. Consider the Nash circuit with resistances ai + diδi in place for the price { } for all the providers, where di [0, 1] for each i. The equivalent resistance ri...j of a group ∈ gi...j of single providers connected in parallel is ri...j = (1 + mi...j)ai...j + mi...jdi...jδi...j where mi...j [1,ηi...j] and di...j [0, 1]. Furthermore, the equivalent latency resistance is ∈ ∈ ci...j2ai...j + ci...jdi...jδi...j 1 for some ci...j [0, ]. ∈ 2 12 Proof. The equivalent resistance of the group i...j is G (2ai...jk + di...jkδi...jk)= (2ai...jk + δi...j di...jkai..jk" ) q q " $ $ k$=# k =2ai...j + di...jδi...j for some di...j [0, 1]. A portion of the power dissipated in the group corresponds to ∈ provider profits, with the remainder corresponding to latency loss. Since more than half the resistance of each branch corresponds to the provider’s price, the latency loss must be less than half the power dissipated. Therefore the effective latency resistance is ci...j2ai...j + ci...jdi...jδi...j where ci...j [0,.5] is a constant. ∈ Lemma 7. Consider the Nash circuit with resistances ai+diδi . The equivalent resistance { } ri...j of a group gi...j is (1 + mi...j)ai...j + di...jmi...jδi...j form some mi...j [0,ηi...j] and di...j [0, 1]. Furthermore, the equivalent latency resistance ∈ ∈ is (1 + mi...j)ci...jai...j + ci...jdi...jmi...jδi...j 1 for some ci...j [0, ]. ∈ 2 Proof. The proof proceeds by induction. Suppose that it has been shown that for each subgroup i...jk i...j, the effective resistance is G ⊂G (1 + mi...jk)ai...jk + di...jkmi...jkδi...jk for some mi...jk [0,ηi...jk] and di...jk [0, 1]. Also suppose that it has been shown that ∈ ∈ the effective latency resistance is ci...jk(1 + mi...jk)ai...jk + ci...jkdi...jkmi...jkδi...jk 1 for some ci...jk [0, ]. Note that both of these facts are true if group i...jk consists of ∈ 2 G single providers by either Lemma 5 or 6. Suppose group gi...j contains subgroups connected in parallel: We claim that the equivalent resistance is (1 + mi...j)ai...j + di...jmi...jδi...j for some mi..j [0,ηi...j] and di...j [0, 1] by the following reasoning. Consider a hypo- ∈ ∈ thetical connection of resistors of size (1 + maxk" (1 + mi...jk")ai...jk k =1, 2, ... connected in parallel. The equivalent resistance of this interconnection is (1 + maxk" (1 + mi...jk")ai...j. Now add a resistor of size mi...jδi...j placed in series on each of the branches. This increases the resistance by an amount not more than mi...jδi...j. Now decrease the resistance on each branch k to be (1 + mi...jk)ai....jk. This will decrease the overall resistance of the interconnection, and also make the hypothetical interconnection match that of i...j. It G should now be possible to find suitable constants mi...j and di...j. The equivalent latency resistance is (1 + mi...j)ci...jai...j + ci...jdi...jmi...jδi...j 1 were ci...j [0, ] is a constant. his can be seen by the following reasoning. The latency ∈ 2 resistance of each subgroup is no more than half the total resistance. Therefore no more than half the total power dissipated group i...j can be due to latency. Thus the equivalent G latency resistance is no more than half the overall equivalent resistance. Suppose group gi...j contains subgroups connected in series: 13 Then the equivalent resistance is ri...j = (1 + mi...jk)ai...jk + di...jkmi...jkδi...jk #k Substuting the definition of δi...jk we have the following relatons ri...j (1 + ηi...jk)ai...jk + ηi...jkδi...jk ≤ #k (1 + ηi...jk)ai...jk + ηi...jk δi...j + ai...jk" ≤ " #k k#=# k ai...jk + ηi...jk ai...jk" + ηi...jkδi...j ≤ % " & #k #k (1 + ηi...j)ai...j + ηi...jδi...j. ≤ Also it is clear that ri...j ai...j. Thus there exists an mi...j [1,ηi...j] and di...j [0, 1] ≥ ∈ ∈ such that ri..j = (1 + mi...j)ai...j + di...jmi...jδi...j. The equivalent latency resistance is no more than half of the above quantity, because the equivalent latency resistance of each subgroup is no more than half the equivalent resistance of each subgroup. 3.4 Utility Function Truncation In this subsection, we argue that given a disutility function U( ), • · a Nash equilibrium of the type characterized by Theorem 2, and Nash equilibrium • flow f, an associated price of anarchy, • we can create a new example with the disutility function replaced by one that is identically equal to U(f) for flows between 0 and f, and then is a decreasing line of slope s for flows more than s. This new example will have a larger price of anarchy than the old example. Therefore we can restrict our search for the worst case price of anarchy to disutility functions of this shape. We begin the development of the argument with the following lemma. Lemma 8. Consider a Nash equilibrium of the game G with flows fi and prices pi , { } { } and total flow f. Let Gl be a new game identical to G except that the disutility function of the new game is a line tangential to the disutility function of G at the point U(f)=d, i.e. Ul(fl)=s (f fl)+d. If the left and right hand derivatives are different at U(f), × − + let Gl have the disutility function Ul(fl)=s (f fl)+d. Then fi and pi describe a − { } { } Nash equilibrium of Gl. Proof. Any provider i with positive flow satisfy the condition (1). While any provider with zero flow must be in a situation where if that provider unilaterally made his price arbitrarily small, his flow would be zero. Now consider flows fi and prices pi in the context of game Gl. These flows and { } { } prices satisfy the Wardrop equilibrium conditions. Furthermore, the best response condi- tion for any player i with positive flow becomes pi + + − + [δi(ON ,s )+ai,δi(ON ,s )+ai] fi ∈ 14 i) ii) d d -s p1 b3 a1 f f1 f2 f3 iii) iv) d d d* b3 a1 f f* User Provider Surplus Surplus Figure 2: i) The Nash equilibrium of the game G. (In this example three providers compete in parallel, and the shaded areas represent each of their profits.) ii) The Nash equilibrium of the game Gl, where the disutility function of G has been linearized. Note that the flow vector is unchanged from the Nash equilibrium of G. iii) The Nash equilibrium of the game Gt, where the disutility function of G has been linearized and “truncated.” iv) The social optimum configuration of the game G. Note that the flow f ∗ >fand that link 2 is not used in the social optimum configuration of this example. which is a weaker condition than the condition that we already know fi and prices pi satisfies. Furthermore any player with zero flow cannot induce a positive flow with an arbitrarily small price, for if he could, he would have been able to in game G. We conclude flows fi and prices pi describe a Nash equilibrium of game Gl. { } { } Having looked at a game with a linearized disutility function in the previous lemma, in the next lemma we consider a game with a “truncated” disutility function. Lemma 9. Consider a Nash equilibrium of the game G with flows fi and prices pi , { } { } and total flow f. Let Gt be a game that is identical to G except that the disutility curve that is modified from the disutility curve in G by “truncating” it in the following way: dxf U(x)= ≤ /d s+(x f) x > f. − − Then flows fi and prices pi describe a Nash equilibrium of game Gt. { } { } Proof. The argument is essentially the same as presented in [8]. Any provider i with positive flow satisfy the condition (1). While any provider with zero flow must be in a situation where if that provider unilaterally made his price arbitrarily small, his flow would be zero. Now consider flows fi and prices pi in the context of game Gl. These flows and { } { } prices satisfy the Wardrop equilibrium conditions. Furthermore, the best response condi- tion for any player i with positive flow becomes pi + + − [δi(ON ,s )+ai,δi(ON , )+ai] fi ∈ ∞ 15 which is a weaker condition than the condition that we already know fi and prices pi satisfies. Furthermore any player with zero flow cannot induce a positive flow with an arbitrarily small price, for if he could, he would have been able to in game G. We conclude flows fi and prices pi describe a Nash equilibrium of game Gt. { } { } Lemma 10. Consider a Nash equilibrium of the game G with flows fi and prices pi . { } { } By Lemmas 9 the same flows and prices describe a Nash equilibrium of game Gt. Let N(G) and S(G) be the social welfare in the Nash equilibrium being considered and in social optimum respectively of the game G. Likewise let N(Gt) and S(Gt) be the social welfare in the Nash equilibrium and social optimum of game Gt. Then S(G ) S(G) t . N(Gt) ≥ N(G) Proof. Let U(G) be the user welfare in the Nash equilibrium of game G. In the Nash equilibrium of game Gt, the prices and flows are the same, and thus the provider welfare will be the same in both cases. However, the user welfare will be 0. Thus we have N(Gt)=N(G) U(G). − Consider a new game Gtx with the same latency functions as G but with a disutility function that has been truncated but not linearized. Therefore dxf Utx(x)= ≤ /U(x) x > f. where d and f are the Nash equilibrium disutility and flow of G, Utx( ) is the disutility · function of Gtx, and U( ) is the disutility function of G. Because we know the disutility · d∗ in social optimum is less than d we can view the optimization problem of finding the optimal flow vector for game Gtx as the same problem as finding the optimal flow vector for the game G but with the constant U(G)= f (U(x) d)dx subtracted offthe objective 0 − function. Thus, the social optimum flow vector will be identical for game G as for game 4 tx Gt. Thus S(Gtx)=S(G) U(G). − Now consider the game Gt where we have both truncated and linearized the disutility function. By convexity, the disutility function of Gt is never smaller than that of Gtx. Thus the solution to the optimization problem of finding the optimal flow vector has an objective function that is never smaller than the objective function that we maximized to find S(Gtx). Consequently S(Gt) S(Gtx)=S(G) U(G). ≥ − Combining our observations, we have S(Gt) S(G) U(G) S(G) − . N(Gt) ≥ N(G) U(G) ≥ N(G) − 3.5 Preliminaries We now compare the social optimum and Nash equilibrium welfare. Let n& be the number of providers that carry flow in a Nash equilibrium of the type characterized by Theorem 2. In social optimum, the number of branches that carry no flow cannot decrease. This can be seen by the following reasoning. The Nash equilibrium solves the Nash circuit where the resistances for provider prices are diδi +ai for constants di [0, 1]. The social optimum ∈ 16 circuit has the same structure, but with the resistances on all branches reduced, thus the social optimum has a higher total flow, and thus U(f ∗) U(f) where f is the total flow ≤ in Nash equilibrium and f ∗ is the total flow in social optimum. Branches are undercut ∗ (carry no flow) when U(f ) bi, thus the Nash equilibrium cannot have more branches ≤ undercut than the social optimum solution. We define the following notation. The vectors of flows in Nash equilibrium and social optimum are T ∗ ∗ ∗ ∗ T F =[f1,f2, ..., fn] , and F =[f1 ,f2 , ..., fn] respectively. We define T T V =[V, V, ...V ] , and b =[b1,b2, ...bn] as an n dimensional vector of all V ’s and the vector of bi’s respectively. The matrices A = diag(a1,a2, ..., an),∆= diag(δ1,δ2, ..., δn),D = diag(d1,d2, ..., dn),M = diag(m1,m2, ..., mn) & are diagonal matrices of the ais ,δi’s, and di’s respectively. It will also be convenient to define 11 ... J = 11 ... . . . . . .. to be a n n matrix of all ones. Also define × −1 t ! tr(A ),α ! 1/s + t, and e ! 1m×1. We state and prove the following lemma. Lemma 11. The following identities hold: 1 1 (2A + sJ)−1 = A¯−1 A−1JA−1 , 2 − t +2s−1 - . ∆=( αA I)−1A. − Proof. The Matrix Inversion Lemma states that (F + UCV )−1 = F −1 F −1U(C−1 + VF−1U)−1VF−1 − where F, U, C, V are arbitrary matrices with only the condition that the matrices are of appropriate dimension and all the inverses in the above expression exist. Let F =2A, T T U = e, V = e , C = s/2 and note that J = ee =1m×m. Then we may apply the Matrix Inversion Lemma in the following way: −1 1 1 1 eT A−1e (2A + sJ)−1 = A−1 A−1e + eT A−1 2 − 4 s 2 - . 1 2 −1 = A−1 A−1e + tr(A−1) eT A−1 2 − s % - . & 1 1 = A−1 A−1JA−1 . 2 − 2s−1 + t - . −1 1 1 To derive the next identity, recall δi = j=# i ai + s . Therefore 7! 1 8 ∆−1 = tr(A−1)+ A−1 s − - . = αI A−1. − Thus A∆−1 = αA I, and therefore ∆A−1 =(αA I)−1. Note by construction, α> − −1 − −1 1/ mini ai so that (αA I) exists. Finally we may conclude that ∆= (αA I) A. − − 17 3.6 Price of Anarchy Analysis In this section, we derive the main results of this report. We find upper bounds for the price of anarchy of the pricing game, where we take the price of anarchy to mean the worst case ratio of the social welfare in a zero-flow zero-price Nash equilibrium to the social optimum welfare. Our bounds are found to be a function of a parameter y we term the conductance ratio. The conductance ratio is the conductance of the most conductive branch divided by the conductance of network as a whole. The conductance ratio is therefore a measure of how concentrated the capabilities of the network are in a single branch. A conductance ratio near 1 means that most of the conductance of the system is concentrated in a single branch. The smaller the conductance ratio, the more that the overall conductance of the system is distributed across multiple branches. Thus, in a sense the conductance ratio reflects the market power or concentration of the system. As one would expect, the PoA bounds that we find increase as the conductance ratio approaches 1. 3.6.1 Simple Parallel-Serial Competition Our result for the simple parallel-serial topology is given by the following theorem. Theorem 3. Consider game G with a simple parallel-serial topology. Consider the fol- lowing ratio max 1/a y = i i , i 1/ai which is the conductance of the most conductive! branch divided by the overall conductance. The price of anarchy for zero-flow zero-price Nash equilibria is no more than 2 2 1 m +2m(1+y)+(y−1) 4 m y m/3 m2(2−y)+m(4−y2 −y)+2(y−1)2 ≤ (3) / y m/3 8m−6my ≥ where m is the maximum number of providers connected serially. Furthermore, the maxi- mum of the above bound occurs when y =1, and consequently the price of anarchy is mo more than 1+m/2. (4) Proof. We suppose that there are n branches where each branch i contains mi competitors connected serially. We consider a zero-flow zero-price Nash equilibrium of game G with flow vector F . Using the arguments presented in Section 3.4, the same prices and flows describe a Nash equilibrium for the game Gl with a linearized disutility function, and Gt with a truncated disutility function. Using the results of Section 3.3, the KVL equations of the Nash circuit are ((M + I)A + sJ + MD∆)F = V b. − The consumer surplus in Nash equilibrium for the game Gl with a linearized disutility function is s F T JF 2 while the provider profit in Nash equilibrium is F T (MA+ MD∆)F. (5) In the game Gt a truncated disutility function, the provider profit remains the same, but the consumer surplus becomes 0. Therefore T Wt = F (MA+ MD∆)F 18 where Wt is the social welfare in Nash equilibrium of game Gt. Now consider the social optimum welfare. As we showed in Lemma 2 an upper bound on the social optimum welfare can be found by evaluating the social optimum circuit with diodes removed. The KVL equations for the social optimum circuit with diodes removed expressed in vector form are (2A + sJ)F ∗ = V b, while the provider profit can be seen to be F ∗T AF ∗, and the − 1 ∗T ∗ consumer surplus for a linear disutility function is 2 F JF . Combining these relations, we have that the upper bound on social welfare in social optimum, with a linear disutility ∗ ( denoted Wl ) is 1 W ∗ = (V b)T (2A + sJ)−1(V b). 9 l 2 − − When we truncate the disutility9 function, the social optimum flows do not change. There- fore an upper bound on the social optimum welfare for a truncated linear disutility function is 1 s W ∗ = (V b)T (2A + sJ)−1(V b) F T JF. t 2 − − − 2 To bound the price9 of anarchy to be no more than an amount z it is sufficient to show ∗ that zWt W is always positive. Algebra reveals that − t ∗ zWt W 9 − t 1 = F T zMA + zMD∆+ sJ F 9 2 − :1 ; F T (M + I)A + sJ + MD∆ (2A + sJ)−1 (M + I)A + sJ + MD∆ F (6) 2 : ; : ; = F T ((z 1/2)M 1/2I)A +(z 1/2)MD∆ F − − − − :1 ; F T (M I)A + MD∆ (2A + sJ)−1 (M + I)A + sJ + MD∆ F, 2 − : ; : ; = F T (z 1)MA+(z 1)MD∆ F − − − :1 ; F T (M I)A + MD∆ (2A + sJ)−1 (M I)A + MD∆ F 2 − − : ; : ; = F T (z 1)MA+(z 1)MD∆ F − − − :1 ; A−1JA−1 F T (M I)A + MD∆ A−1 (M I)A + MD∆ F. 4 − − t +2s−1 − : ;: ;: ; Consider the matrix in the above expression that is left and right multiplied by F T and F respectively. To show the above expression is non-negative, we can either show that this matrix is positive definite, or since F always has positive entries, we can simply show that all entries of the matrix are non-negative. We take this approach. It is easy to verify that the off-diagonal elements are positive, therefore we need to focus on diagonal elements. The diagonal elements are given by −2 1 −1 ai mi(z 1)ai+mi(z 1)diδi (mi 1)ai+midiδi a (mi 1)ai+midiδi . − − − 2 − i − t +2s−1 − < = + , + , + , Some algebra reduces the above expression to −1 −2 −3 N0 + N1s + N2s + N3s ((t + s−1)a 1)2(t +2s−1) − 19 where 1 N =mitai(tai 1)(tai 1+di)z (tai 1) 0 − − − 4 − × 2 2 2 2 2 (mi + 1) t a +2tai(m (di 1) + midi 1) + (mi(di 1) + 1) , i i − − − 2 2 2 N =mi 4(tai 1) + (2 + 3di)(tai 1) + di z (mi + 1) (tai 1) 1 + − − − − −, 3 1 mi+( (mi + 1)di + 2)(tai 1) midi( m, idi + 1), 2 − − 2 5 2 2 5 5 2 3 N =mi(5tai 4 + 2di)z (mi + 1) ta + +( di)m +( di + )mi, 2 − − 4 i 4 4 − i − 2 2 N =2miz 1/2(mi 1) . 3 − − These expressions are monotone increasing in z. Therefore there exists a sufficiently large z to make each of these expressions positive. For each i =0...3 let zi be the unique value for which Ni(zi) = 0. The values of zi can be found to be 1 (m + 1)2x2 + (2m2d +4m +2md)x +4md + m2d2 z = , 0 4 m(x + d)(1 + x) 1 2(m + 1)2x2 + (3md +4m +3m2d)x + m2d2 +2md z = , 1 2 m(4x2 + (2 + 3d)x + d) 1 5(m + 1)2x +4m2d +4m +4md z = , 2 4 m(1 + 5x +2d) 1 (m + 1)2 z = 3 4 m where x ! tai 1, m = mi and d = di. The derivatives w.r.t d of each of the above − expressions are 1 (m2 1)x2 +2m2dx + m2d2 − , 4 m(x + d)2(1 + x) 1 (6m2 6)x3 +( 2m 2 + 8m2d +4m2)x2 + (4m2d +3m2d2)x + m2d2 , − − − , 2 m(4x2 + (2 + 3d)x + d)2 1 (5m2 5)x +2m2 2m − − , 2 m(1 + 5x +2d)2 0 which are all nonnegative. Therefore the highest critical values zi occur when d = 1. Substituting d = 1 we have 1 (m + 1)2x2 + (2m2 +6m)x +4m + m2 z (x, m)= , 0 4 m(1 + x)2 1 2(m + 1)2x2 + (7m +3m2)x + m2 +2m z (x, m)= , 1 2 m(4x + 1)(x + 1) 1 5(m + 1)2x +4m2 +8m z (x, m)= , 2 4 m(3 + 5x) 1 (m + 1)2 z (x, m)= . 3 4 m The quantities z (x, m) z (x, m) and z (x, m) z (x, m) reduce to 1 − 2 1 − 3 1 (m + 13)(m 1)x2 + (3m2 + 12m 5)x +4m +2m2 1 (m2 +4m 5)x + m2 +2m 1 − − , − − 4 m(4x + 1)(x + 1)(3 + 5x) 4 m(4x + 1)(x + 1) respectively which are both positive for m 1. Thus z and z are dominated by z . ≥ 2 3 1 20 Thus the price of anarchy is bounded by the larger of z1 and z0. Critical value z1 minus value z0 is 1 ((m 3)x + m)((m 1)x + m) − − 4 (1 + x)2(4x + 1)m which implies that z is smaller than z whenever x > m/(3 m) or m 3. 1 0 − ≥ The derivative with respect to x of z0 and z1 respectively are 1 (m 1)x + m 1 (2m2 +8m 10)x2 + (4m2 +8m 4)x +2m2 +3m − , − − −2 (1 + x)3m −2 m(4x + 1)2(x + 1)2 which are negative. this shows the bound decreases in x. substituting x = 0 into z1 we get 1 + m/2, which is the upper bound on the price of anarchy in the theorem statement. To get the bound on the price of anarchy in terms of conductance ratio y, we substitute the relation y =1/(x + 1) into the expressions for z1 and z0. This yields 1 m2 +2m(1 + y) + (y 1)2 m2(2 y)+m(4 y2 y) + 2(y 1)2 z (y, m)= − ,z(y, m)= − − − − 0 4 m 1 8m 6my − and z1(y, m) is the larger of the two when y > m/3. 3.6.2 General Topology In this section, we study the general topology case. Theorem 4. The price of anarchy of game G with a general topology, is no more than 2 times the upper bounds in expressions (3) and (4) from Theorem 3. Proof. As for the simple parallel-serial topology, the KVL equations of the Nash circuit can be expressed as ((M + I)A + sJ + MD∆)F = V b. − where A is a diagonal matrix whose diagonal elements are the equivalent resistances of the branches (level 1 groups) of the circuit. Again as in the simple parallel-serial case, the consumer surplus in Nash equilibrium for a linear disutility function is s F T JF. 2 At this point, we use the Lemma 7 which gives an expression for the maximum “latency resistance” of an arbitrarily complex group of providers. Since the provider profit of a group is equivalent to the total power dissipated by the resistors of the group minus the power dissipated by the latency resistance, we find that the total provider profit in Nash equilibrium is no less than 1 1 F T ((M + I)A + MD∆)F F T (MA+ MD∆)F. (7) 2 ≥ 2 1 Note that the left expression is just 2 times expression 5 , which recall was the expression we found for provider profit in the simple parallel-serial case. By the same argument as we used for the simple parallel-serial case, an upper bound on social optimum welfare for a truncated disutility function is 1 s W ∗ = (V b)T (2A + sJ)−1(V b) F T JF. t 2 − − − 2 As in the previous proof,9 we find a bound on the price of anarchy by finding a sufficiently large z to make the below expression non-negative ∗ T 1 1 1 1 zWt W = F (z MA+ zMD∆+ sJ − t 2 2 2 2 − 1 9 F T (M + I)A + sJ + MD∆ (2A + sJ)−1 (M + I)A + sJ + MD∆ F. 2 : ; : ; 21 m=1 player connected serialy m=3 player connected serialy m=5 player connected serialy 1.5 2.5 3.5 1.45 1.4 3 1.35 2 1.3 2.5 1.25 PoA PoA PoA 1.2 2 1.5 1.15 1.1 1.5 1.05 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Conductance Ratio Conductance Ratio Conductance Ratio Figure 3: Simple parallel serial PoA bound for the cases where there are either m = 1, 3, or 5 providers connected serially. The bounds are plotted as a function of the conductance ratio – the ratio of the conductance of the most conductive branch to the conductance of the whole circuit. The points are the PoA of randomly generated example topologies. This expression is similar to expression (6) studied in Theorem 3. The only difference is 1 that every term that has a factor of z has added factor of 2 in the above expression as compared to (6). Therefore any value of z that made (6) non-negative, twice that value will make the above expression non negative. Therefore two times the upper bounds found in Theorem 3 for the price of anarchy found are upper bounds for the price of anarchy for general parallel serial topologies. 4 Result Characterization and Simulated Exam- ples In this section we graph the the PoA bounds found in Theorems 3 and 4. We also compare the bounds to PoA found for randomly generated examples evaluated numerically. Figure 4 shows the price of anarchy bounds for the simple parallel serial topology as given by Theorem 3. The plots are shown with respect to the ratio of the conductance of the most conductive branch in the circuit to the conductance of the overall circuit. A con- ductance ratio near 1 means that most of the conductance of the system is concentrated in a single branch. The smaller the conductance ratio, the more that the overall conductance of the system is distributed across multiple branches. In a sense the conductance ratio reflects the market power or concentration of the system. 5 Conclusion We have seen how a circuit analogy can be used to obtain an upper bound on the price of anarchy for a network routing game with latency, elastic demand, and selfish providers. Furthermore, our bound depends on what may be a useful measure of “market power” in this context: the ratio of the conductance of the most conductive branch of the system to the conductance of the whole system. 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