Shared Resource Management via Reward Schemes Shahar Dobzinski1 and Amir Ronen2 1

Weizmann Institute of Science 2 IBM Resarch, Haifa

Abstract. We study scenarios in which consumers have several alternatives for using a shared resource. We investigate whether rewards can be used to motivate effective usage of the resource. Our goal is to design reward schemes that, in equilibrium, minimize the cost to society and the total sum of rewards. We introduce a generic scheme which does not use any knowledge about the valuations of the √ consumers, yet its cost in equilibrium is always within a factor of α of the cost of the optimal scheme that has complete knowledge of the consumers’ valuations. Here, α is the ratio between the costs of the worst and best alternatives. We show that our scheme is essentially optimal in some settings while in others no good schemes exist.

1

Introduction

In this paper we study the management of resources that are overly demanded. Traditionally, taxes are used to control consumption, see, e.g., London’s or Paris’ congestion charges [3, 17]. In contrast, recent experiments in traffic congestion reduction [12, 11, 16] take a different approach. Instead of charging users for driving during peak hours, drivers who travel during off-peak hours receive monetary compensation. While in a purely quasi-linear world taxes and rewards are equivalent, in practice they may differ greatly, as established by a large body of research in, e.g., psychology [14] and prospect theory [10]. We refer the interested reader to [2] and references within for a thorough discussion on fees vs. rewards in congestion control3 . We study a basic problem of managing the consumption of a single public resource. We assume that there are several alternative consumption methods, such as time slots, locations, etc. The social cost is determined not only by the total consumption but also by the way that this consumption is distributed across alternatives (e.g., it is preferred that factories will consume electricity during off peak hours). Our goal is to develop mechanisms that minimize the sum of the social cost of the consumption and the total payment of the mechanism. Our interest is not limited only to congestion control; Similar settings include basic infrastructures such as electricity that can be consumed in peak or 3

We emphasize that our claim is not that rewards are better than prices; just that they may serve as a powerful tool in some settings.

non-peak hours, fresh air and water, forest preservation, and more. All of which are settings in which a designer – such as a transportation authority – may work with a small number n of big consumers (polluting factories, large truck fleets, major electricity consumers, etc.) on a voluntary basis to improve the efficiency of using the resource4 . Our main contributions are as follows: 1. We present a stylized model of resource management via reward schemes. 2. We introduce a novel solution concept for analyzing the performance of reward schemes: in many settings the designer either does not know the valuations of the consumers or is legally not allowed to use them for designing reward schemes. In these cases, the designer may use only his own costs. The competitive ratio of a scheme R is obtained by considering all possible instances I (i.e., all possible realizations of the valuations of the players) and taking the maximum ratio between the worst equilibrium induced by R and the best equilibrium induced by any other scheme RI . Notice that RI may be designed using knowledge on the valuations of the consumers. 3. We present a generic reward scheme called the square root scheme and show that it is competitive in many settings. In some other settings, we prove that no reward scheme is competitive. We focus on two basic models of consumption: atomic (e.g., truck drives to the city center) and non atomic (e.g., electricity consumption). We now provide a more formal description of our model, the solution concept, and our results. The Model In our formal setup we have n agents (consumers), each is interested in consuming one unit of the public good (a unit could be, for example, a single drive to the city center or 1000KW of electricity). There are m different alternative ways to consume the resource, such as time slots or locations, so i ) such that for each j, the strategy of each agent i is a vector f i = (f1i , . . . , fm fji ≥ 0 and Σj fji = 1. In the atomic model we always assume that fji ∈ {0, 1}. In the non atomic model we do not make this assumption. For each consumer i and alternative j, there is a function uij that depends only on the use of the alternative fj = Σi fji . This function quantifies the gains of consumer i from consuming via alternative j. We assume that uij (fj ) = vji · Lj (fj ), where vji is privately known to consumer i and the attractiveness functions Lj (·) are public information. The attractiveness functions are decreasing monotonically. The utility of consumer i is defined to be Σj fji ·uij (fj )+pif 1 ,...,f n , where pif 1 ,...,f n ≥ 0 is the payment that the consumer receives. 4

E.g., a municipality attempting to incentivize an international company to locate its polluting factory further away from the city.

We assume that the cost to society for every unit of good consumed via alternative j is wj > 0. The direct loss to society is therefore Σj wj · fj . Without loss of generality we assume that w1 ≥ . . . ≥ wm . The ratio α = wwm1 will be essential in our analysis. We assume without loss of generality that wm = 1. In this paper we seek for pif 1 ,...,f n ’s so that in equilibrium the total cost to the society Σj wj · fj + Σi pif 1 ,...,f n is minimized (notice that this cost includes payments). To illustrate this point, observe that the VCG mechanism can be used to minimize the direct loss of the society (Σj wj · fj ). However, it is well known that in the VCG mechanism the amount that the society has pay might be huge, whereas in many settings big subsidies are unacceptable either politically or economically. A set of payment functions pif 1 ,...,f n ’s as above is called a reward scheme or payment scheme. Of particular interest are simple reward schemes where payment to each consumer i depends only on its own consumption f i . In this paper we seek for such payment schemes that are competitive with respect to arbitrarily complicated schemes. Given a reward scheme R and a consumption profile f , we let CR (f ) = Σj wj · fj + Σi pif 1 ,...,f n denote the overall cost to society, i.e. the total loss plus the total payment. The Solution Concept: Robustness in an Incomplete Information Setting Assuming full information of the utilities of the consumers is often unrealistic. For instance, it equips the designer with the unlikely power to accurately estimate the profit of a specific freight forwarder for sending a specific truck at peak hours. In some other cases, the designer might be legally prohibited from using the valuations of the consumers in a reward scheme. On the other extreme, the popular approach in the Algorithmic Mechanism Design community is to design truthful mechanisms. However, since our objective function involves payments, more often than not this leads to impossibility results. See, e.g., the various impossibility results on the design of frugal mechanisms (e.g., [7, 1]). The externalities among the consumers also make the existence of satisfactory truthful mechanisms even less likely5 . In this paper we take a middle way that allows us to design payment schemes with good guarantees for games with incomplete information. The private infori . Suppose that all the private information of the consumer i is v i = v1i , . . . , vm mation v = v 1 , . . . , v n is known. Let R be a reward scheme. Together, we have a complete information game. Let f be an equilibrium in this game. The total cost of f in R equals Σj wj · fj + Σi pif 1 ,...,f n . An optimal solution is a reward scheme O and an equilibrium f 0 in O, such that the total cost of f 0 is minimal among all reward schemes and equilibria. In other words, CO (f 0 ) denotes the 5

For VCG, while the welfare is maximized, it is easy to see that the payment can be huge.

total cost to society of the best equilibrium of the best reward scheme given that the values of the consumers are v. Definition 1. (competitive scheme) A reward scheme R is c-competitive if for every vector of values of the consumers v, equilibrium f in the game defined by v and R, an optimal scheme O for v, and every equilibrium f 0 in the game (f ) defined by O and v it holds that CCOR(f 0 ) ≤ c. Notice our guarantee: if the “market” converges to an equilibrium, then the total cost to society of the solution is not too far away from the total cost of the optimal solution – even had we known the consumers’ valuations in advance. Thus, to some extent we seek for Nash implementation of reward schemes. However, while most work on Nash implementation usually considers full information games or takes a Bayesian approach, we assume private information and our guarantees are worst-case ones. In this sense, our work lies on the border of price of anarchy analysis and algorithmic mechanism design. Related to our work is a series of works on setting tolls in congestion games to improve the efficiency of equilibrium (see, e.g., [6, 8, 9]). One main difference between this line of work and ours is that the aforementioned papers design tolls for a specific complete-information game. In our approach consumers have private information, yet equilibrium is always guaranteed to be approximatelyefficient. Another key difference is that our objective function takes into account only the cost to the society and not the cost to the consumers. Finally, there are also technical differnces in the approximation notion. Our approach is closely related to coordination mechanisms [4], and in particular to coordination mechanisms for congestion games [5]. A coordination mechanism is allowed to change the latencies of some edges in order to improve the price of anarchy in the network. One main difference between our work and the work on coordination mechanisms (besides considering completely different models) is that our benchmark takes into account the effort needed to change the network (in the sense of payments), while in coordination mechanisms changes in the network are “free” as long as the quality of the solution improves. Some Comments on our Modeling Choices A few words are in place about our modeling choices. First, perhaps our most controversial design choice is the separation of the welfare of the consumers from the public welfare. This seems justified to us as in the applications we envision the designer works with a few big consumers on a voluntary basis (note that the consumers can always choose an option that will not give them any rewards) in order to improve the welfare of a large public. For example, consider a big freight company that has to send trucks to the center of the city. Suppose that there are only two alternatives: use

peak hours (and negatively affect the commute time of many drivers) or off peak hours. We think about the wj ’s (the cost of alternative j to the society) as the marginal loss that the rest of the drivers incur from the freight company that uses alternative j. Notice that the freight company can always decline changing and get no rewards. It is thus reasonable for the society to neglect the welfare of such a consumer and focus on the rest of the public as long as voluntary participation is kept. Nevertheless, relaxing this assumption is an interesting future direction. We also assume that society’s loss from a specific alternative is linear in the congestion. This assumption may be considered as more technical than conceptual and we leave relaxing this assumption to future work. A Simple Example In order to make our model and notation more concrete consider the following toy example. We will have only a single consumer (n = 1) and two alternatives with w1 = 9 and w2 = 1. Thus, α = 9. Suppose that the consumer must choose either alternative 1 or alternative 2 but cannot split its consumption (e.g., cannot use half of alternative 1 and half of alternative 2). Notice that the attractiveness functions Lj (·) have no real meaning here since there is only a single consumer. Thus we assume that Lj (.) is always 1. Consider the case were no reward is given. Suppose that consumer that slightly prefers alternative 1 over alternative 2, e.g. v11 = 1 +  and v21 = 1. Thus, he will choose alternative 1, i.e. f11 = 1, and the overall cost will be 9. Paying  for using alternative 2 will yield a total cost of only 1 + . Consider the scheme that gives the consumer a reward of 2 for using alternative 2, and no reward for using alternative 1. We claim that the social cost of the solution in this scheme is always at most a factor of 3 away from the cost of the optimal solution, regardless of the valuation of the consumer. Since this scheme is actually a special case of the square root scheme, to be introduced shortly, we will not formally prove this but instead analyze some interesting instances. Suppose that the customer’s preferences are as before, alternative 2 will be chosen and the overall cost would be 1 + 2 = 3 (1 for using alternative 1 and a reward of 2). However, when the value of the consumer for using alternative 1 is 1 and 1 +  for using alternative 2, the optimal solution is to pay nothing to the consumer, since he prefers the less costly alternative anyway. Similarly to before, the total cost is 1 + 2 = 3 while the optimal cost is 1. The last instance we consider is when the value of the consumer for using alternative 1 is 100 and his value for using alternative 2 is 1. The optimal solution is to pay nothing to the consumer and let him use alternative 1, since we have to pay the consumer at least 99 to prefer alternative 2. In this case the consumer will use alternative 1 and the total cost is 9. In our scheme the total cost is also 9, since the offered reward is too small for the consumer to prefer alternative 2.

Results Our main tool is a payment scheme that we name the square root scheme. We divide the alternatives into two sets: an alternative j is said to be √ cheap if wj ≤ α (recall that w1 = α and wm = 1), and is expensive other√ wise. The reward for choosing a cheap alternative j is defined as pj = α − wj and zero for expensive alternatives. Note that the more attractive an alternative is to society, the higher its payment. We pay consumer i that plays a strategy i ) a payment of Σ f i · p . Notice that the scheme is simple to f i = (f1i , . . . , fm j j j implement: the designer announces only which alternatives are cheap, and the reward for using every cheap alternative. We show that the square root payment scheme produces good results in some settings – in particular it is applicable to the two alternative case (e.g., peak and off-peak hours), both in the atomic and the non-atomic settings6 , and in the case of a single consumer. The later can also be applied to cases where there is no friction between consumers so they can be considered separately. In the non-atomic model, however, we prove these results only with respect to linear schemes – where the payment is proportional to the usage of the alternative. √ Theorem: The square root payment scheme is O( α) competitive in the following settings, both in the atomic model and with respect to linear schemes in the non-atomic model: (1) a single consumer and multiple alternatives, and (2) multiple consumers and two alternatives. The proofs are based on charging arguments comparing the contribution of each consumer to the total cost in the optimal solution and in the equilibrium produced by the scheme. We state this charging argument as one key lemma, and apply it in different settings. Proving the result in the non-atomic model is more involved and requires structural understanding of the equilibrium. For some settings we can show that the competitive ratio cannot be substantially improved: √ Theorem: Every payment scheme is Ω( α)-competitive for a single consumer and many alternatives both in √the atomic and the non-atomic settings. Furthermore, no payment scheme is nα -competitive in the atomic model with n ≥ 2 consumers. When there are at least three alternatives and at least two consumers, we show that there are essentially no good schemes, at least in the atomic case and when the number of consumers is small: -competitive in the Theorem: No non-increasing payment scheme is n−1+α n atomic model when there are at least three alternatives and n ≥ 2 consumers. 6

The two models may look superficially similar, but in fact give rise to different behavior.

Observe that the multi-consumer bounds hold only in the atomic model and are tight only for a small number of consumers. Extending these bounds to nonatomic settings, as well as obtaining tight bound for a large number of consumers, is an open question. On the positive side, for the non-atomic model, we show that when the consumers preferences are identical (but unknown to – or unused by – the designer) an optimal competitive ratio can be achieved: Theorem: In the non-atomic model, when all the consumers have the same √ preferences, the square root scheme is α-competitive with respect to any linear anonymous reward scheme.

2

The Square Root Reward Scheme

A main tool in our constructions is the following square root scheme. Before introducing it, we require some notation. We say that an alternative j is cheap if √ wj ≤ α. Otherwise, the alternative is called expensive. Definition 2. (square root reward scheme). The square root reward scheme is defined as follows. The reward function to each consumer i is identical for all consumers, and depends only on the actions of consumer i. √ α − wj , j is cheap; tj = 0, j is expensive. The payment is then defined by pif = Σj fji · tj . √ Notice that in the atomic case a consumer is rewarded by α − wj when choosing a cheap alternative j and zero otherwise. A concrete example of the square root scheme was given in the introduction. 2.1

A Key Lemma

To facilitate the analysis of our scheme in various settings, we now provide a lemma that will be form the basis of our analysis in each proof. Roughly specking, the lemma relies on two facts. The first is that the overpayment of the square root scheme on cheap alternatives is not so big. On the other hand, if the usage of cheap alternatives is greater in the square root scheme, then the optimal scheme pays at least as the square root scheme. Let C denote the set of cheap alternatives. Recall that tj denotes the threshold of the square root scheme. Definition 3. Let M be the square root scheme and denote by p(·) its payment function. Let M ∗ be any scheme, and p∗ (·) its payment function. We say that M is payment bounded with respect to M ∗ if the following holds for every equilibrium f, f ∗ of M, M ∗ respectively:

– If there exists an expensive alternative e such that fe > fe∗ , then it holds that for every consumer i and cheap alternative j such that fj∗ > fj we have that M ∗ pays consumer i at least as M for using alternative j: p∗i (f ∗ ) ≥ P ∗i i j∈C,f ∗ ≥fj (fj − fj ) · tj . j

In other words, for each cheap alternative j, either the usage of j in f is greater than the usage of j in f ∗ , or M ∗ ’s rewards are at least as M ’s for it. We now use this property to analyze the competitive ratio of the square root scheme. Lemma 1. (key lemma) Let M be the square root scheme and denote by p(·) its payment function. Let M ∗ be any scheme, and p∗ (·) its payment function. If M is payment bounded with respect to M ∗ than for every equilibrium f, f ∗ in √ ∗ (f ∗ ). M, M ∗ respectively, CM (f ) ≤ α · CM Proof. Let C and P E denote the setsPof cheap and expensive alternatives, respectively. Let fC = j∈C fj , fE = j∈E fj denote the total usage of cheap and expensive alternatives respectively. We divide the proof into two different cases. In the first one, the usage of cheap alternatives in f is bigger than the the usage of cheap alternatives in f ∗ . We will then consider the complement case. – Case 1: fC ≥ fC∗ . The square root scheme pays nothing on the expensive √ alternatives. On each cheap alternative j we the cost is [( α − wj ) + wj ] · √ fj = α · fj . Thus, CM (f ) ≤ α · fE +

√

α · fC ≤

√

√ √ √ √ α · ( α · fE + fC ) ≤ α · ( α · fE∗ + fC∗ ) ≤ α · opt

The third inequality holds since fC − fC∗ = fE∗ − fE so we only “shifted √ mass” from the cheap to the expensive alternatives (i.e., from weight α to weight α). – Case 2: fC < fC∗ . In this case fE > fE∗ so there exists at least one expensive alternative e such that fe > fe∗ . Thus, by the payment bound condition, for every cheap alternative j, if fj∗ ≥ fj then M ∗ pays at least as M for using alternative j. Given a consumer i, we can thus decompose i’s payment to P 0i ∗i i 0i ∗ j tj · (fj − fj ) such that tj ≥ tj on every cheap alternative j where fj ≥ i∗ P fj −fji fj . Let t0j denote the weighted average reward (over consumers) · f ∗ −f i j

j

t0ij . In particular, t0j ≥ tj for all the cheap alternatives for which fj∗ ≥ fj . We now divide the total usage of both schemes into three groups and show the competitive ratio in each group separately. 1. S1 is composed of usage of expensive alternatives of size fE∗ from each of the equilibria. On f ∗ this comprises from all usage of expensive alternatives. On f this covers all except fE − fE∗ of it.

2. S2 will be a consumption of size fE − fE∗ = fC∗ − fC . From the square root scheme M we will take this mass from the expensive consumption. From M ∗ we will take this mass7 only from alternatives j ∈ C such that fj∗ ≥ fj and from consumers i such that fj∗i ≥ fji . From each such consumer we will take P up to fj∗i − fji . This is feasible as, fC∗ − fC = P P ∗ ∗ ∗ j∈C,fj∗ ≥fj fj − fj + j∈C,fj∗
3

Atomic Consumption

This section assumes that each consumer chooses a single alternative and cannot divide his usage between several alternatives. Examples include a company that can build a new factory either close to the city or in a more remote site, a single 7

By taking mass of size Pm we mean that we are given a set S of pairs i, j where i is a consumer and j alternative s.t. (i,j)∈S fji ≥ m and we take up to m of it, for example by letting c s.t. P c · (i,j)∈S fji = m and taking c · fji from every such pair.

drive that can to be made either in peak hours or in off-peak hours, etc. From a technical perspective, in this case each fji is in {0, 1}. We first prove the existence of Nash equilibrium under some mild conditions8 . A scheme is non-increasing if the reward for using an alternative j does not increase with fj . A scheme is alternative-based if the reward for using alternative j depends only on fj . The square root scheme maintains both properties. Proposition 1. If the reward scheme is alternative-based and non-increasing then a pure Nash equilibrium exists in the induced complete information game. The proof can be found in the appendix. In the sequel we analyze the power and limitations of reward schemes. We start with analyzing the two-alternative case. 3.1

Two Alternatives

Theorem 1. If m = 2 the square root payment scheme is

√

α-competitive.

Proof. Recall that by our convention alternative 1 is the expensive alternative and alternative 2 is the cheap one. Let M ∗ be some scheme. According to Lemma 3 it suffices to show that the square root scheme M is payment bounded w.r.t M ∗ . Let f, f ∗ denote equilibria in M and in M ∗ , respectively. If f1 ≤ f ∗1 then Lemma 3 trivially holds for f and f ∗ , therefore assume that f1 > f1∗ . Thus, f2 < f2∗ . Since the attractiveness functions are non-increasing L1 (f1∗ ) ≥ L1 (f1 ), L2 (f2∗ ) ≤ L2 (f2 ), our goal is to show that if consumer i uses alternative 2 in f ∗ and alternative 1 in f , then M ∗ pays to consumer i at least as √ the possible payment of M to i, which is α − 1. Let pij , p∗i j denote the payment to i when choosing j in each scheme respectively (fixing the actions of the other consumers). Since M pays nothing for using alternative 1 but the consumer still uses it, pi2 + v2i · L2 (f2 ) ≤ v1i · L1 (f1 ). In M ∗ the consumer prefers alternative 2 and therefore: ∗i i ∗ i ∗ ∗i i i i p∗i 2 ≥ p1 + v1 · L1 (f1 ) − v2 · L2 (f2 ) ≥ p1 + v1 · L1 (f1 ) − v2 · L2 (f2 ) ≥ p2 =

Thus the conditions of Lemma 3 are met and we have that Cf (M ) ≤ Cf (M ∗ ), and the competitive ratio follows. 

√

α·

We now show that under mild conditions, this bound is essentially tight. An attractiveness function Lj is bounded away from 0 if for every if for any 8

To see that in general the existence of a Nash equilibrium is not guaranteed, consider a scheme for two alternatives and two players that pays a huge amount for one consumer to use any alternative alone, and a huge payment for the second consumer to use any alternative with the second consumer.

√

α − 1.

number of consumers n0 and any value t ≥ 0 there exists some v such that v · Lj (n0 ) = t (i.e., Lj (n0 ) > 0). The following theorem (proof in the appendix) provides bounds which are tight up to a constant for the case of m = 2 and for the single consumer case. We will later provide stronger bounds when both the number of alternatives and the number of consumers is big. Theorem 2. Consider the case where m ≥ 2. For all attractiveness functions that are bounded away from 0, and for all  > 0, no deterministic payment (n−1)+α √ . Furthermore, no scheme can provide a competitive ratio better than (n−1)+ α randomized scheme can provide a competitive ratio better than 3.2

(n−1)+α √ . 2(n−1)+ α

Multiple Alternatives

We begin with a lower bound for multiple alternatives and consumers. Our bound is meaningful only for small number of players, and we do not know how to extend it further. After proving the lower bound we will show that if there is a single consumer the square root scheme does provide a good competitive ratio when there are multiple alternatives. Proofs appear in the appendix. Theorem 3. When m ≥ 3, n ≥ 2 and the attractiveness functions that are bounded away from 0, no non-increasing reward scheme provides a competitive ratio better than n−1+α . n √ Theorem 4. When n = 1, the square root payment scheme is α-competitive.

4

Non Atomic Consumption

In this section we focus on consumption that can be split between different alternatives (e.g. a total of 1000KW has to be consumed, but some of it can be consumed in off-peak hours). From a technical point of view, the difficulty of analysis in this model stems from the fact that the attractiveness functions are functions of the overall load on the alternatives (including the consumer’s own usage), so the marginal utility of a consumer from using an alternative depends not just on the total load, but also on his own usage of that alternative. Due to lack of space we postpone this section to Appendix A.

5

Discussion and Open Questions

We studied a basic problem of managing a single shared resource via positive incentives. We focused on the case where a central designer is interested in

motivating a few big consumers to change their demand while minimizing the cost. Various concrete open questions are mentioned in the body of this paper. Perhaps the most important one is understanding the setting where there are multiple alternatives and many consumers – our lower bounds hold only when the number of consumers is small and good schemes may exist. Many extensions of our model almost suggest themselves. For example, what happens if the consumers have combinatorial preferences over each set of alternatives (two drives in peak hours worth more than a drive in peak hours and a drive in off-peak hours). Also, broadening the set of attractiveness functions and the set of society’s cost functions is a natural extension. One can also take a graph-theoretic view of our model by considering two nodes (source and target) connected by m parallel edges, each corresponding to another alternative. Analyzing different graph topologies seems like a worthy research avenue.

References ´ Tardos. Frugal path mechanisms. In SODA’02. 1. A. Archer and E. 2. E. Ben-Elia and D. Ettema. Carrots versus sticks: Rewarding commuters for avoiding the rush-houra study of willingness to participate. Transport policy, 16(2):68–76, 2009. 3. P. Bonsall, J. Shires, J. Maule, B. Matthews, and J. Beale. Responses to complex pricing signals: Theory, evidence and implications for road pricing. Transportation Research Part A: Policy and Practice, 41(7):672–683, 2007. 4. George Christodoulou, Elias Koutsoupias, and Akash Nanavati. Coordination mechanisms. In Automata, Languages and Programming, pages 345–357. Springer, 2004. 5. George Christodoulou, Kurt Mehlhorn, and Evangelia Pyrga. Improving the price of anarchy for selfish routing via coordination mechanisms. In Algorithms–ESA 2011. 6. R. Cole, Y. Dodis, and T. Roughgarden. Pricing network edges for heterogeneous selfish users. In STOC, pages 521–530. ACM, 2003. 7. E. Elkind, A. Sahai, and K. Steiglitz. Frugality in path auctions. In SODA’04. 8. L. Fleischer, K. Jain, and M. Mahdian. Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games. In FOCS, pages 277–285. IEEE, 2004. 9. Dimitris Fotakis, George Karakostas, and Stavros G. Kolliopoulos. On the existence of optimal taxes for network congestion games with heterogeneous users. In SAGT’10. 10. D. Kahneman and A. Tversky. Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, pages 263–291, 1979. 11. J. Knockaert, M. Bliemer, Ettema D., Joksimovic D., Mulder A, Rouwendal J., and van Amelsfort D. Experimental design and modelling Spitsmijden. Bureau Spitsmijden, The Hague., 2007. 12. John Markoff. Incentives for drivers who avoid traffic jams. New York Times, 2012. 13. I. Milchtaich. Congestion games with player-specific payoff functions. Games and economic behavior, 13(1):111–124, 1996. 14. R.A. Rescorla. A pavlovian analysis of goal-directed behavior. American Psychologist, 42(2):119, 1987. 15. JB Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica: Journal of the Econometric Society, pages 520–534, 1965. 16. Nofar Sinai. Haaretz TheMarker, 2012. 17. Wikiepdia. London congestion charge, 2012.

A

Non Atomic Consumption

In this section we focus on consumption that can be split between different alternatives (e.g. a total of 1000KW has to be consumed, but some of it can be consumed in off-peak hours). From a technical point of view, the difficulty of analysis in this model stems from the fact that the attractiveness functions are functions of the overall load on the alternatives (including the consumer’s own usage), so the marginal utility of a consumer from using an alternative depends not just on the total load, but also on his own usage of that alternative. We assume that the attractiveness functions Lj (fj ) are smooth, strictly decreasing, and concave (for example, functions of the form 1 − poly(fj ) where the coefficients of the polynomial are non-negative satisfy these assumptions). We are not able to prove competitive ratios with respect to the optimal scheme. Instead, we limit ourselves to some natural classes: Definition 4. A scheme is called linear if for every consumer i there exist nonnegative constants pi1 , . . . , pim such that the reward of consumer i for each strati ) is Σ f i · pi . egy f i = (f1i , . . . , fm j j j For  linear schemes we get that the utility of consumer i equals: ui (f ) = i i i j fj · vj · Lj (fj ) + pj .

P

Given a linear scheme, let the utility of alternative j be Uji (fji , fj−i ) = fji · P (vji · Lj (fj ) + pj ). Let ui (fji , fj−i ) = j Uji (fji , fj−i ). The marginal utility of an alternative j for consumer i equals dfdi Uji (fji , fj−i ). This quantity is central to j

our equilibrium analysis. Definition 5. A scheme is called anonymous if the payment functions are identical for all consumers. √ The square root scheme is linear and anonymous with pij = max(0, α − wj ). The structure of the rest of this section is as follows: We start with some structural observations on the equilibria of games resulting from linear schemes. While for general schemes, the existence of Nash equilibrium is not guaranteed, its existence for linear schemes is due to [15]. Then, we show that the square √ root scheme provides a competitive ratio of α with respect to linear schemes in several cases: when there are two alternatives, in the single consumer case, and when there are multiple consumers with identical preferences. In the last case we also require that the scheme is anonymous. Constructing schemes with good competitive ratio when there are multiple alternatives and consumers with different preferences remains an open problem.

A Simple Example Consider a setting with two alternatives and only one consumer i. Let v1i = v2i = 1, and Lj (fj ) = 1 − fj (since is only one consumer, we omit fj−i ). Consider a linear scheme: each alternative is associated with a reward pij , and the consumer receives a total reward of Σj fji · pij . The marginal utilities in this case are given by dfdi Uji (fji ) = pij + Lj (fji ) + fji · L0j (fji ) = pij + 1 − 2fji . j

As we shall see, if there is a point in which the marginal utilities are equal, then this is an equilibrium point. The marginal utilities are strictly decreasing in fji . Thus there is no more than a single such point. When there are no payments this point is by symmetry (1/2, 1/2). When, for instance, pi1 = 0 and pi2 = 1, we get that 2 − 2f2i = 1 − 2f1i so f i = (0.25, 0.75) is the unique equilibrium. A.1

Some Basic Observations on Linear Schemes

Lemma 2. (monotonicity of marginal utilities) For any linear scheme, the marginal utilities are strictly decreasing in fji and fj−i and are strictly increasing in pij . Proof. d d Uj (fji , fj−i ) = pij + vji · i (fji · Lj (fj )) i dfj dfj Thus the marginal utility is strictly increasing in pij . Using simple derivation we get: ! d d −i i −i i i i i Uj (fj , fj ) = pj + vj · Lj (fj ) + fj · i (Lj (fj + fj )) dfji dfj   d i i i = pj + vj · Lj (fj ) + fj · (Lj (fj )) dfj Lj (·) is strictly decreasing in fj and hence in both fji and fj−i . As for fji · d (Lj (fj )): if fj−i increases, then by concavity dfdj (Lj (fj )) is weakly decreasdf i j

ing so it can only becomes more negative and thus so is the product. Similarly, if fji increases then since fji is positive and dfdj (Lj (fj )) can only get more negative, the value of the expression decreases. Claim. Let f = (f 1 , . . . f n ) be an equilibrium. For any consumer i, if j is in the support of i then for any alternative j 0 , it holds that dfdi Uji (fji , fj−i ) ≥ j

d U i (f i , f −i ). dfji0 j 0 j 0 j 0

Proof. Since the Uji (·)’s are smooth, they are locally approximately linear. If the inequality does not hold consumer i is better off slightly increasing fji0 and reducing fji , and therefore f is not an equilibrium. Proposition 2. In any linear scheme there exists a pure Nash equilibrium in the resulting game. Proof. Rosen [15] defined a concave game as a game in which each player chooses a point xi ∈ Rm , the set of points x = (x1 , . . . , xn ) is convex and closed and the Uji ’s are continuous in x and concave in xi . He showed that at least one pure Nash equilibrium exists in any such game. In our game the strategy set is convex and closed and by the smoothness of the attractiveness functions, U i are continuous in f . To apply Rosen’s theorem we need to prove concavity: Claim. Uji (f i , f −i ) is concave in f i . Proof. ui (f i , f −i ) =

  i · v · L (f ) + pi . It is thus sufficient to show f i j j j j j

P

that each factor fji · vi · Lj (fj ) + fji · pj is concave in f i , i.e., its derivative is strictly decreasing. This is immediate from the fact that Lj (·) is decreasing and concave. P i i −i The consumer’s utility is thus a sum Ui = Uj (fj , fj ) of concave funci tions and is hence concave in fj . This meets all conditions of Rosen’s theorem. A.2

Two Alternatives

This subsection analyzes the competitive ratio of the square root scheme when there are only two alternatives. Towards this end, we start by analyzing the structural properties of equilibrium in this case. Then, we use these structural properties to give a bound on the competitive ratio of the square root scheme. Lemma 3. Consider a two-alternative setting. Let f be an equilibrium in some linear scheme P. For every consumer i at least one of the following holds: d U i (f i , f −i ) > dfdi U2i (f2i , f2−i ) meaning that f1i = 1. df1i 1 1 1 2 For all f i , dfdi U1i (f1i , f1−i ) < dfdi U2i (f2i , f2−i ) meaning that f2i = 1. 1 2 Given f −i , there is a unique value f i = (f1i , f2i ) such that dfdi U1 (f1i , f1−i ) 1 d i , f −i ). U (f 2 i 2 2 df2

1. For all f i , 2. 3.

=

Proof. The first and second cases are a direct application of Claim A.1. As for the third case, the value must be unique due to the strict monotonicity of the d d marginal utilities: define R(x) = dx U1i (x, f1−i ) − dx U2i (1 − x, f2−i ). Since one term is strictly decreasing in x and the other one is strictly increasing in x, there exists a single point for which R(x) = 0. This is f i , where the marginal utilities become equal. Next, we show that if there are two alternatives then there is a unique equilibrium. We remind the reader that there exists at least one Nash equilibrium. Lemma 4. (uniqueness of equilibrium) Let f and f 0 be two equilibria under the same two-alternative linear reward scheme. Then, f and f 0 are identical. Proof. We require the following lemma: Lemma 5. Let f, f 0 be two equilibria in the same two-alternative linear reward scheme. Then, for every alternative j, fj = fj0 . Proof. Suppose by contradiction that for some alternative j, fj < fj0 . This means that for the second alternative j 0 , fj 0 > fj0 0 and Lj 0 (fj 0 ) > Lj 0 (fj0 0 ). This also implies that there must be at least one consumer i such that fji < fj0i . But then vji · dfdi (fji · Lj (fj )) strictly increases for j when moving from f 0 to j

f and strictly decreases for j 0 . This means that fji = 1. This is a contradiction, since f i = 1 and since we assumed that fji < fj0i . According to Lemma 5, fj = fj0 for each alternative j. We thus need to show that fji = fj0i for every consumer i and alternative j. The proof is almost identical to the proof of Lemma 5. Assume by contradiction that there is a consumer i and alternative j such that fji < f 0 ij . There is also an alternative j 0 such that fji0 > f 0 ij 0 . This, if j maximized the derivative in f , than its derivative is strictly higher than j 0 in f˜ (because of the strict monotonicity of the marginal utilities in fji , fj ) – contradicting the fact that j 0 is in the support. We are now ready to analyze the competitive ratio of the square root scheme: Theorem 5. If m = 2, the square root scheme provides a competitive ratio of √ α with respect to any linear reward scheme. Proof. Let M ∗ be any linear scheme with payments p∗ (·) and M be the square root scheme with payments p(·). Let f ∗ be an equilibrium in M ∗ and f be an equilibrium in M . We need to show that M is payment bounded with respect

to M ∗ . To show this, we need to show that if f2∗ ≥ f2 and f2∗i > f2i , then √ p∗i α − 1. Then we will be able to apply Lemma 3. 2 ≥ By Lemma 3, for each consumer either the derivatives of the marginal utilities in both alternatives are equal in f ∗ , or the consumer fully uses alternative 2. By Lemma 2, the marginal utilities are decreasing in fj and fji and increas√ ing in pj . Thus, if by pi∗ α − 1 then dfdi U2i (f2 , f2−i ) ≥ dfd∗i U2i (f2∗ , f2∗−i ). 2 < 2

2

Since f1 ≤ f1∗ and pi1 = 0, we have that the marginal utility of alternative 1 is smaller in the square root scheme. But then dfdi U2i (f2i , f2−i ) > dfdi U1i (f1i , f1−i ) 2 1 contradicting Lemma 3. Thus, by the Lemma 3, the competitive ratio is as stated. √ We conjecture that the square root scheme is O( α)-competitive with respect to all schemes, not just linear ones. A.3

Multiple Alternatives

This section analyses the competitive ratio of the square root scheme against linear schemes when there are more than two alternatives. We consider two special cases: when n = 1, and, with respect to anonymous schemes, when there are multiple consumers with identical preferences. √ Theorem 6. The square root scheme is α-competitive with respect to linear schemes when there is only a single consumer. Proof. Let M ∗ be any linear scheme, p∗ (·) its payment function and f ∗ an equilibrium in that scheme. Similarly, let M be the square root scheme, p(·) its payment function and f an equilibrium. To apply Lemma 3 we need to show that M is payment bounded with respect to M ∗ . Suppose that there exists an expensive alternative e such that fe∗ < fe . Recall that by Claim A.1 all alternatives in the support have the same marginal utility. Let d and d∗ denote the marginal utilities in the support Pof the square root scheme and the optimal scheme, respectively. Let f = C j∈C fj , fE = P f denote the total usage of cheap and expensive alternatives respectively. j∈E j Claim. d∗ > d. Proof. Recall that fe∗ < fe . Since the payment for e in the square root scheme is 0, the strict monotonicity of the marginal utilities implies that dfde Ue∗ (fe∗ ) > d dfe Ue (fe ). By Claim A.1, all alternatives in the support have the same marginal utility. Therefore, d∗ = dfde Ue∗ (fe∗ ) > dfde Ue (fe ) = d.

Corollary 1. Let j ∈ C be such that fj∗ > fj . Then, p∗j > pj . Proof. By the above dfdj Uj∗ (fj∗ ) > dfdj Uj (fj ). The corollary follows since the marginal utilities (Lemma 2) are decreasing in fj and increasing pj . Thus, the square root scheme is payment bounded w.r.t. M ∗ and the competitive ratio follows. Theorem 7. When all the consumers have the same preference vector v i , the √ square root scheme is α-competitive with respect to any linear anonymous reward scheme. Proof. We first analyze the structure of equilibrium in this case. Then we will use this structure to argue about the competitive ratio. We first show that in equilibrium all consumers have the same marginal utilities in their support. Claim. Consider a linear anonymous reward scheme and let f be an equilibrium in this scheme. Let i, l be two consumers and j, k be two alternatives in their support respectively. Then dfdi Uji (fji , fj−i ) = dfdl Ukl (fkl , fk−l ). j

k

Proof. Recall that for each consumer, the marginal utilities of the alternatives in their support are equal. Denote the consumers’ marginal values by di and dl . Assume by contradiction that di > dl . Recall that dfdi Uj (fji , fj−i ) = pj + vj · j   d i Lj (fj ) + fj · dfj (Lj (fj )) . Since the payments and the valuations are identical for both consumers, different marginal utilities imply the that allocation to the players is different as well. Thus, there exists an alternative j in the support of i such that fji > fjl . Since j is on i’s support:     d d i l pj +vj · Lj (fj ) + fj · Lj (fj ) > dl ≥ pj +vj · Lj (fj ) + fj · Lj (fj ) dfj dfj implying that: fji ·

d d Lj (fj ) > fjl · Lj (fj ) dfj dfj

But since the derivatives are non-positive and fji > fjl we get a contradiction. Corollary 2. Consider a linear anonymous reward scheme and let f be an equilibrium. Then all consumers have the same consumption in f . Proof. Otherwise, there exist two consumers i, l s.t. fji > fjl for some alternative j in i’s support. Similarly to Claim A.3, this implies that: fji ·

d d (Lj (fj )) ≥ fjl · Lj (fj ) dfj dfj

But since dfdj Lj (fj ) is non-positive, the above implies that this expression is zero. Since Lj (·) is concave and strictly decreasing this can only happen if fj = 0. A contradiction to the assumption that j is in the support of i. Claim. Let f, f 0 be two equilibria of a linear anonymous reward scheme. Then, f = f 0. Proof. Otherwise, there exists an alternative j such that fj < fj0 . From the monotonicity of the marginal utilities (Lemma 2), !   d d (Lj (fj )) > pij +vji · Lj (fj0 ) + fj0 /n · 0 (Lj (fj0 )) . pij +vji · Lj (fj ) + fj /n · dfj dfj Thus, the marginal utilities in the support of f are strictly greater than those of f 0 . However, there also exists another alternative k such that fk > fk0 that will prove exactly the opposite - a contradiction. To conclude, there exists a unique equilibrium and in this equilibrium all the agents have the same consumption f /n and reward function. To complete the proof, we now follow the same arguments of the single consumer setting (proof of theorem 6).

B

Missing Proofs

Proof of Proposition 1 (sketch) Milchtaich [13] introduces player-specific congestion games: n players share a common set of r strategies. The payoff the i’th player receives for playing the j’th strategy is a monotonically non-increasing function that depends only only the number of players playing strategy j. It is shown that player-specific congestion games always have a pure Nash equilibrium. Observe that for alternative-based and non-increasing payment schemes, our game is a player-specific congestion game, hence a pure Nash equilibrium always exists. Proof of Theorem 2 We consider a setting with m alternatives. Let w2 = . . . = wm = 1 and w1 = α. We first prove the theorem for deterministic schemes, then we extend it to randomized ones. In all the instances that we consider there are n consumers. We will have a set I, |I| = n − 1 of consumers that will have the same valuation in all the instances we consider. For every alternative j and consumer r ∈ I we have that vjr · Lj (1) < , for some  > 0. The only consumer that will

have a different valuation in each of the instances we consider is consumer i ∈ / I. For the rest of the proof we fix some payment scheme P and analyze the performance of this payment scheme. We construct two instances in which the total cost of the optimal solution is √ at least n − 1 + α. Thus, if some consumer uses alternative 1 we get that the total cost is at least n − 1 + α, which gives us the lower bound. In particular, observe that consumers in I are almost indifferent to which alternative they are using in the sense that if any of them is offered a payment of  to use some alternative j and 0 to all other alternatives, each of them will prefer alternative j regardless of the number of other consumers that are using j and the attractiveness functions. Thus we may assume the reward scheme offers  to each of the consumers in I to use alternative m, and 0 for all other alternatives. This implies that each of the consumers in I will use alternative m. Now, for consumer i ∈ / I: by the above discussion all consumers in I use alternative m. Consider the payment in P that this consumer is offered for using each alternative j. Fixing consumers in I, let pj be the payment for i for using alternative j, and let j ∗ = arg maxj pj . Denote by vj the value of consumer i for using alternative j (fixing the consumers in I). √ First, suppose that pj ∗ ≥ α − 1. Let the values of the consumer be such that vm , . . . , v1 ≤ , for arbitrarily small  > 0 (we can choose such values by the bounded away from 0 assumption). Let x be the alternative used by the consumer in the scheme. Denote by alg the cost of the solution. We claim that √ alg ≥ (n − 1) · (1 + ) + α − . The contribution of consumers in I to the cost is (n − 1) · (1 + ), thus we only have to bound the contribution of consumer i. The profit of consumer i for using any alternative j is vj + pj . In particular, √ the profit of consumer i for using alternative j ∗ is at least pj ∗ ≥ α − 1. Since vj ≤  for any alternative j, and since the profit of the chosen alternative √ √ j 0 is at least α − 1 (since the profit is at least α − 1 when using j ∗ ), we √ have that pj 0 ≥ α − 1 − . Thus the cost of consumer i in P is at least √ √ 1 + α − 1 −  = α − . In contrast, consider the payment scheme that pays  to consumer i for using alternative m and 0 to any other alternative. The consumer prefers to use alternative m, and thus the total cost is n(1 + ). This √ gives the bound for the case where pj ∗ ≥ α − 1. √ The complement case is when pj ∗ < α − 1. Consider an instance where √ v2 = . . . = vm = 1 and v1 = α − 1 (we can choose such values by the bounded away from 0 assumption). There exists a solution with total cost (n − √ √ 1) · (1 + ) + α: pay α − 1 to the consumer to use any alternative except alternative 1. However, the cost of the solution induced by P is at least (n − 1) · (1 + ) + α, since consumer i prefers to use alternative 1. This concludes the proof for deterministic schemes.

We now extend this proof to randomized schemes. By Yao’s principle, it is enough to provide a distribution over the inputs for which every deterministic (n−1)·(1+)+α √ . Our distribuscheme provide a competitive ratio no better than 2(n−1)·(1+)+ α tion has two instances in its support, in both each alternative j has the same cost wj . With probability 21 we will have the instance in which vm , . . . , v1 ≤ , and √ with probability 21 the instance in which v2 = . . . = vm = 1 and v1 = α − 1. Following the deterministic case, every deterministic payment scheme provides (n−1)·(1+)+α √ . for at least one of these two ina competitive ratio of at least (n−1)·(1+)+ α stances, hence the expected competitive ratio of every deterministic scheme for (n−1)·(1+)+α √ , as needed. this distribution is no better than 2(n−1)·(1+)+ α

Proof of Theorem 3 Consider the following setting with m ≥ 3 alternatives and n consumers. Let wm = . . . = w2 = 1, w1 = α. We will have a set I = {3, . . . , n}, |I| = n − 2 of consumers that will have the same valuation in all the instances we consider. For each alternative j and each consumer r ∈ I we have that vji · Lj (1) ≤ , for some arbitrarily small  > 0. The only consumers that will have different valuations are consumers 1 and 2. For the rest of the proof we fix some payment scheme P and analyze the performance of this payment scheme. We will show that the optimal cost in the instance we construct is no much more than n. Thus, any solution in that instance in which at least one consumer uses alternative 1 has a cost of (n − 1) + α, and this will prove the lower bound. We use the fact that consumers in I are almost indifferent to which alternative they are using in the sense that if any of them is offered a payment of  to use some alternative j and 0 to all other alternatives, each of these consumers will prefers alternative j regardless of the number of other consumers that are using j and the attractiveness functions. Indeed, we offer  to each of the consumers in I to use alternative m, and 0 for all other alternatives. Thus, each of the consumers in I will use alternative m. Since the actions of consumers in I are fixed, we denote by vji (1) the value of each consumer 1 and 2 for using alone alternative j ∈ {1, . . . , m − 1} and by vji (2) his value for using these alternatives together with the other consumer. Slightly abusing notation, for alternative m we let vji (1) and vji (2) denote the value of consumers 1 and 2 for using alternative m only with consumers in I and all consumers together, respectively. Similarly, let pij (k) denote the payment in P for consumer i ∈ {1, 2} for using alternative j without the other consumer (when k = 1) or with him (k = 2). Let j 0 ∈ arg max p2j (1).

We use the following valuations of the consumers. For consumer 1: v11 (1) = p1j 0 (2), vj10 (1) = p1j 0 (2), and 0 for all other values. For consumer 2: vj2 (1) = 0, for every alternative j. We now claim that in P it is a Nash equilibrium for consumer 1 to use alternative 1, and consumer 2 to use alternative j 0 . Consumer 2 clearly does not want to deviate since his payment (and thus, his total utility) decreases when using any other alternative. As for consumer 1, his utility from using alternative j is p1j (2), and has the same utility for using alternative j (here we break ties in favor of alternative 1). On the other hand, consider a scheme where consumer 2 receives a payment of δ for using some alternative j 6= j 0 (regardless of the number of consumers using j) and 0 for using any other alternative. The scheme also pays nothing to the consumer 1 regardless of his actions. We now calculate the cost of this solution. First, observe that the consumer indeed uses alternative j, the only alternative for which his utility is positive. Now observe that consumer 1 uses alternative j 0 (we break ties in favor of alternative j 0 here). The cost of the optimal solution is at most (n − 2) ·  + 2 + δ. Proof of Theorem 4 Consider some scheme M ∗ . By Lemma 3 it suffices to show that the square root scheme M is payment bounded w.r.t. M ∗ . Let f denote and equilibrium in M and f ∗ denote an equilibrium in M ∗ . To prove that Lemma 3 holds, we have consider a case where there exists an expensive alternative e s.t. fe > fe∗ . Thus, fe = fe1 = 1. If the consumer uses an expensive alternative in f ∗ , then √ we immediately get our α bound, since by definition the cost of any expensive alternative is α and the cost of the square root scheme is at most α. Thus assume that there exists a cheap alternative j such that fj∗1 = 1. We √ need to show that p1 (f ∗ ) ≥ α − wj . For each alternative j, denote by p∗j , pj the payment in f ∗ , f when the consumer uses j in M ∗ and M , respectively. We also denote by vj the value of the consumer for using alternative j (this does not depend on the congestion, as there is only a single consumer). In f the consumer uses alternative e and does not receive any payment and thus pj ≤ ve − vj . On the other hand, in f ∗ the consumer uses alternative j: p∗j ≥ p∗e + ve − vj ≥ p∗e + pj √ ≥ α − wj Thus, by Lemma 3, CM (f ) ≤

√

∗ (f ∗ ) for every scheme M ∗ . α · CM