14.15J/6.207J Networks: Applications of Game Theory to Networks
14.15J/6.207J Networks: Applications of Game Theory to Networks
14.15J/6.207J Networks: Applications of Game Theory to Networks
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6.207/14.15: <strong>Networks</strong>Lecture 12: <strong>Applications</strong> <strong>of</strong> <strong>Game</strong> <strong>Theory</strong> <strong>to</strong> <strong>Networks</strong>Daron Acemoglu and Asu OzdaglarMITOc<strong>to</strong>ber 21, 20091
<strong>Networks</strong>: Lecture 12IntroductionOutlineTraffic equilibrium: the Pigou exampleGeneral formulation with single origin-destination pairMulti-origin-destination traffic equilibriaCongestion games and a<strong>to</strong>mic traffic equilibriaPotential functions and potential gamesNetwork cost-sharingStrategic network formation.Reading:EK, Chapter 8.Jackson, Chapter 6.2
<strong>Networks</strong>: Lecture 12IntroductionMotivationMany games are played over networks, in the sense that playersinteract with others linked <strong>to</strong> them through a network-like structure.Alternatively, in several important games, the actions <strong>of</strong> playerscorrespond <strong>to</strong> 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 <strong>of</strong> networks is typically a game-theoretic(strategic) problem.In this lecture, we take a first look at some <strong>of</strong> these problems,focusing on traffic equilibria and formation <strong>of</strong> networks.3
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaThe Pigou Example <strong>of</strong> Traffic EquilibriumRecall the following simple example from lecture 9, where a unit mass <strong>of</strong>traffic is <strong>to</strong> be routed over a network:delay depends on congestion1 unit <strong>of</strong> trafficno congestion effectsSystem optimum (minimizing aggregate delay) is <strong>to</strong> 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 <strong>of</strong> this large (non-a<strong>to</strong>mic) game, also referred<strong>to</strong> 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 <strong>of</strong>Ceq(x WE ) = l i (x iWE)x iWE= 1 + 0 = 1 > 3 .4i4
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaThe Wardrop EquilibriumWhy the Wardrop equilibrium?It is nothing but a Nash equilibrium in this game, in view <strong>of</strong> the factthat it is non-a<strong>to</strong>mic—each player is infinitesimal. Thus, taking thestrategies <strong>of</strong> others as given is equivalent <strong>to</strong> taking aggregates, here<strong>to</strong>tal traffic on different routes, as given.Therefore, the Wardrop equilibrium (or the Nash equilibrium <strong>of</strong> alarge game) is a convenient modeling <strong>to</strong>ol when each participant inthe game is small.A small technical detail: so far we <strong>of</strong>ten <strong>to</strong>ok the set <strong>of</strong> players, I, <strong>to</strong>be a finite set. But in fact nothing depends on this, and in non-a<strong>to</strong>micgames, I is typically taken <strong>to</strong> be some interval in R, e.g., [0, 1].5
<strong>Networks</strong>: 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
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaMore General Traffic Model: NotationLet us start with a single origin-destination pair.Directed network N = (V , E ).P denotes the set <strong>of</strong> 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 <strong>to</strong>tal traffic <strong>to</strong> 1 and in the context <strong>of</strong> the gametheoretic formulation here, I = [0, 1]. We also assume that all trafficis homogeneous. Each mo<strong>to</strong>rist wishes <strong>to</strong> minimize delay.7
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaMore General Traffic Model (continued)We denote a routing pattern by the vec<strong>to</strong>r x. If it satisfies the twoconstraints above, it is a feasible routing pattern.The <strong>to</strong>tal delay (latency) cost <strong>of</strong> a routing pattern x is:C (x) = x i l i (x i ),i∈Ethat is, it is the sum <strong>of</strong> latencies l i (x i ) for each link i ∈ E multipliedby the flow over this link, x i , summed over all links E .8
<strong>Networks</strong>: Lecture 12Socially Optimal RoutingWardrop EquilibriaSocially optimal routing, defined as the routing pattern minimizingaggregate delay, is given by x S that is a solution <strong>to</strong> the followingproblemminimize x i l i (x i )i∈Esubject <strong>to</strong> 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 Equilibrium<strong>Networks</strong>: Lecture 12Wardrop EquilibriaWhat is a Wardrop equilibrium?Since it is a Nash equilibrium, it has <strong>to</strong> be the case that for eachmo<strong>to</strong>rist their routing choice must be optimal.This implies that if a mo<strong>to</strong>rist 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
<strong>Networks</strong>: Lecture 12Characterizing Wardrop EquilibriaWardrop EquilibriaTheoremA feasible routing pattern x WE is a Wardrop equilibrium if and only if it isa solution <strong>to</strong> x iminimizel i (z) dzi∈E0subject <strong>to</strong> 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
Pro<strong>of</strong><strong>Networks</strong>: Lecture 12Wardrop EquilibriaRewrite the minimization problem asminimizei∈E i∈p xp0l i (z) dzsubject <strong>to</strong> 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 <strong>to</strong> x p are WEx i ≥ λi∈pl iwith the complementary slackness, i.e., with equality wheneverx WE > 0. p12
Pro<strong>of</strong> (continued)<strong>Networks</strong>: Lecture 12Wardrop EquilibriaHere λ is the Lagrange multiplier on the constraintp∈Px p = 1.The Lagrange multiplier will be equal <strong>to</strong> 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 <strong>of</strong> equalitiesi∈p l i (xWEi) =i∈p l i (xWEi) admits unique solution, establishinguniqueness.13
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaInefficiency <strong>of</strong> the EquilibriumWe saw from the Pigou example that the Wardrop equilibrium fails <strong>to</strong>minimize <strong>to</strong>tal 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 <strong>of</strong> traffic14
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaInefficiency <strong>of</strong> 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
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaInefficiency <strong>of</strong> 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 <strong>of</strong> anarchy is nowiCsystem(x S ) k+1= (1 + k) − k + 1 − (1 + k) −1/k .Ceq(x WE )This limits <strong>to</strong> 0 as k → ∞ (the first term tends <strong>to</strong> zero, while the lastterm limits <strong>to</strong> 1).Thus the equilibrium can be “arbitrarily” inefficient relative <strong>to</strong> thesocial optimum.16
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaFurther Paradoxes <strong>of</strong> Decentralized Equilibrium: Braess’sParadoxIdea: Addition <strong>of</strong> an intuitively helpful route negatively impactsnetwork users.Paradoxical, since the addition <strong>of</strong> another route should help traffic. Infact, the addition <strong>of</strong> 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 <strong>of</strong> the paradox.17
<strong>Networks</strong>: Lecture 12Wardrop EquilibriaFurther Paradoxes <strong>of</strong> Decentralized Equilibrium: Braess’sParadox (continued)1/2x11x11 unit <strong>of</strong>traffic1 unit <strong>of</strong>traffic01 x1x1/2C eq = 1/2 (1/2+1) + 1/2 (1/2+1) = 3/2C = 3/2sysC eq = 1 + 1 = 2C = 3/2sys18
<strong>Networks</strong>: Lecture 12Multiple Origin-Destination PairsMultiple Origin-Destination PairsThe above model is straightforward <strong>to</strong> generalize <strong>to</strong> multipleorigin-destination pairs.Suppose that there are K such pairs (some <strong>of</strong> them having possiblythe same origin or destination).Origin-destination pair j has <strong>to</strong>tal traffic r j .Let us denote the set <strong>of</strong> paths for origin-destination pair j by P j , andnow P = ∪ j P j .Then the socially optimal routing pattern is a solution <strong>to</strong>minimize x i l i (x i )i∈Esubject <strong>to</strong> 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
<strong>Networks</strong>: 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 <strong>to</strong> x iminimizel i (z) dzi∈E0subject <strong>to</strong> 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
<strong>Networks</strong>: Lecture 12Congestion <strong>Game</strong>sCongestion <strong>Game</strong>sA Wardrop equilibrium presumes that there are so many players thatthey all take aggregates as given.This “non-a<strong>to</strong>mic” player assumption is a good approximation formany situations. But in others, players may be large and may thusnaturally take in<strong>to</strong> account their effect on the <strong>to</strong>tal amount <strong>of</strong> trafficon a particular path.For example, United Airlines would naturally take in<strong>to</strong> account thecongestion implications <strong>of</strong> using Washing<strong>to</strong>n Dulles as a hub.This would be a special case <strong>of</strong> a congestion game.21
<strong>Networks</strong>: Lecture 12Congestion <strong>Game</strong>s (continued)Congestion <strong>Game</strong>sCongestion Model: C = I, M, (S i ) i∈I , (c j ) j∈M whereI = {1, 2, · · · , I } is the set <strong>of</strong> players.M = {1, 2, · · · , m} is the set <strong>of</strong> resources.S i ⊂ M is the set <strong>of</strong> resource combinations (e.g., links or commonresources) that player i can take/use. A strategy for player i iss i ∈ S i , corresponding <strong>to</strong> the resources that this player is using.c j (k) is the benefit for the negative <strong>of</strong> the cost <strong>to</strong> 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 <strong>of</strong> users <strong>of</strong> resource j under strategies s.How do we analyze congestion games? To do so, we will introduce amore general class <strong>of</strong> games called potential games.j∈s i22
Potential <strong>Game</strong>s<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sA finite game (or a game with a finite number <strong>of</strong> 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 <strong>to</strong> as the potential function.The potential function has a natural analogy <strong>to</strong> “energy” in physicalsystems. It will be useful both for locating pure strategy Nashequilibria and also for the analysis <strong>of</strong> “myopic” dynamics in the nextlecture.23
Potentials<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sA 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
<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sPotential <strong>Game</strong>sA finite game G is called an ordinal (exact) potential game if itadmits an ordinal (exact) potential.In what follows, we refer <strong>to</strong> 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 <strong>of</strong> players isa potential game if it admits a continuous potential function.25
<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sPure Strategy Nash Equilibria in Potential <strong>Game</strong>sTheoremEvery potential game has at least one pure strategy Nash equilibrium.Pro<strong>of</strong>: The global maximum <strong>of</strong> an ordinal potential function is a purestrategy Nash equilibrium. To see this, suppose that s ∗ corresponds<strong>to</strong> 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 <strong>to</strong> local maxima.26
<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sExamples <strong>of</strong> Ordinal Potential <strong>Game</strong>sExample: Cournot competition.I firms choose quantity q i ∈ (0, ∞)The pay<strong>of</strong>f 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
<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sExamples <strong>of</strong> Exact Potential <strong>Game</strong>sExample: 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
<strong>Networks</strong>: Lecture 12Congestion and Potential <strong>Game</strong>sPotential <strong>Game</strong>sTheoremEvery congestion game is a potential game and thus has a pure strategyNash equilibrium.Pro<strong>of</strong>: For each j define k¯jias the usage <strong>of</strong> resource j excludingplayer i, i.e.,k¯i= I [j ∈ s i ] ,ji =iwhere I [j ∈ s i ] is the indica<strong>to</strong>r for the event that j ∈ s i .With this notation, the utility difference <strong>of</strong> player i from twostrategies s i and s i(when others are using the strategy pr<strong>of</strong>ile 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
Pro<strong>of</strong> Continued<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sNow 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
Pro<strong>of</strong> Continued<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sTherefore: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 Sharing<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sConsider the problem <strong>of</strong> sharing the cost <strong>of</strong> some network resourcesamong participants.Directed graph N = (V , E ) with edge cost c e ≥ 0, k players.Each player i has a set <strong>of</strong> nodes T i he wants <strong>to</strong> connect.A strategy <strong>of</strong> player i set <strong>of</strong> edges S i ⊂ E such that S i connects <strong>to</strong> allnodes in T i .Cost sharing mechanism: All players using an edge split the costequallyGiven a vec<strong>to</strong>r <strong>of</strong> player’sstrategies S = (S 1 , . . . , S k ), the cost <strong>to</strong>agent i is C i (S) =e∈S i(c e /x e ), where x e is the number <strong>of</strong> agentswhose strategy contains edge e.32
<strong>Networks</strong>: Lecture 12Potential <strong>Game</strong>sNetwork 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 <strong>of</strong> 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
<strong>Networks</strong>: Lecture 12Network FormationA Simple <strong>Game</strong> <strong>of</strong> Network FormationThe model <strong>of</strong> network cost sharing can be viewed as a specificinstance <strong>of</strong> “endogenous network formation”. However, the mostinteresting aspect <strong>of</strong> network formation, the benefits as well as thecost <strong>of</strong> forming links, are absent from this model.Let us now consider a simple game <strong>of</strong> network formation.There are I symmetric players. The set <strong>of</strong> possible graphs (networks)is denoted by G, and corresponds <strong>to</strong> all possible configurations <strong>of</strong> linksbetween the I players.The utility function <strong>of</strong> player i isu i : G → R,and assigns a utility level <strong>to</strong> every possible network configuration.34
<strong>Networks</strong>: Lecture 12Network FormationA Simple <strong>Game</strong> <strong>of</strong> 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 <strong>of</strong> a direct connection;d i (g) is the degree <strong>of</strong> 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
<strong>Networks</strong>: 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 <strong>of</strong> a link requires “agreement” from two parties.An alternative proposed by Jackson and Wolinsky (1996) “AStrategic Model <strong>of</strong> Social and Economic <strong>Networks</strong>” is <strong>to</strong> look at theconcept <strong>of</strong> pairwise stability.36
<strong>Networks</strong>: Lecture 12Network FormationPairwise Stability (continued)Pairwise stability requires that there are no pr<strong>of</strong>itable deviations by apair <strong>of</strong> 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
<strong>Networks</strong>: 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 <strong>to</strong> the star and are distance equal <strong>to</strong> 2from all other nodes (<strong>of</strong> which there are I − 2).The utility <strong>of</strong> the star node isu i (g) = (I − 1) [b (1) − c] .38
<strong>Networks</strong>: Lecture 12Network FormationPairwise Stability and Efficiency in Network Formation(continued)Thus <strong>to</strong>tal utility isU = (I − 1) {[b (1) − c + (I − 2) b (2)] + [b (1) − c]} .Given symmetry, it is sufficient <strong>to</strong> check that <strong>to</strong>tal utility cannot beincreased by adding or subtracting one link.Removing one link would lead <strong>to</strong> a new network with <strong>to</strong>tal 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
<strong>Networks</strong>: 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 <strong>to</strong> <strong>to</strong>tal 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 <strong>of</strong> the fact that b (1) < c.−2 [b (1) − c + (I − 2) b (2)]= 2 (b (1) − c) − 2b (2) < 0,40
<strong>Networks</strong>: Lecture 12Network FormationPairwise Stability and Efficiency in Network Formation(continued)However, the star network is not pairwise stable, since the utility <strong>of</strong>the star isu i (g) = (I − 1) [b (1) − c] < 0,again in view <strong>of</strong> the fact that b (1) < c.This example shows how efficient network formation is difficult <strong>to</strong>achieve 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 <strong>to</strong> have <strong>to</strong>o few links. Here the equilibrium notion, captured bythe pairwise stability concept, reflects this intuition.41
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