14.15J/6.207J Networks: Applications of Game Theory to Networks

6.207/14.15: NetworksLecture 12: Applications of Game Theory to NetworksDaron Acemoglu and Asu OzdaglarMITOctober 21, 20091

Networks: Lecture 12IntroductionOutlineTraffic equilibrium: the Pigou exampleGeneral formulation with single origin-destination pairMulti-origin-destination traffic equilibriaCongestion games and atomic traffic equilibriaPotential functions and potential gamesNetwork cost-sharingStrategic network formation.Reading:EK, Chapter 8.Jackson, Chapter 6.2

Networks: Lecture 12IntroductionMotivationMany games are played over networks, in the sense that playersinteract with others linked to them through a network-like structure.Alternatively, in several important games, the actions of playerscorrespond to a path in a given network.The most important examples are choosing a route in a traffic problemor in a data routing problem.Other examples are cost sharing in network-like structures.Finally, the formation of networks is typically a game-theoretic(strategic) problem.In this lecture, we take a first look at some of these problems,focusing on traffic equilibria and formation of networks.3

Networks: Lecture 12Wardrop EquilibriaThe Pigou Example of Traffic EquilibriumRecall the following simple example from lecture 9, where a unit mass oftraffic is to be routed over a network:delay depends on congestion1 unit of trafficno congestion effectsSystem optimum (minimizing aggregate delay) is to split traffic equallybetween the two routes, giving 1 1 3min Csystem(x S ) = l i (x iS)x iS= + = .x 1+x 2≤1 4 2 4iInstead, the Nash equilibrium of this large (non-atomic) game, also referredto as Wardrop equilibrium, is x 1 = 1 and x 2 = 0 (since for any x 1 < 1,l 1 (x 1 ) < 1 = l 2 (1 − x 1 )), giving an aggregate delay ofCeq(x WE ) = l i (x iWE)x iWE= 1 + 0 = 1 > 3 .4i4

Networks: Lecture 12Wardrop EquilibriaThe Wardrop EquilibriumWhy the Wardrop equilibrium?It is nothing but a Nash equilibrium in this game, in view of the factthat it is non-atomic—each player is infinitesimal. Thus, taking thestrategies of others as given is equivalent to taking aggregates, heretotal traffic on different routes, as given.Therefore, the Wardrop equilibrium (or the Nash equilibrium of alarge game) is a convenient modeling tool when each participant inthe game is small.A small technical detail: so far we often took the set of players, I, tobe a finite set. But in fact nothing depends on this, and in non-atomicgames, I is typically taken to be some interval in R, e.g., [0, 1].5

Networks: Lecture 12Wardrop EquilibriaMore General Traffic ModelLet us now generalize the Pigou example. In the general model, thereare several origin-destination pairs and multiple paths linking thesepairs220 03x 1x+10 022Cost = 2x1 + 2x3=86

Networks: Lecture 12Wardrop EquilibriaMore General Traffic Model: NotationLet us start with a single origin-destination pair.Directed network N = (V , E ).P denotes the set of paths between origin and destination.x p denotes the flow on path p ∈ P.Each link i ∈ E has a latency function l i (x i ), wherex i = x p .{p∈P|i∈p}Here the notation p ∈ P|i ∈ p denotes the paths p that traverse linki ∈ E .The latency function captures congestion effects. Let us assume forsimplicity that l i (x i ) is nonnegative, differentiable, and nondecreasing.We normalize total traffic to 1 and in the context of the gametheoretic formulation here, I = [0, 1]. We also assume that all trafficis homogeneous. Each motorist wishes to minimize delay.7

Networks: Lecture 12Wardrop EquilibriaMore General Traffic Model (continued)We denote a routing pattern by the vector x. If it satisfies the twoconstraints above, it is a feasible routing pattern.The total delay (latency) cost of a routing pattern x is:C (x) = x i l i (x i ),i∈Ethat is, it is the sum of latencies l i (x i ) for each link i ∈ E multipliedby the flow over this link, x i , summed over all links E .8

Networks: Lecture 12Socially Optimal RoutingWardrop EquilibriaSocially optimal routing, defined as the routing pattern minimizingaggregate delay, is given by x S that is a solution to the followingproblemminimize x i l i (x i )i∈Esubject to x p = x i , for all i ∈ E ,{p∈P|i∈p}x p = 1 and x p ≥ 0 for all p ∈ P.p∈P9

Wardrop EquilibriumNetworks: Lecture 12Wardrop EquilibriaWhat is a Wardrop equilibrium?Since it is a Nash equilibrium, it has to be the case that for eachmotorist their routing choice must be optimal.This implies that if a motorist k ∈ I is using path p, then there doesnot exist path p such that l i (x i ) > l i (x i ).i∈pPut differently, x must be such that (1) For all p, p ∈ P with x p , x p > 0, l i (x i ) = l i (x i ).i∈p i∈p (2) For all p, p ∈ P with x p > 0 and x p = 0, l i (x i ) ≥ l i (x i ).i∈p i∈pi∈p10

Networks: Lecture 12Characterizing Wardrop EquilibriaWardrop EquilibriaTheoremA feasible routing pattern x WE is a Wardrop equilibrium if and only if it isa solution to x iminimizel i (z) dzi∈E0subject to x p = x i , for all i ∈ E ,{p∈P|i∈p}x p = 1 and x p ≥ 0 for all p ∈ P.p∈PMoreover, if each l i is strictly increasing, then x WE is unique.Note that by Weierstrass’s Theorem, a solution exists, and thus aWardrop equilibrium always exists.11

ProofNetworks: Lecture 12Wardrop EquilibriaRewrite the minimization problem asminimizei∈E i∈p xp0l i (z) dzsubject to x p = 1 and x p ≥ 0 for all p ∈ P.p∈PSince each l i is nondecreasing, this is a convex program. Therefore,first-order conditions are necessary and sufficient.First-order conditions with respect to x p are WEx i ≥ λi∈pl iwith the complementary slackness, i.e., with equality wheneverx WE > 0. p12

Proof (continued)Networks: Lecture 12Wardrop EquilibriaHere λ is the Lagrange multiplier on the constraintp∈Px p = 1.The Lagrange multiplier will be equal to the lowest cost path, whichthen implies the result that for all p, p ∈ P with xpWE , x WE > 0,p i∈p l i (xWEi) =i∈p l i (xWEi). And clearly, for paths with xWEp = 0,the cost can be higher.Finally,if each l i isstrictly increasing, then the set of equalitiesi∈p l i (xWEi) =i∈p l i (xWEi) admits unique solution, establishinguniqueness.13

Networks: Lecture 12Wardrop EquilibriaInefficiency of the EquilibriumWe saw from the Pigou example that the Wardrop equilibrium fails tominimize total delay—hence it is inefficient.In fact, it can be arbitrarily inefficient. Consider the following figure,which is the same as the Pigou example, except with a differentlatency on path 1.1 unit of traffic14

Networks: Lecture 12Wardrop EquilibriaInefficiency of the Equilibrium (continued)In this example, socially optimal routing again involvesl 1 (x 1 ) + x 1 l 1 (x 1 ) = l 2 (1 − x 1 ) + (1 − x 1 ) l 2 (1 − x 1 )k kx 1 + kx 1 = 1Therefore, the system optimum sets x 1 = (1 + k) −1/k andx 2 = 1 − (1 + k) −1/k , so thatk+1min Csystem(x S S S) = l i (x i )x i = (1 + k) − k + 1 − (1 + k) −1/k .x 1 +x 2 ≤1i15

Networks: Lecture 12Wardrop EquilibriaInefficiency of the Equilibrium (continued)The Wardrop equilibrium again has x 1 = 1 and x 2 = 0 (since onceagain for any x 1 < 1, l 1 (x 1 ) < 1 = l 2 (1 − x 1 )). ThusCeq(x WE WE WE) = l i (x i )x i = 1 + 0 = 1.Therefore, the Price of anarchy is nowiCsystem(x S ) k+1= (1 + k) − k + 1 − (1 + k) −1/k .Ceq(x WE )This limits to 0 as k → ∞ (the first term tends to zero, while the lastterm limits to 1).Thus the equilibrium can be “arbitrarily” inefficient relative to thesocial optimum.16

Networks: Lecture 12Wardrop EquilibriaFurther Paradoxes of Decentralized Equilibrium: Braess’sParadoxIdea: Addition of an intuitively helpful route negatively impactsnetwork users.Paradoxical, since the addition of another route should help traffic. Infact, the addition of a link can never increase aggregate delay in thesocial optimum.But the situation is different in a Wardrop equilibrium.This idea was first introduced in transportation networks by Braess.Hence the name of the paradox.17

Networks: Lecture 12Wardrop EquilibriaFurther Paradoxes of Decentralized Equilibrium: Braess’sParadox (continued)1/2x11x11 unit oftraffic1 unit oftraffic01 x1x1/2C eq = 1/2 (1/2+1) + 1/2 (1/2+1) = 3/2C = 3/2sysC eq = 1 + 1 = 2C = 3/2sys18

Networks: Lecture 12Multiple Origin-Destination PairsMultiple Origin-Destination PairsThe above model is straightforward to generalize to multipleorigin-destination pairs.Suppose that there are K such pairs (some of them having possiblythe same origin or destination).Origin-destination pair j has total traffic r j .Let us denote the set of paths for origin-destination pair j by P j , andnow P = ∪ j P j .Then the socially optimal routing pattern is a solution tominimize x i l i (x i )i∈Esubject to x p = x i , i ∈ E ,{p∈P|i∈p}xp = r j , j = 1, . . . , k, and x p ≥ 0 for all p ∈ P.p∈P j19

Networks: Lecture 12Multiple Origin-Destination PairsWardrop Equilibrium with Multiple Origin-Destination PairsEssentially the same characterization theorem for Wardrop equilibriumapplies with multiple origin-destination pairs.TheoremA feasible routing pattern x WE is a Wardrop equilibrium if and only if it isa solution to x iminimizel i (z) dzi∈E0subject to x p = x i , i ∈ E ,{p∈P|i∈p}x p = r j , j = 1, . . . , k, and x p ≥ 0 for all p ∈ P.p∈P jMoreover, if each l i is strictly increasing, then x WE is uniquely defined.20

Networks: Lecture 12Congestion GamesCongestion GamesA Wardrop equilibrium presumes that there are so many players thatthey all take aggregates as given.This “non-atomic” player assumption is a good approximation formany situations. But in others, players may be large and may thusnaturally take into account their effect on the total amount of trafficon a particular path.For example, United Airlines would naturally take into account thecongestion implications of using Washington Dulles as a hub.This would be a special case of a congestion game.21

Networks: Lecture 12Congestion Games (continued)Congestion GamesCongestion Model: C = I, M, (S i ) i∈I , (c j ) j∈M whereI = {1, 2, · · · , I } is the set of players.M = {1, 2, · · · , m} is the set of resources.S i ⊂ M is the set of resource combinations (e.g., links or commonresources) that player i can take/use. A strategy for player i iss i ∈ S i , corresponding to the resources that this player is using.c j (k) is the benefit for the negative of the cost to each user who usesresource j if k users are using it.Define congestion game I, (S i ), (u i ) with utilitiesu i (s i , s −i ) = c j (k j ),where k j is the number of users of resource j under strategies s.How do we analyze congestion games? To do so, we will introduce amore general class of games called potential games.j∈s i22

Potential GamesNetworks: Lecture 12Potential GamesA finite game (or a game with a finite number of players but withinfinite strategy spaces) is a potential game [ordinal potential game,exact potential game] if there exists a function Φ : S → R such thatΦ (s i , s −i ) gives information about u i (s i , s −i ) for each i ∈ I.If so, Φ is referred to as the potential function.The potential function has a natural analogy to “energy” in physicalsystems. It will be useful both for locating pure strategy Nashequilibria and also for the analysis of “myopic” dynamics in the nextlecture.23

PotentialsNetworks: Lecture 12Potential GamesA function Φ : S → R is called an ordinal potential function for thegame G if for each i ∈ I and all s −i ∈ S −i ,u i (x, s −i )−u i (z, s −i ) ≥ 0 iff Φ(x, s −i )−Φ(z, s −i ) ≥ 0, for all x, z ∈ S i ,andu i (x, s −i )−u i (z, s −i ) > 0 iff Φ(x, s −i )−Φ(z, s −i ) > 0, for all x, z ∈ S i .A function Φ : S → R is called an exact potential function for thegame G if for each i ∈ I and all s −i ∈ S −i ,u i (x, s −i ) − u i (z, s −i ) = Φ(x, s −i ) − Φ(z, s −i ), for all x, z ∈ S i .24

Networks: Lecture 12Potential GamesPotential GamesA finite game G is called an ordinal (exact) potential game if itadmits an ordinal (exact) potential.In what follows, we refer to ordinal potential games as potentialgames, and only add the “exact” qualifier when this is necessary.A game G with infinite strategy space and finite number of players isa potential game if it admits a continuous potential function.25

Networks: Lecture 12Potential GamesPure Strategy Nash Equilibria in Potential GamesTheoremEvery potential game has at least one pure strategy Nash equilibrium.Proof: The global maximum of an ordinal potential function is a purestrategy Nash equilibrium. To see this, suppose that s ∗ correspondsto the global maximum. Then, for any i ∈ I, we have, by definition,Φ(s ∗ i, s−i∗ ) − Φ(s, s−i∗ ) ≥ 0 for all s ∈ S i . But since Φ is a potentialfunction,u i (s i ∗ , s ∗ −i ) ≥ 0 iff −i )−Φ(s, s ∗−i )−u i (s, s ∗ Φ(s i ∗ , s ∗ −i ) ≥ 0, for alls ∈ S i .Therefore, u i (s ∗ i, s−i∗ ) − u i (s, s−i∗ ) ≥ 0 for all s ∈ S i and for all i ∈ I.Hence s ∗ is a pure strategy Nash equilibrium.Note, however, that there may also be other pure strategy Nashequilibria corresponding to local maxima.26

Networks: Lecture 12Potential GamesExamples of Ordinal Potential GamesExample: Cournot competition.I firms choose quantity q i ∈ (0, ∞)The payoff function for player i given by u i (q i , q −i ) = q i (P(Q) − c). IWe define the function Φ(q 1 , · · · , q I ) =i=1 q i (P(Q) − c).Note that for all i and all q −i ,u i (q i , q −i ) − u i (q i , q −i ) > 0 iff Φ(q i , q −i ) − Φ(q i , q −i ) > 0 for all q i , q i >Φ is therefore an ordinal potential function for this game.27

Networks: Lecture 12Potential GamesExamples of Exact Potential GamesExample: Cournot competition (again).Suppose now that P(Q) = a − bQ and costs c i (q i ) are arbitrary.We define the functionIIIIΦ ∗ 2(q 1 , · · · , q n ) = a q i − b q i − b q i q l − c i (q i ).It can be shown that for all i and all q −i ,i=1 i=1 1≤i 0.Φ is an exact potential function for this game.28

Networks: Lecture 12Congestion and Potential GamesPotential GamesTheoremEvery congestion game is a potential game and thus has a pure strategyNash equilibrium.Proof: For each j define k¯jias the usage of resource j excludingplayer i, i.e.,k¯i= I [j ∈ s i ] ,ji =iwhere I [j ∈ s i ] is the indicator for the event that j ∈ s i .With this notation, the utility difference of player i from twostrategies s i and s i(when others are using the strategy profile s −i ) isu i (s i , s −i ) − u i (s i , s −i ) = c j (k¯j i + 1) − c j (k¯j i + 1).j∈s ij∈s i29

Proof ContinuedNetworks: Lecture 12Potential GamesNow consider the function⎡ ⎤k jΦ(s) = ⎣ c j (k)⎦ .j∈ i ∈I s i k=1We can also write⎡ ⎤k¯i j ⎢ Φ(s i , s −i ) = ⎣ cj(k) ⎥ ⎦ + c j (k¯ji + 1).j∈ s i k=1 j∈s ii =i30

Proof ContinuedNetworks: Lecture 12Potential GamesTherefore:Φ(s i , s −i ) − Φ(s i , s −i ) =⎡k¯i j⎡ ⎤ k¯i j⎢ ⎥⎣ c j (k)⎦ + c j (k¯j i + 1)= c j (k¯ji + 1) − c j (k¯ji + 1)⎤⎢ ⎣ c j (k)⎦ ⎥ + c j (k¯ji + 1)j∈ s i k=1 j∈s ii =i−j∈s ij∈ s i k=1 j∈sii =ij∈s i= u i (s i , s −i ) − u i (s i , s −i ).31

Network Cost SharingNetworks: Lecture 12Potential GamesConsider the problem of sharing the cost of some network resourcesamong participants.Directed graph N = (V , E ) with edge cost c e ≥ 0, k players.Each player i has a set of nodes T i he wants to connect.A strategy of player i set of edges S i ⊂ E such that S i connects to allnodes in T i .Cost sharing mechanism: All players using an edge split the costequallyGiven a vector of player’sstrategies S = (S 1 , . . . , S k ), the cost toagent i is C i (S) =e∈S i(c e /x e ), where x e is the number of agentswhose strategy contains edge e.32

Networks: Lecture 12Potential GamesNetwork Cost Sharing (continued)The game here involves each player simultaneously choosing S i ⊂ E .This immediately implies:PropositionThe network cost-sharing game is a potential game and thus has a purestrategy Nash equilibrium.In fact, network cost sharing games typically have several equilibria.As a trivial example, in a network consisting of two links with equal(or similar) costs and I players who could all use either link, there aretwo pure strategy equilibria: all players using link 1 or all players usinglink 2.33

Networks: Lecture 12Network FormationA Simple Game of Network FormationThe model of network cost sharing can be viewed as a specificinstance of “endogenous network formation”. However, the mostinteresting aspect of network formation, the benefits as well as thecost of forming links, are absent from this model.Let us now consider a simple game of network formation.There are I symmetric players. The set of possible graphs (networks)is denoted by G, and corresponds to all possible configurations of linksbetween the I players.The utility function of player i isu i : G → R,and assigns a utility level to every possible network configuration.34

Networks: Lecture 12Network FormationA Simple Game of Network Formation (continued)More specifically, suppose that for any network g ∈ G, we haveu i (g) = b ( ij (g)) − d i (g) c,where:j=ic is the cost of a direct connection;d i (g) is the degree of player i in the network g; ij (g) is the distance between player i and player j in the network g,with the convention that ij (g) = ∞ if the two players are notconnected;b : N → R is a benefit function depending on the distance; we assumethat b (·) is strictly decreasing and b (∞) = 0.A network g ∈ G is “efficient” if there does not exist g ∈ G such that⎡ ⎤ ⎡ ⎤IIU = ⎣ b ( ij (g )) − d i (g ) c⎦ > U = ⎣ b ( ij (g)) − d i (g) c⎦ .i=1 j=i i=1 j=i35

Networks: Lecture 12Network FormationPairwise StabilityWe can next study the equilibrium networks that will emerge in thisgame. For example, as in the network cost sharing game, we couldlook for the simultaneous move games and study the pure strategyNash equilibria. However, as shown in that discussion, there are manysuch equilibria, and multiplicity problem becomes worse when theformation of a link requires “agreement” from two parties.An alternative proposed by Jackson and Wolinsky (1996) “AStrategic Model of Social and Economic Networks” is to look at theconcept of pairwise stability.36

Networks: Lecture 12Network FormationPairwise Stability (continued)Pairwise stability requires that there are no profitable deviations by apair of agents by either adding a new link or by removing an existinglink.More formally, a network g is pairwise stable if12For all {i, j} ∈ g, u i (g) ≥ u i (g − {i, j}) and u j (g) ≥ u j (g − {i, j});andFor all {i, j} ∈/ g, if u i (g + {i, j}) > u i (g), thenu j (g + {i, j}) < u j (g).37

Networks: Lecture 12Network FormationPairwise Stability and Efficiency in Network FormationSuppose thatb (1) < c < b (1) + (I − 2) b (2) .Then, the efficient network is a star network.To see this, note that in a star network all nodes except the star haveutilityu i (g) = b (1) − c + (I − 2) b (2) ,since they are connected only to the star and are distance equal to 2from all other nodes (of which there are I − 2).The utility of the star node isu i (g) = (I − 1) [b (1) − c] .38

Networks: Lecture 12Network FormationPairwise Stability and Efficiency in Network Formation(continued)Thus total utility isU = (I − 1) {[b (1) − c + (I − 2) b (2)] + [b (1) − c]} .Given symmetry, it is sufficient to check that total utility cannot beincreased by adding or subtracting one link.Removing one link would lead to a new network with total utilityThereforeU = (I − 2) {[b (1) − c + (I − 3) b (2)] + [b (1) − c]} .U − U = −2 [b (1) − c] − 2 (I − 2) b (2) < 0,since, by assumption, c < b (1) + (I − 2) b (2).39

Networks: Lecture 12Network FormationPairwise Stability and Efficiency in Network Formation(continued)Now imagine adding one more link. This would not change thedistance for all but two players and thus lead to total utilityThenU = (I − 3) [b (1) − c + (I − 2) b (2)] + (I − 1) [b (1) − c]+2 {2 [b (1) − c] + (I − 3) b (2)} .U − U = 2 {2 [b (1) − c] + (I − 3) b (2)}in view of the fact that b (1) < c.−2 [b (1) − c + (I − 2) b (2)]= 2 (b (1) − c) − 2b (2) < 0,40

Networks: Lecture 12Network FormationPairwise Stability and Efficiency in Network Formation(continued)However, the star network is not pairwise stable, since the utility ofthe star isu i (g) = (I − 1) [b (1) − c] < 0,again in view of the fact that b (1) < c.This example shows how efficient network formation is difficult toachieve in general.Intuitively, in this case, forming a link creates a positive externalityon others because it reduces the distance between other players.This externality is not internalized and thus equilibrium networks willtend to have too few links. Here the equilibrium notion, captured bythe pairwise stability concept, reflects this intuition.41

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