Local Smoothness and the Price of Anarchy in ... - Semantic Scholar
Local Smoothness and the Price of Anarchy in Atomic Splittable Congestion Games Tim Roughgarden∗ Abstract We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upperbound side, we define the framework of “local smoothness”, which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worstcase POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. 1
Introduction
Congestion games play a central role in the theory of worst-case approximation guarantees for game-theoretic equilibria. In the standard model [15], there is a ground set of resources, and each player selects a subset of them (e.g., a path in a network). Each resource has a univariate cost function that depends on the load induced by the ∗ Stanford University, Department of Computer Science. Supported in part by NSF CAREER Award CCF-0448664, an ONR Young Investigator Award, an ONR PECASE Award, an AFOSR MURI grant, and an Alfred P. Sloan Fellowship. Email: [email protected] † Stanford University, Department of Computer Science. Supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD). Email: fschopp@ uni-paderborn.de
Florian Schoppmann† players that use it, and each player strives to minimize the sum of the resources’ costs in its chosen strategy (given the strategies chosen by the other players). We study the splittable variant of congestion games, where each player has a weight wi and a list of available pure strategies (each a subset of resources), and each player chooses how to split fractionally its weight over its pure strategies. The splittable model is more appropriate than the traditional “unsplittable” model in some applications, such as multipath routing in networks. Indeed, in the computer networking literature, the splittable model was studied a decade prior to the unsplittable model (beginning with [12]). In this paper, we resolve the worst-case price of anarchy (POA) [11] — the ratio between the sum of players’ costs in a Nash equilibrium and in a minimumcost outcome — of splittable congestion games. Prior to this work, no tight bounds on the POA in splittable congestion games were known, even for the simplest non-trivial special case of affine cost functions. In contrast, tight bounds for essentially all classes of cost functions were proved for the nonatomic model (with a continuum of players) and the unsplittable model some years ago [1, 8, 17, 19]. We make two distinct contributions. On the upper-bound side, we define the framework of “local smoothness”, which provides a sufficient condition for a game to have a bounded POA. This framework refines the smoothness paradigm introduced in [17] for games with convex strategy sets, intuitively by requiring certain inequalities only for nearby pairs of outcomes, rather than for all pairs of outcomes as in [17]. Standard smoothness arguments provably cannot establish tight bounds on the POA in atomic splittable congestion games, whereas local smoothness arguments can (as we show). Further, we prove that every POA bound derived via local smoothness applies automatically, without any quantitative degradation, to every correlated equilibrium of the game (and hence also to every mixed Nash equilibrium). Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to all of the coarse correlated equilibria of a game (or equivalently, to all sequences of repeated play in which
all players experience vanishing per-step regret; see [4]). Extending POA bounds to more general equilibrium concepts is important because it weakens the rationality assumptions under which the bounds are valid. An upper bound that applies only to pure Nash equilibria presumes that players reach one. A bound that applies more generally to correlated equilibria does not require players to converge to anything: if a game is played repeatedly and each player has vanishing per-step “swap regret” (see, e.g., [3]), then the bound applies to their time-averaged cost. Our second contribution is a very general lower bound. For a set L of allowable resource cost functions, we denote by γ(L) the smallest upper bound on the POA that is provable via a local smoothness argument. We prove that for every set L that satisfies mild technical conditions, the worst-case POA in atomic splittable congestion games with cost functions in L is exactly γ(L). Thus, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. The technical challenge in proving our lower bound stems from its generality: we need to exhibit a worstcase splittable congestion game for a set L of cost functions without knowing anything about L! Our high-level approach is to exhibit an example for which all of the inequalities used in the upper bound proof are tight, in the spirit of “complementary slackness” arguments in linear programming. This goal translates to a labyrinth of restrictions on a candidate worst-case splittable congestion game — on the allowable cost functions, on the resource loads in equilibrium and optimal outcomes, and on the relative use of a resource by different players in an equilibrium. Nevertheless, we show that all of these conditions can be met simultaneously and thus there are splittable congestion games with POA arbitrarily close to our upper bound of γ(L). For concreteness, Table 1 illustrates our exact bounds for the special case of bounded-degree polynomials with non-negative coefficients. (The necessary calculations are not immediately obvious and are given in Section 6.) The worst-case price of anarchy in splittable congestion games is generally strictly larger than that in nonatomic congestion games and strictly less than that in unsplittable congestion games. 1.1 Related Work. See [18, §4.8] for general references on splittable congestion games; here, we focus only on the prior research most relevant to the present work. Splittable congestion games seem more difficult to reason about than other congestion game models. For example, it was only shown recently that Nash equilibria need not be unique in such games [2]. Splittable
Table 1: Price of anarchy with polynomial cost functions
Degree
Atomic splittable
Atomic unsplittable (weighted) [1]
1 2 3 4 5 6 7 8
1.500 2.549 5.063 11.09 26.32 66.88 180.3 512.0
2.618 9.909 47.82 277.0 1,858 14,099 118,926 1,101,126
1.333 1.626 1.896 2.151 2.394 2.630 2.858 3.081
Θ( logd d )d+1
Θ( logd d )
d
√ d+1 d+1 ) 2
( 1+
Nonatomic [19]
congestion games also exhibit counterintuitive behavior, like the fact that fusing two players into one — seemingly, increasing the amount of cooperation in the game — can increase the cost of a game’s Nash equilibrium [5]. Finally, two independent proofs claimed that the worstcase price of anarchy in splittable congestion games is never worse than that in nonatomic congestion games [9, 16]. Cominetti et al. [7] showed, however, that these proofs are valid only in symmetric games — where all players have the same weight and the same set of strategies — and adapted an example in [5] to refute the general claims. The first upper bounds on the POA in general splittable congestion games were given in [7]. These bounds are derived using a special case of our local smoothness framework in which one of our two parameters (λ) is fixed at 1. This restricted approach yields finite upper bounds on the worst-case POA only for cost functions that are polynomials with degree at most 3 and nonnegative coefficients. Later Harks [10] used what we are calling local smoothness arguments to derive significantly better upper bounds. The upper bounds in [10] are equivalent to ours (as in, e.g., Table 1), but are given in a more complicated form that renders intractable the problems of constructing matching lower bounds and giving exact closed-form expressions for polynomial cost functions. Also, prior to our work, there were no known upper bounds on the POA for any equilibrium concept more general than pure Nash equilibria. The only lower bounds known prior to the present work follow from the counterexamples in [7]. As an example comparison, for bounded-degree polynomial cost functions, our new lower bounds are exponentially larger (in the degree d) than those in [7].
2
The Model
Atomic Splittable Congestion Games. In an atomic splittable congestion game, a set E of resources has to be shared between n ∈ N players. Each resource e ∈ E exhibits a load-dependent cost, defined by its cost function `e : R≥0 → R≥0 . Each player i ∈ [n] := {1, . . . , n} has a set Pi ⊆ 2E \ ∅ of pure strategies available. A fractional strategy of player i is any distribution of its weight wi ∈ R>0 among the pure strategies available to it, i.e., the player P i’s set of (fractional) strategies is Si := {~xi ∈ RPi | p∈Pi xip = wi }. A strategy profile is a vector ~x = (~xi )i∈[n] of all players’ strategies. Resource Cost Functions. Following standard terminology, we say a cost function ` is semi-convex if x · `(x) is convex. In this work, we always assume that cost functions are non-decreasing, continuously differentiable, and semi-convex (the latter two conditions are necessary for existence and characterization of Nash equilibria, see below). We say that a set of cost functions L is scaleinvariant if ` ∈ L implies that σ · `(τ · x) ∈ L for every σ, τ > 0. Load. Given aP strategy profile ~x and a resource e ∈ E, we define xie := p∈Pi : e∈p xip as the load player i puts P on resource e and xe := i∈[n] xie as the total load on e. We also use the abbreviating notation ~xe := (xie )i∈[n] . Cost and Equilibria. Given a strategy profile ~x, the P cost of player i is defined as ci (~x) := e∈E xie ·`e (xe ). We are interested in equilibria of the game, i.e., states where no player can reduce its (expected) cost by unilaterally deviating. To make this notion precise, we consider the following hierarchy of equilibrium concepts (see, e.g., [20] for more details and context). A (pure) Nash equilibrium — the most restricted concept — is a strategy profile ~x such that for every player i and every fractional strategy ~y i it holds that ci (~x) ≤ ci (~y i , ~x−i ), where ~x−i denotes the strategies chosen by the players other than i in ~x. It is known that (pure) Nash equilibria always exist in atomic splittable congestion games [12, 14]. In general, an equilibrium is a probability distribution P over the set of strategy profiles (we can regard a pure Nash equilibrium as a point mass). We say that P is a mixed Nash equilibrium if all players’ strategies are stochastically independent and for all players i and all fractional strategies ~y i of player i, we have (2.1)
E~x∼P [ci (~x)] ≤ E~x∼P [ci (~y i , ~x−i )] .
following chain of inclusions for the sets of equilibria in a game: Nash ⊆ mixed Nash ⊆ correlated ⊆ coarse correlated. Since cost functions are differentiable and semiconvex, a necessary and sufficient condition for a Nash equilibrium is that for every player i, its marginal costs for all used pure strategies are equal and at most that of every unused pure strategy. That is, for all players i ∈ [n], and all p, p0 ∈ Pi with xip > 0 it must hold that P P i xe ) ≤ e∈p0 `ie (~xe ), where `ie (~xe ) := `e (xe ) + e∈p `e (~ xie · `0e (xe ). This condition can alternatively be stated as a variational inequality: For every P player i ∈ [n] and for every strategy ~y i it holds that e∈E `ie (~xe )·(yei −xie ) ≥ 0. The overall measure for the quality P of a strategy profile ~x is its social cost SC(~x) := x). Clearly, i∈[n] ci (~ P SC(~x) = e∈E xe · `e (xe ). Price of Anarchy. The price of anarchy of an equilibrium concept in a game is the largest ratio between the (expected) social cost of an equilibrium and that of a minimum-cost strategy profile. 3
Local Smoothness
This section presents a “local” refinement of the smoothness framework in [17]. This refinement can lead to better upper bounds on the price of anarchy for games with convex strategy sets, and in particular permits optimal upper bounds for atomic splittable congestion games. Bounds proved using local smoothness extend automatically to the correlated equilibria of a game; but in contrast to standard smoothness bounds, they do not extend to the coarse correlated equilibria of a game. By a cost-minimization game, we mean a finite set of players, a strategy set Si for each player i, and a cost function ci for each player that maps outcomes (i.e., strategy profiles) to the nonnegative reals. Roughgarden [17] generalized several previous works [13, 6, 1] by defining a (λ, µ)-smooth cost-minimization game as one that satisfies n X (3.1) ci (~y i , ~x−i ) ≤ λ · SC(~y ) + µ · SC(~x) i=1
for every pair ~x, ~y of outcomes. Every coarse correlated equilibrium of a (λ, µ)-smooth game has expected cost at most λ/(1 − µ) times the cost of an optimal outcome [17]. For the rest of this section, we consider costminimization games for which each strategy set Si is a convex subset of some Euclidean space Rmi and each cost function ci is continuously differentiable with a bounded derivative.1 The rough intuition behind local smoothness
We say that P is a correlated equilibrium if for all players i and all functions δ : Si → Si it holds that E~x∼P [ci (~x)] ≤ E~x∼P [ci (δ(~xi ), ~x−i )]. Finally, P is a 1 The atomic splittable congestion games that we consider, coarse correlated equilibrium if (2.1) holds for all players which have continuously differentiable cost functions and compact i and all strategies ~y i . To summarize, we have the convex strategy sets, satisfy these assumptions.
convex, ~x is a well-defined strategy for every between 0 and 1. In the limit as goes to zero, E~x∼P [ 1 (ci (~x ) − ci (~x))] tends to E~x∼P [∇i ci (~x)T (~y i −~xi )], which is strictly negative by assumption.3 Thus, there is a sufficiently small > 0 such that E~x∼P [ci (~x )] < E~x∼P [ci (~x)], which contradicts the assumption that P is a correlated Definition 3.1 (Locally Smooth Games). A cost- equilibrium. t u minimization game is locally (λ, µ)-smooth with respect Example 3.3 (Local Smoothness Does Not to the outcome ~y if for every outcome ~x, Bound All Coarse Correlated Equilibria). n X Consider the cost-minimization game defined by N = T i i ci (~x) + ∇i ci (~x) (~y − ~x ) {1, 2}, S1 = S2 = [0, 1], and c1 (~x) = c2 (~x) = (x1 − x2 )2 + (3.2) i=1 ε, where ε > 0. Note that the game is an identical≤ λ · SC(~y ) + µ · SC(~x) . interest game and has continuously differentiable convex cost functions. The sole purpose of ε is to ensure that i i In Definition 3.1, ∇i ci := (∂ci /∂x1 , . . . , ∂ci /∂xmi ) the optimum has strictly positive social cost. Let P be i denotes the gradient of ci with respect to ~x . a randomized strategy profile that chooses (0, α) and We next prove that if a game is locally (λ, µ)-smooth (1, 1 − α) with equal probability, where α > 0. Elemenwith respect to an optimal outcome with µ < 1, then tary calculations verify that this is a coarse correlated the expected cost of every correlated equilibrium (and equilibrium with expected social cost 2α2 + 2ε when hence every pure and mixed Nash equilibrium) is at most α ≤ 1/4. Further calculations show that for every profile λ/(1 − µ) times that of an optimal outcome. ~x and every optimal profile ~y (i.e., y1 = y2 ) it holds that P2 x)(yi −xi ) = −2(x1 −x2 )2 = − SC(~x)+SC(~y ). Theorem 3.2 (Local Smoothness Bounds All i=1 ∇i ci (~ Correlated Equilibria). Let P be a correlated Consequently, the game is (1, 0)-smooth with respect to ~y . equilibrium of a cost-minimization game. If the game is The corresponding approximation factor of λ/(1 − µ) = 1 locally (λ, µ)-smooth with respect to the outcome ~y , then obviously does not apply to all coarse correlated equilibλ ria. E~x∼P [SC(~x)] ≤ 1−µ · SC(~y ). is to require the constraint (3.1) only for outcomes ~y that are “arbitrarily close to” ~x. Since dropping constraints increases the set of feasible values for λ and µ, this idea has the potential to yield improved upper bounds.2 Formally, we implement this idea as follows.
Proof. The key claim is that E~x∼P ∇i ci (~x)T (~y i − ~xi ) ≥ 0
4
A Locally Smooth Upper Bound
We now instantiate the local smoothness framework of Section 3 for atomic splittable congestion games. We 2 for every player i. Assuming the claim is true, we can first need a simple observation. Define κ(x, y) as y /4 if x ≥ y/2 and x(y − x) otherwise. complete the proof by using (3.2) and the linearity of expectation (twice) to derive Lemma 4.1. x, y ≥ 0. For all ~x, ~y ∈ P Let n ∈ N andP Rn≥0 with x = x and i i i yi = y it holds that P E~x∼P [SC(~x)] 2 y · x − x ≤ κ(x, y). i i i i n X (3.3) ≤ E~x∼P ci (~x) + ∇i ci (~x)T (~y i − ~xi ) Proof. Let xmax = maxi xi and note that X i=1 yi · xi − x2i ≤ y · xmax − x2max ≤ E~x∼P [λ · SC(~y ) + µ · SC(~x)] i
and then rearrange the terms. To prove for contradiction the key claim, suppose that E~x∼P ∇i ci (~x)T (~y i − ~xi ) < 0. For brevity, define ~x := ((1 − ) · ~xi + · ~y i , ~x−i ). Since strategy sets are 2 To see why standard smoothness arguments cannot prove optimal upper bounds on the POA of atomic splittable congestion games, note that the strategy sets in a splittable game contain those of its unsplittable counterpart. Thus, for a fixed set of cost functions, the requirement (3.1) is only more constraining in splittable games, and the best-provable upper bound can only be larger. But, as Table 1 shows, the worst-case POA in atomic splittable games is generally smaller than that in the corresponding class of unsplittable games.
=
2 y2 y − − xmax . 4 2
t u
Proposition 4.2. Let L be a class of allowable cost functions. If (4.1) y · `(x) + κ(x, y) · `0 (x) ≤ λ · y · `(y) + µ · x · `(x) 3 This can be formally justified using the dominated convergence theorem: Since cost functions are continuously differentiable with bounded derivatives, there is some M < ∞ so that Rfor all ~ x∈S we have | 1 (ci (~ x ) − ci (~ x))| < M . Hence, lim&0 1 (ci (~ x ) − R ci (~ x)) dP (~ x) = ∇i ci (~ x)T (~ yi − ~ xi ) dP (~ x).
for every ` ∈ L and x, y ≥ 0, then every atomic splittable 5 A Matching Lower Bound for All congestion game with cost functions in L is locally (λ, µ)Scale-Invariant Classes of Cost Functions smooth with respect to every outcome. We now show that for every scale-invariant set of cost functions L, the worst-case price of anarchy of atomic Proof. In an atomic splittable congestion game, we have splittable congestion games with cost functions in L is for all strategy profiles ~x, ~y , exactly γ(L). n We first need two important technical lemmas that X (4.2) ci (~x) + ∇i ci (~x)T (~y i − ~xi ) shows that γ(L) can be arbitrarily approximated by the i=1 intersection of two curves g`1 ,x1 ,y1 (µ) and g`2 ,x2 ,y2 (µ), X X where one is non-increasing and one is non-decreasing. = xie · `e (xe ) + yei · `ie (~xe ) − xie · `ie (~xe ) i∈[n] e∈E
5.1 Approximating γ(L) by Two Curves g`,x,y . X X 0 i i i 2 = ye · `e (xe ) + `e (xe ) · ye · xe − (xe ) Define ΓL : R0,`(y)>0
g`,x,y (µ) .
Figure 2 in Section 6 provides plots of the functions ΓL when L contains only linear functions and constants. ≤ [λ · ye · `e (ye ) + µ · xe · `e (xe )] It is easy to see that a set of cost functions L must e∈E be restricted in order for γ(L) < ∞. If that is the case, = λ · SC(~y ) + µ · SC(~x) . the following Lemma 5.1 shows that the infimum of ΓL is Inequality (4.3) is due to Lemma 4.1, and inequality always attained on a closed interval (hence, the infimum (4.4) follows from the hypothesis (4.1). t u is in fact also a minimum). Define (4.4)
X
h`,x,y := (y − x) · `(x) + κ(x, y) · `0 (x) . We now define the quantity γ(L) as, intuitively, the best upper bound on the POA that is provable using A simple calculation shows that the derivative of g `,x,y (µ) Proposition 4.2. Formally, we first define with respect to µ always has the same sign as h`,x,y . y · `(x) + κ(x, y) · `0 (x) − µ · x · `(x) Lemma 5.1. Let L be a set of cost functions such that g`,x,y (µ) := y · `(y) · (1 − µ) γ(L) < ∞. Then, Γ−1 L ({γ(L)}) is a non-empty closed interval [s, t] ⊆ [0, 1). If L contains an unbounded for every cost function ` and values x ≥ 0, y > 0 with function, then s > 0. `(y) > 0 (in short, we say an admissible triple `, x, y).
The constraint (4.1) is equivalent to g`,x,y (µ) ≤ Given a set of cost functions L, we then define γ(L) := inf
µ∈[0,1)
sup `∈L x≥0,y>0,`(y)>0
λ 1−µ .
g`,x,y (µ) ,
with the interpretation that sup ∅ = 1. Theorem 3.2, Proposition 4.2, and the definition of γ(L) imply the following generic upper bound.4 Corollary 4.3. For every set L of cost functions and every atomic splittable congestion game with cost functions in L, the price of anarchy of correlated equilibria is at most γ(L). 4 We can ignore triples `, x, y in which y = 0 or `(y) = 0 for the following reason. If y = 0 then inequality (4.1) is guaranteed by µ ≥ 0 alone. If `(y) = 0 and ξ denotes max `−1 ({0}), then if (4.1) holds for all y > ξ it also holds for y = ξ (by continuity), and hence for all y ∈ [0, ξ] (since the left-hand side of (4.1) is monotonic in y).
Proof. For convenience, fix Γ = ΓL . The proof proceeds in several steps. x→∞
1. For all ` ∈ L and any µ < 0, we have g`,x,1 (µ) −−−−→ x→∞ ∞. If ` is unbounded, then also g`,x,1 (0) −−−−→ ∞. 2. For all ` ∈ L and y > x > 0, we have h`,x,y > 0 and µ→1
µ→1
therefore g`,x,y (µ) −−−→ ∞. Hence, Γ(µ) −−−→ ∞.
3. Let (µi )i ⊂ [0, 1) be an arbitrary sequence with limn→∞ Γ(µn ) = γ(L). Since (µi )i is bounded, we may assume w.l.o.g. (Bolzano-Weierstraß) that (µi ) converges to some µ, where due to the previous limit argument we have µ ∈ [0, 1). By definition, we have Γ(µ) ≥ γ(L). Now all g`,x,y are continuous, so Γ(µ) < ∞ and for all ε > 0 there is a δ > 0 so that for all z ∈ (µ − δ, µ + δ) it holds that Γ(z) ≥ Γ(µ) − ε. Hence, Γ(µ) ≤ γ(L), i.e., even equality holds. Moreover, the set Γ−1 ({γ(L)}) is non-empty and compact.
4. Suppose that µ1 , µ2 ∈ [0, 1) with µ1 < µ2 and Γ(µ1 ) = Γ(µ2 ) = γ(L). Since all g`,x,y are monotone, it must hold that also Γ(z) = γ(L) for all z ∈ [µ1 , µ2 ]. Consequently, Γ−1 ({γ(L)}) is a closed interval [s, t] ⊆ [0, 1), and s > 0 if L contains unbounded functions. t u
5.2 The Construction. An exact lower bound requires the inequalities (3.3), (4.3), and (4.4) from the proofs of Theorem 3.2 and Proposition 4.2 to be asymptotically tight. Motivated by Lemma 5.2, the plan for our construction is as follows. We construct a family of instances that contain only two groups of resources, one with cost Lemma 5.2. Let L be a set of cost functions. For every function `1 and one with `2 . Each instance needs to γ b < γ(L), there are µ < 1 and admissible triples `1 , x1 , y1 possess a Nash equilibrium ~x so that the load on all resources of group i ∈ {1, 2} is xi , yet there must be some and `2 , x2 , y2 so that other strategy profile ~y where the load is only yi on each λ g`1 ,x1 ,y1 (µ) = g`2 ,x2 ,y2 (µ) > γ b and resource of group i. Suppose now that g`i ,xi ,yi (µ) = 1−µ . By definition of h , we then have ` ,x ,y sgn(h`1 ,x1 ,y1 ) = − sgn(h`2 ,x2 ,y2 ) . i i xi · `i (xi ) = λ · yi · `i (yi ) + µ · xi · `i (xi ) − h`i ,xi ,yi . Proof. Suppose first that γ(L) < ∞. Due to Lemma 5.1, ΓL attains its minimum γ(L) on some closed interval [s, t]. Now note that for every ξ ∈ (s, t) there are `, x, y We hence need sgn(h`1 ,x1 ,y1 ) = − sgn(h`2 ,x2 ,y2 ) and the so that g`,x,y (ξ) > γ b. Consequently, there are `1 , x1 , y1 number of resources in groups 1 and 2 to be chosen so that the sum, over all resources, of the h`i ,xi ,yi -terms so that one of the following holds. x) λ vanishes. Then SC(~ SC(~ y ) = 1−µ as needed. • g`1 ,x1 ,y1 (s) > γ b and h`1 ,x1 ,y1 ≥ 0 (i.e., g`1 ,x1 ,y1 is Tightness of inequality (4.3) alone gives another non-decreasing) constraint: Due to continuity, for every δ > 0 there are `2 , x2 , y2 so that g`2 ,x2 ,y2 (s−δ) > γ(L) and h`2 ,x2 ,y2 < 0. The Remark 5.3. Suppose both ~x and ~y in Lemma 4.1 are claim follows by choosing δ small enough so that sorted in descending order (without loss of generality). As n → ∞, the inequality is asymptotically tight when g`1 ,x1 ,y1 (s − δ) ≥ γ b. x1 = min{ y , x}, x2 = · · · = xn = o(1), and y1 = y, b and h`1 ,x1 ,y1 ≤ 0 (i.e., g`1 ,x1 ,y1 is y = · · · = y2 = 0. The left-hand side of the inequality • g`1 ,x1 ,y1 (t) > γ 2 n non-increasing) 1 is maximized when x2 = · · · = xn = x−x n−1 (see Cominetti The argument for the first case can be adapted et al. [7, Theorem 3.1]). correspondingly. In view of this observation, for each resource e of group i In the remainder of the proof, suppose now that γ(L) = there needs to be one player who in the Nash equilibrium ∞. Let puts load min{ y2i , xi } on resource e, and all other players only put an infinitesimal load on e. In the following, we µ∗ = sup{µ < 1 | ∃ admissible triple `, x, y with show that all of the above conditions can indeed be met g`,x,y ≥ γ(L) and h`,x,y < 0} . simultaneously. Then µ∗ ∈ [0, 1] because for any µ < 0 we could Theorem 5.4. Let λ ∈ R, µ < 1. Moreover, let `1 , `2 choose an arbitrary `, a large enough x, and a small be cost functions and x1 , x2 ≥ 0 and y1 , y2 > 0. If −µ·x·`(x) enough y so that g`,x,y (µ) ≥ (1−µ)·y·`(y) ≥ γ(L) and x2 < y2 or `02 (x2 ) = 0, define r := `2 (x2 ) + x2 · `02 (x2 ). 2 2 y2 0 h`,x,y = (y − x) · `(x) + y4 · `0 (x) < 0. Now, the definition Otherwise, define r := `2 (x2 ) + 2 · `2 (x2 ). Suppose that of µ∗ ensures the following: `1 (x1 ) = `2 (x2 ) = 1 , • There are `1 , x1 , y1 with h`1 ,x1 ,y1 ≥ 0 and λ g`1 ,x1 ,y1 (µ) = g`2 ,x2 ,y2 (µ) = 1−µ , and limµ→µ∗ g`1 ,x1 ,y1 (µ) > γ b. h`2 ,x2 ,y2 = −h`1 ,x1 ,y1 · r ≥ 0 . If µ∗ = 1, this follows by the definition of the func-
tions g`,x,y . Otherwise, it follows from ΓL (µ∗ ) = ∞ Then, there is an infinite family of atomic splittable but the fact that g`,x,y (µ∗ ) ≤ γ(L) for all `, x, y congestion games with cost functions in {σ1 `1 , `2 : σ1 ≥ with h`,x,y < 0. λ . 1} and with limiting price of anarchy at least 1−µ ∗ • For every µ < µ , there are `2 , x2 , y2 with Proof. We construct a family of instances determined h`2 ,x2 ,y2 < 0 and g`2 ,x2 ,y2 (µ) > γ b. by two scaling parameters n, p2 ∈ N. All other variables The claim follows by choosing µ < µ∗ large enough. t u (described in Table 2) are functions of n, p2 . For
convenience, we also define hi := h`i ,xi ,yi for i ∈ {1, 2}, and we use the notation 1 := 2 and 2 := 1. Figure 1 illustrates our construction. Before we start, note that the theorem’s assumptions imply that h1 ≤ 0 and hence x1 > y21 . Table 2: Symbols used in description of lower-bound construction Symbol
Meaning (load refers to load in Nash equilibrium)
n pi qi ti
number of players per group size of “optimal” strategies in group i size of “non-optimal” strategies in group i number of “non-optimal” strategies for each player in group i load each player from group i puts on its “optimal” strategy load each player from group i puts on its “nonoptimal” strategies load each player from group i puts on each “optimal” strategy of group i scaling factor for cost functions in group i
αi βi γi σi
Resources. There are two groups of resources, with group i ∈ {1, 2} consisting of n · pi resources that we denote by (i, 0), . . . , (i, n · pi − 1). A good intuition is to think of two circles. Resources in group i have the cost function σi · `i , where σ1 will be determined later and σ2 := 1. Players and Strategies. There will be two groups of atomic players, with group i ∈ {1, 2} consisting of n players denoted by (i, 0), . . . , (i, n − 1). Each player (i, j) has one “optimal” strategy Pi,j,0 available. If xi ≥ y2i and `0i (xi ) > 0, player (i, j) has also ti := pi ·(n−1) “nonqi optimal” strategies Pi,j,1 , . . . , Pi,j,ti available. Finally, players from group 2 can also choose all “optimal” strategies in group 1, i.e., P1,0,0 , . . . , P1,n−1,0 . Formally: Pi,j,0 := {(i, j · pi ), . . . , (i, (j + 1) · pi − 1)} , and
Pi,j,k := {(i, (j + 1) · pi + (k − 1) · qi ), . . . ,
(i, (j + 1) · pi + k · qi − 1)} for k ≥ 1 .
The weight of each player in group i will be wi := 1 αi + ti · βi + n · γi , where γ1 := −h and γ2 := 0 n (by construction, players from group 1 cannot use any resources in group 2). The Equilibrium. Consider the strategy profile where each atomic player (i, j) uses strategy Pi,j,0 with load αi and each of the strategies Pi,j,1 , . . . , Pi,j,ti −1 with load βi (if xi < y2i or `0i (xi ) = 0 then βi is necessarily 0). Moreover, each player in group 2 also uses each of the n “optimal” strategies in group 1 with load γ1 . In the
S 1,j + 1,1
S 1,j + 1,0
S 2,j,1
S 1,j,1
S 2,j,0
S 1,j,0
S 2,j,s2 S 2,j,s2 – 1
S 1,j,s1 S 1,j + 1,s1 n · p1 resources with cost σ1 · �1 (·)
n · p2 resources with cost �2 (·)
Figure 1: Illustration of construction with p1 = 3, q1 = 4 and p2 = 2, q2 = 3
following, we show that we can choose values for the variables from Table 2 so that the following six conditions are satisfied: 1. The load on each resource of group i is exactly xi . That is, αi + (n − 1) · βi + n · γi = xi , xi − αi − n · γi (5.1) βi = . n−1
i.e.,
2. Each player is faced with equal marginal costs for all its strategies. The first condition for players in group 2 is p1 · σ1 · (`1 (x1 ) + γ1 · `01 (x1 ))
(5.2)
= p2 · σ2 · (`2 (x2 ) + α2 · `02 (x2 )) .
Moreover, if xi ≥ both groups)
yi 2
and `0i (xi ) > 0, then also (for
pi · (`i (xi ) + αi · `0i (xi ))
(5.3)
= qi · (`i (xi ) + βi · `0i (xi )) .
3. If `0i (xi ) > 0, then for each resource in group i there is one player who puts load min{ y2i , xi } ± o(1) on it whereas all other players put load o(1) on it. If i = 2 and x2 ≤ y22 , there is nothing to show because α2 = x2 . Otherwise, plugging `0i (xi ) = 4(xi −yi +hi ) and (5.1) into (5.3) yields y2 i
−1
q · (x − n · γ ) i i i αi = + 4 · (xi − yi + hi ) (n − 1) · pi yi2 ·
· 1+
qi pi
qi (n − 1) · pi
−1 .
n,p2 →∞
Now αi −−−−−→ if
(5.4)
n,p2 →∞
yi 2
and βi −−−−−→ 0 are fulfilled
qi p2 →∞ 2xi − yi + 2hi −−−−→ , i.e., pi yi 2xi − yi + 2hi . qi := pi · yi
4. In the “optimal” strategy profile, where each player from group i only uses its “optimal” strategy in group i, the load on any resource in group i is yi + o(1). n,p2 →∞
We need wi −−−−−→ yi . First note that if x2 ≥ and `02 (x2 ) > 0, then
(5.5)
h2 h2 · y2 = y2 r y2 + 22 · `02 (x2 ) y2 2h2 = · . 2 2x2 − y2 + 2h2 y2 2
or `02 (x2 ) = 0, then
h2 n · γ1 = −h1 = r (y2 − x2 ) · (`2 (x2 ) + x2 · `02 (x2 )) = `2 (x2 ) + x2 · `02 (x2 ) = y2 − x2 .
Now, consider i ∈ {1, 2}. • If xi ≥ y2i and `0i (xi ) > 0, then we have due to (5.4) and (5.5) that wi = αi + ti · βi + n · γi pi = αi + · (xi − αi − n · γi ) + n · γi q i 2xi − yi − 2n · γi n,p2 →∞ yi −−−−−→ · 1+ + n · γi 2 2xi − yi + 2hi = yi .
• If `01 (x1 ) = 0, then w1 = α1 = x1 − n · γ1 = x1 + h1 = x1 + (y1 − x1 ) = y1 .
• If x2 ≤ y22 or `02 (x2 ) = 0, then w2 = α2 +n·γ1 = x2 + (y2 − x2 ) = y2 . λ ( 1−µ
5. The social cost is − o(1)) times as in the “optimal” strategy profile. The social cost contributed by any resource in group i in the equilibrium is σi · xi · `i (xi )
= σi · (λ · yi · `i (yi ) + µ · xi · `i (xi ) − hi ) ,
We want that
y2 2
n · γ1 = −h1 =
If, on the other hand, x2 ≤
λ where the equality is due to g`i ,xi ,yi (µ) = 1−µ and the definition of hi . P Let Φ(n, p2 ) := i=1,2 n · pi · σi · yi · `i (yi ) be a lower bound on the social cost in the “optimal” strategy profile, which we denote by SCopt (n, p2 ). Note that Φ(n, p2 ) ≥ SCopt (n, p2 ) because, in the “optimal” strategy profile, the actual load on each resource of group i is larger than yi . Moreover, let Λ(n, p2 ) be the sum of the (σP i · hi )-terms, over all resources. That is, Λ(n, p2 ) := i=1,2 n · pi · σi · hi .
λ SCeq (n, p2 ) n,p2 →∞ −−−−−→ . Φ(n, p2 ) 1−µ n,p2 →∞
n,p2 →∞
2) Since SCΦ(n,p opt (n,p ) −−−−−→ 1 (as wi −−−−−→ yi ), we 2 then also have
λ SCeq (n, p2 ) n,p2 →∞ −−−−−→ opt 1−µ SC (n, p2 ) as needed. W.l.o.g., we may assume that Λ(n, p2 ) ≤ 0 because we are done otherwise. We are also done if n,p2 →∞ Λ(n, p2 ) −−−−−→ 0, which motivates us to set p1 := dp2 · re. We have Φ(n, p2 ) ≥ n · p2 · (r · σ1 · y1 · `1 (y1 ) + σ2 · y2 · `2 (y2 )) and |Λ(n, p2 )| ≤ n · p2 · h2 · |σ2 − σ1 | . Consequently, n,p2 →∞
Λ(n,p2 ) Φ(n,p2 )
n,p2 →∞
−−−−−→ 0 provided that
σ1 −−−−−→ 1. (Recall that always σ2 = 1.) P Let SCeq (n, p2 ) := i=1,2 n · pi · σi · xi · `i (xi ) be the social cost in the equilibrium. By definition, SCeq (n, p2 ) = λ · Φ(n, p2 ) + µ · SCeq (n, p2 ) + Λ(n, p2 ) , i.e., SCeq (n, p2 ) λ Λ(n, p2 ) = + Φ(n, p2 ) 1 − µ Φ(n, p2 ) · (1 − µ) λ n,p2 →∞ −−−−−→ . 1−µ 6. All parameters are feasible, i.e., n, pi , qi , ti ∈ N,
αi , βi , γi ≥ 0,
σi > 0 .
We argue that all six conditions can indeed be satisfied simultaneously: Suppose the scaling parameters n, p2 ∈ N are given. This determines p1 , qi , αi , and βi (in this order) according to conditions 5, 3, and 1, respectively, and then σ1 according to (5.2) of condition 2. Now, conditions 1–3 imply also condition 4, as shown above. By the theorem’s assumptions and our definitions, n,p2 →∞ we have σi −−−−−→ 1 so that also condition 5 is met. Finally, we need to show condition 6: Since we may assume without loss of generality that pi · (n − 1) is a multiple of qi (because pi and qi do not depend on n), we have ti ∈ N. The only other non-obvious part of condition 6 is that β1 ≥ 0. This holds because n,p2 →∞ x1 − α1 −−−−−→ x1 − y21 and n · γ1 = −h1 = x1 − y 2 ·`0 (x )
y1 − 1 14 1 < x1 − y21 . This verifies the construction and completes the proof. t u We now have everything to formally state the main result of this section:
Corollary 5.5. Let L be a scale-invariant set of cost functions. Then, the worst-case price of anarchy in atomic splittable congestion games with cost functions in L is exactly γ(L). Proof. The upper bound is due to Corollary 4.3. For the lower bound, it suffices to show that for any two triples `1 , x1 , y1 and `2 , x2 , y2 “output” by Lemma 5.2 we can find triples `b1 , x b1 , yb1 and `b2 , x b2 , yb2 with that we can feed the lower-bound construction of Theorem 5.4 and that induce the same functions g`,x,y . We start with a simple observation. Let ` be a cost b := σ · `(τ · x), which function and σ, τ > 0. Define `(x) belongs to L by scale-invariance. Then, `b0 (x) = σ · (`(τ · = g`,τ ·x,τ ·y and x))0 = σ · τ · `0 (τ · x). Consequently, g`,x,y b τ · h`,x,y = σ · h . `,τ ·x,τ ·y b We can assume that `i (xi ) > 0 because otherwise g`i ,xi ,yi = 0. This cannot happen provided we use γ b > 1 in Lemma 5.2. Consequently, we can set 1 `b2 (x) := `2 (x · `2 (x), x b2 = x2 , yb2 = y2 and `b1 (x) := 2) x1 1 b1 = τ , yb1 = yτ1 . Here, we have the `1 (x1 ) · `1 (τ · x), x freedom to choose τ as needed. t u Remark 5.6. Since each player’s pure strategies in the lower-bound construction are disjoint, the construction can be transformed easily into a (directed) network congestion game — orient both circles, give each player its own source and sink vertices, and paths corresponding to its pure strategies in the construction above. Remark 5.7. A “lucky case” in the lower-bound construction occurs when h`i ,xi ,yi = 0, i.e., when g`i ,xi ,yi is a constant. Then, only one circle of resources is needed
and the scale-invariance hypothesis can be dropped. Appendix A.1 gives an example. When h`i ,xi ,yi = 0, a price of anarchy of γ(L) can even arise with singleton strategies (this, however, only with scale-invariance). Appendix A.2 details the construction. The lucky case occurs, e.g., when L contains only monomials (see Section 6). 6
Polynomials
This section determines the exact price of anarchy — that is, evaluates the parameter γ(L) — when the cost functions are polynomials of degree at most d ∈ N and with non-negative coefficients. We therefore introduce the following notation: For d ∈ N, let Pd denote this set of cost functions (with degree bound d). Moreover, we write X d to denote the monomial function x 7→ xd (where X 0 is the constant function 1), and we let Md := {X d , X d−1 , . . . , X 0 } be the set of all monomials of degree at most d. We define Ψd as the unique positive √ d−1 real x with xd + d·x4 = xd+1 , i.e., Ψd := 12 (1 + d + 1). ∗ To save work, we define g`,x,y , h∗`,x,y , Γ∗L , and γ ∗ (L) as in Section 4 and Section 5.1, respectively, but replace 2 κ(x, y) by y4 . We start with three lemmas to simplify γ ∗ (Pd ). In the end, it will turn out that γ(Pd ) = γ ∗ (Pd ). Lemma 6.1. Let µ ∈ (0, 1) and d ≥ 1. Define g : d−1 R≥0 → R, g(x) := xd + d·x4 − µ · xd+1 . Then, it holds that g has exactly one global maximum, at p d + d2 + d · µ · (d2 − 1) ξ= . 2µ · (d + 1)
Moreover, ξ is the only local extremum on R>0 .
Proof. We first show that x = 0 is not a global maximum. 1 1 > 14 = g(0). If d > 1, If d = 1, then g( 2µ ) = 12 + 4µ then g(Ψd ) = (1 − µ) · Ψd > 0 = g(0). Now since limx→∞ g(x) = −∞, g is continuous, and we know that g attains values strictly larger than g(0) somewhere on R>0 , it suffices to show that there is a unique local extremum on R>0 . For x > 0, we have as necessary first-order condition for a local extremum that d−1 (6.1) g 0 (x) = dxd−2 x + − µ(d + 1)xd = 0 . 4 Indeed, ξ is the unique real value for x that satisfies (6.1). t u Remark 6.2. Note that Lemma 6.1 gives a closed form ∗ for the function µ 7→ supx≥0 gX d ,x,1 (µ), which is the ∗ envelope function of all gX d ,x,y . In Figure 2, this curve is shown as a thick line for the case d = 1. In detail, the envelope function is here µ 7→
1+µ . 4 · µ · (1 − µ)
gX 1 ,2,1
γ({X 1 , X 0 }) = 1.5
2
gX 0 ,0,1
γ({X 1 }) ≈ 1.46
gX 1 ,1.3,1 gX 1 ,Ψ1 ,1 gX 1 ,1.1,1
1 gX 1 ,0.6,1 gX 1 ,0.5,1
0
0.5
1
Figure 2: Functions g`,x,y when ` is the identity or a constant function.
Lemma 6.3. Let d ∈ N. Then,
Consequently,
∗ γ ∗ (Pd ) = γ ∗ (Md ) = min max g`,x,1 (µ) . µ∈(0,1) `∈Md x≥0
∗
γ (Pd ) =
inf
(λ,µ)∈R×(0,1)
Proof. We can rewrite (6.2) (
λ ∀` ∈ Pd , x ≥ 0, y > 0 : γ(Pd ) = inf (λ,µ)∈R×(0,1) 1 − µ ) 2 0 y · `(x) + y ·`4 (x) − µ · x · `(x) λ≥ . y · `(y) Now observe that the condition (6.3)
∀` ∈ Pd , x ≥ 0, y > 0 : λ≥
y · `(x) +
y 2 ·`0 (x) 4
− µ · x · `(x) y · `(y)
is fulfilled if and only if the inequality holds for all monomials. Moreover, when ` is constant, the inequality boils down to λ ≥ 1 − µ · xy . Consequently, (6.3) is equivalent to (6.4)
y · xr +
y 2 ·r·xr−1 4 y r+1
− µ · xr+1
and λ ≥ 1 .
Now the first condition in (6.4) is equivalent to (6.5)
∀r ∈ [d], x ≥ 0 : λ ≥ xr +
∗ = min max g`,x,1 (µ) . µ∈(0,1) `∈Md x∈R≥0
Here the last equality can be seen as follows: Lemma 6.1 implies that γ(Pd ) < ∞ and that for each µ ∈ (0, 1) ∗ the supremum Γ∗Pd (µ) is attained by some g`,x,1 (µ). Moreover, Lemma 5.1 implies that the infimum γ ∗ (Pd ) is indeed attained by some ΓPd (µ) with µ ∈ (0, 1). t u Lemma 6.4. Let d ∈ N. Then: 1. γ ∗ ({X d }) = Ψd+1 . d 2. γ ∗ ({X 1 , X 0 }) = 32 . If d ≥ 2, then γ ∗ ({X d , X 0 }) = γ ∗ ({X d }) = Ψd+1 . d 3. If L is one of {X d } or {X d , X 0 }, then γ(L) = γ ∗ (L).
∀r ∈ [d], x ≥ 0, y > 0 : λ≥
= γ ∗ (Md )
λ ∀r ∈ [d], x ≥ 0 : 1−µ λ ≥ gX r ,x,1 (µ) and 1−µ λ 0 ≥ gX ,0,1 (µ) 1−µ
r · xr−1 − µ · xr+1 . 4
4. γ(Pd ) = γ({X d , X 0 }). Proof. For x > 0 define µx :=
d · (4x + d − 1) . (d + 1) · 4x2
Now, for any ξ it holds that first order condition (6.1) for ∗ gX By Lemma 6.1, d ,x,1 (µξ ). a global maximum on R≥0 . ∗ maxx∈R≥0 {gX d ,x,1 (µξ )}.
ξ fulfills the necessary Corollary 6.5. Suppose that L is one of the following local extrema of x 7→ sets of cost functions. we get that ξ is even 1. If L is the set of linear functions, then γ(L) = Ψ21 ≈ ∗ Hence, gX d ,ξ,1 (µξ ) = 1.457,
1. Fix ξ := Ψd . Note that Ψ2d = Ψd + µξ =
d 4
2. If L = P1 , then γ(L) =
and hence
3 2
> Ψ21 ,
3. If L = Pd and d ∈ N≥2 , then γ(L) = Ψd+1 . d
d · (4Ψd + d − 1) ∈ (0, 1) . (d + 1) · (4Ψd + d)
Acknowledgments. We thank Kshipra Bhawalkar, So far, we have shown that γ ∗ ({X d }) ≤ Martin Gairing, Uri Nadav, and Tobias Harks for d+1 ∗ gX . Since h∗X d ,ξ,1 = 0, it holds helpful discussions, and an anonymous reviewer for very d ,ξ,1 (µξ ) = Ψd d+1 ∗ ∗ that gX for insightful remarks. d ,ξ,1 is a constant, so Γ{X d } (µ) ≥ Ψd all µ ∈ (0, 1) and we have indeed γ ∗ ({X d }) = Ψd+1 . d 3 2
2. Consider first the case d = 1: Fix ξ := and note that µξ = 13 ∈ (0, 1). We have that 1 3 1 ∗ ∗ gX Consequently, also 0 ,0,1 ( 3 ) = 2 = gX d ,ξ,1 ( 3 ). 3 ∗ d 0 γ ({X , X }) = 2 . Otherwise, if d ≥ 2, choose again ξ := Ψd . It holds that √ d+1 1+ d+1 d+1 ∗ gX d ,ξ,1 (µξ ) = Ψd = 2 √ 2 2 · (d + 1) 1+ d+1 √ > · 2 d+1+ d+1 1 ∗ = = gX 0 ,0,1 (µξ ) . 1 − µξ
As for (1.), we hence have γ ∗ ({X d , X 0 }) = Ψd+1 . d y 2,
we have that κ(x, y) = x · (y − x) = 2 2 − − x ≤ y4 . Therefore, for every admissible ∗ triple `, x, y we have g`,x,y ≤ g`,x,y (pointwise), 1 x where equality holds if y ≥ 2 . Hence, if ξ ≥ 12 , then also gX d ,ξ,1 (µξ ) = maxx∈R≥0 {gX d ,x,1 (µξ )}.
3. For x < 2
y 4
y 2
4. The derivative of gX r ,ξ,1 (µ) with respect to r is r−1
r·ξ ∂ ξ r + 4 − µ · ξ r+1 ∂r 1−µ ξ r−1 = + ln(ξ) · gX r ,ξ,1 (µ) , 4(1 − µ)
which is positive if ξ > 1 and gX r ,ξ,1 (µ) ≥ 0. Consequently, if ξ > 1, then gX d ,ξ,1 (µξ ) = max {gX r ,x,1 (µξ )} . r∈[d] x∈R≥0
t u
Corollary 5.5, Lemma 6.3, and Lemma 6.4 immediately imply:
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Appendix A
Details to Remark 5.7 from Section 5 (Lower Bound)
A.1 Construction Example. As an example for our construction, consider the lucky case mentioned in Remark 5.7 where only a single group of resources is needed. The admissible triple `, x, y with `(z) = z 3 , x = 32 , y = 1 constitutes a lucky case. It is easy to verify
that y2 0 h`,x,y = (y − x) · `(x) + · ` (x) 4 3 = x2 · (y − x) · x + = 0. 4 λ . Let λ, µ such that g`,x,y (µ) = 1−µ The family of instances is as follows. There are n players and n resources, each with cost function `. The players’ “optimal” strategies have size p = 1, whereas their “non-optimal” strategies have size q = 2. Each player thus has t = n−1 2 “non-optimal” strategies. We consider the strategy profile where every player puts load 3 2 −1 n+5 α = [ 12 + n−1 ] · [1 + n−1 ] = 2·(n+1) on its “optimal” x−α 1 and β = n−1 = n+1 on all its “non-optimal” strategies. Then:
1. The load on each resource is exactly α + (n − 1) · β = x . 2. Each player is faced with equal marginal costs for all its strategies, because p · (`(x) + α · `0 (x)) = x2 · (x + 3 · α) 6n + 18 = x2 · 2 · (n + 1)
= x2 · 2 · (x + 3 · β)
= q · (`(x) + β · `0 (x)) .
3. For each resource, there is one player who puts load α = 12 ± o(1) on it whereas all other players put load β = o(1) on it. 4. In the “optimal” strategy profile, where each player only uses its “optimal” strategy, the load on any resource is 1 + o(1), because each player has weight α+t·β =α+
n−1 n→∞ −−−−→ 1 . 2 · (n + 1)
λ 5. The social cost is ( 1−µ − o(1)) times as in the “optimal” strategy profile. This holds because each resource contributes cost
x · `(x) = λ · y · `(y) + µ · x · `(x) , where the equality is due to g`,x,y (µ) = the definition of h`,x,y .
λ 1−µ
and
Together with the upper bound from Section 6, this construction alone immediately shows that the price of anarchy for atomic splittable congestion games with polynomial cost functions of degree at most 3 is exactly ( 32 )4 = 5.0625.
Player Weights:
Resource Level:
y 2
y · τ1
0
1
(x − y2 ) · τ l
y · τl
y · τ2
2
l–1
l
Figure 3: Illustration of construction with singleton strategies
y A.2 Construction with Singleton Strategies. In = `j (x · τ j ) + · τ j · `0j (x · τ j ) . 2 the lucky case mentioned in Remark 5.7, a price of anarchy of γ(L) can even arise with singleton strategies. By plugging in that ` (z) = 1 · `( z ) and `0 (z) = j j σj τj The lucky case occurs, e.g., when L contains only 1 0 z · ` ( ), this is equivalent to j j j σ ·τ τ monomials (see Section 6). i y y 1 h Theorem A.1. Let λ ∈ R, µ < 1. Moreover, let L `(x) + · τ · `0 (x) = · `(x) + · `0 (x) , 2 σ 2 be a scale-invariant set of cost functions, ` ∈ L, and x ≥ y > 0. Suppose that i.e., g`,x,y (µ) =
λ 1−µ
and h`,x,y = 0 .
σ=
`(x) + y2 · `0 (x) k→∞ y `0 (x) 2x − y − − − − → 1 + · = , y 0 `(x) + 2 · τ · ` (x) 2 `(x) y
Then, there is an infinite family of atomic splittable congestion games with singleton strategies, with cost where the last equality follows from h`,x,y = 0. Consefunctions in L, and with limiting price of anarchy at k→∞ λ quently, k · τ · σ1 −−−−→ 1, and the social cost contributed least 1−µ . k→∞ by all resources on level j ∈ [l]0 is k j · x · τ j · `(x) −−−→ σj − Proof. We define a family of singleton congestion games, x · `(x). represented by full k-ary trees of height l. To simplify Now consider the profile where each player uses only our presentation, assume that the root node and each the strategy farther away from the root. The social cost leaf node have self-loops. Then, each edge corresponds contributed by all resources on level j ∈ [l − 1] is y · `(y). to a player, and each node in the tree corresponds to The root resource on level 0 contributes y2 ·`( y2 ), and level a resource. The strategies of a players are its (at most y y y y l l `(x+ 2 ) k→∞ two) incident nodes. An illustration of the construction l contributes k ·(x+ 2 )·τ · σl −−−−→ (x+ 2 )·`(x+ 2 ), which is constant in l. is shown in Figure 3. Consequently, as both k, l → ∞, we have that the Let σ, τ > 0 be values to be determined later (dependent on k and l). The cost function for resources ratio of social cost in the Nash equilibrium divided by the x·`(x) t on level j is `j (z) := σ1j · `( τzj ). Note that the root social cost in the other profile goes to y·`(y) = g`,x,y (µ).u resource has cost function `0 = `. We say a player is in level j ∈ [n] if its edge is between resource levels j − 1 and j. The weight of each player in level j is y · τ j . The player who only has the root resource as strategy has weight y2 , and the players who only have a leaf resource as strategy have weight (x − y2 ) · τ l . We first show that we can choose σ and τ such that the profile in which each player splits its weight equally (i.e., each player on level j puts load y2 · τ j on both of its strategies) is a Nash equilibrium. Let τ := 2x−y y·k , so that the equilibrium load on each resource of level j ∈ [l]0 is y y j j+1 = x · τ j . We need that each player 2 ·τ +k· 2 ·τ faces equal marginal costs on each of its strategies, i.e., for players on all levels j ∈ [l] that y `j−1 (x · τ j−1 ) + · τ j · `0j−1 (x · τ j−1 ) 2
Introduction
Congestion games play a central role in the theory of worst-case approximation guarantees for game-theoretic equilibria. In the standard model [15], there is a ground set of resources, and each player selects a subset of them (e.g., a path in a network). Each resource has a univariate cost function that depends on the load induced by the ∗ Stanford University, Department of Computer Science. Supported in part by NSF CAREER Award CCF-0448664, an ONR Young Investigator Award, an ONR PECASE Award, an AFOSR MURI grant, and an Alfred P. Sloan Fellowship. Email: [email protected] † Stanford University, Department of Computer Science. Supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD). Email: fschopp@ uni-paderborn.de
Florian Schoppmann† players that use it, and each player strives to minimize the sum of the resources’ costs in its chosen strategy (given the strategies chosen by the other players). We study the splittable variant of congestion games, where each player has a weight wi and a list of available pure strategies (each a subset of resources), and each player chooses how to split fractionally its weight over its pure strategies. The splittable model is more appropriate than the traditional “unsplittable” model in some applications, such as multipath routing in networks. Indeed, in the computer networking literature, the splittable model was studied a decade prior to the unsplittable model (beginning with [12]). In this paper, we resolve the worst-case price of anarchy (POA) [11] — the ratio between the sum of players’ costs in a Nash equilibrium and in a minimumcost outcome — of splittable congestion games. Prior to this work, no tight bounds on the POA in splittable congestion games were known, even for the simplest non-trivial special case of affine cost functions. In contrast, tight bounds for essentially all classes of cost functions were proved for the nonatomic model (with a continuum of players) and the unsplittable model some years ago [1, 8, 17, 19]. We make two distinct contributions. On the upper-bound side, we define the framework of “local smoothness”, which provides a sufficient condition for a game to have a bounded POA. This framework refines the smoothness paradigm introduced in [17] for games with convex strategy sets, intuitively by requiring certain inequalities only for nearby pairs of outcomes, rather than for all pairs of outcomes as in [17]. Standard smoothness arguments provably cannot establish tight bounds on the POA in atomic splittable congestion games, whereas local smoothness arguments can (as we show). Further, we prove that every POA bound derived via local smoothness applies automatically, without any quantitative degradation, to every correlated equilibrium of the game (and hence also to every mixed Nash equilibrium). Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to all of the coarse correlated equilibria of a game (or equivalently, to all sequences of repeated play in which
all players experience vanishing per-step regret; see [4]). Extending POA bounds to more general equilibrium concepts is important because it weakens the rationality assumptions under which the bounds are valid. An upper bound that applies only to pure Nash equilibria presumes that players reach one. A bound that applies more generally to correlated equilibria does not require players to converge to anything: if a game is played repeatedly and each player has vanishing per-step “swap regret” (see, e.g., [3]), then the bound applies to their time-averaged cost. Our second contribution is a very general lower bound. For a set L of allowable resource cost functions, we denote by γ(L) the smallest upper bound on the POA that is provable via a local smoothness argument. We prove that for every set L that satisfies mild technical conditions, the worst-case POA in atomic splittable congestion games with cost functions in L is exactly γ(L). Thus, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. The technical challenge in proving our lower bound stems from its generality: we need to exhibit a worstcase splittable congestion game for a set L of cost functions without knowing anything about L! Our high-level approach is to exhibit an example for which all of the inequalities used in the upper bound proof are tight, in the spirit of “complementary slackness” arguments in linear programming. This goal translates to a labyrinth of restrictions on a candidate worst-case splittable congestion game — on the allowable cost functions, on the resource loads in equilibrium and optimal outcomes, and on the relative use of a resource by different players in an equilibrium. Nevertheless, we show that all of these conditions can be met simultaneously and thus there are splittable congestion games with POA arbitrarily close to our upper bound of γ(L). For concreteness, Table 1 illustrates our exact bounds for the special case of bounded-degree polynomials with non-negative coefficients. (The necessary calculations are not immediately obvious and are given in Section 6.) The worst-case price of anarchy in splittable congestion games is generally strictly larger than that in nonatomic congestion games and strictly less than that in unsplittable congestion games. 1.1 Related Work. See [18, §4.8] for general references on splittable congestion games; here, we focus only on the prior research most relevant to the present work. Splittable congestion games seem more difficult to reason about than other congestion game models. For example, it was only shown recently that Nash equilibria need not be unique in such games [2]. Splittable
Table 1: Price of anarchy with polynomial cost functions
Degree
Atomic splittable
Atomic unsplittable (weighted) [1]
1 2 3 4 5 6 7 8
1.500 2.549 5.063 11.09 26.32 66.88 180.3 512.0
2.618 9.909 47.82 277.0 1,858 14,099 118,926 1,101,126
1.333 1.626 1.896 2.151 2.394 2.630 2.858 3.081
Θ( logd d )d+1
Θ( logd d )
d
√ d+1 d+1 ) 2
( 1+
Nonatomic [19]
congestion games also exhibit counterintuitive behavior, like the fact that fusing two players into one — seemingly, increasing the amount of cooperation in the game — can increase the cost of a game’s Nash equilibrium [5]. Finally, two independent proofs claimed that the worstcase price of anarchy in splittable congestion games is never worse than that in nonatomic congestion games [9, 16]. Cominetti et al. [7] showed, however, that these proofs are valid only in symmetric games — where all players have the same weight and the same set of strategies — and adapted an example in [5] to refute the general claims. The first upper bounds on the POA in general splittable congestion games were given in [7]. These bounds are derived using a special case of our local smoothness framework in which one of our two parameters (λ) is fixed at 1. This restricted approach yields finite upper bounds on the worst-case POA only for cost functions that are polynomials with degree at most 3 and nonnegative coefficients. Later Harks [10] used what we are calling local smoothness arguments to derive significantly better upper bounds. The upper bounds in [10] are equivalent to ours (as in, e.g., Table 1), but are given in a more complicated form that renders intractable the problems of constructing matching lower bounds and giving exact closed-form expressions for polynomial cost functions. Also, prior to our work, there were no known upper bounds on the POA for any equilibrium concept more general than pure Nash equilibria. The only lower bounds known prior to the present work follow from the counterexamples in [7]. As an example comparison, for bounded-degree polynomial cost functions, our new lower bounds are exponentially larger (in the degree d) than those in [7].
2
The Model
Atomic Splittable Congestion Games. In an atomic splittable congestion game, a set E of resources has to be shared between n ∈ N players. Each resource e ∈ E exhibits a load-dependent cost, defined by its cost function `e : R≥0 → R≥0 . Each player i ∈ [n] := {1, . . . , n} has a set Pi ⊆ 2E \ ∅ of pure strategies available. A fractional strategy of player i is any distribution of its weight wi ∈ R>0 among the pure strategies available to it, i.e., the player P i’s set of (fractional) strategies is Si := {~xi ∈ RPi | p∈Pi xip = wi }. A strategy profile is a vector ~x = (~xi )i∈[n] of all players’ strategies. Resource Cost Functions. Following standard terminology, we say a cost function ` is semi-convex if x · `(x) is convex. In this work, we always assume that cost functions are non-decreasing, continuously differentiable, and semi-convex (the latter two conditions are necessary for existence and characterization of Nash equilibria, see below). We say that a set of cost functions L is scaleinvariant if ` ∈ L implies that σ · `(τ · x) ∈ L for every σ, τ > 0. Load. Given aP strategy profile ~x and a resource e ∈ E, we define xie := p∈Pi : e∈p xip as the load player i puts P on resource e and xe := i∈[n] xie as the total load on e. We also use the abbreviating notation ~xe := (xie )i∈[n] . Cost and Equilibria. Given a strategy profile ~x, the P cost of player i is defined as ci (~x) := e∈E xie ·`e (xe ). We are interested in equilibria of the game, i.e., states where no player can reduce its (expected) cost by unilaterally deviating. To make this notion precise, we consider the following hierarchy of equilibrium concepts (see, e.g., [20] for more details and context). A (pure) Nash equilibrium — the most restricted concept — is a strategy profile ~x such that for every player i and every fractional strategy ~y i it holds that ci (~x) ≤ ci (~y i , ~x−i ), where ~x−i denotes the strategies chosen by the players other than i in ~x. It is known that (pure) Nash equilibria always exist in atomic splittable congestion games [12, 14]. In general, an equilibrium is a probability distribution P over the set of strategy profiles (we can regard a pure Nash equilibrium as a point mass). We say that P is a mixed Nash equilibrium if all players’ strategies are stochastically independent and for all players i and all fractional strategies ~y i of player i, we have (2.1)
E~x∼P [ci (~x)] ≤ E~x∼P [ci (~y i , ~x−i )] .
following chain of inclusions for the sets of equilibria in a game: Nash ⊆ mixed Nash ⊆ correlated ⊆ coarse correlated. Since cost functions are differentiable and semiconvex, a necessary and sufficient condition for a Nash equilibrium is that for every player i, its marginal costs for all used pure strategies are equal and at most that of every unused pure strategy. That is, for all players i ∈ [n], and all p, p0 ∈ Pi with xip > 0 it must hold that P P i xe ) ≤ e∈p0 `ie (~xe ), where `ie (~xe ) := `e (xe ) + e∈p `e (~ xie · `0e (xe ). This condition can alternatively be stated as a variational inequality: For every P player i ∈ [n] and for every strategy ~y i it holds that e∈E `ie (~xe )·(yei −xie ) ≥ 0. The overall measure for the quality P of a strategy profile ~x is its social cost SC(~x) := x). Clearly, i∈[n] ci (~ P SC(~x) = e∈E xe · `e (xe ). Price of Anarchy. The price of anarchy of an equilibrium concept in a game is the largest ratio between the (expected) social cost of an equilibrium and that of a minimum-cost strategy profile. 3
Local Smoothness
This section presents a “local” refinement of the smoothness framework in [17]. This refinement can lead to better upper bounds on the price of anarchy for games with convex strategy sets, and in particular permits optimal upper bounds for atomic splittable congestion games. Bounds proved using local smoothness extend automatically to the correlated equilibria of a game; but in contrast to standard smoothness bounds, they do not extend to the coarse correlated equilibria of a game. By a cost-minimization game, we mean a finite set of players, a strategy set Si for each player i, and a cost function ci for each player that maps outcomes (i.e., strategy profiles) to the nonnegative reals. Roughgarden [17] generalized several previous works [13, 6, 1] by defining a (λ, µ)-smooth cost-minimization game as one that satisfies n X (3.1) ci (~y i , ~x−i ) ≤ λ · SC(~y ) + µ · SC(~x) i=1
for every pair ~x, ~y of outcomes. Every coarse correlated equilibrium of a (λ, µ)-smooth game has expected cost at most λ/(1 − µ) times the cost of an optimal outcome [17]. For the rest of this section, we consider costminimization games for which each strategy set Si is a convex subset of some Euclidean space Rmi and each cost function ci is continuously differentiable with a bounded derivative.1 The rough intuition behind local smoothness
We say that P is a correlated equilibrium if for all players i and all functions δ : Si → Si it holds that E~x∼P [ci (~x)] ≤ E~x∼P [ci (δ(~xi ), ~x−i )]. Finally, P is a 1 The atomic splittable congestion games that we consider, coarse correlated equilibrium if (2.1) holds for all players which have continuously differentiable cost functions and compact i and all strategies ~y i . To summarize, we have the convex strategy sets, satisfy these assumptions.
convex, ~x is a well-defined strategy for every between 0 and 1. In the limit as goes to zero, E~x∼P [ 1 (ci (~x ) − ci (~x))] tends to E~x∼P [∇i ci (~x)T (~y i −~xi )], which is strictly negative by assumption.3 Thus, there is a sufficiently small > 0 such that E~x∼P [ci (~x )] < E~x∼P [ci (~x)], which contradicts the assumption that P is a correlated Definition 3.1 (Locally Smooth Games). A cost- equilibrium. t u minimization game is locally (λ, µ)-smooth with respect Example 3.3 (Local Smoothness Does Not to the outcome ~y if for every outcome ~x, Bound All Coarse Correlated Equilibria). n X Consider the cost-minimization game defined by N = T i i ci (~x) + ∇i ci (~x) (~y − ~x ) {1, 2}, S1 = S2 = [0, 1], and c1 (~x) = c2 (~x) = (x1 − x2 )2 + (3.2) i=1 ε, where ε > 0. Note that the game is an identical≤ λ · SC(~y ) + µ · SC(~x) . interest game and has continuously differentiable convex cost functions. The sole purpose of ε is to ensure that i i In Definition 3.1, ∇i ci := (∂ci /∂x1 , . . . , ∂ci /∂xmi ) the optimum has strictly positive social cost. Let P be i denotes the gradient of ci with respect to ~x . a randomized strategy profile that chooses (0, α) and We next prove that if a game is locally (λ, µ)-smooth (1, 1 − α) with equal probability, where α > 0. Elemenwith respect to an optimal outcome with µ < 1, then tary calculations verify that this is a coarse correlated the expected cost of every correlated equilibrium (and equilibrium with expected social cost 2α2 + 2ε when hence every pure and mixed Nash equilibrium) is at most α ≤ 1/4. Further calculations show that for every profile λ/(1 − µ) times that of an optimal outcome. ~x and every optimal profile ~y (i.e., y1 = y2 ) it holds that P2 x)(yi −xi ) = −2(x1 −x2 )2 = − SC(~x)+SC(~y ). Theorem 3.2 (Local Smoothness Bounds All i=1 ∇i ci (~ Correlated Equilibria). Let P be a correlated Consequently, the game is (1, 0)-smooth with respect to ~y . equilibrium of a cost-minimization game. If the game is The corresponding approximation factor of λ/(1 − µ) = 1 locally (λ, µ)-smooth with respect to the outcome ~y , then obviously does not apply to all coarse correlated equilibλ ria. E~x∼P [SC(~x)] ≤ 1−µ · SC(~y ). is to require the constraint (3.1) only for outcomes ~y that are “arbitrarily close to” ~x. Since dropping constraints increases the set of feasible values for λ and µ, this idea has the potential to yield improved upper bounds.2 Formally, we implement this idea as follows.
Proof. The key claim is that E~x∼P ∇i ci (~x)T (~y i − ~xi ) ≥ 0
4
A Locally Smooth Upper Bound
We now instantiate the local smoothness framework of Section 3 for atomic splittable congestion games. We 2 for every player i. Assuming the claim is true, we can first need a simple observation. Define κ(x, y) as y /4 if x ≥ y/2 and x(y − x) otherwise. complete the proof by using (3.2) and the linearity of expectation (twice) to derive Lemma 4.1. x, y ≥ 0. For all ~x, ~y ∈ P Let n ∈ N andP Rn≥0 with x = x and i i i yi = y it holds that P E~x∼P [SC(~x)] 2 y · x − x ≤ κ(x, y). i i i i n X (3.3) ≤ E~x∼P ci (~x) + ∇i ci (~x)T (~y i − ~xi ) Proof. Let xmax = maxi xi and note that X i=1 yi · xi − x2i ≤ y · xmax − x2max ≤ E~x∼P [λ · SC(~y ) + µ · SC(~x)] i
and then rearrange the terms. To prove for contradiction the key claim, suppose that E~x∼P ∇i ci (~x)T (~y i − ~xi ) < 0. For brevity, define ~x := ((1 − ) · ~xi + · ~y i , ~x−i ). Since strategy sets are 2 To see why standard smoothness arguments cannot prove optimal upper bounds on the POA of atomic splittable congestion games, note that the strategy sets in a splittable game contain those of its unsplittable counterpart. Thus, for a fixed set of cost functions, the requirement (3.1) is only more constraining in splittable games, and the best-provable upper bound can only be larger. But, as Table 1 shows, the worst-case POA in atomic splittable games is generally smaller than that in the corresponding class of unsplittable games.
=
2 y2 y − − xmax . 4 2
t u
Proposition 4.2. Let L be a class of allowable cost functions. If (4.1) y · `(x) + κ(x, y) · `0 (x) ≤ λ · y · `(y) + µ · x · `(x) 3 This can be formally justified using the dominated convergence theorem: Since cost functions are continuously differentiable with bounded derivatives, there is some M < ∞ so that Rfor all ~ x∈S we have | 1 (ci (~ x ) − ci (~ x))| < M . Hence, lim&0 1 (ci (~ x ) − R ci (~ x)) dP (~ x) = ∇i ci (~ x)T (~ yi − ~ xi ) dP (~ x).
for every ` ∈ L and x, y ≥ 0, then every atomic splittable 5 A Matching Lower Bound for All congestion game with cost functions in L is locally (λ, µ)Scale-Invariant Classes of Cost Functions smooth with respect to every outcome. We now show that for every scale-invariant set of cost functions L, the worst-case price of anarchy of atomic Proof. In an atomic splittable congestion game, we have splittable congestion games with cost functions in L is for all strategy profiles ~x, ~y , exactly γ(L). n We first need two important technical lemmas that X (4.2) ci (~x) + ∇i ci (~x)T (~y i − ~xi ) shows that γ(L) can be arbitrarily approximated by the i=1 intersection of two curves g`1 ,x1 ,y1 (µ) and g`2 ,x2 ,y2 (µ), X X where one is non-increasing and one is non-decreasing. = xie · `e (xe ) + yei · `ie (~xe ) − xie · `ie (~xe ) i∈[n] e∈E
5.1 Approximating γ(L) by Two Curves g`,x,y . X X 0 i i i 2 = ye · `e (xe ) + `e (xe ) · ye · xe − (xe ) Define ΓL : R0,`(y)>0
g`,x,y (µ) .
Figure 2 in Section 6 provides plots of the functions ΓL when L contains only linear functions and constants. ≤ [λ · ye · `e (ye ) + µ · xe · `e (xe )] It is easy to see that a set of cost functions L must e∈E be restricted in order for γ(L) < ∞. If that is the case, = λ · SC(~y ) + µ · SC(~x) . the following Lemma 5.1 shows that the infimum of ΓL is Inequality (4.3) is due to Lemma 4.1, and inequality always attained on a closed interval (hence, the infimum (4.4) follows from the hypothesis (4.1). t u is in fact also a minimum). Define (4.4)
X
h`,x,y := (y − x) · `(x) + κ(x, y) · `0 (x) . We now define the quantity γ(L) as, intuitively, the best upper bound on the POA that is provable using A simple calculation shows that the derivative of g `,x,y (µ) Proposition 4.2. Formally, we first define with respect to µ always has the same sign as h`,x,y . y · `(x) + κ(x, y) · `0 (x) − µ · x · `(x) Lemma 5.1. Let L be a set of cost functions such that g`,x,y (µ) := y · `(y) · (1 − µ) γ(L) < ∞. Then, Γ−1 L ({γ(L)}) is a non-empty closed interval [s, t] ⊆ [0, 1). If L contains an unbounded for every cost function ` and values x ≥ 0, y > 0 with function, then s > 0. `(y) > 0 (in short, we say an admissible triple `, x, y).
The constraint (4.1) is equivalent to g`,x,y (µ) ≤ Given a set of cost functions L, we then define γ(L) := inf
µ∈[0,1)
sup `∈L x≥0,y>0,`(y)>0
λ 1−µ .
g`,x,y (µ) ,
with the interpretation that sup ∅ = 1. Theorem 3.2, Proposition 4.2, and the definition of γ(L) imply the following generic upper bound.4 Corollary 4.3. For every set L of cost functions and every atomic splittable congestion game with cost functions in L, the price of anarchy of correlated equilibria is at most γ(L). 4 We can ignore triples `, x, y in which y = 0 or `(y) = 0 for the following reason. If y = 0 then inequality (4.1) is guaranteed by µ ≥ 0 alone. If `(y) = 0 and ξ denotes max `−1 ({0}), then if (4.1) holds for all y > ξ it also holds for y = ξ (by continuity), and hence for all y ∈ [0, ξ] (since the left-hand side of (4.1) is monotonic in y).
Proof. For convenience, fix Γ = ΓL . The proof proceeds in several steps. x→∞
1. For all ` ∈ L and any µ < 0, we have g`,x,1 (µ) −−−−→ x→∞ ∞. If ` is unbounded, then also g`,x,1 (0) −−−−→ ∞. 2. For all ` ∈ L and y > x > 0, we have h`,x,y > 0 and µ→1
µ→1
therefore g`,x,y (µ) −−−→ ∞. Hence, Γ(µ) −−−→ ∞.
3. Let (µi )i ⊂ [0, 1) be an arbitrary sequence with limn→∞ Γ(µn ) = γ(L). Since (µi )i is bounded, we may assume w.l.o.g. (Bolzano-Weierstraß) that (µi ) converges to some µ, where due to the previous limit argument we have µ ∈ [0, 1). By definition, we have Γ(µ) ≥ γ(L). Now all g`,x,y are continuous, so Γ(µ) < ∞ and for all ε > 0 there is a δ > 0 so that for all z ∈ (µ − δ, µ + δ) it holds that Γ(z) ≥ Γ(µ) − ε. Hence, Γ(µ) ≤ γ(L), i.e., even equality holds. Moreover, the set Γ−1 ({γ(L)}) is non-empty and compact.
4. Suppose that µ1 , µ2 ∈ [0, 1) with µ1 < µ2 and Γ(µ1 ) = Γ(µ2 ) = γ(L). Since all g`,x,y are monotone, it must hold that also Γ(z) = γ(L) for all z ∈ [µ1 , µ2 ]. Consequently, Γ−1 ({γ(L)}) is a closed interval [s, t] ⊆ [0, 1), and s > 0 if L contains unbounded functions. t u
5.2 The Construction. An exact lower bound requires the inequalities (3.3), (4.3), and (4.4) from the proofs of Theorem 3.2 and Proposition 4.2 to be asymptotically tight. Motivated by Lemma 5.2, the plan for our construction is as follows. We construct a family of instances that contain only two groups of resources, one with cost Lemma 5.2. Let L be a set of cost functions. For every function `1 and one with `2 . Each instance needs to γ b < γ(L), there are µ < 1 and admissible triples `1 , x1 , y1 possess a Nash equilibrium ~x so that the load on all resources of group i ∈ {1, 2} is xi , yet there must be some and `2 , x2 , y2 so that other strategy profile ~y where the load is only yi on each λ g`1 ,x1 ,y1 (µ) = g`2 ,x2 ,y2 (µ) > γ b and resource of group i. Suppose now that g`i ,xi ,yi (µ) = 1−µ . By definition of h , we then have ` ,x ,y sgn(h`1 ,x1 ,y1 ) = − sgn(h`2 ,x2 ,y2 ) . i i xi · `i (xi ) = λ · yi · `i (yi ) + µ · xi · `i (xi ) − h`i ,xi ,yi . Proof. Suppose first that γ(L) < ∞. Due to Lemma 5.1, ΓL attains its minimum γ(L) on some closed interval [s, t]. Now note that for every ξ ∈ (s, t) there are `, x, y We hence need sgn(h`1 ,x1 ,y1 ) = − sgn(h`2 ,x2 ,y2 ) and the so that g`,x,y (ξ) > γ b. Consequently, there are `1 , x1 , y1 number of resources in groups 1 and 2 to be chosen so that the sum, over all resources, of the h`i ,xi ,yi -terms so that one of the following holds. x) λ vanishes. Then SC(~ SC(~ y ) = 1−µ as needed. • g`1 ,x1 ,y1 (s) > γ b and h`1 ,x1 ,y1 ≥ 0 (i.e., g`1 ,x1 ,y1 is Tightness of inequality (4.3) alone gives another non-decreasing) constraint: Due to continuity, for every δ > 0 there are `2 , x2 , y2 so that g`2 ,x2 ,y2 (s−δ) > γ(L) and h`2 ,x2 ,y2 < 0. The Remark 5.3. Suppose both ~x and ~y in Lemma 4.1 are claim follows by choosing δ small enough so that sorted in descending order (without loss of generality). As n → ∞, the inequality is asymptotically tight when g`1 ,x1 ,y1 (s − δ) ≥ γ b. x1 = min{ y , x}, x2 = · · · = xn = o(1), and y1 = y, b and h`1 ,x1 ,y1 ≤ 0 (i.e., g`1 ,x1 ,y1 is y = · · · = y2 = 0. The left-hand side of the inequality • g`1 ,x1 ,y1 (t) > γ 2 n non-increasing) 1 is maximized when x2 = · · · = xn = x−x n−1 (see Cominetti The argument for the first case can be adapted et al. [7, Theorem 3.1]). correspondingly. In view of this observation, for each resource e of group i In the remainder of the proof, suppose now that γ(L) = there needs to be one player who in the Nash equilibrium ∞. Let puts load min{ y2i , xi } on resource e, and all other players only put an infinitesimal load on e. In the following, we µ∗ = sup{µ < 1 | ∃ admissible triple `, x, y with show that all of the above conditions can indeed be met g`,x,y ≥ γ(L) and h`,x,y < 0} . simultaneously. Then µ∗ ∈ [0, 1] because for any µ < 0 we could Theorem 5.4. Let λ ∈ R, µ < 1. Moreover, let `1 , `2 choose an arbitrary `, a large enough x, and a small be cost functions and x1 , x2 ≥ 0 and y1 , y2 > 0. If −µ·x·`(x) enough y so that g`,x,y (µ) ≥ (1−µ)·y·`(y) ≥ γ(L) and x2 < y2 or `02 (x2 ) = 0, define r := `2 (x2 ) + x2 · `02 (x2 ). 2 2 y2 0 h`,x,y = (y − x) · `(x) + y4 · `0 (x) < 0. Now, the definition Otherwise, define r := `2 (x2 ) + 2 · `2 (x2 ). Suppose that of µ∗ ensures the following: `1 (x1 ) = `2 (x2 ) = 1 , • There are `1 , x1 , y1 with h`1 ,x1 ,y1 ≥ 0 and λ g`1 ,x1 ,y1 (µ) = g`2 ,x2 ,y2 (µ) = 1−µ , and limµ→µ∗ g`1 ,x1 ,y1 (µ) > γ b. h`2 ,x2 ,y2 = −h`1 ,x1 ,y1 · r ≥ 0 . If µ∗ = 1, this follows by the definition of the func-
tions g`,x,y . Otherwise, it follows from ΓL (µ∗ ) = ∞ Then, there is an infinite family of atomic splittable but the fact that g`,x,y (µ∗ ) ≤ γ(L) for all `, x, y congestion games with cost functions in {σ1 `1 , `2 : σ1 ≥ with h`,x,y < 0. λ . 1} and with limiting price of anarchy at least 1−µ ∗ • For every µ < µ , there are `2 , x2 , y2 with Proof. We construct a family of instances determined h`2 ,x2 ,y2 < 0 and g`2 ,x2 ,y2 (µ) > γ b. by two scaling parameters n, p2 ∈ N. All other variables The claim follows by choosing µ < µ∗ large enough. t u (described in Table 2) are functions of n, p2 . For
convenience, we also define hi := h`i ,xi ,yi for i ∈ {1, 2}, and we use the notation 1 := 2 and 2 := 1. Figure 1 illustrates our construction. Before we start, note that the theorem’s assumptions imply that h1 ≤ 0 and hence x1 > y21 . Table 2: Symbols used in description of lower-bound construction Symbol
Meaning (load refers to load in Nash equilibrium)
n pi qi ti
number of players per group size of “optimal” strategies in group i size of “non-optimal” strategies in group i number of “non-optimal” strategies for each player in group i load each player from group i puts on its “optimal” strategy load each player from group i puts on its “nonoptimal” strategies load each player from group i puts on each “optimal” strategy of group i scaling factor for cost functions in group i
αi βi γi σi
Resources. There are two groups of resources, with group i ∈ {1, 2} consisting of n · pi resources that we denote by (i, 0), . . . , (i, n · pi − 1). A good intuition is to think of two circles. Resources in group i have the cost function σi · `i , where σ1 will be determined later and σ2 := 1. Players and Strategies. There will be two groups of atomic players, with group i ∈ {1, 2} consisting of n players denoted by (i, 0), . . . , (i, n − 1). Each player (i, j) has one “optimal” strategy Pi,j,0 available. If xi ≥ y2i and `0i (xi ) > 0, player (i, j) has also ti := pi ·(n−1) “nonqi optimal” strategies Pi,j,1 , . . . , Pi,j,ti available. Finally, players from group 2 can also choose all “optimal” strategies in group 1, i.e., P1,0,0 , . . . , P1,n−1,0 . Formally: Pi,j,0 := {(i, j · pi ), . . . , (i, (j + 1) · pi − 1)} , and
Pi,j,k := {(i, (j + 1) · pi + (k − 1) · qi ), . . . ,
(i, (j + 1) · pi + k · qi − 1)} for k ≥ 1 .
The weight of each player in group i will be wi := 1 αi + ti · βi + n · γi , where γ1 := −h and γ2 := 0 n (by construction, players from group 1 cannot use any resources in group 2). The Equilibrium. Consider the strategy profile where each atomic player (i, j) uses strategy Pi,j,0 with load αi and each of the strategies Pi,j,1 , . . . , Pi,j,ti −1 with load βi (if xi < y2i or `0i (xi ) = 0 then βi is necessarily 0). Moreover, each player in group 2 also uses each of the n “optimal” strategies in group 1 with load γ1 . In the
S 1,j + 1,1
S 1,j + 1,0
S 2,j,1
S 1,j,1
S 2,j,0
S 1,j,0
S 2,j,s2 S 2,j,s2 – 1
S 1,j,s1 S 1,j + 1,s1 n · p1 resources with cost σ1 · �1 (·)
n · p2 resources with cost �2 (·)
Figure 1: Illustration of construction with p1 = 3, q1 = 4 and p2 = 2, q2 = 3
following, we show that we can choose values for the variables from Table 2 so that the following six conditions are satisfied: 1. The load on each resource of group i is exactly xi . That is, αi + (n − 1) · βi + n · γi = xi , xi − αi − n · γi (5.1) βi = . n−1
i.e.,
2. Each player is faced with equal marginal costs for all its strategies. The first condition for players in group 2 is p1 · σ1 · (`1 (x1 ) + γ1 · `01 (x1 ))
(5.2)
= p2 · σ2 · (`2 (x2 ) + α2 · `02 (x2 )) .
Moreover, if xi ≥ both groups)
yi 2
and `0i (xi ) > 0, then also (for
pi · (`i (xi ) + αi · `0i (xi ))
(5.3)
= qi · (`i (xi ) + βi · `0i (xi )) .
3. If `0i (xi ) > 0, then for each resource in group i there is one player who puts load min{ y2i , xi } ± o(1) on it whereas all other players put load o(1) on it. If i = 2 and x2 ≤ y22 , there is nothing to show because α2 = x2 . Otherwise, plugging `0i (xi ) = 4(xi −yi +hi ) and (5.1) into (5.3) yields y2 i
−1
q · (x − n · γ ) i i i αi = + 4 · (xi − yi + hi ) (n − 1) · pi yi2 ·
· 1+
qi pi
qi (n − 1) · pi
−1 .
n,p2 →∞
Now αi −−−−−→ if
(5.4)
n,p2 →∞
yi 2
and βi −−−−−→ 0 are fulfilled
qi p2 →∞ 2xi − yi + 2hi −−−−→ , i.e., pi yi 2xi − yi + 2hi . qi := pi · yi
4. In the “optimal” strategy profile, where each player from group i only uses its “optimal” strategy in group i, the load on any resource in group i is yi + o(1). n,p2 →∞
We need wi −−−−−→ yi . First note that if x2 ≥ and `02 (x2 ) > 0, then
(5.5)
h2 h2 · y2 = y2 r y2 + 22 · `02 (x2 ) y2 2h2 = · . 2 2x2 − y2 + 2h2 y2 2
or `02 (x2 ) = 0, then
h2 n · γ1 = −h1 = r (y2 − x2 ) · (`2 (x2 ) + x2 · `02 (x2 )) = `2 (x2 ) + x2 · `02 (x2 ) = y2 − x2 .
Now, consider i ∈ {1, 2}. • If xi ≥ y2i and `0i (xi ) > 0, then we have due to (5.4) and (5.5) that wi = αi + ti · βi + n · γi pi = αi + · (xi − αi − n · γi ) + n · γi q i 2xi − yi − 2n · γi n,p2 →∞ yi −−−−−→ · 1+ + n · γi 2 2xi − yi + 2hi = yi .
• If `01 (x1 ) = 0, then w1 = α1 = x1 − n · γ1 = x1 + h1 = x1 + (y1 − x1 ) = y1 .
• If x2 ≤ y22 or `02 (x2 ) = 0, then w2 = α2 +n·γ1 = x2 + (y2 − x2 ) = y2 . λ ( 1−µ
5. The social cost is − o(1)) times as in the “optimal” strategy profile. The social cost contributed by any resource in group i in the equilibrium is σi · xi · `i (xi )
= σi · (λ · yi · `i (yi ) + µ · xi · `i (xi ) − hi ) ,
We want that
y2 2
n · γ1 = −h1 =
If, on the other hand, x2 ≤
λ where the equality is due to g`i ,xi ,yi (µ) = 1−µ and the definition of hi . P Let Φ(n, p2 ) := i=1,2 n · pi · σi · yi · `i (yi ) be a lower bound on the social cost in the “optimal” strategy profile, which we denote by SCopt (n, p2 ). Note that Φ(n, p2 ) ≥ SCopt (n, p2 ) because, in the “optimal” strategy profile, the actual load on each resource of group i is larger than yi . Moreover, let Λ(n, p2 ) be the sum of the (σP i · hi )-terms, over all resources. That is, Λ(n, p2 ) := i=1,2 n · pi · σi · hi .
λ SCeq (n, p2 ) n,p2 →∞ −−−−−→ . Φ(n, p2 ) 1−µ n,p2 →∞
n,p2 →∞
2) Since SCΦ(n,p opt (n,p ) −−−−−→ 1 (as wi −−−−−→ yi ), we 2 then also have
λ SCeq (n, p2 ) n,p2 →∞ −−−−−→ opt 1−µ SC (n, p2 ) as needed. W.l.o.g., we may assume that Λ(n, p2 ) ≤ 0 because we are done otherwise. We are also done if n,p2 →∞ Λ(n, p2 ) −−−−−→ 0, which motivates us to set p1 := dp2 · re. We have Φ(n, p2 ) ≥ n · p2 · (r · σ1 · y1 · `1 (y1 ) + σ2 · y2 · `2 (y2 )) and |Λ(n, p2 )| ≤ n · p2 · h2 · |σ2 − σ1 | . Consequently, n,p2 →∞
Λ(n,p2 ) Φ(n,p2 )
n,p2 →∞
−−−−−→ 0 provided that
σ1 −−−−−→ 1. (Recall that always σ2 = 1.) P Let SCeq (n, p2 ) := i=1,2 n · pi · σi · xi · `i (xi ) be the social cost in the equilibrium. By definition, SCeq (n, p2 ) = λ · Φ(n, p2 ) + µ · SCeq (n, p2 ) + Λ(n, p2 ) , i.e., SCeq (n, p2 ) λ Λ(n, p2 ) = + Φ(n, p2 ) 1 − µ Φ(n, p2 ) · (1 − µ) λ n,p2 →∞ −−−−−→ . 1−µ 6. All parameters are feasible, i.e., n, pi , qi , ti ∈ N,
αi , βi , γi ≥ 0,
σi > 0 .
We argue that all six conditions can indeed be satisfied simultaneously: Suppose the scaling parameters n, p2 ∈ N are given. This determines p1 , qi , αi , and βi (in this order) according to conditions 5, 3, and 1, respectively, and then σ1 according to (5.2) of condition 2. Now, conditions 1–3 imply also condition 4, as shown above. By the theorem’s assumptions and our definitions, n,p2 →∞ we have σi −−−−−→ 1 so that also condition 5 is met. Finally, we need to show condition 6: Since we may assume without loss of generality that pi · (n − 1) is a multiple of qi (because pi and qi do not depend on n), we have ti ∈ N. The only other non-obvious part of condition 6 is that β1 ≥ 0. This holds because n,p2 →∞ x1 − α1 −−−−−→ x1 − y21 and n · γ1 = −h1 = x1 − y 2 ·`0 (x )
y1 − 1 14 1 < x1 − y21 . This verifies the construction and completes the proof. t u We now have everything to formally state the main result of this section:
Corollary 5.5. Let L be a scale-invariant set of cost functions. Then, the worst-case price of anarchy in atomic splittable congestion games with cost functions in L is exactly γ(L). Proof. The upper bound is due to Corollary 4.3. For the lower bound, it suffices to show that for any two triples `1 , x1 , y1 and `2 , x2 , y2 “output” by Lemma 5.2 we can find triples `b1 , x b1 , yb1 and `b2 , x b2 , yb2 with that we can feed the lower-bound construction of Theorem 5.4 and that induce the same functions g`,x,y . We start with a simple observation. Let ` be a cost b := σ · `(τ · x), which function and σ, τ > 0. Define `(x) belongs to L by scale-invariance. Then, `b0 (x) = σ · (`(τ · = g`,τ ·x,τ ·y and x))0 = σ · τ · `0 (τ · x). Consequently, g`,x,y b τ · h`,x,y = σ · h . `,τ ·x,τ ·y b We can assume that `i (xi ) > 0 because otherwise g`i ,xi ,yi = 0. This cannot happen provided we use γ b > 1 in Lemma 5.2. Consequently, we can set 1 `b2 (x) := `2 (x · `2 (x), x b2 = x2 , yb2 = y2 and `b1 (x) := 2) x1 1 b1 = τ , yb1 = yτ1 . Here, we have the `1 (x1 ) · `1 (τ · x), x freedom to choose τ as needed. t u Remark 5.6. Since each player’s pure strategies in the lower-bound construction are disjoint, the construction can be transformed easily into a (directed) network congestion game — orient both circles, give each player its own source and sink vertices, and paths corresponding to its pure strategies in the construction above. Remark 5.7. A “lucky case” in the lower-bound construction occurs when h`i ,xi ,yi = 0, i.e., when g`i ,xi ,yi is a constant. Then, only one circle of resources is needed
and the scale-invariance hypothesis can be dropped. Appendix A.1 gives an example. When h`i ,xi ,yi = 0, a price of anarchy of γ(L) can even arise with singleton strategies (this, however, only with scale-invariance). Appendix A.2 details the construction. The lucky case occurs, e.g., when L contains only monomials (see Section 6). 6
Polynomials
This section determines the exact price of anarchy — that is, evaluates the parameter γ(L) — when the cost functions are polynomials of degree at most d ∈ N and with non-negative coefficients. We therefore introduce the following notation: For d ∈ N, let Pd denote this set of cost functions (with degree bound d). Moreover, we write X d to denote the monomial function x 7→ xd (where X 0 is the constant function 1), and we let Md := {X d , X d−1 , . . . , X 0 } be the set of all monomials of degree at most d. We define Ψd as the unique positive √ d−1 real x with xd + d·x4 = xd+1 , i.e., Ψd := 12 (1 + d + 1). ∗ To save work, we define g`,x,y , h∗`,x,y , Γ∗L , and γ ∗ (L) as in Section 4 and Section 5.1, respectively, but replace 2 κ(x, y) by y4 . We start with three lemmas to simplify γ ∗ (Pd ). In the end, it will turn out that γ(Pd ) = γ ∗ (Pd ). Lemma 6.1. Let µ ∈ (0, 1) and d ≥ 1. Define g : d−1 R≥0 → R, g(x) := xd + d·x4 − µ · xd+1 . Then, it holds that g has exactly one global maximum, at p d + d2 + d · µ · (d2 − 1) ξ= . 2µ · (d + 1)
Moreover, ξ is the only local extremum on R>0 .
Proof. We first show that x = 0 is not a global maximum. 1 1 > 14 = g(0). If d > 1, If d = 1, then g( 2µ ) = 12 + 4µ then g(Ψd ) = (1 − µ) · Ψd > 0 = g(0). Now since limx→∞ g(x) = −∞, g is continuous, and we know that g attains values strictly larger than g(0) somewhere on R>0 , it suffices to show that there is a unique local extremum on R>0 . For x > 0, we have as necessary first-order condition for a local extremum that d−1 (6.1) g 0 (x) = dxd−2 x + − µ(d + 1)xd = 0 . 4 Indeed, ξ is the unique real value for x that satisfies (6.1). t u Remark 6.2. Note that Lemma 6.1 gives a closed form ∗ for the function µ 7→ supx≥0 gX d ,x,1 (µ), which is the ∗ envelope function of all gX d ,x,y . In Figure 2, this curve is shown as a thick line for the case d = 1. In detail, the envelope function is here µ 7→
1+µ . 4 · µ · (1 − µ)
gX 1 ,2,1
γ({X 1 , X 0 }) = 1.5
2
gX 0 ,0,1
γ({X 1 }) ≈ 1.46
gX 1 ,1.3,1 gX 1 ,Ψ1 ,1 gX 1 ,1.1,1
1 gX 1 ,0.6,1 gX 1 ,0.5,1
0
0.5
1
Figure 2: Functions g`,x,y when ` is the identity or a constant function.
Lemma 6.3. Let d ∈ N. Then,
Consequently,
∗ γ ∗ (Pd ) = γ ∗ (Md ) = min max g`,x,1 (µ) . µ∈(0,1) `∈Md x≥0
∗
γ (Pd ) =
inf
(λ,µ)∈R×(0,1)
Proof. We can rewrite (6.2) (
λ ∀` ∈ Pd , x ≥ 0, y > 0 : γ(Pd ) = inf (λ,µ)∈R×(0,1) 1 − µ ) 2 0 y · `(x) + y ·`4 (x) − µ · x · `(x) λ≥ . y · `(y) Now observe that the condition (6.3)
∀` ∈ Pd , x ≥ 0, y > 0 : λ≥
y · `(x) +
y 2 ·`0 (x) 4
− µ · x · `(x) y · `(y)
is fulfilled if and only if the inequality holds for all monomials. Moreover, when ` is constant, the inequality boils down to λ ≥ 1 − µ · xy . Consequently, (6.3) is equivalent to (6.4)
y · xr +
y 2 ·r·xr−1 4 y r+1
− µ · xr+1
and λ ≥ 1 .
Now the first condition in (6.4) is equivalent to (6.5)
∀r ∈ [d], x ≥ 0 : λ ≥ xr +
∗ = min max g`,x,1 (µ) . µ∈(0,1) `∈Md x∈R≥0
Here the last equality can be seen as follows: Lemma 6.1 implies that γ(Pd ) < ∞ and that for each µ ∈ (0, 1) ∗ the supremum Γ∗Pd (µ) is attained by some g`,x,1 (µ). Moreover, Lemma 5.1 implies that the infimum γ ∗ (Pd ) is indeed attained by some ΓPd (µ) with µ ∈ (0, 1). t u Lemma 6.4. Let d ∈ N. Then: 1. γ ∗ ({X d }) = Ψd+1 . d 2. γ ∗ ({X 1 , X 0 }) = 32 . If d ≥ 2, then γ ∗ ({X d , X 0 }) = γ ∗ ({X d }) = Ψd+1 . d 3. If L is one of {X d } or {X d , X 0 }, then γ(L) = γ ∗ (L).
∀r ∈ [d], x ≥ 0, y > 0 : λ≥
= γ ∗ (Md )
λ ∀r ∈ [d], x ≥ 0 : 1−µ λ ≥ gX r ,x,1 (µ) and 1−µ λ 0 ≥ gX ,0,1 (µ) 1−µ
r · xr−1 − µ · xr+1 . 4
4. γ(Pd ) = γ({X d , X 0 }). Proof. For x > 0 define µx :=
d · (4x + d − 1) . (d + 1) · 4x2
Now, for any ξ it holds that first order condition (6.1) for ∗ gX By Lemma 6.1, d ,x,1 (µξ ). a global maximum on R≥0 . ∗ maxx∈R≥0 {gX d ,x,1 (µξ )}.
ξ fulfills the necessary Corollary 6.5. Suppose that L is one of the following local extrema of x 7→ sets of cost functions. we get that ξ is even 1. If L is the set of linear functions, then γ(L) = Ψ21 ≈ ∗ Hence, gX d ,ξ,1 (µξ ) = 1.457,
1. Fix ξ := Ψd . Note that Ψ2d = Ψd + µξ =
d 4
2. If L = P1 , then γ(L) =
and hence
3 2
> Ψ21 ,
3. If L = Pd and d ∈ N≥2 , then γ(L) = Ψd+1 . d
d · (4Ψd + d − 1) ∈ (0, 1) . (d + 1) · (4Ψd + d)
Acknowledgments. We thank Kshipra Bhawalkar, So far, we have shown that γ ∗ ({X d }) ≤ Martin Gairing, Uri Nadav, and Tobias Harks for d+1 ∗ gX . Since h∗X d ,ξ,1 = 0, it holds helpful discussions, and an anonymous reviewer for very d ,ξ,1 (µξ ) = Ψd d+1 ∗ ∗ that gX for insightful remarks. d ,ξ,1 is a constant, so Γ{X d } (µ) ≥ Ψd all µ ∈ (0, 1) and we have indeed γ ∗ ({X d }) = Ψd+1 . d 3 2
2. Consider first the case d = 1: Fix ξ := and note that µξ = 13 ∈ (0, 1). We have that 1 3 1 ∗ ∗ gX Consequently, also 0 ,0,1 ( 3 ) = 2 = gX d ,ξ,1 ( 3 ). 3 ∗ d 0 γ ({X , X }) = 2 . Otherwise, if d ≥ 2, choose again ξ := Ψd . It holds that √ d+1 1+ d+1 d+1 ∗ gX d ,ξ,1 (µξ ) = Ψd = 2 √ 2 2 · (d + 1) 1+ d+1 √ > · 2 d+1+ d+1 1 ∗ = = gX 0 ,0,1 (µξ ) . 1 − µξ
As for (1.), we hence have γ ∗ ({X d , X 0 }) = Ψd+1 . d y 2,
we have that κ(x, y) = x · (y − x) = 2 2 − − x ≤ y4 . Therefore, for every admissible ∗ triple `, x, y we have g`,x,y ≤ g`,x,y (pointwise), 1 x where equality holds if y ≥ 2 . Hence, if ξ ≥ 12 , then also gX d ,ξ,1 (µξ ) = maxx∈R≥0 {gX d ,x,1 (µξ )}.
3. For x < 2
y 4
y 2
4. The derivative of gX r ,ξ,1 (µ) with respect to r is r−1
r·ξ ∂ ξ r + 4 − µ · ξ r+1 ∂r 1−µ ξ r−1 = + ln(ξ) · gX r ,ξ,1 (µ) , 4(1 − µ)
which is positive if ξ > 1 and gX r ,ξ,1 (µ) ≥ 0. Consequently, if ξ > 1, then gX d ,ξ,1 (µξ ) = max {gX r ,x,1 (µξ )} . r∈[d] x∈R≥0
t u
Corollary 5.5, Lemma 6.3, and Lemma 6.4 immediately imply:
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Appendix A
Details to Remark 5.7 from Section 5 (Lower Bound)
A.1 Construction Example. As an example for our construction, consider the lucky case mentioned in Remark 5.7 where only a single group of resources is needed. The admissible triple `, x, y with `(z) = z 3 , x = 32 , y = 1 constitutes a lucky case. It is easy to verify
that y2 0 h`,x,y = (y − x) · `(x) + · ` (x) 4 3 = x2 · (y − x) · x + = 0. 4 λ . Let λ, µ such that g`,x,y (µ) = 1−µ The family of instances is as follows. There are n players and n resources, each with cost function `. The players’ “optimal” strategies have size p = 1, whereas their “non-optimal” strategies have size q = 2. Each player thus has t = n−1 2 “non-optimal” strategies. We consider the strategy profile where every player puts load 3 2 −1 n+5 α = [ 12 + n−1 ] · [1 + n−1 ] = 2·(n+1) on its “optimal” x−α 1 and β = n−1 = n+1 on all its “non-optimal” strategies. Then:
1. The load on each resource is exactly α + (n − 1) · β = x . 2. Each player is faced with equal marginal costs for all its strategies, because p · (`(x) + α · `0 (x)) = x2 · (x + 3 · α) 6n + 18 = x2 · 2 · (n + 1)
= x2 · 2 · (x + 3 · β)
= q · (`(x) + β · `0 (x)) .
3. For each resource, there is one player who puts load α = 12 ± o(1) on it whereas all other players put load β = o(1) on it. 4. In the “optimal” strategy profile, where each player only uses its “optimal” strategy, the load on any resource is 1 + o(1), because each player has weight α+t·β =α+
n−1 n→∞ −−−−→ 1 . 2 · (n + 1)
λ 5. The social cost is ( 1−µ − o(1)) times as in the “optimal” strategy profile. This holds because each resource contributes cost
x · `(x) = λ · y · `(y) + µ · x · `(x) , where the equality is due to g`,x,y (µ) = the definition of h`,x,y .
λ 1−µ
and
Together with the upper bound from Section 6, this construction alone immediately shows that the price of anarchy for atomic splittable congestion games with polynomial cost functions of degree at most 3 is exactly ( 32 )4 = 5.0625.
Player Weights:
Resource Level:
y 2
y · τ1
0
1
(x − y2 ) · τ l
y · τl
y · τ2
2
l–1
l
Figure 3: Illustration of construction with singleton strategies
y A.2 Construction with Singleton Strategies. In = `j (x · τ j ) + · τ j · `0j (x · τ j ) . 2 the lucky case mentioned in Remark 5.7, a price of anarchy of γ(L) can even arise with singleton strategies. By plugging in that ` (z) = 1 · `( z ) and `0 (z) = j j σj τj The lucky case occurs, e.g., when L contains only 1 0 z · ` ( ), this is equivalent to j j j σ ·τ τ monomials (see Section 6). i y y 1 h Theorem A.1. Let λ ∈ R, µ < 1. Moreover, let L `(x) + · τ · `0 (x) = · `(x) + · `0 (x) , 2 σ 2 be a scale-invariant set of cost functions, ` ∈ L, and x ≥ y > 0. Suppose that i.e., g`,x,y (µ) =
λ 1−µ
and h`,x,y = 0 .
σ=
`(x) + y2 · `0 (x) k→∞ y `0 (x) 2x − y − − − − → 1 + · = , y 0 `(x) + 2 · τ · ` (x) 2 `(x) y
Then, there is an infinite family of atomic splittable congestion games with singleton strategies, with cost where the last equality follows from h`,x,y = 0. Consefunctions in L, and with limiting price of anarchy at k→∞ λ quently, k · τ · σ1 −−−−→ 1, and the social cost contributed least 1−µ . k→∞ by all resources on level j ∈ [l]0 is k j · x · τ j · `(x) −−−→ σj − Proof. We define a family of singleton congestion games, x · `(x). represented by full k-ary trees of height l. To simplify Now consider the profile where each player uses only our presentation, assume that the root node and each the strategy farther away from the root. The social cost leaf node have self-loops. Then, each edge corresponds contributed by all resources on level j ∈ [l − 1] is y · `(y). to a player, and each node in the tree corresponds to The root resource on level 0 contributes y2 ·`( y2 ), and level a resource. The strategies of a players are its (at most y y y y l l `(x+ 2 ) k→∞ two) incident nodes. An illustration of the construction l contributes k ·(x+ 2 )·τ · σl −−−−→ (x+ 2 )·`(x+ 2 ), which is constant in l. is shown in Figure 3. Consequently, as both k, l → ∞, we have that the Let σ, τ > 0 be values to be determined later (dependent on k and l). The cost function for resources ratio of social cost in the Nash equilibrium divided by the x·`(x) t on level j is `j (z) := σ1j · `( τzj ). Note that the root social cost in the other profile goes to y·`(y) = g`,x,y (µ).u resource has cost function `0 = `. We say a player is in level j ∈ [n] if its edge is between resource levels j − 1 and j. The weight of each player in level j is y · τ j . The player who only has the root resource as strategy has weight y2 , and the players who only have a leaf resource as strategy have weight (x − y2 ) · τ l . We first show that we can choose σ and τ such that the profile in which each player splits its weight equally (i.e., each player on level j puts load y2 · τ j on both of its strategies) is a Nash equilibrium. Let τ := 2x−y y·k , so that the equilibrium load on each resource of level j ∈ [l]0 is y y j j+1 = x · τ j . We need that each player 2 ·τ +k· 2 ·τ faces equal marginal costs on each of its strategies, i.e., for players on all levels j ∈ [l] that y `j−1 (x · τ j−1 ) + · τ j · `0j−1 (x · τ j−1 ) 2
