On the performance of approximate equilibria in congestion games
aa r X i v : . [ c s . G T ] M a y On the performance of approximate equilibria in congestion games
George Christodoulou ∗ Elias Koutsoupias † Paul G. Spirakis ‡ Abstract
We study the performance of approximate Nash equilibria for linear congestion games. Weconsider how much the price of anarchy worsens and how much the price of stability improves as afunction of the approximation factor ǫ . We give (almost) tight upper and lower bounds for both theprice of anarchy and the price of stability for atomic and non-atomic congestion games. Our resultsnot only encompass and generalize the existing results of exact equilibria to ǫ -Nash equilibria, butthey also provide a unified approach which reveals the common threads of the atomic and non-atomic price of anarchy results. By expanding the spectrum, we also cast the existing results in anew light. For example, the Pigou network, which gives tight results for exact Nash equilibria ofselfish routing, remains tight for the price of stability of ǫ -Nash equilibria. A central concept in Game Theory is the notion of equilibrium and in particular the notion of Nashequilibrium. Algorithmic Game Theory has studied extensively and with remarkable success the com-putational issues of Nash equilibria. As a result, we understand almost completely the computationalcomplexity of exact Nash equilibria (they are PPAD-complete for games described explicitly [11, 5] andPLS-complete for games described succinctly [13]). The results established a long suspected drawbackof Nash equilibria, namely that they cannot be computed effectively, thus upgrading the importanceof approximate Nash equilibria. We don’t understand completely the computational issues of approx-imate Nash equilibria [15, 13, 12, 26], but they provide a more reasonable equilibrium concept: Itmakes sense to assume that an agent is willing to accept a situation that is almost optimal to him.In another direction, a large body of research in Algorithmic Game Theory concerns the degree ofperformance degradation of systems due to the selfish behavior of its users. Central to this area is thenotion of price of anarchy (PoA) [14, 19] and the price of stability (PoS) [1]. The first notion comparesthe social cost of the worst-case equilibrium to the social optimum, which could be obtained if everyagent followed obediently a central authority. The second notion is very similar but it considers thebest Nash equilibrium instead of the worst one.A natural question then is how the performance of a system is affected when its users are ap-proximately selfish: What is the approximate price of anarchy and the approximate price of stability ?Clearly, by allowing the players to be almost rational (within an ǫ factor), we expand the equilibriumconcept and we expect the price of anarchy to get worse. On the other hand, the price of stabilityshould improve. The question is how they change as functions of the parameter ǫ . This is exactly thequestion that we address in this work.We study two fundamental classes of games: the class of congestion games [20, 17] and the classof non-atomic congestion games [16]. The latter class of games includes the selfish routing gameswhich played a central role in the development of the price of anarchy [21, 22]. The former classplayed also an important role in the development of the area of the price of anarchy, since it relatesto the task allocation problem, which was the first problem to be studied within the framework ∗ Max-Planck-Institut f¨ur Informatik, Saarbr¨ucken, Germany. Email: { gchristo } @mpi-inf.mpg.de † Department of Informatics, University of Athens. Email: [email protected] ‡ Computer Engineering and Informatics Department, Patras University, Greece. Email: { spirakis } @cti.gr l ( f ) = 1 + ǫl ( f ) = f Figure 1: The Pigou network.of the price of anarchy [14]. Although the price of anarchy and stability of these games for exactequilibria was established long ago [21, 8, 7, 2]—and actually Tim Roughgarden [23, 25] addressedpartially the question for the price of anarchy of approximate equilibria—our results add an unexpectedunderstanding of the issues involved.While these two classes of games are conceptually very similar, dissimilar techniques were employedto answer the questions concerning the PoA and PoS. Moreover, the qualitative aspects of the answerswere quite different. For instance, the Pigou network of two parallel links captures the hardest networksituation for the price of anarchy for the selfish routing. In fact, Roughgarden [24] proved that thePigou network is the worst case scenario for a very broad class of delay functions. On the other hand,the lower bound for congestion games is different and somewhat more involved [2, 8].For the selfish routing games, the price of stability is not different than the price of anarchy becausethese games have a unique Nash (or Wardrop as it is called in these games) equilibrium. On the otherhand, for the atomic congestion games, the problem proved more challenging [7, 4]. New techniquesexploiting the potential of these games needed in order to come up with an upper bound. The lowerbound is quite complicated and, unlike the selfish routing case, it has a dependency on the number ofplayers (it attains the maximum value at the limit).The main difference between the two classes is the “integrality” of atomic congestion games: Incongestion games, when a player considers switching to another strategy, he has to take into accountthe extra cost that he will add to the edges (or facilities) of the new strategy. The number of playerson the new edges increases by one and this changes the cost. On the other hand, in the selfish routinggames the change of strategies has no additional cost. A simple—although not entirely rigorous—wayto think about it, is to consider the effects of a tiny amount of flow that ponders whether to changepath: it will not really affect the flow on the new edges (at least for continuous cost functions).Is integrality the reason which lies behind the difference of these two classes of games? It seemsso for the exact case. But our work could be interpreted as revealing that the uniqueness of the Nashequilibrium is also an important factor. Because when we move to the wider class of ǫ -Nash equilibria,the uniqueness is dropped and the problems look quite similar qualitatively; the integrality differenceis still there, but it only manifests itself in different quantitative or algebraic differences. Our work encompasses and generalizes some fundamental results in the area of the price of anarchy[21, 8, 7, 2] (see also the recently published book [18] for background information). Our techniquesnot only provide a unifying approach but they cast the existing results in a new light. For instance,the Pigou network (Figure 1) is still the tight example for the price of stability, but not the price ofanarchy. Instead for the price of anarchy, the network of Figure 2 is tight; in fact, this network is tightonly for ǫ ≤
1; a more complicated network is required for larger ǫ .We consider ǫ approximate Nash equilibria. We use the multiplicative definition of approximateequilibria : In congestion games, a player does not switch to a new strategy as long as his current costis less than 1 + ǫ times the new cost. In the selfish routing games, we use exactly the same definition:the flow is at an equilibrium when the cost on its paths is less than 1 + ǫ times the cost of every2 1’2 2’3 3’ f γf γf γ Figure 2: Lower bound for selfish routing. There are 3 distinct edge latency functions: l ( f ) = f , l ( f ) = γ (a constant which depends on ǫ ), l ( f ) = 0 (omitted in the picture). There are 3 commoditiesof rate 1 with source i and destination i ′ . The two paths for the first commodity are shown in boldlines.alternate path. There have been other definitions for approximate Nash equilibria in the literature.The most-studied is the additive case [15, 11]. In [6], they consider approximate equilibria of themultiplicative case and they study convergence issues for congestion games. Our definition differsslightly and our results can be naturally adapted to the definition of [6].There is a large body of work on the price of anarchy in various models [18]. More relevant toour work are the following publications: In [2, 8], it is proved that the price of anarchy of congestiongames for pure equilibria is . Later in [7], it is showed that the ratio 5 / n − n +1 [8],where n is the number of players. For weighted congestion games, the price of anarchy is 1 + φ ≈ . √ /
3. For the selfish routingparadigm, the price of anarchy (and of stability) for linear latencies is 4 / ǫ ≤
1. We extend this to every positive ǫ using different techniques.In this work, we give (almost) tight upper and lower bounds for the PoA and PoS of atomic andnon-atomic linear congestion games. Our results are summarized in the Table 1 (where atomic refersto congestions games). Atomic Non-atomicAnarchy (1 + ǫ ) z +3 z +12 z − ǫ where z = ⌊ ǫ + √ ǫ + ǫ ⌋ (Section 3) (1 + ǫ ) (especially for ǫ ≤ ǫ )3 − ǫ )(Section 4)Stability √ ǫ + √ (Section 5) − ǫ )(1+ ǫ ) (Section 6)Table 1: The upper bounds (with pointers to relevant sections).The results in the above table include the upper bounds. We have matching lower bounds exceptfor the atomic PoS (and for non-integral values ǫ > ǫ ≥
1, which means that the optimal is a 1-Nash equilibrium, for both theatomic and non-atomic case. Also, the price of anarchy is approximately (1 + ǫ )(3 + ǫ ) and (1 + ǫ ) for large ǫ , the atomic and non-atomic case respectively. The price of anarchy for ǫ ≤ ǫ is small. For ǫ ≤ / ǫ )2 − ǫ ǫ )3 − ǫ Stability √ ǫ + √ − ǫ )(1+ ǫ ) Table 2: The upper bounds for ǫ ≤ / ǫ -Nash equilibria.Remarkably, the parameter ǫ appears only in the linear part of the (quadratic) potential function.Our approach is similar to [8, 7], but it is much more involved technically and requires a deeperunderstanding of the potential function issues involved. We want also to draw attention to our tech-niques in bounding the approximate price of anarchy for the selfish routing which differ considerablyfrom the techniques of [21] and others [18]. The main difference is that we move from a domain withunique equilibrium to a domain with a set of solutions. A congestion game [20], also called an exact potential game [17], is a tuple (
N, E, ( S i ) i ∈ N , ( f e ) e ∈ E ),where N = { , . . . , n } is a set of n players, E is a set of facilities, S i ⊆ E is a set of pure strategies forplayer i : a pure strategy A i ∈ S i is simply a subset of facilities and l e is a cost (or latency) function,one for each facility e ∈ E . The cost of player i for the pure strategy profile A = ( A , . . . , A n ) is c i ( A ) = P e ∈ A i l e ( n e ( A )), where n e ( A ) is the number of players who use facility e in the strategyprofile A . Definition 1.
A pure strategy profile A is an ǫ equilibrium iff for every player i ∈ Nc i ( A ) ≤ (1 + ǫ ) c i ( A i , A − i ) , ∀ A i ∈ S i (1)We believe that the multiplicative definition of approximate equilibria makes more sense in theframework that we consider. This is because the costs of the players usually vary in this setting and auniform ǫ does not make much sense. Given that the price of anarchy is a ratio, we need a definitionthat is insensitive to scaling.The social cost of a pure strategy profile A is the sum of the players cost SC ( A ) = Sum ( A ) = X i ∈ N c i ( A )The pure approximate price of anarchy, is the social cost of the worst case ǫ equilibrium over theoptimal social cost P oA = max A is a ǫ -Nash SC ( A ) opt , while the pure approximate price of stability, is the social cost of the worst case ǫ equilibrium overthe optimal social cost P oS = min A is a ǫ -Nash SC ( A ) opt . G = ( V, E ) be a directed graph, where V is a set of vertices and E is a set of edges. In thisnetwork we consider k commodities: source-node pairs ( s i , t i ) with i = 1 . . . k , that define the sourcesand destinations. The set of simple paths in every pair ( s i , t i ) is denoted by P i , while with P = ∪ ki =1 P i we denote their union. A flow f , is a mapping from the set of paths to the set of nonnegative reals f : P → R + . For a given flow f , the flow on an edge is defined as the sum of the flows of all the pathsthat use this edge f e = P P ∈P ,e ∈ P f P . We relate with every commodity ( s i , t i ) a traffic rate r i , as thetotal traffic that needs to move from s i to t i . A flow f is feasible, if for every commodity { s i , t i } , thetraffic rate equals the flow of every path in P i , r i = P P ∈P i f P . Every edge introduces a delay in thenetwork. This delay depends on the load of the edge and is determined by a delay function, l e ( · ). Aninstance of a routing game is denoted by the triple ( G, r, l ). The latency of a path P , for a given flow f , is defined as the sum of all the latencies of the edges that belong to P , l P ( f ) = P e ∈ P l e ( f e ). Thesocial cost that evaluates a given flow f , is the total delay due to fC ( f ) = X P ∈P l P ( f ) f P . The total delay can also be expressed via edge flows C ( f ) = P e ∈ E l e ( f e ) f e . From now on, when we are talking about flows, we mean feasible flows. In [3, 10], it is shownthat there exists a (unique) equilibrium flow, known as Wardrop equilibrium[27]. In analogy to theirdefinition, we define the ǫ Wardrop equilibrium flows, as follows
Definition 2.
A feasible flow f , is an ǫ -Nash (or Wardrop) equilibrium, if and only if for everycommodity i ∈ { , . . . , k } and P , P ∈ P i with f P > l P ( f ) ≤ (1 + ǫ ) l P ( f ).In this work we restrict our attention to linear latency functions : l e ( x ) = a e x + b e , where a e and b e are nonnegative constants. Our results naturally extend to mixed and correlated equilibria. We alsobelieve that they can be also extended to more general latency functions such as polynomials. In this section we study the dependency on the parameter ǫ , of the price of anarchy for the case ofatomic congestion games. For large ǫ the price of anarchy is roughly (1 + ǫ ) . The same holds thenon-atomic case as we are going to establish in the next section.We will need the following arithmetic lemma. Lemma 1.
For every α, β, z ∈ N : β ( α + 1) ≤ z + 1 α + z + 3 z + 12 z + 1 β Proof.
Consider the function f ( α, β ) which we obtain when we subtract the left part of the statement’sinequality from the right part and multiply the result by 2 z + 1.5 ( α, β ) = a + ( z + 3 z + 1) β − (2 z + 1) β ( α + 1)= (cid:18) α − z + 12 β (cid:19) + (8 z + 3) β − (8 z + 4) β . For β = 0, and for any β ≥ f ( α, β ) is clearly positive. For β = 1 it takes the form f ( α,
1) =( α − z )( α − z − ≥
0, and the lemma follows.Our first theorem gives an upper bound for the price of anarchy for congestion games; this is tight,as we are going soon to establish. This result generalizes the bound in [2, 8] to approximate equilibria.The proof is for linear latency functions of the form l e ( x ) = x , but it can be easily extended to latenciesof the form l e ( x ) = a e x + b e , with nonnegative a e , b e . Theorem 1 (Atomic-PoA-Upper-Bound) . For any positive real ǫ , the approximate price of anarchyof general congestion games with linear latencies is at most (1 + ǫ ) z + 3 z + 12 z − ǫ , where z ∈ N is the maximum integer with z z +1 ≤ ǫ (or equivalently for z = ⌊ ǫ + √ ǫ + ǫ ⌋ ).Proof. Let A = ( A , . . . , A n ) be an ǫ -approximate pure Nash, and P = ( P , . . . , P n ) be the optimumallocation. From the definition of ǫ -equilibria (Inequality (1)) we get X e ∈ A i n e ( A ) ≤ (1 + ǫ ) X e ∈ P i ( n e ( A ) + 1) . If we sum up for every player i and use Lemma 1, we get Sum ( A ) = X i ∈ N c i ( A )= X i ∈ N X e ∈ A i n e ( A )= X e ∈ E n e ( A ) ≤ (1 + ǫ ) X e ∈ E n e ( P ) ( n e ( A ) + 1) ≤ ǫ z + 1 X e ∈ E n e ( A ) + (1 + ǫ )( z + 3 z + 1)2 z + 1 X e ∈ E n e ( P )= 1 + ǫ z + 1 Sum ( A ) + (1 + ǫ )( z + 3 z + 1)2 z + 1 opt . From this we obtain the theorem
Sum ( A ) ≤ (1 + ǫ ) z + 3 z + 12 z − ǫ opt . The above is a typical proof in this work. All our upper bound proofs have similar form. Theproofs of the price of stability are more challenging however, as they require the use of appropriategeneralizations of the potential function. We now show that the above upper bound is tight.6 heorem 2 (Atomic-PoA-Lower-Bound) . For any real positive ǫ , there are instances of congestiongames with linear latencies, for which the approximate price of anarchy of general congestion gameswith linear latencies, is at least (1 + ǫ ) z + 3 z + 12 z − ǫ , where z ∈ N is the maximum integer with z z +1 ≤ ǫ .Proof. Let z ∈ N be the maximum integer with z z +1 ≤ ǫ . We will construct an instance with z + 2players and 2 z + 4 facilities. There are two types of facilities: • z + 2 facilities of type α , with latency l e ( x ) = x and • z + 2 facilities of type β with latency l e ( x ) = γx = ( z +1) − (1+ ǫ )( z +2)(1+ ǫ )( z +1) − z x .Player i has two alternative pure strategies, S i and S i . • The first strategy is to play the two facilities α i and β i , i.e. S i = { α i , β i } . • The second strategy is to play every facility of type α except for α i and z + 1 facilities of type β starting at facility β i +1 . More precisely, the second strategy has the facilities S i = { α , . . . , α i − , α i +1 , . . . , α z +2 , β i +1 , . . . , β i +1+ z } , where the indices may require computations ( mod z + 2).First we prove that playing the second strategy S = ( S , . . . , S n ) is a ǫ -Nash equilibrium. Thecost of player i is c i ( S ) = ( z + 1) + γz , as there are exactly z + 1 players using facilities of type α and exactly z players using facilities of type β . If player i unilaterally switches to the other available strategy S i he has cost c i ( S i , S − i ) = ( z + 2) + γ ( z + 1) = c i ( S )1 + ǫ , which shows that S is an ǫ -Nash equilibrium.The optimum allocation is the strategy profile S , where every player has cost c i ( S ) = 1 + γ andso the price of anarchy is c i ( S ) c i ( S ) = ( z + 1) + γz γ = (1 + ǫ ) z + 3 z + 12 z − ǫ . Notice that the parameter z is an integer because it expresses a number of facilities.The above theorems (lower and upper bound) employ, for any positive real ǫ , an integer z ( ǫ ), whichis the maximum integer that satisfies z z +1 ≤ ǫ . So for ǫ ∈ [0 , / z ( ǫ ) = 1 and the price of anarchyis ǫ )2 − ǫ , for ǫ ∈ [1 / , / z ( ǫ ) = 2 and the price of anarchy is ǫ )4 − ǫ and so on. Roughly the priceof anarchy grows as (1 + ǫ )(3 + ǫ ). 7 Selfish Routing – PoA
In this section we estimate the price of anarchy for non-atomic congestion games and consequentlyfor its special case, the selfish routing. Our results generalize the results in [21, 22] to the case ofapproximate equilibria. The proof has the same form with the proof of the atomic case in the previoussection.Again, we will need an arithmetic lemma. The main change now is that we deal with continuousvalues instead of integrals.
Lemma 2.
For every reals α, β, λ it holds, βα ≤ λ α + λβ , whereProof. Simply because α + 4 λ β − λαβ = ( α − λβ ) ≥ Theorem 3 (Selfish-PoA-Upper-Bound) . For any positive real ǫ , and for every λ ≥ , the approximateprice of anarchy of non-atomic congestion games with linear latencies is at most λ (1 + ǫ )4 λ − − ǫ . Proof.
Let f be an ǫ -approximate Nash flow, and f ∗ be the optimum flow (or any other feasible flow).From the definition of approximate Nash equilibria (Inequality (1)), we get that for every path p withnon-zero flow in f and any other path p ′ : X e ∈ p l e ( f e ) ≤ (1 + ǫ ) X e ∈ p ′ l e ( f ∗ e ) . We sum these inequalities for all pairs of paths p and p ′ weighted with the amount of flow of f and f ∗ on these paths. X p,p ′ f p f ∗ p ′ X e ∈ p l e ( f e ) ≤ (1 + ǫ ) X p,p ′ f p f ∗ p ′ X e ∈ p ′ l e ( f ∗ e ) X p ′ f ∗ p ′ X e ∈ E l e ( f e ) f e ≤ (1 + ǫ ) X p f p X e ∈ E l e ( f ∗ e ) f ∗ e ( X p ′ f ∗ p ′ ) X e ∈ E l e ( f e ) f e ≤ (1 + ǫ )( X p f p ) X e ∈ E l e ( f ∗ e ) f ∗ e But P p f p = P p ′ f ∗ p ′ is equal to the total rate for the feasible flows f and f ∗ . Simplifying, we get X e ∈ E l e ( f e ) f e ≤ (1 + ǫ ) X e ∈ E l e ( f e ) f ∗ e . This is the generalization to approximate equilibria of the inequality established by Beckmann, McGuire,and Winston [3] for exact Wardrop equilibria.Since we consider linear functions of the form l e ( f e ) = a e f e + b e , we get X e ∈ E (cid:0) a e f e + b e f e (cid:1) ≤ (1 + ǫ ) X e ∈ E a e f e f ∗ e + (1 + ǫ ) X e ∈ E b e f ∗ e . X e ∈ E (cid:0) a e f e + b e f e (cid:1) ≤ (1 + ǫ ) X e ∈ E a e (cid:18) λ f e + λf ∗ e (cid:19) + (1 + ǫ ) X e ∈ E b e f ∗ e . from which we get X e ∈ E (cid:18) a e (cid:18) − (1 + ǫ ) 14 λ (cid:19) f e + b e f e (cid:19) ≤ λ (1 + ǫ ) X e ∈ E a e f ∗ e + (1 + ǫ ) X e ∈ E b e f ∗ e , and for λ ≥ λ − − ǫ λ SC ( f ) ≤ (1 + ǫ ) λSC ( f ∗ ) . This gives price of anarchy at most of 4 λ (1 + ǫ )4 λ − − ǫ , for every λ ≥ λ (1+ ǫ )4 λ − − ǫ of the theorem is minimized for λ = (1 + ǫ ) / ǫ ≥ λ = 1 when ǫ ≤ λ = 1 and λ = (1 + ǫ ) /
2. Thefirst corollary was proved before in [23, 25] using different techniques.
Corollary 1.
For any nonnegative real ǫ ≤ , the approximate price of anarchy of non-atomic con-gestion games with linear latencies is at most ǫ )3 − ǫ . Corollary 2.
For any positive real ǫ ≥ , the approximate price of anarchy of non-atomic congestiongames with linear latencies is at most (1 + ǫ ) . We now show that the above upper bounds are tight. To be precise, we show that Corollary 1 istight and that Corollary 2 is partially tight—only for integral values of ǫ .The following theorem for the case of ǫ ≤ ǫ >
1, in Theorem 5.
Theorem 4 (Selfish-PoA-Lower-Bound for ǫ ≤ . For any nonnegative real ǫ ≤ , there are instancesof congestion games with linear latencies, for which the approximate price of anarchy of general con-gestion games with linear latencies, is at least ǫ )3 − ǫ . Proof.
We will construct an instance with 3 commodities, each of them with unit flow, and 6 facilities(a slightly more involved network case appears in Figure 2. There are two types of facilities: • α , with latency l ( x ) = x and • β with constant latency l ( x ) = γ = − ǫ )1+ ǫ .Commodity i has two alternative pure strategies, S i and S i .9 The first strategy is to choose both the facilities α i and β i , i.e. S i = { α i , β i }• As a second alternative, players of commodity i may choose every facility of type α except for α i ; we denote this set by S i = { α − i } .First we prove that playing the second strategy S = ( S , S , S ) is a ǫ -Nash equilibrium. The cost ofevery player in commodity i is c i ( S ) = 4, as there are exactly z + 1 players using facilities of type α and exactly z players using facilities of type β .If player i unilaterally switches to the other available strategy S i he gets c i ( S i , S − i ) = 2 + γ = c i ( S )1 + ǫ and so S is an ǫ − approximate equilibrium.In the optimum case, the players use strategy profile S , where commodity i has cost c i ( S ) = 1+ γ and so the price of anarchy is c i ( S ) c i ( S ) = 41 + γ = 4(1 + ǫ )3 − ǫ . For larger ǫ ( ǫ > Theorem 5 (Selfish-PoA-Lower-Bound for ǫ ≥ . For any real positive ǫ , there are instances ofcongestion games with linear latencies, for which the approximate price of anarchy of general congestiongames with linear latencies, is at least (1 + ǫ ) z ( z + 1)2 z − ǫ = (1 + ǫ ) z + z z − ǫ , where z = ⌊ ǫ ⌋ . Proof.
Let z = ⌊ ǫ ⌋ . We will construct an instance with z + 2 commodities and 2 z + 4 facilities.There are two types of facilities: • z + 2 facilities of type α , with latency 1 and • z + 2 facilities of type β with latency γ = ( z +1) − (1+ ǫ )( z +1)(1+ ǫ ) z − z .Commodity i has two alternative pure strategies, S i and S i . • The first strategy is to choose both the facilities α i and β i , i.e. S i = { α i , β i } . • As a second alternative, commodity i may choose every facility of type α except for α i and z facilities of type β as defined in the following: S i = { α , . . . , α i − , α i +1 , . . . , α z +2 , β i +1 , . . . , β z + i +1 } , where the indices are computed ( mod z + 2).First we prove that playing the second strategy S = ( S , . . . , S n ) is a ǫ -Nash equilibrium. The costof commodity i is c i ( S ) = ( z + 1) + γz , as there are exactly z + 1 commodities using facilities of type α and exactly z players using facilitiesof type β . 10f commodity i unilaterally switches to the other available strategy S i , its cost becomes c i ( S i , S − i ) = ( z + 1) + γz = c i ( S )1 + ǫ , which shows that S is an ǫ − approximate equilibrium.The optimum is the strategy profile S , where every commodity has cost c i ( S ) = 1 + γ . It followsthat the price of anarchy is c i ( S ) c i ( S ) = ( z + 1) + γz γ = (1 + ǫ ) z ( z + 1)2 z − ǫ . An upper bound of the price of stability is perhaps more difficult to obtain because we have to find away to characterize the best ǫ -Nash equilibrium. We don’t know how to do this, so we use an indirectapproach: We identify a property such that every profile satisfying the property is guaranteed to be an ǫ -Nash equilibrium. We then upper bound the price of anarchy of all profiles satisfying this property.To this end, we generalize the notion of potential [17]; a characteristic property of congestion gamesis that they possess a potential function.We define the ǫ -potential function of a profile A to be qΦ ǫ ( A ) = 12 X e ∈ E ( a e n e ( A ) + b e ) n e ( A ) + 12 1 − ǫ ǫ X e ∈ E ( a e + b e ) n e ( A ) . For ǫ = 0, this reduces to the classical potential function for congestion games. More importantly, itgeneralizes the following interesting property to ǫ -Nash equilibrium. Theorem 6.
Any profile A which is a local minimum of Φ ǫ , is an ǫ -Nash equilibrium.Proof. Consider a profile A = ( A , . . . , A n ). We want to compute the change in the ǫ -potential functionwhen player i changes from strategy A i to a strategy P i ∈ S i . The resulting profile ( P i , A − i ) has n e ( P i , A − i ) = n e ( A ) + 1 , e ∈ P i − A i n e ( A ) − , e ∈ A i − P i n e ( A ) , otherwise.From this we can compute the differenceΦ ǫ ( P i , A − i ) − Φ ǫ ( A ) = X e ∈ P i − A i (cid:18) a e n e ( A ) + 11 + ǫ ( a e + b e ) (cid:19) − X e ∈ A i − P i (cid:18) a e n e ( A ) + 11 + ǫ ( − a e ǫ + b e ) (cid:19) . We can rewrite this asΦ ǫ ( P i , A − i ) − Φ ǫ ( A ) = X e ∈ P i (cid:18) a e n e ( A ) + 11 + ǫ ( a e + b e ) (cid:19) − X e ∈ P i ∩ A i a e − (2) X e ∈ A i (cid:18) a e n e ( A ) + 11 + ǫ ( − a e ǫ + b e ) (cid:19) . A is a local minimum of Φ ǫ . This translates to Φ ǫ ( P i , A − i ) ≥ Φ ǫ ( A )for all i . The cost for player i before the change is c i ( A ) = P e ∈ A i ( a e n e ( A ) + b e ) and after thechange is c i ( P i , A − i ) = P e ∈ P i ( a e n e ( P i , A − i ) + b e ). We want to show that A is an ǫ -Nash equilibrium: c i ( A ) ≤ (1 + ǫ ) c i ( P i , A − i ).The ǫ -potential consists of two parts that can be used to bound the cost of player i at profile A and ( P i , A − i ): c i ( A ) = X e ∈ A i ( a e n e ( A ) + b e ) ≤ X e ∈ A i (1 + ǫ ) (cid:18) a e n e ( A ) + 11 + ǫ ( − a e ǫ + b e ) (cid:19) , (which holds because n e ( A ) ≥ e ∈ A i ), and c i ( P i , A − i ) = X e ∈ P i ( a e ( n e ( A ) + 1) + b e ) − X e ∈ P i ∩ A i a e ≥ X e ∈ P i (cid:18) a e n e ( A ) + 11 + ǫ ( a e + b e ) (cid:19) − X e ∈ P i ∩ A i a e (which holds for ǫ ≥ c i ( A ) ≤ (1 + ǫ ) c i ( P i , A − i ). Consequently, A is an ǫ -Nash equilibrium.First we present an easy upper bound, that uses only the previous theorem. Proposition 1.
For linear congestion games, the price of stability is at most ǫ .Proof. Let A be the allocation that minimizes the ǫ potential Φ ǫ , and let P be the optimum allocation.We have Φ ǫ ( A ) ≤ Φ ǫ ( P )and so Sum ( A ) + 1 − ǫ ǫ X e ∈ E ( a e + b e ) n e ( A ) ≤ Sum ( P ) + 1 − ǫ ǫ X e ∈ E ( a e + b e ) n e ( P ) , (3)from which we get Sum ( A ) ≤
21 + ǫ Sum ( P ) . The previous theorem gives us an easy way to bound the price of stability. Clearly this is nottight: for ǫ = 0, it doesn’t provide us the right answer 1 + √ / Lemma 3.
For integers α, β and for γ = ( √ )( e − √ ) e +3+2 √ γβ + 1 − γǫ ǫ β − γ − ǫ ǫ α + (1 − γ ) βα ≤ (cid:0) √ − (cid:1) ( e − e + 3 + 2 √ α + 2 3 + √ e + 3 + 2 √ β roof. Let f be the expression that we take if we substitute γ = ( √ )( e − √ ) e +3+2 √ , and then substractthe first part from the second and divide by ( √ − ) ( e − e +3+2 √ . We can study f as a function of integers α and β . f ( α, β ) = 1 / (cid:16) √ − b − b √ a (cid:17) + 1 / (cid:16) √ (cid:17) (cid:16) β − − √ (cid:17) . We want to prove that f ( α, β ) ≥
0, for every α, β ∈ N . One can easily verify that f ( α,
0) = (cid:16) a + 2 √ (cid:17) a ≥ ,f ( α,
1) = α ( α − ≥ , while for β ≥ Theorem 7 (Atomic-PoS-Upper-Bound) . For any positive real ǫ ≤ , the approximate price of stabilityof general congestion games with linear latencies is at most √ √ ǫ . Proof.
Let A be the allocation that minimizes the ǫ potential Φ ǫ , and let P be the optimum allocation.Since A is a local minimum of Φ ǫ , if we sum (2) for all players i , we get X e ∈ E n e ( A ) (cid:18) a e n e ( A ) + 11 + ǫ ( − a e ǫ + b e ) (cid:19) ≤ X e ∈ E n e ( P ) (cid:18) a e n e ( A ) + 11 + ǫ ( a e + b e ) (cid:19) − X i ∈ N X e ∈ P i ∩ A i a e . For simplicity let’s assume a e = 1 , b e = 0, although the results hold in general. We get X e ∈ E (cid:18) n e ( A ) − ǫ ǫ n e ( A ) (cid:19) ≤ X e ∈ E n e ( P ) (cid:18) n e ( A ) + 11 + ǫ (cid:19) (4)If we multiply (3) with γ and (4) with (1 − γ ), for γ = ( √ )( e − √ ) e +3+2 √ and add them, we get X e ∈ E n e ( A ) ≤ γβ + 1 − γǫ ǫ X e ∈ E n e ( P ) − γ − ǫ ǫ X e ∈ E n e ( A ) + (1 − γ ) X e ∈ E n e ( P ) n e ( A ) ≤ (cid:0) √ − (cid:1) (1 − ǫ )3 ǫ + 3 + 2 √ X e ∈ E n e ( A ) + 6 + 2 √ ǫ + 3 + 2 √ X e ∈ E n e ( P )and so X e ∈ E n e ( A ) ≤ √ √ ǫ X e ∈ E n e ( P ) . Theorem 6 implies that the socially optimal allocation is 1 − equilibrium. So for ǫ ≥
1, trivially theprice of stability is 1. The following theorem shows that this is tight, in the sense that the social costof the best (1 − δ )-approximate equilibrium, is strictly greater than the social optimum.13 heorem 8. There exist instances of congestion games, (even with two parallel links), where a theoptimum allocation is not a (1 − δ ) − approximate equilibrium, for any arbitrarily small positive δ . Thismeans that the price of stability for (1 − δ ) -approximate equilibria is strictly greater than 1.Proof. Consider a game with two facilities ( parallel links) e , e and n players. The facilities havelatencies l e ( x ) = (2 n − · x − γ , for some arbitrary small positive γ and l e ( x ) = x .Consider the allocation P , that is produced when one player, (say the first), chooses the first linkand the rest of the players use the second link. This has cost Sum ( P ) = 2 n − − γ + ( n − , which is optimal: Any other allocation, in which k = 1 players use the first link and n − k the second,has cost (2 n − − γ ) k + ( n − k ) ≥ (2 n − − γ ) + ( n − . In strategy profile P , the first player has cost 2 n − − γ , while the rest of the players have cost n − n . This means that opt is a (1 − γn )-approximate equilibrium. Therefore, for any δ , there is an instance with sufficientlylarge number of players n ( δ ), where opt is not a (1 − δ )-approximate equilibrium.We now give an almost matching lower bound for the price of stability. The upper and lowerbounds are not equal but they match at the extreme values of ǫ = 0 and ǫ = 1. For ǫ = 0, we get theknown price of stability [7, 4]. The price of stability decreases as a function of ǫ , and drops to 1 for ǫ = 1. Theorem 9 (Atomic-PoS-Lower-Bound) . There are linear congestion games whose approximate Nashequilibria (even their dominant equilibria as the proof reveals) have price of stability of the
Sum socialcost approaching ǫ + θǫ + 3 ǫ + 2 ǫ + θ + θǫ ǫ + 5 θǫ + 6 ǫ + 4 ǫ − θǫ + 2 θǫ , where θ = √ ǫ + 3 + ǫ + 2 ǫ .Proof. We describe a game of n + λ players with parameters α , β , and λ which we will fix later toobtain the desired properties. Each player i has two strategies A i and P i , where the strategy profile( A , . . . , A n ) will be the equilibrium and ( P , . . . , P n ) will have optimal social cost.There are also λ players that have fixed strategies; they don’t have any alternative. They play afixed facility f λ .There are 3 types of facilities: • n facilities α i , i = 1 , . . . , n , each with cost function l ( x ) = αx . Facility α i belongs only tostrategy P i . • n ( n −
1) facilities β ij , i, j = 1 , . . . , n and i = j , each with cost l ( x ) = βx . Facility β ij belongsonly to strategies A i and P j . • f λ with unit cost l ( x ) = x .We will first compute the cost of every player and every strategy profile. By symmetry, we need onlyto consider the cost cost A ( k ) of player 1 and the cost cost P ( k ) of player N of the strategy profile( A , . . . , A k , P k +1 , . . . , P n ). Therefore, cost A ( k ) = (2 n − k − β + ( λ + k ) . cost P ( k ) = α + ( n + k − β. We now want to select the parameters α and β so that the strategy profile ( A , . . . , A n ) is (1 + ǫ )-dominant. Equivalently, at every strategy profile ( A , . . . , A k , P k +1 , . . . , P n ), player i , i = 1 , . . . , k ,has no reason to switch to strategy P i , because it’s (1 + ǫ ) times less profitable. We use dominantstrategies because it is easier to guarantee that there is no other equilibrium. This is expressed by theconstraints (1 + ǫ ) · cost A ( k ) ≤ cost P ( k − , for every k = 1 , . . . , n . (5)All these constraints are linear in k and they are satisfied by equality when α = (1 + ǫ ) (2 nǫ − ǫ + ǫλ + n + 2 λ + 1)2 + ǫ and β = 1 + ǫ ǫ , as one can verify with straightforward, albeit tedious, substitution.In summary, for the above values of the parameters α and β , we obtain the desired property thatthe strategy profile ( A , . . . , A n ) is a (1 + ǫ )-dominant strategy. If we increase α by any small positive δ , inequality (5) becomes strict and the (1 + ǫ )-dominant strategy is unique (and therefore uniqueNash equilibrium).We now want to select the value of the parameter m so that the price of anarchy of this equilibriumis as high as possible. The price of anarchy is cost A ( N ) + λ ( λ + n ) cost P (0) + λ which for the above values of α and β can be simplified to3 n + 2 n ǫ − n − nǫ + 4 nλ + 2 nλǫ + 2 λ + ǫλ n ǫ − nǫ + 3 nλǫ + 2 n + 2 nλ + 2 n ǫ − nǫ + nǫ λ + 2 λ + ǫλ . If we optimize the parameter λ and take the limit of n to infinity we get the theorem. Here we follow the ideas of the previous section to define an appropriate potential function for ǫ -Nashequilibrium for the selfish routing problem or more generally non-atomic congestion games. It is easierto deal with the more general case of non-atomic congestion games rather than the selfish routing case,since we don’t have to concern ourselves with the underlying network. In fact, our approach revealshow little we really need to establish results that encompass many influential results in the literature.Consider a flow f for the selfish routing with flow f e through every edge e . We define the ǫ -potentialfunction Φ ǫ ( f ) = X e ∈ E (cid:18) a e f e + 11 + ǫ b e f e (cid:19) . We will show that the global minimum of Φ ǫ ( f ) is an ǫ -Nash equilibrium: Since this is the unique 1 + ǫ Nash Equilibrium of this game, the terms price of anarchy and price of stability areequivalent. heorem 10. In a non-atomic congestion game, the flow f which minimizes the ǫ -potential functionis an ǫ -Nash equilibrium. Furthermore, for any other flow f ′ the following inequality holds: X e ∈ E (cid:18) a e f e + 11 + ǫ b e f e (cid:19) ≤ X e ∈ E (cid:18) a e f e f ′ e + 11 + ǫ b e f ′ e (cid:19) . Proof.
Consider a flow f and two paths p and p ′ of the same commodity. Suppose that the flow f onpath p is positive. We want to compute how much Φ ǫ ( f ) changes when we shift a small amount δ > p to path p ′ . More precisely, if f ′ denotes the new flow, we compute the followinglimit lim δ → Φ ǫ ( f ′ ) − Φ ǫ ( f ) δ = X e ∈ p ′ (cid:18) a e f e + 11 + ǫ b e (cid:19) − X e ∈ p (cid:18) a e f e + 11 + ǫ b e (cid:19) (6)If f minimizes Φ ǫ , then the above quantity is nonnegative. But we can bound the cost of paths p and p ′ with the two terms of this quantity as follows: l p ( f ) = X e ∈ p ( a e f e + b e ) ≤ (1 + ǫ ) X e ∈ p ( a e f e + 11 + ǫ b e )and l p ′ ( f ) = X e ∈ p ′ ( a e f e + b e ) ≥ X e ∈ p ′ ( a e f e + 11 + ǫ b e ) . It follows that l p ( f ) ≤ (1 + ǫ ) l p ′ ( f ), which implies that f is an ǫ -Nash equilibrium.For the second part, we observe that the expression (6), which is nonnegative for f which minimizesΦ ǫ , implies that for every path p on which f is positive and every other path p ′ we must have X e ∈ p (cid:18) a e f e + 11 + ǫ b e (cid:19) ≤ X e ∈ p ′ (cid:18) a e f e + 11 + ǫ b e (cid:19) . Consider now another flow f ′ which satisfies the rate constraints for the commodities and let us sumthe above inequalities for all paths p and p ′ weighted with the amount of flow in f and f ′ . Moreprecisely: X p,p ′ f p f ′ p ′ X e ∈ p (cid:18) a e f e + 11 + ǫ b e (cid:19) ≤ X p,p ′ f p f ′ p ′ X e ∈ p ′ (cid:18) a e f e + 11 + ǫ b e (cid:19)X p ′ f ′ p ′ X e ∈ E (cid:18) a e f e + 11 + ǫ b e f e (cid:19) ≤ X p f p X e ∈ E (cid:18) a e f e f ′ e + 11 + ǫ b e f ′ e (cid:19) But P p ′ f ′ p ′ = P p f p is equal to the sum of the rates for all commodities. If we remove from theexpression this common factor, the second part of the theorem follows.From Lemma 2, if we substitute λ with 1 / (1 + ǫ ), we get that for any reals α, β , and ǫ ∈ [0 , αβ ≤ ǫ α + 11 + ǫ β . (7) Theorem 11 (Selfish-PoS-Upper-Bound) . The price of stability is at most − ǫ )(1+ ǫ ) . roof. Let f be the potential minimizer of Φ ǫ and f ∗ be the optimum flow. From Theorem (10) and(7) we get that X e ∈ E a e f e + 11 + ǫ b e f e ≤ X e ∈ E a e ( 1 + ǫ f e ǫ f ∗ e ) + 11 + ǫ b e f ∗ e or X e ∈ E a e − ǫ f e + 11 + ǫ b e f e ≤
11 + ǫ C ( f ∗ ) , and since 1 / (1 + ǫ ) ≥ (3 − ǫ ) /
4, we get3 − ǫ C ( f ) ≤
11 + ǫ C ( f ∗ ) , which gives the desired result: C ( f ) ≤ − ǫ )(1 + ǫ ) C ( f ∗ ) . We now establish that the Pigou network (extended to take into account the parameter ǫ , Figure 1)gives tight results. Theorem 12 (Selfish-PoS-Lower-Bound) . The price of stability is at least − ǫ )(1+ ǫ ) .Proof. Consider the Pigou network of Figure 1. There is a unit of flow that wants to move from s to t . Clearly, the only (1 + ǫ )-Wardrop flow is to choose the lower edge, for ǫ <
1. This gives a socialcost of 1.On the other hand the optimum is to route (1 + ǫ ) / − ǫ ) / (1+ ǫ )(1 − ǫ )2 + (1+ ǫ )2 (1+ ǫ )2 = (1+ ǫ )(3 − ǫ )4 , andso the price of stability is − ǫ )(1+ ǫ ) as needed. Acknowledgements
The authors would like to thank Ioannis Caragiannis for many helpful discus-sions and Tim Roughgarden for useful pointers to literature.
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