# Market Design & Analysis for a P2P Backup System

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5 arge the bounds on his suppy are, because he has negative utiity for giving up his resources. To make this more forma, we et K i = w i X i with K i S U D A, denote the vector of resources that the user keeps, i.e., his endowment minus the suppy he gives up. We can specify the user s preference reation over a the resources he keeps, and the services he consumes: i (K is, K iu, K id, K ia, Y ib, Y iσ, Y ir. We make the foowing assumption (cf. [11], chapters 1-3: Assumption 1. Each user s preferences are (i compete, (ii transitive, (iii continuous, (iv stricty convex, and (v monotone. Note that strict convexity requires margina rates of substitution between two goods, i.e., the user needs to get more and more of one good when we take away 1 unit of another good. This is a reasonabe assumption for the user s resources and services because it represents a preference for diversification. Monotonicity means that a commodities are goods, i.e., if we give users more of any of the commodities, they are at east as we off as before. Given compete, transitive, and continuous preferences, there exists a utiity function u i(k i, Y i = u i(k is, K iu, K id, K ia, Y ib, Y iσ, Y ir that represents the preference reation and this utiity function is continuous (cf. [11], p Prices and Fow Constraints The system can avoid non-inear prices and support an equiibrium with inear prices as ong as the user specifies a suppy vector within the sack region that the system aows for each user s suppy. The ony resource that is not subject to these sack regions, or imits, is avaiabiity: as ong as the user s avaiabiity is arger than zero, the other resources can be used. To simpify the pricing mode, we introduce three new composite resources S, U, and D, incorporating the user s avaiabiity into the other resources in the foowing way: X iu U = X iu X ia X id D = X id X ia X is S = ϕ(x is, X ia X is X ia overhead factor Note that this notation denotes composite and not vector quantities. The definitions for the composite resources upoad and downoad bandwidth are straightforward: we mutipy the bound on bandwidth the user suppies (e.g., 300 KB/S with the user avaiabiity [0, 1] and then mutipy it with 24 hours, 60 minutes and 60 seconds, to cacuate how many KBs we can actuay send to this user per day. The definition of X is is a itte more intricate because the user s avaiabiity does not enter ineary into the cacuation. However, it enters monotonicay, i.e., more avaiabiity is aways better. Here, it suffices to know that the server can compute this function ϕ and convert a user s space and avaiabiity suppy into the new composite resource; further detais are beyond this paper. We can now define user i s suppy vector for the three composite resources: X i = (X is, X iu, X id. The advantage of using these composite resources is that now, the suppy from different users with different avaiabiities is comparabe. For exampe, 1 unit of S from agent i with avaiabiity 0.5 is now equivaent to 1 unit of S from agent j with avaiabiity 0.9. Obviousy, internay agent i has to give much more space to make up for his ower avaiabiity, but in terms of bookkeeping, we can now operate directy with composites. We define the aggregate suppy vector for the composite resources as X = i Xi, and anaogousy for Y, x and y. We make the foowing we-known observation (cf. [11], chapter 3 that wi be usefu ater: Observation 1. The suppy and demand functions X i and Y i are homogeneous of degree zero. This impies that the aggregate suppy and demand functions, i.e., X and Y are aso homogeneous of degree zero. Now we get to the pricing aspect of the system. We use p = (p S, p U for the prices for suppied composite resources, and q = (q B, q Σ, q R for the demanded services. We require that in steady state, users can pay for their consumption with their suppy. We can express this fow constraint formay: X i p = Y i q. (1 At the same time, the server aocates enough work to user i such that the user s current suppy x i is enough to pay for the demand y i, which eads to a second fow constraint: x i p = y i q. (2 4.2 Production Functions We have aready mentioned the important roe of the server in our market, i.e., that of combining resources from different suppiers into a vauabe bunde. Formay, the server is the ony producer in our market. 3 For each service, we have a production function that defines how many input resources are needed to produce one unit of that service: Backup: f B : S U D B Storage: f Σ : S U D Σ Retrieva: f R : S U D R Note that these production functions are defined via the impementation of our system, i.e., the particuar production technoogy that we impemented. For exampe, they are defined via the particuar erasure coding agorithm that is being used, by the frequency of repair operations, etc. Thus, we can now specify a series of properties that these production functions guarantee due to our impementation: System Property 1. Production functions are fixed and the same for a users. System Property 2. The production functions a exhibit constant returns to scae (they are homogeneous of degree 1, i.e., {B, Σ, R} : f (k a, k b, k c = k f (a, b, c k R. System Property 3. Each production function is bijective, and thus we can take the inverses: B : B S U D Σ : Σ S U D R : R S U D Given the inverse functions for the individua services backup, storage, and retrieva, we can define an inverse function for a three-dimensiona service vector (b, σ, r B Σ R: (b, σ, r = B (b + Σ (σ + R (r (3 3 Note that this is what aows us to define an exchange economy despite the fact that production is happening in the market. For more detais see [11], pp

6 Property 1 hods because of the way we have defined the composite resources, with any differences between the agents avaiabiities aready considered. Property 2 (CRTS hods approximatey for fie sizes above a certain threshod (approx. 1MB due to the properties of the erasure coding agorithm. 4 Property 3, the bijectivity of production, hods, because for each service unit, there is ony one way to produce it. For exampe, to backup one fie fragment, the erasure coding agorithm tes us exacty how many suppier fragments we need, and the server tes us how much repair and testing traffic we can expect on average per fragment. The foowing system s property comes from the UI design and is motivated by keeping the money fow in the system constant. This property is ony possibe because the server is the ony producer in our system and production functions are fixed (Property 1, exhibit constant returns to scae (Property 2, and are bijective (Property 3: System Property 4. We charge the consumers exacty the amount we pay the suppiers, i.e., for demand vector y i, we charge user i exacty: y i q = (y p. Using Property 4, we can now re-write the fow constraints for agent i as: X i p = (Y i p and x i p = (y i p Thus, from now on, we can omit the price vector q for demanded services and ony need to consider price vector p. 5 Effectivey, we can treat the whoe market as an exchange economy for the composite resources S, U, and D, assuming users ony engage in exchange of suppied resources because everyone has access to the same production technoogy (cf. [11], pp Remember that the UI automaticay cacuates and adjusts the maximum demand vector Y i for user i based on the user s suppy bound X i. In practice, the maximum income is divided by the current average income of the user, and the resuting factor is mutipied with the user s current demand, giving us the maximum demand the user can afford: System Property 5. The system uses a inear demand prediction mode for the cacuation of a user s maximum demand Y i : Y i = Xi p yi = λi yi x i p To faciitate the equiibrium anaysis in the next section, we make the foowing simpifying assumption: Assumption 2. We assume that with a arge number of users, a inear demand prediction is aso correct for the aggregate demand vectors, i.e.: λ : Y = λ y 4 Very sma fies are an exception and need specia treatment in the impementation, because they are more expensive to be produced (again due to the erasure coding. We take care of this in the impementation by charging users more when they are backing up sma fies (essentiay we have two sets of prices, one for norma fies and one for sma fies. 5 Going forward, pease remember that mutipications with p are aways dot products, and thus p showing up on the eft and the right side of an equation does not cance out. This assumption is justified because in practice, the system wi have thousands or miions of users. Let n denote the number of users in the economy, et Y n = n i=1 Yi, y n = n i=1 yi, and et µ(λi denote the mean of the distribution of the λ i s. Given that the λ i s are independent from the y i s, it foows from the strong aw of arge numbers, that if the number of users n is arge enough, then Y n is ineary predictabe by µ(λ i y n aong each dimension to any additive error. More specificay, for any ε and δ 0, for arge enough n: P r[ Y n µ(λ i y n ε] 1 δ. 5. EQUILIBRIUM ANALYSIS A rea-word instance of the P2P backup appication woud have thousands if not miions of users. Thus, the underying market woud be arge enough so that no individua agent had a significant effect on market prices. Consequenty, users can be modeed as price-taking agents and a genera equiibrium mode is suitabe to anayze this market. 5.1 The Buffer Equiibrium A standard equiibrium concept in genera equiibrium theory is the Warasian equiibrium which requires that demand equas suppy such that the market cears. Certainy we want to have enough suppy to satisfy current demand, i.e., we want that: x = (y. But remember that users are not constanty adjusting x i. Instead, they choose maximum bounds on their suppy X i via the UI. But given that the maximum suppy X i is generay arger than x i, it is not a very strong requirement to have x = (y. In particuar, we do not ony want to baance the market now, but we want to guarantee that the backup system can aso satisfy demand Y in the future, which impies that we must aways have some excess suppy of a resources. Ideay, we want to maximize the buffer between the current usage of resources, i.e., (y and the maximum suppy of resources, i.e., X. We wi use this size of the buffer repeatedy and thus define it more formay: Definition 1. (Size of the Suppy-Side Buffer The size of the suppy-side buffer is the smaest ratio, over a resources, of maximum suppy to current demand: λ = min {S,U,D} X (y The reason for having a suppy side buffer is that we want to be safe, i.e., we want to be sure that we can satisfy new incoming requests. More specificay, we want to make sure that as demand increases from its current state y to the maximum state we aow the users Y, we wi aways have enough suppy to satisfy this demand. More formay: Definition 2. (Safety Property The safety property of the system is that we aways have enough suppy to satisfy increasing demand, i.e.: (4 y Y : X (y (5 Using the bijectivity and the CRTS properties of the production functions, it is easy to show the foowing emma:

7 Lemma 1. If X = (Y, then the safety property is satisfied. In words, when the bound on aggregate suppy of a resources equas the amount of resources needed to produce the projected service vector Y, then we can guarantee the safety property. When we have reached this state of the system, we say we have reached the buffer equiibrium: Definition 3. (Buffer Equiibrium A Buffer equiibrium is a price vector p = (p S, p U an aggregate suppy vector X(p and an aggregate demand vector Y (p such that: X(p = (Y (p i.e., it is a Warasian equiibrium defined on the suppy and demand bounds chosen by the users. We ca this equiibrium the buffer equiibrium because the extent to which X is above (y, i.e., the size of the buffer, determines the eve of safety in the system. 5.2 Equiibrium Existence In this section, we wi prove that a buffer equiibrium exists under some reasonabe assumptions. To do so, we first introduce some new notation and prove two Lemmas before we get to the actua theorem. We et L = {S, U, D} and we use to index a particuar composite resources. We define the vector-vaued function Z(p to measure the reative buffer for each individua resource in the foowing way: ( Z (p = X (p (y(p L X (p (y(p In words, the first term represents the average suppy to demand ratio, in our case averaged over the three goods storage space, upoad and downoad bandwidth. The second term represents the suppy to demand ratio of the particuar good. Thus, Z (p represents how far the buffer between suppy and demand for good is away from the average buffer. If Z is negative, then the buffer between suppy and demand for good is reativey high and shoud be decreased; if Z is positive, then the buffer between suppy and demand for good is reative ow and shoud be increased. If the buffer is the same for a goods, we have reached the equiibrium. Thus, we can prove the foowing Lemma: Lemma 2. If Z(p = 0, then the market has reached a buffer equiibrium and p is the equiibrium price vector. Proof. If Z(p = 0 then: ( : X (p (y(p L = X (p (y(p λ > 1 s.t. : X (p = λ (y(p Now, due to Assumption 2 we know that δ : Y = δ y. Thus: (6 : X (p = λ ( 1 Y (p δ (7 : X (p = λ 1 δ (Y (p (8 : X (p = λ (Y (p for λ = λ 1 δ (9 X(p = λ (Y (p (10 But from the fow constraints (Eqn. 4 we aso know that: X(p p = (Y (p p (11 Equations (10 and (11 can ony both be true if λ = 1. Thus, it foows that: X(p = (Y (p which is the definition of the buffer equiibrium. Next we show that Z( has a series of nice properties: Lemma 3. The function Z( has the foowing properties: (i Z( is continuous. (ii Z( is homogeneous of degree zero. (iii p : Z (p = 0. (iv If p n p, where p 0, p > 0 and p k = 0 for some k, then for n sufficienty arge: Z k (p n = max{z S (p n, Z U (p n, Z D (p n }. Proof. Property (i, the continuity of Z( foows directy from the continuity of the user preferences (which is why X(p and y(p are continuous and the continuity of the inverse production functions. Property (ii, the homogeneity of degree zero foows because X(p and y(p are homogeneous of degree zero. Property (iii, foows directy from the definition of Z( : ( Z (p = 3 X (p (y(p L {S,U,D} X(p (y(p = 0 Finay, property (iv: as the price of resource k {S, U, D} goes towards zero, due to users strongy monotone preferences for suppy resources, they wi suppy ess and ess of that resource, and suppy more of the other resources instead, at east of resource whose price is bounded away from zero. However, because of the bunde constraints, the users cannot reduce their suppy of resource k towards zero. Let γ > 1 denote the sack factor we aow users when setting their preferences. The reevant constraints, ower-bounding the suppy for resource k, are: L \ {k} : X ik 1 λ k (Y (Y X i As p n p with p k = 0, for n arge enough, p n wi be sufficienty cose to zero, such that each user i chooses to suppy the minima amount of resource k that is possibe. Thus, at east with respect to one of the other resources, the sack constraint wi be binding, i.e.,: { 1 X ik = max λ k (Y (Y X i, 1 λ k (Y } fm 1 (Y Xim This does not say that the constraint wi be binding for the same resource or m for every user. However, for every user, one of the constraints wi be biding and thus, every user wi contribute east to the suppy side buffer for resource k. Consequenty, the tota suppy side buffer for good k wi be minima (i.e., Z k (p n wi be maxima, which impies that Z k (p n = max{z S (p n, Z U (p n, Z D (p n }.

8 Theorem 1. A buffer equiibrium exists in the P2P exchange economy, given that users preferences are continuous and stricty convex, monotone w.r.t. service products as we as strongy monotone w.r.t. to suppy resources. Proof. We have shown in Lemma 2 that once we have found a price vector p such that Z(p = 0, we have reached a buffer equiibrium. Furthermore, in Lemma 3 we have shown four properties of Z(. Equipped with these two resuts, the remainder of our proof foows using techniques from the standard equiibrium existence proof for the Warasian Equiibrium (see [11] page 586. We omit the detais here due to space constraints, but we want to briefy point out where changes in the proof are necessary. First, note that we are not working with the excess demand function of this economy and instead use the function Z( that measures the reative buffer size for each resource. Thus, in step 1 of the proof in [11], we cannot use Waras aw and instead use property (iii of Lemma 3. Second, in step 4 of the proof in [11], when proving upper hemicontinuity of the fixed-point correspondence, we cannot use the resut that the excess demand for one resources goes to infinity when its price goes towards zero. Instead, we use property (iv of Lemma 3. One might wonder what prevented the direct appicabiity of standard theorems regarding equiibrium existence (e.g., [11] page 585. It turns out that the standard resuts were not directy appicabe for three reasons. Most importanty, we do not assume that users have strongy monotone preferences w.r.t. service products. The consequence of this is that as the price of one resources goes towards zero, it is not necessariy the case that the demand for service products produced from that resource go towards infinity. Second, we do not have a pure exchange economy and have to take the production functions into account. Those exhibit constant returns to scae which impies that production sets are neither stricty convex nor bounded above, which compicates the anaysis significanty (cf. [11] page 583. Third, each user s suppy is subject to the bunde constraints, i.e., a user s suppy cannot drop beow or go above certain imits. Due to these three factors, we needed sighty different machinery to prove equiibrium existence in our economy. 5.3 Equiibrium Uniqueness Without any further restrictions on the user s preferences, we cannot say anything about the uniqueness of the buffer equiibrium (cf. Sonnenschein-Mante-Debreu Theorem, [11], pp , because the substitution effect and the weath effect coud either go in the same direction or in opposite directions.the gross substitutes assumption resoves this probem, by assuming that in gross terms, taking substitution and weath effect into account, the resources are substitutes. We assume that this is the case for the suppy resources: Assumption 3. (Suppy Resources are Gross Substitutes We assume that the aggregate suppy function X(p satisfies the gross substitutes condition [1], i.e., whenever p and p are such that, for some k, p k > p k and p = p for k, we have X (p < X (p for k. The standard equiibrium uniqueness proof for Warasian equiibria reies on the assumption that the aggregate excess demand function satisfies the gross substitutes condition for a commodities. However, we ony want to make that assumption w.r.t. suppy resources where this seems very reasonabe because as the price for one resource decreases, this means that the reative price for another resource increases, and thus users woud be happy to suppy more of the more costy resources now. However, for the demanded services, the gross substitutes assumption is most certainy vioated. For exampe, if the price for storage woud increase, it is not reasonabe to assume that now users woud store fewer fies onine, but instead consume more backup and retrieva operations. Thus, we cannot make the gross substitutes assumption for a commodities. Instead, we wi make the foowing assumption w.r.t consumed services: Assumption 4. (Services are Perfect Compements We assume that the aggregate demand function Y (p satisfies the perfect compements condition. A consequence of the perfect compements condition is that price changes affect a dimensions of the aggregate demand vector equay. For an individua user, the Leontief utiity function woud induce the perfect compements property. However, note that we do not require individua users to have demand functions that satisfy the perfect compements condition. It is a much weaker assumption, and much more reasonabe due to the aw of arge numbers, to assume that the aggregate demand function satisfies it. Theorem 2. The buffer equiibrium is unique (up to normaization, given that the aggregate suppy function satisfies the gross substitutes property (Assumption 3, and that the aggregate demand function satisfies the perfect compements property (Assumption 4. Proof. The fact that we have different assumptions on the suppy and demand side of our economy compicates the uniqueness proof. When prices go up for good, it is not a priori cear what happens to the buffer equiibrium. To get a better hande on this, we first separate the suppy and demand aspects by introducing yet another aternative description of the buffer equiibrium: X = (Y (12 ( (X S, X U, X D = 1 1 (Y, f (Y, f (13 S U D ( 1, X U, X D = (1, U X S X S D S S (14 ( XU, X ( D U X S X S D = 0 S S (15 We define a new vector-vaued function g(p = ( g U (p, g D (p : ( XU g U (p = (Y U X S (Y S ( XD and g D (p = X S (Y D (Y S which naturay eads to a new equiibrium definition: Definition 4. (Buffer Equiibrium [1. Aternative] A buffer equiibrium is a price vector p and g(p such that ( 0 g(p = 0 Thus, we have simpified the probem of finding equiibrium prices to finding the root of the function g(p. Showing uniqueness of the buffer equiibrium is now equivaent to,

9 showing that g(p = 0 has at most one (normaized soution. Now, et s assume that g(p = 0, i.e., p is an equiibrium price vector. We show that for any p, g(p 0 uness p and p are coinear, i.e., uness p = λp for some λ > 0. Note that because X(p and Y (p are homogeneous of degree zero, g( is aso homogeneous of degree zero. Thus, we can assume that p p and p = p for some. We now ater the price vector p to obtain p in two steps, owering (or keeping unatered the price of resources k one at a time. Because of Assumption 4 (the aggregate demand function satisfies the perfect compements condition, a price change affects a dimensions of the demand function equay, i.e., µ R : Y (p = µ Y (p. Because the production function is bijective and exhibits constant returns to scae, this impies that (Y (p = µ (Y (p. Thus, 1 (Y (p f U U = (Y (p (Y (p S S (Y (p, i.e., changes in the demand function Y ( due to price changes do not affect g(. Thus, we ony have to pay attention to changes in the suppy function X. Here, we need to differentiate the foowing 3 cases:. Case 1: =storage. By gross substitution (see Assumption 3, the suppy of good S cannot decrease in any step, and, because p p, it wi actuay increase in at east one step. In turn, the suppy of U and D wi stay the same or decrease because of homogeneity of degree zero. Thus, the first term in the g( functions wi decrease, whie the second term stays constants, and thus, g(p < g(p. Case 2: =upoad bandwidth. By gross substitution, the suppy of good U cannot decrease in any step, and, because p p, it wi actuay increase in at east one step. The suppy of S and D on the other hand wi stay the same or decrease. Thus, the first term in g U ( wi increase, whie the second term stays constants. Thus, g U (p > g U (p (note, we do not even need to consider g D (p in this case. Case 3: =downoad bandwidth. By gross substitution, the suppy of good D cannot decrease in any step, and, because p p, it wi actuay increase in at east one step. The suppy of S and U on the other hand wi stay the same or decrease. Thus, the first term in g D ( wi increase, whie the second term stays constants. Thus, g D (p > g D (p (again, we do not even need to consider g U (p is this case. In summary, in a three cases we estabished that g(p g(p which concudes the equiibrium uniqueness proof. 5.4 Limited Controabiity of the Buffer Size So far we have shown under what conditions the buffer equiibrium exists and when it is unique. We know from Lemma 1 that when the system is in the buffer equiibrium, then the safety property is guaranteed, i.e., we aways have enough suppy to satisfy demand as it increases from y towards Y. But what happens if the system is out of equiibrium? Note that in practice, users do not permanenty adjust their settings, and thus price changes wi ony affect suppy and demand with a significant deay. Consequenty, it woud be desirabe to have a arge enough buffer between current demand and maximum suppy, such that even if the system is out of equiibrium, we can satisfy new incoming demand. For exampe, it seems ike desirabe goa to have at east twice as much suppy as current demand, i.e., X 2 (y. Unfortunatey, the uniqueness of the buffer equiibrium has an immediate consequence regarding the imited controabiity of the buffer equiibrium: Coroary 1. (Limited Controabiity of the Market Given Property 4, and Assumptions 3 and 4, the market operator cannot infuence the size of the buffer in the buffer equiibrium. Given this imited controabiity, it is natura to ask what buffer size to expect in equiibrium. It turns out that, in equiibrium, the suppy side buffer is uniquey determined via the individua demand side buffers of a users. Proposition 1. In the buffer equiibrium, given Assumption 2, the size of the suppy buffer equas the size of the demand buffer. Proof. X = (Y (16 X = (λ y (17 X = λ (y (18 In words, the size of the buffer depends on how forwardooking the agents are. If on average the users give themseves a 25% buffer on the demand side (e.g., a user has currenty backed up 20GB and sets the siders in such a position that his/her maximum onine backup space is 25GB, then we woud aso have a 25% buffer on the suppy side, i.e., X = 1.25 (y. Now we turn to the question why the market operator cannot infuence the size of the suppy side buffer, i.e., which system properties or which assumptions we made in our market economy are the imiting ones. Remember that the imited controabiity of the buffer equiibrium was a coroary of the uniqueness property, which reied on two assumption, namey gross substitutabiity of suppied resources, and that services are perfect compements. It turns out, however, that the imited controabiity remains even without those assumptions, strengthening the resut from Coroary 1: Proposition 2. Given system property 4, if each individua user i has a imited panning horizon in that he chooses not to give himsef more than a demand side buffer of λ i, then there exists a Λ such that the market operator cannot achieve a buffer equiibrium with buffer size Λ. Proof. For the proof we construct a simpe counterexampe. We aow price changes to affect x, X, y and Y and in particuar we do not make the gross substitutabiity assumption or the perfect compements assumption. We choose a Λ such that i : Λ > λ i. And we et λ i = max i λ i. Now: i : Y i = λ i y i (19 Y = i λ i y i (20 Y λ i y i (21 i Y λ i y i (22 i Y λ i y (23 (y λ i (y (24 X λ i (y (25 Thus, the buffer between suppy and demand woud be ess or equa to λ i which by assumption was stricty ess than the buffer Λ that the market operator desired.

11 Definition 5. (The Price Update Agorithm { 1 for = S p t+1 = p t p t pt 1 g (p t g (p t 1 g (p t for = U, D For the impementation of the price update agorithm in our system we took care of a few specia cases (e.g., exacty reaching the equiibrium such that terms cance out. Due to space constraints, we omit the detais here. 6.2 Theoretica Convergence Anaysis In this section we prove convergence of the price update agorithm under quite mid assumptions. We begin with the anaysis of the convergence of the foowing iteration rue: x (k+1 = x (k D(x (k 1 F (x (k (29 where F is a function F : R n R n and D is the diagona sub-matrix of the Jacobian J of F. We define the matrix L by the rue J(x = D(x + L(x, i.e., L comprises of the off-diagona partia derivatives in the Jacobian. For this iteration rue, the foowing theorem hods (proof is omitted due to space constraints: Theorem 3. Let F be a continuousy differentiabe function. Suppose that in the iteration rue given by equation (29, x (0 is chosen cose enough to a root x of F, J(x is non-singuar, J and D are Lipschitz continuous, and L(x = 0. Then the successive iterations x (k produced by the iteration rue converge to x, and the rate of convergence is at east Q-inear. 6 The probem one faces when trying to appy the secant method to higher dimensions is that the system of equations provided by J k (x (k x (k 1 F (x (k F (x (k 1 (where J k is the current estimate of the Jacobian is under determined. However, if one uses the diagona approximation to the Jacobian, then the system is fuy determined. What Theorem 3 says is that under certain conditions, using the diagona sub-matrix of the Jacobian instead of the fu Jacobian in the given iteration rue, sti eads to convergence to a root of the function. Equipped with Theorem 3, it is now easy to prove that the price update agorithm given in Definition 5 converges to a buffer equiibrium. We ony need to consider the update agorithm for resource prices p U and p D because the price for space remains constant at 1. Consider the function g(, and as before, J is the Jacobian of g(, D is the diagona submatrix of J, and L is defined by the rue J(x = D(x+L(x. Coroary 2. Consider the price update agorithm given in Definition 5. If g( is a continuousy differentiabe function, p (0 is chosen cose enough to a root p of g(, the Jacobian J(p is non-singuar, J and D are Lipschitz continuous, and L(p = 0, then the price update agorithm converges to an equiibrium price vector p, and the rate of convergence is at east Q-inear. 6 We can in fact prove that the iteration rue exhibits faster than Q-inear convergence: just ike Broyden s method, its convergence is ocay Q-superinear. However, showing this resut requires a more intricate argument and we defer this to a future extended version of this paper. Proof. We have shown in Section 5.3 that if we find a price vector p such that g(p = 0, then we have reached a buffer equiibrium. Thus, we ony have to show that the price update agorithm converges to a root of the function g(. Now, note that the price update agorithm provided in Definition 5 defines a quasi-newton iteration rue that uses the diagona sub-matrix of the Jacobian of the function g(, equivaent to the iteration rue given in equation (29. By Theorem 3, that iteration rue converges ocay to a root of g(, and the rate of convergence is at east quadratic. One might wonder how restrictive the conditions of Theorem 3 and Coroary 2 are. The condition that the matrices J and D be Lipschitz continuous puts upper bounds on how fast the partia derivatives of the function can change. It turns out that one can reax this assumption to just that of J and D being Lipschitz continuous in a neighborhood of the root without affecting the concusions of the theorem and coroary. Loca Lipschitz continuity near the neighborhood of the root seems ike a pausibe condition for g( to satisfy because it is hard to envision wid changes in the function near an equiibrium point. The non-singuarity of J(p means that our function does not have a higher order zero at the equiibrium point. If this assumption fais, then the Newton method is known to converge ony ineary to the root. It is ikey that our agorithm woud sti converge, but we do not have a proof of this. The oca convergence of our method is an aspect we share with a Newton s methods operating in mutipe dimensions, and this is the most worrisome property as we as the hardest to get a hande on. If J(p 1 and Lipschitz constants of J and D around p are a sma, then the basin of convergence is arge. However, it seems that ony experimenta evidence can vaidate whether this assumption is reasonabe in our situation. 6.3 Experimenta Convergence Anaysis In this section we anayze the convergence of the price update agorithm experimentay via simuations. To make sure a unique buffer equiibrium exists, we must specify utiity functions that satisfy the properties of Theorems 1 and 2. It is we known that the CES (constant easticity of substitution utiity function with the foowing form u(x 1, x 2 = (α 1x ρ 1 + α2xρ 2 p 1 satisfies the gross substitutes condition for 0 < ρ < 1 (cf. [11], p 612. Furthermore, the utiity function u(x 3, x 4 = min(α 3 x τ 3, α 4 x τ 4 impies that x 3 and x 4 are perfect substitutes (whie sti maintaining stricty convex preferences. Thus, a whoe cass of utiity functions satisfying the necessary assumptions is given by: u i (K i, Y i = (α 1 K ρ S + α 2K ρ U + α 3K ρ D 1 ρ + min(β 1 Y τ B, β 2 Y τ Σ, β 3 Y τ R Because this utiity function is too compex to derive suppy and demand functions anayticay, we used the NLPsover MINOS which is abe to sove the utiity maximization probem via reduced-gradient methods very efficienty. Experimenta Set-up. We impemented a market simuation with 100 agents, each initiaized with a different utiity function. For each agent we randomy picked an endowment vector w i, a current demand vector y i, and the parameters α 1, α 2, α 3, β 1, β 2, β 3, ρ and τ. After each time step, we used MINOS to sove 100 utiity maximization probems and fed the new suppy and demand vectors back into the simuation to cacuate the next price iteration. We stopped the experiment once g U + g D

12 Experimenta Resuts. Our findings from this experiment confirm the resuts from the theoretica convergence anaysis. When the initia price vector was chosen cose enough to the equiibrium price vector, then the price update agorithm converged very quicky. However, when a price vector far away from the equiibrium prices was chosen, the agorithm sometimes never converged, which was expected. Whie these preiminary resuts are encouraging, we are are much more detaied experimenta anaysis of the price update agorithm in the future. For exampe, we wi study how the agorithm reacts to demand and suppy shocks, what happens when the users update their settings with different deays, what happens when a subset of the users becomes price-insensitive (cf. [13], discussion about price discoverabiity, or what happens if individua users suddeny drop out of the market or join the market. 7. CONCLUSION In this paper, we have presented the design and anaysis of a nove resource exchange market underying a P2P backup appication. At a times, for the mode formuation and the theoretica anaysis, the focus was on the actua impemented system which we have successfuy tested in apha version. During the design phase for this market, we foowed the hidden market design paradigm [14]. The resut is a system that hides many of the market compexities and aows the users to interact with the market in a very indirect way. In contrast to existing work in this area where users are generay required to constanty update their suppy and demand, in our mode users choose bounds on their maximum suppy in return for being aowed to consume a certain maximum amount of backup services in the future. In this setting, we have proved the existence and uniqueness of a buffer equiibrium under reasonabe assumptions. We have introduced a price update agorithm and have shown that it converges Q-ineary, given that initia prices are chosen cose enough to the equiibrium. However, we are panning a more extensive experimenta anaysis to fuy understand the behavior of the agorithm. The most surprising resut is the imited controabiity of the size of the suppy side buffer in equiibrium. We have shown that suppy and demand prices woud have to be decouped to enabe more contro. In future research, we wi study how to design new UIs that aow for this decouping without presenting fase information to the users. In ongoing work we are augmenting the market design with a payment mechanism that wi provide for robustness against strategic deviations from users that manipuate the protoco by reprogramming their software cient. Furthermore, we are extending our current design to aow for rea monetary payments. On the one side, users wi then be abe to pay for their consumption of services by either providing their own resources or by paying with rea money, and on the other side, users wi then aso be abe to earn rea money by suppying their resources. We beieve that the market design presented in this paper has appicabiity beyond P2P backup systems. For exampe, significant research efforts currenty go into the deveopment of smart grids [6]. To effectivey invove the individua consumers of eectricity in these new energy markets, the deveopment of new user interfaces and hidden market designs wi be necessary. Hopefuy, the ideas we presented in this paper wi inspire other researchers to deveop simiar market designs for nove appications in many other domains. Acknowedgements We are thankfu to David Parkes, Kama Jain, and Aex White for very hepfu discussions on this work. Furthermore, we thank seminar participants from Harvard EconCS, HBS, Humbodt University Berin, and NetEcon 09 for very usefu feedback. Seuken gratefuy acknowedges the support of a Microsoft Research PhD Feowship. 8. REFERENCES [1] J. Aexander S. Keso and V. P. Crawford. Job Matching, Coaition Formation, and Gross Substitutes. Econometrica, 50(6: , [2] C. Aperjis and R. Johari. A Peer-to-Peer System as an Exchange Economy. In Proceedings from the Workshop on Game Theory for Communications and Networks (GameNets, Pisa, Itay, October [3] W. J. Boosky, J. R. Douceur, and J. Howe. The Farsite Project: A Retrospective. SIGOPS Operating Systems Review, 41(2:17 26, [4] C. G. Broyden, J. E. Dennis Jr., and J. J.Moré. On the Loca and Superinear Convergence of Quasi-Newton Methods. J. Inst. Math. App., 12: , [5] L. P. Cox and B. D. Nobe. Samsara: Honor Among Thieves in Peer-to-Peer Storage. In Proceedings of the nineteenth ACM symposium on Operating systems principes (SOSP, pages , Boton Landing, NY, [6] U. S. D. O. Energy. Grid 2030: A Nationa Vision for Eectricity s Second 100 Years [7] M. J. Freedman, C. Aperjis, and R. Johari. Prices are Right: Managing Resources and Incentives in Peer-Assisted Content Distribution. In Proceedings of the 7th Internationa Workshop on Peer-to-Peer Systems, Tampa Bay, FL, February [8] J. Kubiatowicz, D. Binde, Y. Chen, S. Czerwinski, P. Eaton, D. Gees, R. Gummadi, S. Rhea, H. Weatherspoon, W. Weimer, C. Wes, and B. Zhao. Oceanstore: An Architecture for Goba-Scae Persistent Storage. In Proceedings of the Ninth internationa Conference on Architectura Support for Programming Languages and Operating Systems (ASPLOS, [9] K. Lai, L. Rasmusson, E. Adar, L. Zhang, and B. A. Huberman. Tycoon: An Impementation of a Distributed, Market-based Resource Aocation System. Mutiagent Grid Systems, 1(3: , [10] J. Li and C. Zhang. Distributed Hosting of Web Content with Erasure Coding and Unequa Weight Assignment. In Proceedings of the IEEE Internationa Conference on Mutimedia Expo, pages 27 30, Taipei, June [11] A. Mas-Coe, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, [12] S. Seuken, D. Chares, M. Chickering, and S. Puri. Market Design and Anaysis for a P2P Backup System. In Proceedings of the Workshop on the Economics of Networks, Systems, and Computation (NetEcon, Stanford, CA, Juy [13] S. Seuken, K. Jain, D. Tan, and M. Czerwinski. Hidden Markets: UI Design for a P2P Backup Appication. In Proceedings of the Conference on Human Factors in Computing Systems (CHI, Atanta, GA, Apri [14] S. Seuken, D. C. Parkes, and K. Jain. Hidden Market Design. In Proceedings of the 24th Conference on Artificia Inteigence (AAAI, Atanta, GA, Juy [15] L. Waras. Eéments d économie poitique pure; ou, théorie de a richesse sociae (Eements of pure economics; or, the theory of socia weath. Corbaz, Lausanne, [16] F. Ygge and H. Akkermans. Power Load Management as a Computationa Market. In Proceedings of the 2nd Internationa Conference on Muti-Agent Systems (ICMAS, pages , Kyoto, Japan, 1996.

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