Page 1 of 40 Asymptotics of Efficiency Loss in Competitive Market Mechanisms Jia Yuan Yu and Shie Mannor Department of Electrical and Computer Engineering McGill University Montréal, Québec, Canada jia.yu@mail.mcgill.ca and shie@ece.mcgill.ca 24th November 2005
Page 2 of 40 1. Introduction How to use network resources efficiently? Need to capture heterogeneous demand (e.g., sending emails, downloading large files) and prioritize the service. centralized scheme of allocating re- Approach 1: sources. Problem: communication complexity overhead. How to determine true priority (gaming the system)? Approach 2: market mechanism (bidding followed by proportional allocation). The resulting game has an unique Nash equilibrium [Kel97], but is the equilibrium efficient? This is the question we try to answer. We will look for asymptotic results for a large populations of random users.
Page 3 of 40 2. 1. Definitions and assumptions. 2. Modeling large and heterogeneous population of users. 3. Social optimum outcome. 4. Resource allocation mechanism and Nash equilibrium outcome. 5. Known results.
Page 4 of 40 2.1. Utility functions A set of users 1,..., n of a given communication link. Link capacity c (infinitely divisible). User i is characterized by the utility function u i. Assumption 2.1. The utility functions u i : i. are concave and continuously differentiable, ii. are strictly increasing, with u i (0) = 0. Remark 1. The continuous derivative assumption makes analysis easier. The strictly increasing condition ensures that all the resource c is allocated.
Examples Page 5 of 40 Linear (e.g., downloading a large file) Saturate at some point (e.g., voice conversation) High marginal utility at 0 (e.g., emergency calls)
Page 6 of 40 2.2. Modeling a heterogeneous population of users As n increases, more users enter while other users stay. Their utility functions are drawn i.i.d. from a set of utility functions Ω. We use capital U i denote random utility functions. Let v be the supremum among all u i(0), i.e., v = sup u Ω {u (0)}. Assumption 2.2. There is no single user with significantly greater market influence than the rest.
Page 7 of 40 Worded differently: Assumption 2.3. The probabilities satisfy 1. if v <, then for every ɛ > 0, there exists a δ > 0 such that 2. if v =, then Pr ( U i(0) [v ɛ, v] ) > δ, Pr(U i(0) = ) > δ.
Page 8 of 40 Examples Modulation satisfying Assumption 2.3: U i (x) = S i x, where S i has bounded support. U i (x) = S i log(1 + x). U i (x) = S i x, (Si need not have bounded support since all slopes at 0 are infinite). Remark 2. Allowed: S i taking finite number of values.
Page 9 of 40 2.3. Social optimum An allocation x 1,..., x n achieves social optimality if it solves: SYSTEM max x 1,...,x n subject to n u i (x i ) i=1 n x i c, i=1 (aggregate utility) x i 0, i = 1,..., n. (1) A central agent with knowledge of every user s utility function can implement this.
Picture Page 10 of 40
Page 11 of 40 2.4. Market mechanism for resource allocation 1. User i submits a bid b i, 2. the price for the commodity is set to n i=1 λ = b i, c 3. each user receives an amount x i = b i /λ of the commodity. Remark 3. Kelly [Kel97] introduced this mechanism in the context of pricing of network resources, but it can also be attributed to earlier work by Shubik [Shu73] and Shapley [Sha76] in economics.
Page 12 of 40 2.5. Nash equilibrium The objective of user i is to maximize his surplus, i.e., ( ) max b i u i subject to b i 0. b i b i + j i b j c b i Assumption 2.4. User i knows the sum of everyone else s bids j i b j. Assumption 2.5. Every user is price-anticipating. Remark 4. A price-taking assumption makes sense only when there are so many users that each user has negligible influence on the market outcome. This alternative leads to a competitive equilibrium.
Page 13 of 40 2.6. Known results Existence of unique Nash equilibrium (Hajek and Gopalakrishnan [HG04]). The loss of efficiency is at most 25% (Johari and Tsitsiklis [JT04]). Definition 2.1. Let x i and y i denote socially optimal and Nash equilibrium allocations. The loss of efficiency is n i=1 loe 1 u i(y i ) n u i=1 i(x i ).
Page 14 of 40 3. loe 0 for single link and inelastic supply (constant c). Convergence rate. loe 0 for single link and inelastic supply under different assumptions than 2.3. loe 0 for single link and elastic supply. Results extend to general network topology.
Page 15 of 40 3.1. Single link and inelastic supply Theorem 3.1 (Convergence of loss of efficiency). With Assumptions 2.1 and 2.3, the loss of efficiency tends to 0 with probability 1 as the number of users tends to infinity. Intuition Assumption 2.3 guarantees that there is always a significant number of users that value the resource highly. Since the amount of resource c is fixed, with enough users with high valuation, the influence of each individual on the market outcome becomes limited. We end up with perfect competition and high efficiency.
Page 16 of 40 Proof outline Hajek and Gopalakrishnan [HG04]: the Nash equilibrium allocation is the solution of GAME max y 1,...,y n subject to n [ ( 1 y ) i U i (y i ) + 1 C C n y i C, i=1 i=1 y i 0, i = 1,..., n. Notice similarity with SYSTEM (1). yi 0 ] U i (z) dz (2)
Page 17 of 40 Complementary slackness conditions for social optimum: u i(x (n) i ) = λ (n) for all active users (i such that x (n) i > 0), u i(0) λ (n) for all inactive users (x (n) i = 0), n i=1 x(n) i = c, Complementary slackness conditions for Nash equilibrium: ) (1 y(n) i u c i(y (n) i ) = µ (n) for all active users, u i(0) µ (n) for all inactive users, n i=1 y(n) i = c.
Page 18 of 40 Observe that the sequence µ (n) is monotone increasing because adding users can only increase the price. Moreover, µ (n) v <, so lim n µ (n) exists. Suppose µ (n) < v ɛ, infinite number of active users as n, active users receive non-negligible y i, but, c is finite! Contradiction. µ (n) v and y i 0, the optimality conditions coincide in the limit loe 0.
Simulation results Scalar modulated linear utility functions. Scalars distributed uniformly over [0, 1]. Page 19 of 40 loe versus n for S i x utility functions and S i sampled uniformly in [0, 1]. The error bars have width equal to one standard deviation.
Simulation results Scalar modulated square root utility functions. Page 20 of 40 loe versus n for S i x utility functions and Si sampled uniformly in [0, 1].
Page 21 of 40 3.2. Convergence rate Suppose that the exists a unique utility function w( ) that is much higher than the rest, i.e., for some δ > 0, ( 1 δ ) w (δ) > u (0), u( ) w( ). c Suppose we have enough (i.e., c/δ ) users with the same utility function w( ), (by the optimality conditions) all the capacity will be allocated evenly among those users in both the social optimum and Nash equilibrium outcomes, loe = 0.
Page 22 of 40 By a counting argument, we get Pr(loe (n) > 0) c/δ ( ) n Pr(U i = w) k( 1 Pr(U i = w) ) n k. (3) k k=0 This probability approaches zero exponentially fast for large enough n (by the Hoeffding Inequality). Theorem 3.2 (Convergence rate). For a fixed ɛ > 0, let us choose δ > 0 such that ψ Pr(U i(δ) v ɛ) > 0. Then, for n c/δ /ψ, ( Pr loe (n) > ɛ ) } (nψ c/δ )2 exp { 2. v n
Page 23 of 40 3.3. Increasing capacity What if we allow the capacity to increase as the number of users increases? Rate of convergence result hold for sub-linearly scaled capacity c(n) o(n). Active users still tend to receive a small fraction of the scarce resource c(n). Almost sure convergence result holds for superlinearly scaled capacity c(n) ω(n). Abundant resource; every user receives enough resource to saturate its utility function. n loe (n) i=1 = 1 u i(y (n) n i ) a.s. n u 1 β i=1 i i=1 i(x (n) n i ) β = 0. i=1 i
Page 24 of 40 3.4. Counter-examples Question: What happens when we relax some assumptions? Recall assumption that no single user with significant market influence. Consider linear utility functions U i (x) = S i x, where S i are drawn i.i.d. over support [0, ). Find distribution for S i such that a few users are dominant. Definition 3.1 (Order statistics). The largest and second-largest elements of a sequence of random variables S 1,..., S n are the first and second order statistics and are denoted by α (n) 1 and α (n) 2. The theory of extreme value distributions deals with the limiting distributions (as n ) of order statistics.
Page 25 of 40 Extreme value theorem Theorem 3.3 (Extremal types, [LLR83]). Let α (n) 1 = max (S 1,..., S n ), where S i are independent random variables drawn from a distribution F (z). If α (n) 1 converges in distribution after any linear scaling, then ( ) Pr l n α (n) 1 + m n z G(z), n where G is one of the three extreme value distributions: (Gumbel) G 1 (z) = e e z, z R, { e z θ, z > 0, (Fréchet) G 2 (z; θ) = 0, z 0, { e ( z) θ, z 0, (Weibull) G 3 (z; θ) = 1, z > 0.
Page 26 of 40 Necessary and sufficient conditions Theorem 3.4 (Domains of attraction, [LLR83]). Let F (z) be the distribution function of the sequence of i.i.d. random variables {S i } and z F = sup{z F (z) < 1}. Then, F D(G 1 ) if and only if there exists a strictly positive function g(t) such that 1 F (t + g(t)z) lim t zf 1 F (t) = e z, for all z R. (4) F D(G 2 ) if and only if z F = and there exists θ > 0 such that 1 F (tz) lim t 1 F (t) = z θ, for all z > 0. (5)
Page 27 of 40 Lemma 3.5 (Bound on LOE as a function of α 2 /α 1 ). Suppose that the utility functions are of the form u i (x) = s i x, s i > 0 and for n 2. The efficiency loss can be bounded from below as follows: ) (1 α(n) 2 loe (n) α(n) 2 α (n) 1 α (n) 1 ), n 2. (1 + α(n) 2 α (n) 1 Proof is algebraic. It relies on the simple form of optimization problem (1) for linear utility functions.
Page 28 of 40 Lower-bound on loe as a function of α (n) 2 /α (n) 1.
Page 29 of 40 Empirical joint relative frequency of loe and α 2 /α 1 out of 5 10 4 experiments for S i x utility functions, S i sampled with Pareto distribution, and n = 10 3 users.
Page 30 of 40 Corollary 3.6 (Necessary condition for zero loss of efficiency). A necessary condition for the loss of efficiency to tend almost surely to 0 is: lim inf n Pr ( α (n) 2 α (n) 1 = 0 or 1 ) = 1.
Example: Pareto distribution Page 31 of 40 F (z) = 1 κ z θ, θ, κ > 0, with support [κ 1/θ, ). If S i has a Pareto distribution over (0, ), then Assumption 2.3 is violated: we have v =, but Pr(U i(0) = ) = 0. It belongs to the domain of attraction of the Fréchet distribution.
By scaling α (n) 1 by l n = (κ n) 1/θ, we obtain the following convergence in distribution [LLR83]: Pr ( (κ n) 1/θ α (n) 1 }{{} ρ (n) 1 z ) n F ρ1 (z; θ). Page 32 of 40 Similarly, scale α (n) 2 to get ρ (n) 2. We can also obtain the limiting (n ) joint p.d.f. for ρ 1 and ρ 2 [LLR83]. By conditioning, and evaluating probability integrals, we find that α (n) 2 /α (n) 1 is bounded away from 1 with positive probability.
Simulations Page 33 of 40 loe versus n for S i x utility functions and S i sampled with Pareto distribution (θ = 2).
Page 34 of 40 We can generalize this result for the class of heavy tail distributions (Theorem 3.4). 1 F (tz) lim t 1 F (t) = z θ, for all z > 0. It s an open question whether the same holds for light tail distributions.
Simulations Page 35 of 40 loe versus n for S i x utility functions, S i sampled with exponential distribution.
Page 36 of 40 3.5. Relation with classic results from economics The resource allocation problem that we consider is a special case of the market or exchange economies described in the literature. Noncooperative equilibria of markets tend to be inefficient with few participants, but efficient with many insignificant participants (c.f. Cournot s work [Cou38] on the notion of perfect competition among a large number of producers). Example from continuum economics (i.e., continuum [0, 1] of traders): Dubey et al. [DMCS80]: under standard conditions, every noncooperative equilibria of continuum markets are efficient. How well does the continuum-of-participants assumption reflects markets with large, but finite, numbers of participants?
Page 37 of 40 Comparison with our work Our work offers a new take on an old problem. Our motivation lies mainly in modeling heterogeneity, as opposed to large populations. We avoid the continuum of traders assumption, where each individual trader is insignificant by construction: summation over the users is replaced by integration, which makes individuals negligible. We adopt a probabilistic line of analysis. We consider a random population of users and provide convergence rates for a moderate population sizes.
Page 38 of 40 Further related works Green [Gre80] shows that in a sequence of replica markets, the equilibria of the finite games converge to the equilibrium of the continuum representation. Postlewaite and Schmeidler [PS78] consider a different notion of efficiency and show that under some conditions on the initial distribution of resources, every Nash equilibrium is efficient when the number of players is large. Our method of sampling random participants is similar in spirit to that of Palfrey and Srivastava [PS86].
Page 39 of 40 4. Decentralization can be efficient (if no agent has too much influence). In such cases, price-taking behaviour is optimal (no need to be too smart). Results extend to elastic supply, general networks.
References [Cou38] A. A. Cournot. Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris, 1838. [DMCS80] P. Dubey, A. Mas-Colell, and M. Shubik. Efficiency properties of strategic market games: An axiomatic approach. Journal of economic theory, 1980. Page 40 of 40 [Gre80] [HG04] [JT04] [Kel97] [LLR83] [PS78] [PS86] [Sha76] E. J. Green. Noncooperative price taking in large dynamic markets. Journal of economic theory, 22:155 182, 1980. B. Hajek and G. Gopalakrishnan. A framework for studying demand in hierarchical networks. Preliminary draft, 2004. R. Johari and J. N. Tsitsiklis. Efficiency loss in a network resource allocation game. Mathematics of Operations Research, 29(3):407 435, 2004. F. P. Kelly. Charging and rate control for elastic traffic. European transactions on telecommunications, 8:33 37, 1997. M. R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and related properties of random sequences and processes. Springer-Verlag, Berlin, 1983. A. Postlewaite and D. Schmeidler. Approximate efficiency of non-walrasian nash equilibria. Econometrica, 46(1):127 35, 1978. T. R. Palfrey and S. Srivastava. Private information in large economies. Journal of economic theory, 39:34 58, 1986. L. S. Shapley. Noncooperative general exchange. In S. A. Y. Lin, editor, Theory and Measurement of Economic Externalities, pages 155 175. Academic Press, New York, 1976. [Shu73] M. Shubik. Commodity money, oligopoly, credit and bankruptcy in a general equilibrium model. Western Economic Journal, 1973.