Towards an efficient market mediator for divisible resources

Performance Evaluation 127–128 (2018) 212–234

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Performance Evaluation journal homepage: www.elsevier.com/locate/peva

Towards an efficient market mediator for divisible resources ∗

Mao Zou a,b , Richard T.B. Ma c , , Yinlong Xu a,b a b c

School of Computer Science and Technology, University of Science and Technology of China, China Anhui Province Key Laboratory of High Performance Computing (USTC), China School of Computing, National University of Singapore, Singapore

article

info

Article history: Available online 22 October 2018 Keywords: Divisible resource Allocation and acquisition Market efficiency Auction design Price differentiation

a b s t r a c t Auction-based markets of divisible resources have proliferated over recent years. One fundamental problem facing every market mediator is how to achieve market efficiency for optimal social welfare, especially when a limited number of agents forms a monopolistic or oligopolistic market, because each agent’s selfish strategic behavior may lead to serious degradation in efficiency. In general, it is difficult for a market mediator to achieve efficiency since agents’ preferences are hidden information that they are unwilling to reveal due to security and privacy concerns. Therefore, the design of auction mechanisms should align the selfish behavior of agents with the altruistic objective of social welfare and allow the mediator to elicit necessary private information during the auction process. In this paper, we consider a market of divisible resource consisting of agents on both sides of demand and supply. We design an adaptive auction framework for a market mediator to achieve efficient resource allocation and acquisition. Our novel design generalizes demand/supply function bidding mechanisms by introducing price differentiation via tunable parameters. We design algorithms that enable the mediator and agents to jointly run the market in an adaptive fashion: the mediator sends market signals to agents; each agent submits her bid based on the signals in a distributed manner; the mediator adjusts tunable parameters based on bids and update market signals. We also design an adaptive algorithm to dynamically determine the optimal amount of resource that needs to be transacted so as to maximize social welfare, if not known a priori. By utilizing our market mechanisms, the market mediator will be able to reach an efficient market outcome under Nash equilibrium. © 2018 Elsevier B.V. All rights reserved.

1. Introduction With the development of the Internet and information technology, markets have emerged as a new paradigm for managing resources and have much proliferated over recent years in complex systems such as online marketplaces [1], cloud services [2] and energy markets [3]. In general, a resource market consists of agents that are buyers and sellers of the resource and a market mediator that facilitates the transactions between buyers and sellers. In particular, auction-based markets have gained much popularity in recent years since auctions are known to be more flexible and agile compared to standard fixed-price mechanisms. For example, eBay runs auction-based e-commerce mechanisms, Amazon EC2 [2] uses auctions to sell spot instances, and electricity markets use day-ahead auctions to match dynamic demand and supply [4]. In such an auction-based market as shown in Fig. 1, agents can adjust their bids and the mediator can update auction parameters and send market signals to agents dynamically; and therefore, the overall efficiency ∗ Corresponding author. E-mail addresses: [email protected] (M. Zou), [email protected] (R.T.B. Ma), [email protected] (Y. Xu). https://doi.org/10.1016/j.peva.2018.09.012 0166-5316/© 2018 Elsevier B.V. All rights reserved.

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213

Fig. 1. Resource market model with a market mediator.

and flexibility could be significantly improved. Furthermore, concerns regarding the computational tractability of auctions have also been greatly alleviated by software that can calculate complex optimal strategies in a timely fashion. We focus on markets for divisible resources including computational resources and energy resources, where the use of auction mechanisms for resource management are prevalent. For example, auctions have been designed for resource management in virtualized data centers [5–7]. Auction-based systems have also been deployed for allocating and acquiring computing resources such as bandwidth, CPU cycles, and storage [8,9]. In these scenarios, the platform, i.e., data centers or systems, serve as the market mediator and users and cluster jobs are the agents. Electricity markets have largely relied on auction mechanisms to match power supply to demand by having utility companies submit supply functions and consumers participating in demand response programs [10,4,11]. Here the electricity grid serves as the mediator and the utility companies and consumers are the agents. One fundamental problem facing every market mediator is how to achieve efficiency, i.e., maximization of social welfare. Although welfare economists argue that individual self-interested actions could lead to efficient market outcomes due to Adam Smith’s invisible hand [12], this argument works only for competitive markets with a large population of agents on both sides, where each agent is aware of her insignificant market power and will act as a price-taker. Nevertheless, in markets with only a few agents such as monopolistic or oligopolistic markets, each agent will be price-anticipating and such selfish behavior may lead to serious efficiency loss. Since agents’ preferences such as value and cost functions are hidden information and agents are often unwilling to reveal them due to security and privacy concerns, it is in general very difficult for the mediator to achieve efficiency. Consequently, the design of auction mechanisms should align the selfish behavior of agents with the altruistic objective of social welfare and allow the mediator to elicit necessary private information during the auction process. In this paper, we design a unified framework for both the demand-side and supply-side auction mechanisms. Our novel approach generalizes demand/supply function bidding mechanisms by introducing price differentiation via tunable parameters into the auctions. As is illustrated in Fig. 1, the mediator and agents jointly run the market through rounds of iterations, where each iteration consists of the following three steps: (1) the mediator sends market signals such as differentiated prices to agents, (2) each agent submits her bid based on the received signals in a distributed manner, and (3) the mediator adjusts tunable parameters based on submitted bids of agents and update the corresponding market signals. The resource allocation and acquisition outcomes are determined by the agents’ bids in a steady-state characterized by Nash equilibrium. We summarize our findings and results as follows.

• We propose a unified adaptive auction framework for resource allocation and acquisition, where each agent is allowed to submit a one-dimensional bid that parameterizes her demand/supply function in Section 3.3. The mediator decides (1) the amount of resource allocated to/acquired from all the agents based on their bids, and (2) the corresponding payment transfers based on tunable price parameters. • We characterize the existence and uniqueness of Nash equilibrium by casting them into convex optimization problems and equivalent variational inequalities in Theorem 2. • We design a feedback control mechanism that enables the mediator to dynamically parameterize the auctions and effectively differentiate agents based on observable information under Nash equilibrium in Section 4.1. We prove the convergence to the maximum social welfare in Theorem 4 based on equilibrium dynamics derived by variational inequality techniques in Lemma 1. • To facilitate agents to reach the Nash equilibrium under any fixed auction parameters, we design a distributed bidding framework under which the mediator sends market signals to influence agents while each agent bids to maximize utility accordingly in Section 4.3. We prove the convergence of the proposed algorithm to the Nash equilibrium in Theorem 5.

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• We combine the ideas of distributed bidding and feedback control and design an integrated bidding-control framework in Section 4.2: each agent performs one-step bidding distributedly and the mediator updates auction parameters via one-step feedback control in turn. In practice, this integrated algorithm exhibits fast convergence. • We design an adaptive algorithm in Section 5 to dynamically determine the amount of resource that needs to be transacted to maximize social welfare, if not known a priori. We also characterize the payment transfers between the mediator and agents under the optimal resource level in Theorem 6. We believe our auction framework can help mediators manage divisible resources efficiently as it aligns the incentives of agents with social welfare. We believe that it can also foster further research: a simple price differentiation scheme can achieve efficiency while incurring only marginal computation cost. 2. Related work The last decades have witnessed a dramatic advancement in research towards designing auction mechanisms for divisible resources. The celebrated VCG (Vickrey–Clarke–Groves) mechanism is a generic mechanism for achieving efficiency [13]. However, it requires each agent to report her full information and also the mediator to conduct completely centralized decision-making, which renders it impractical. To overcome these drawbacks, a line of research focuses on designing and characterizing auctions with one-dimensional bids [14–21,4,6]. By restricting the strategy space, this line of research improves the efficiency and scalability of the auctions. Among these prior works, many studied auctions for resource allocation. Kelly proposed in the seminal work [14] to allocate resource among agents in proportion to their bids and showed that social welfare is maximized if agents are pricetakers. Hayek and Gopalakrishnan [22] proved that a unique equilibrium exists when agents are price-anticipating. Johari and Tsitsiklis [16] showed that the worst efficiency loss of Kelly mechanism at Nash equilibrium is 25%. Nguyen et al. [20] proposed weighted proportional allocation of resources. As for resource procurement, Johari and Tsitsiklis [19] proposed a scalar-parameterized supply function bidding mechanism and showed bounded efficiency loss under price anticipation. Li et al. [4], Xu et al. [23] and Kamyab et al. [24] applied supply function bidding mechanisms for demand response in smart grids. Our work is closely related to this line of research: we generalize the Kelly mechanism [14] and the supply function bidding mechanism [19] by introducing price differentiation, and we employed the techniques in [22] to conduct equilibrium analysis. Nevertheless, there are two major differences between our work and prior works. First, our auction mechanism manages both resource allocation and acquisition in a unified framework, while most prior works can only be applied to one of the scenarios. Second, we achieve full efficiency through market algorithms while most prior works focused on characterizing and improving the efficiency loss and did not consider algorithmic issues. Several prior works adopted price differentiation to achieve efficiency. Maheswaran and Basar [25] proposed a method of generating efficient auctions for allocating divisible resources. Yang and Hajek [17] proposed the VCG–Kelly mechanism that adapted the VCG mechanism to scalar strategy spaces and achieve efficiency. Johari [18] also adapted the VCG mechanism and proposed a class of efficient mechanisms called the SSVCG mechanisms. However, these results only apply in the resource allocation scenario and have more complicated payment structure than ours. Ma et al. [26] introduced price differentiation via tunable parameters into the Kelly mechanism and show that efficiency can be achieved by choosing appropriate parameters. Following this design, Yang et al. [21] designed a feedback control algorithm to determine the appropriate parameters. We adopt this price differentiation scheme in our auction framework. Compared to their works, our work consider both resource allocation and acquisition in a unified manner, and provide market algorithms with convergence guarantees. Ma [6] considered resource allocation and consolidation in virtualized data center resources, and proposed a feedback control mechanism to achieve efficient outcomes. Our work, however, have different auction designs that take the individual rationality of agents into consideration. Moreover, we design market operation algorithms that allow the mediator and agents to jointly run the market in a distributed manner. 3. Auction-based market design 3.1. Market model We assume there are two sets M = {1, . . . , M } and N = {1, . . . , N } of agents who demand and supply divisible resource respectively in the market. The market mediator enables and facilitates the allocation and acquisition of resource through auction mechanisms. On the demand side, each agent i ∈ M has a corresponding value function vi (di ) that measures her value in monetary units when di amount of resource is allocated to her. Likewise, on the supply side, each agent j ∈ N is associated with a cost function ci (si ) that measures the cost of supplying si amount of resource. We make a common assumption of diminishing marginal returns and increasing marginal costs, characterized by the following assumption. Assumption 1. The value function vi (·) is continuous, strictly increasing and concave over [0, +∞) for each agent i ∈ M, and the cost function cj (·) is continuous with cj (sj ) = 0 if sj ≤ 0, strictly increasing and convex over [0, +∞) for each agent j ∈ N.

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An agent’s demand or supply of resource is associated with a payment transfer between the agent and the market mediator. That is, when an agent i ∈ M is allocated di amount of resource and makes a payment ti to the market mediator, her utility is ui = vi (di ) − ti ; when an agent j ∈ N supplies sj amount of resource and receives a payment tj from the market mediator, her utility is uj = tj − cj (sj ). We further assume that the agents are utility-maximizing and individually rational: an agent always chooses an optimal action and will participate in economic activities if and only if she can earn a positive utility by doing so. 3.2. Resource allocation and acquisition We denote the target volume of resource transactions in the market by X . We are particularly interested in two market scenarios. First, in computing resources management, it is very likely that the platform has pre-configured resource quotas for a group of users or jobs. Besides, some markets such as electricity market exhibit inelastic short run demand that can be predicted very accurately. Under these cases, the market participants can reach a consensus on the target resource level X . In the second market scenario, the scale of transactions between the sides of demand and supply would have considerable social impacts, and the mediator would like to regulate on the resource level X so as to maximize social welfare. In resource allocation and acquisition, the objective of the mediator is to maximize social welfare. For the demand side, this is equivalent to maximizing the aggregate value of agents as follows:

MAX-VALUE maximize d

∑

vi (di )

i∈M

{ subject to

M ∑

M

d∈D≜

d ∈ R+ |

(1)

} .

di = X

i=1

For the supply side, welfare maximization is equivalent to minimizing the aggregate cost of agents as follows:

MIN-COST minimize s

subject to

∑

cj (sj )

j∈N

s∈S≜

⎧ ⎨

N ∑

⎩

j=1

s ∈ RN+ |

sj = X

⎫ ⎬

(2)

.

⎭

By Assumption 1, both MAX-VALUE and MIN-COST always admit a feasible solution. If an allocation d solves MAX-VALUE, we say that demand-side efficiency is achieved, and if a supply s solves MIN-COST, we say that supply-side efficiency is achieved. If a consensus on the target resource level X is known a priori, once both demand-side and supply-side efficiency are achieved, we can conclude that the overall social welfare of the market is maximized. Otherwise, the mediator needs to determine the optimal level of resource that maximizes social welfare as follows:

MAX-WELFARE maximize (d,s,X )

subject to

∑

vi (di ) −

i∈M

∑

∑

cj (sj )

j∈N

di =

∑

sj = X ,

i∈M

j∈N

X ≥ 0,

d ∈ RM +,

(3)

and s ∈ RN+ .

If a tuple (X ∗ , d, s) solves MAX-WELFARE, we call X ∗ the optimal resource level at which global efficiency is achieved. In general, the selfish behavior of agents lead to market inefficiency, i.e., suboptimal social welfare. Since the agents’ value and cost functions are hidden information to the mediator, it is hard to directly enforce efficient resource allocation and acquisition. Therefore, the main goal for auction mechanism design is to align the agents’ self-interests with the objective of social welfare without knowing the private information of agents. Next, we present our adaptive auction design that achieves optimal outcomes using one-dimensional strategy space and price differentiation. 3.3. Auction design and competition game We establish a unified auction framework for both the demand and supply sides of the market, which generalizes the demand/supply function bidding mechanisms by introducing price differentiation. On each side of the market, the mediator and the agents run the market through the following procedure:

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• each agent submits a scalar bid that parameterizes her demand (or supply) function, and the mediator determine a baseline market-clearing price1 that clears the market;

• the mediator decides the amount of resource allocated to (or acquired from) each agent by the market clearing price and the agent’s bid indicating her demand or supply function;

• the mediator sets the effective price of per unit resource for each agent as the baseline market-clearing price scaled by an agent-dependent price weight, and charges (or pays) her accordingly. By setting a different price weight for each agent, the auction mechanism implements price differentiation among the agents. In particular, we design the demand and supply side auctions as follows. Demand-Side Auction: We restrict each agent i to submit a scalar bid bi that parameterizes her demand function in the form of D(µ; bi ), which denotes the amount of resource she demands given the baseline market price µ. Based on the collective bids b of agents, the mediator determines the baseline market-clearing price µ(b) at which the aggregate demand equals the target resource level, i.e.,

∑

D (µ(b); bi ) = X .

(4)

i∈M

If we denote the price weights as p ∈ RM ++ , the effective price of per unit resource for each agent i is pi µ(b). As a result, each agent i makes a payment pi µ(b)D(µ(b); bi ) to the mediator and has a utility ui (b) = vi (D (µ(b); bi )) − pi µ(b)D(µ(b); bi ).

(5)

In particular, we adopt the family of demand functions in the form D(µ; b) ≜ b/µ, where b ∈ R+ represents an agent’s willingness-to-demand and µ is ∑the baseline market price of per unit resource. Given the collective bids b of all agents, Eq. (4) yields µ(b) = B/X , where B ≜ k∈M bk . Thus the amount of resource allocated to each agent i equals di = D(µ(b); bi ) =

bi

X, B which follows a proportional allocation rule. Moreover, by Eq. (5), the utility of agent i given the bids of all agents b is ui (b) = vi

(

bi B

) X

− pi bi ,

if b−i ̸ = 0,

(6)

where the second term pi bi is her payment to the mediator. Supply-Side Auction: Likewise, we restrict each agent j to submit a scalar bid wj that parameterizes her supply function S(ν; wj ), which denotes the amount of resource she is willing to supply at the baseline market price ν . Given the bids w, the mediator determines the baseline market-clearing price ν (w) at which the aggregate supply equals the target resource level, i.e.,

∑

S(ν (w); wj ) = X .

(7)

j∈ N

If we denote the price weights as q ∈ RN++ , the effective price of per unit resource for agent j is qj ν (w). As a result, agent j receives payment qj ν (w)S(ν (w); wj ) from the mediator and obtains a utility uj (w) = qj ν (w)S(ν (w); wj ) − cj (S(ν (w); wj )).

(8)

In particular, we adopt the family of supply functions in the form S(ν; w ) = X − w/ν , where the bid w ∈ R+ can be interpreted as an agent’s unwillingness-to-supply and ν is∑ the baseline market price of per unit resource. Given the bids w of agents, Eq. (7) yields ν (w) = W /[(N − 1)X ], where W ≜ k∈N wk . Thus, each agent j will contribute sj amount of resource, where sj = S(ν (w); wi ) = X −

wj

(N − 1)X . W Moreover, by Eq. (8), the utility of agent j given other agents’ bids w is

( uj (w) = qj

W N −1

) ] [ wj − wj − cj 1 − (N − 1) X , if w−i ̸= 0, W

(9)

where the first term is the payment she receives from the mediator. Nash Equilibrium: We assume that the auction rules, the demand and supply function forms and the price weights are publicly known to all agents. By choosing price weights p, q, the mediator creates competitive games on both sides of the market. We denote the demand game as (M, p) and the supply game as (N , q). We further define the Nash equilibrium as follows. 1 We call this market-clearing price baseline because each agent is eventually charged (or paid) an effective price, i.e, this baseline market clearing price scaled by her price weight.

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Definition 1. A bidding profile b ∈ RM ++ is a Nash Equilibrium of the game (M, p) if and only if for each agent i ∈ M, ui (bi , b−i ) ≥ ui (b′i , b−i ) for all b′i ≥ 0. Likewise, a bidding profile w ∈ RN++ is a Nash Equilibrium of the game (N , q) if and only if for each agent j ∈ N , ui (wj , w−j ) ≥ ui (wj′ , w−j ) for all wi′ ≥ 0. Remarks: We restrict the agents to submit a scalar-parameterized demand or supply function. In this way, the agents can express their preferences while restricting information revelation, and the mediator can easily balance demand and supply by setting the baseline market-clearing price on both sides of the market. Notice that our choices of demand and supply functions have complementary rather than symmetric forms. As we will see more clearly in the next subsection, this design allows us to characterize and analyze the Nash equilibrium on both sides in a unified manner. Finally, we discuss the impacts of price differentiation. For the demand game, when there is no price differentiation, i.e., p = 1, we recover the Kelly mechanism [14] that is known to suffer up to 25% efficiency loss at Nash equilibrium when agents are price anticipating [16]. By choosing appropriate price weights p, we can close this efficiency gap while also inheriting the merits of Kelly mechanism including the proportional allocation rule. As for the supply game, when q = 1, we recover the supply function bidding scheme proposed by Johari and Tsitsiklis [19], the aggregate costs of which may go 1 up to (1 + N − ) · OPT at Nash equilibrium. By price differentiation among agents, we can also close this efficiency gap by 2 choosing appropriate price weights q. 3.4. Characterization of Nash equilibrium The following theorem guarantees the existence and uniqueness of Nash equilibrium of the demand and the supply game. Theorem 1. For the demand-side game (M, p), if M > 1, there exists a unique Nash equilibrium, and the corresponding allocation d(q) has at least two positive components. For the supply-side game (N , q), if N > 2, there always exists a Nash equilibrium, and any equilibrium induces the same supply vector s(q). Furthermore, if s(q) has at least two positive components, the Nash equilibrium is unique. The above theorem proves the existence of Nash equilibrium and the uniqueness of equilibrium allocation and supply. Next, we would like to gain a deeper understanding of agents’ strategic behavior at Nash equilibrium and the effects of competition among agents, which will guide us in the design of market algorithms that help agents arrive at Nash equilibrium and guide the mediator to tune the price weights p, q so as to achieve efficiency. To this end, we study the equilibrium allocation and supply and characterize them as the solutions of convex optimization problems and equivalent Variational Inequalities (VI)2 [27]. Definition 2. For the demand-side games (M, p) and the supply-side game (N , q), we define the effective value function of each agent i ∈ M under market competition and the effective cost function of each agent j ∈ N under market competition, respectively, as

vˆi (di , pi ) ≜

di

∫

pi

0

cˆj (sj , qj ) ≜

∫

1

sj

1 qj

0

( z) vi′ (z) 1 − dz and

(10)

X

[

cj′ (z) 1 +

z (N − 2)X

]

dz .

(11)

Theorem 2. For the demand-side game (M, p), the equilibrium allocation d(p) is the unique solution to the convex optimization: NASH-DEMAND(p) maximize d

∑

vˆi (di , pi ) s.t. d ∈ D.

(12)

i∈M

For the supply-side game (N , q), the equilibrium supply s(q) is the unique solution to the convex optimization problem: NASH-SUPPLY(q) minimize s

∑

cˆi (ci , qi ) s.t.

s ∈ S.

(13)

i∈N

Furthermore, d(p) and s(q) are the unique solution to VI(D, F ) and VI(S , G), where each component of F (d, p) and G(s, q) is defined as

( ) ∂ 1 ′ di Fi (d, p) ≜ − vˆi (di , pi ) = − vi (di ) 1 − ∂d pi X

and

2 Given a closed convex set K ⊂ Rn and a continuous mapping F : Rn → Rm , the variational inequality problem, denoted VI(K, F ), is to find a vector y ∈ K such that F (y)t (y − x) ≥ 0, ∀x ∈ K.

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M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234

Gj (s, q) ≜

[ ] sj ∂ 1 cˆj (sj , qj ) = cj′ (sj ) 1 + . ∂s qj (N − 2)X

Theorem 2 indicates that at Nash equilibrium, the strategic behavior of agents can be viewed as maximizing the aggregate effective value or minimizing the aggregate effective cost. We remark that the effective value and cost functions capture both the influence of price weights and competition among agents. Take the demand-side game (M, p) as an example, agent i’s marginal effective value is

( ) di 1 ′ ∂ vˆi (di , pi ) = vi (di ) 1 − , ∂d pi X where the first term pi−1 captures the negative effects of price weights p on agents’ gain from allocated resource, and the d rest part vi′ (di )(1 − Xi ) indicates that agents’ marginal gain are reduced due to competition. In Section 4, we will exploit the structure of NASH-DEMAND(p) and NASH-SUPPLY(q) to design market algorithms. Their equivalent VI forms will play an important role in the convergence analysis in Section 4.1. Corollary 1. The equilibrium allocation d(p) and equilibrium supply s(q) are homogeneous functions of degree 0, i.e., d(kp) = d(p),

s(kq) = s(q),

∀k > 0.

Corollary 1 indicates that the equilibrium allocation and supply are dependent on the relative values among the price weights rather than the absolute values. This property allows us to scale the price weights without impacting the resulting market state. 4. Market mechanism and algorithms In Section 3, we have presented the design of auction mechanism on both sides of the market and further characterized the existence and uniqueness of Nash equilibrium. Nonetheless, two interesting questions remain unanswered:

• how to choose the price weights p and q so as to implement an efficient Nash equilibrium that maximizes social welfare?

• how to enable and facilitate the agents to reach such a Nash equilibrium given fixed price weights p and q? Despite the characterization of Nash equilibrium, the above questions are nontrivial in that agents have privacy and security concerns and are unwilling to reveal their value and cost functions. Thus, our goal is to answer the two questions without the mediator knowing the value and cost functions. To this end, we design the following market mechanism and algorithms that enable the mediator to influence the agents’ bidding and achieve efficient outcomes. Fig. 2 illustrates the market operating mechanisms to resolve the aforementioned two questions. The upper figure shows that the mediator sends price weights in rounds of iterations and update them accordingly based on the reached Nash equilibrium in the previous iteration. The key is the design of a feedback control algorithm that converges to the Nash equilibrium that maximizes social welfare. The lower figure shows that the mediator sends market signals to agents in rounds of iterations, upon which they calculate their optimal bids. The key is to decouple an optimization problem into the interactions between the mediator and agents, and construct the corresponding market signals and decoupled optimal bidding problem for each agent. In the rest of this section, we first provide conditions for the games (M, p) and (N , q) to implement efficient allocation and supply, based on which we design a feedback control algorithm with convergence guarantee that adaptively adjusts the price weights p, q to achieve market efficiency. Second, by exploiting the structure of the convex optimization problems NASH-DEMAND(p) and NASH-SUPPLY(q), we design a distributed bidding algorithm with strong convergence guarantee that does not rely on the characteristics of the underlying value or cost functions. Lastly, we design an algorithm that integrates the above bidding and control procedures to achieve faster convergence in practice. 4.1. Feedback control algorithm We study how the market mediator should adjust the price weights p, q algorithmically so that efficiency is achieved on both sides of the market. For the demand-side game (M, p), if the resulting allocation d = d(p) is efficient, i.e., d solves MAX-VALUE, for any agents i ̸= k ∈ M that submit positive bids, the KKT conditions [28] imply that vi′ (di ) = vk′ (dk ). Moreover, the KKT conditions of NASH-DEMAND(p) imply 1 pi

( ) ( ) di 1 dk vi′ (di ) 1 − = vk′ (dk ) 1 − . X

pk

Combining the two equations, we have X − di (p) pi

=

X − dk (p) pk

.

X

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219

Fig. 2. Market operating mechanisms (demand-side) — feedback control (upper) and distributed bidding (lower).

Similar arguments can be made on the supply side. Based on this observation, we have the following theorem that characterizes the conditions for Nash equilibrium to be efficient. Theorem 3. Assume M > 1 and N > 2. Suppose d∗ > 0 and s∗ > 0 solves optimization problem MAX-VALUE and MIN-COST respectively. Then the price weights p, q achieve efficiency , i.e., d(p) = d∗ and s(q) = s∗ , if and only if X − di (p) (N − 1)X

pi

= ∑

k

pk

(N − 2)X + sj (q)

,

∀i ∈ M and

qj

= ∑

(N − 1)2 X

k

qk

,

∀j ∈ N .

(14)

(15)

One particularly desirable property of Eqs. (14) and (15) is that the private marginal values and costs do not explicitly appear therein and the market outcome and the price weights are decoupled. This inspires us to design a feedback control scheme (Algorithm 1) to achieve efficiency on both sides without knowing the value and cost functions. We take the following steps to study the convergence properties of Algorithm 1. First we use the variational inequalities VI(D, F ) and VI(S , G) define in Theorem 2 to derive the dynamics of Nash equilibrium under the changes of price weights. Then we formulate the iterative process in Algorithm 1 as dynamical systems in continuous time, which is standard in previous work [14,20,6]. Finally we use the dynamics result to prove the convergence of Algorithm 1. Lemma 1. Suppose M > 1 and N > 2, and the value functions vi (·) and cost functions ci (·) are twice differentiable. Then M M on the demand side, for any price weights p ∈ RM ++ , the equilibrium allocation d(p) : R++ ↦ → R+ satisfies the following dynamics:

∂ di (p) = ∂ pj

(

)

αi

∑M

k=1

αk

− 1{j=i} αj βj ,

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Algorithm 1 Feedback Control Algorithm to Achieve Efficiency 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

procedure CONTROL-DEMAND(p1 , δ, ϵ ) k←1 repeat sends pki to each agent i ∈ M wait until agents reach a Nash equilibrium bk dk ← (bk /∥bk ∥1()X ) X − dk N −1

pk+1 ← pk + δ

−

k←k+1 until ∥pk+1 − pk ∥1 < ϵ end procedure procedure CONTROL-SUPPLY(q1 , δ, ϵ ) k←1 repeat sends qkj to each agent j ∈ N

▷ new weights

wait until agents reach a Nash equilibrium wk

16:

sk : skj ← X − ∑

wjk

qk+1 ← qk + δ

17:

19: 20:

▷ equilibrium allocation

pk X ∥pk ∥1

15:

18:

▷ new weights

k

(n wn

(N − 1)X

(N −2)X +sk (N −1)2

−

▷ equilibrium supply qk

∥qk ∥1

X

)

k←k+1 until ∥qk+1 − qk ∥1 < ϵ end procedure

where αi , βi are defined as

]−1 ∂2 v ˆ (d , p ) 1{di >0} and i i i ∂ d2i ∂2 2 ′ βi ≜ − vˆi (di , pi ) = p− i vi (di )(1 − di /X ). ∂ pi ∂ di

αi ≜ −

[

On the supply side, for any price weights q ∈ RN++ , the equilibrium allocation s(q) : RN++ ↦ → RN+ satisfies the following dynamics:

∂ si (q) = ∂ qj

(

)

αi

∑N

k=1

αk

− 1{j=i} αj βj ,

where αi , βi are defined as

]−1 ∂2 ˆ c (s , q ) 1{si >0} and i i i ∂ s2i [ ] ∂2 si 2 ′ βi ≜ cˆi (si , qi ) = −q− c (s ) 1 + . i i i ∂ qi ∂ si (N − 2)X

αi ≜

[

We use the dynamics to prove that the dynamical systems corresponding to the iterative process in Algorithm 1 will converge to the equilibrium allocation and supply. Theorem 4. Under the assumptions of Lemma 1, suppose d∗ > 0 and s∗ > 0 solves optimization problem MAX-VALUE and MIN-COST respectively. We define di (t) ≜ di (p(t)) as the equilibrium allocation under p at time t. If p is governed by the feedback of d(p(t)) based on the differential equation d dt

pi (t) =

X − di (t) N −1

pi (t)

−∑

k

pk (t)

X,

then for any p(0), the price weights trajectory p(t) satisfies lim p(t) = p∗ ,

t →∞

where d∗ = d(p∗ ).

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221

Fig. 3. Market operating mechanism (demand-side) — integrated feedback control and distributed bidding.

Similarly, if we define sj (t) ≜ sj (q(t)) as the equilibrium supply under q at time t, and q is governed by d dt

qj (t) =

(N − 2)X + sj (t) (N −

1)2

qj (t)

−∑

k

qk (t)

X,

then for any q(0), the price weights trajectory q(t) satisfies lim q(t) = q∗ ,

t →∞

where

s∗ = s(q∗ ).

Theorem 4 guarantees that if we choose an appropriate step size δ , Algorithm 1 will converge to efficient outcome. In general, increasing δ below a certain threshold will lead to faster convergence. However, when δ becomes too large, the price weights updates may result in negative weights and cause serious oscillation. Step size adaptation schemes [29] may ease the search process for effective δ . 4.2. Integrated distributed bidding and control By Theorems 4 and 5, if we choose appropriate step size δ for adjusting price weights, our market operating mechanisms are guaranteed to converge to efficient equilibrium allocation and supply. However, a drawback of this approach is that it may suffer slow convergence due to its nested iterations. To this end, we design an integrated procedure as shown in Fig. 3 to solicit the bids of agents and perform the feedback control simultaneously. In particular, during each iteration,

• the mediator announces the current price weights and market signals to agents; • agents perform the same one-step bidding as before; • the mediator updates the price weights and market signals based on the newly received bids accordingly. 4.3. Distributed bidding algorithm In general, it would be difficult for agents to reach Nash equilibrium without the help of the market mediator. Furthermore, the agents would also like to reduce the information got revealed during the bidding process. In light of this, we focus on a distributed algorithmic paradigm in which the mediator and the agents jointly run the market in an alternating manner. During each iteration,

• the mediator sends market signals such as estimated baseline market price and estimated allocation to the agents; • each agent bids in response to her received signals; • the mediator gathers the bids and updates these market signals accordingly. We make several remarks on this paradigm. From a global perspective, the mediator and the agents alternate in updating their responses to their observations of the environment. The mediator influences agents’ behavior via market signals, and when agents’ bids are gathered, she implicitly learns part of agents’ hidden information and updates the market signals. The agents bid in a distributed manner, i.e., each agent makes decision based solely on her received signals, regardless of the states and moves of other market participants. We use the demand game (M, p) to illustrate how to translate the aforementioned paradigm into a concrete algorithm by exploiting the convex optimization problem NASH-DEMAND(p). As discussed above, our design must meet two principles: (1) the mediator and the agents make moves in an alternating fashion, and (2) the agents bid in a distributed manner. To this

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end, we introduce an auxiliary variable z ∈ RM and transform NASH-DEMAND(p) into the following equivalent optimization problem: maximize

− g(z) +

d, z

∑

vˆi (di , pi ) (16)

i∈M

subject to

0 ≤ di ≤ X ,

∀i,

and d − z = 0,

where g(z) = 0 if ∥z∥1 = X and g(z) = +∞ otherwise. The augmented Lagrangian Lρ (d, z, µ ) of (16) can be expressed as Lρ = −g(z) +

∑

1

vˆi (di , pi ) − µ T (d − z) − ρ∥d − z∥22 2

i∈M

] ∑[ 1 2 = vˆi (di , pi ) − µi di − ρ (di − zi ) − g(z) + µ T z,

(17)

2

i∈M

where µ is the Lagrangian multiplier whose economic interpretation is the shadow price of resource, and ρ > 0 is a penalty parameter. Intuitively, we would like to maximize Lρ (d, z, µ ) so as to solve (16), which involves maximizing Lρ in the direction of z and d respectively. If we view the auxiliary variable z and the shadow price µ as the market signals and the allocation d as agents’ decisions, then the maximization of Lρ suggests a natural decomposition of the moves of the mediator and the agents. Moreover, the maximization of Lρ with respect to d is separable in each di , which readily suggest a distributed bidding scheme. The above intuitive idea of maximizing augmented Lagrangian Lρ in the direction of d and z directly links to the Alternating Direction Method of Multipliers (ADMM) [30] algorithm, which consists of the iterations

• dk = arg max Lρ (d, zk , µ k ); 0≤d≤X

• zk+1 = arg max Lρ (dk , z, µ k ); ∥z∥1 =X

• µ k+1 = µ k + ρ (dk − zk+1 ). It is clear from the above steps that the ADMM is alternating direction when updating the primal variables d and z. Putting the above in our market context, this means that the mediator and agents alternate in updating their responses to their observations of the environment. The updates of d can be carried out by each agent i in a distributed manner, and the mediator updates the auxiliary variable z and shadow price µ based on agents’ bids. The update on shadow price µ is in effect the dual update step in ADMM. The iterations of bidding and updates are illustrated in Fig. 2. We further discuss the implementation of the updates in our market scenario and the economic interpretation. The auxiliary variable z can be viewed as the mediator’s estimation of equilibrium allocation, and the shadow price µ maintains a separate approximation of the baseline market-clearing price for each agent. Each agent i makes its decision based on her received signals (zik , µki ) by calculating dki = arg max 0≤d≤X

1

vˆi (d, pi ) − µki d − ρ (d − zik )2 , 2

(18)

and bids bki = µki dki to the mediator. Notice that since µi is a market price signal, vˆi (d, pi ) − µki d is in fact the effective utility of agent i. Thus, the bidding rule encourages each agent i to maximize her effective utility while being punished for deviating from the estimated equilibrium allocation zik . This bidding rule is most desirable since it encapsulates the agent’s price-anticipating behavior. The mediator updates her estimation of equilibrium allocation z and ∑shadow price µ . The update of z is equivalent to setting zk+1 as the projection of dk + µ k /ρ on the affine space {z ∈ RM | i zi = X }, i.e., zk+1 =

] 1 [ X − 1T (dk + µ k /ρ ) 1 + dk + µ k /ρ. M

Finally, the mediator updates the shadow price µ . All the above discussion readily applies to the supply game with modest modifications and we avoid the duplication here. We summarize our design of the distributed bidding framework in Algorithm 2, which terminates when both the primal and dual residual meet precision criteria. We also prove that Algorithm 2 has strong convergence guarantees. Theorem 5. Denote the baseline market-clearing price for (M, p) and (N , q) as µ(p) and ν (q), respectively. Under Assumption 1, if the penalty parameter ρ > 0, Algorithm 2 converges to:

{dk } → d(p),

{µki } → µ(p), ∀i ∈ M,

{s } → s(q),

{νjk } → ν (q), ∀j ∈ N .

k

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223

Theorem 5 states that Algorithm 2 inherits the strong convergence guarantee of ADMM algorithms: as long as the agents have concave value functions and convex cost functions, the algorithm arrives at Nash equilibrium under any positive penalty ρ , regardless of the initialization of the auxiliary vector z and Lagrangian multiplier µ (or ν ).

Algorithm 2 ADMM-Based Distributed Bidding Algorithm 1: 2: 3: 4: 5: 6: 7: 8:

procedure BID-DEMAND(p, ρ, z1 , µ1 , ϵpri , ϵdual ) mediator broadcast ρ to agents mediator sends pi to each agent i ∈ M k←1 repeat mediator sends (zik , µki ) to each agent i ∈ M for i ← 1, M do in parallel dki ← arg max vˆi (d, pi ) − µki d − ( 21 ρ )(d − zik )2

▷ penalty ▷ weights

0≤d≤X

9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:

bid bki ← µki dki to the mediator end for mediator updates dk : dki ← bki /µki , ∀i. [ ] zk+1 ← M1 X − 1T (dk + µ k /ρ ) 1 + dk + µ k /ρ µ k+1 ← µ k + ρ (dk − zk+1 ) k←k+1 ρ until X1 ∥dk − zk+1 ∥1 < ϵpri , X ∥zk − zk+1 ∥1 < ϵdual end procedure procedure BID-SUPPLY(q, ρ, z1 , ν 1 , ϵpri , ϵdual ) mediator broadcast ρ to agents mediator sends qj to each agent j ∈ N k←1 repeat mediator sends (zjk , νjk ) to each agent j ∈ N for j ← 1, N do in parallel skj ← arg max νjk s − cˆj (s, qj ) − 12 ρ (s − zjk )2

▷ shadow price

▷ penalty ▷ weights

0≤s≤X

25: 26: 27: 28: 29: 30: 31: 32:

bid wjk ← νjk (X − skj ) to the mediator end for mediator updates sk : skj ← X − wjk /νjk , ∀j. X − 1T (sk − ν k /ρ ) 1 + sk − ν k /ρ ν k+1 ← ν k − ρ (sk − zk+1 ) k←k+1 ρ until X1 |sk − zk+1 ∥1 < ϵpri , X ∥zk − zk+1 ∥1 < ϵdual end procedure zk+1 ←

1 N

[

]

▷ shadow price

Numerical results indicate that Algorithm 2 converge with high accuracy in tens of iterations given proper ρ and are insensitive to the choice of initial z, µ and ν . We remark that faster convergence can be achieved in practice by employing schemes that vary the penalty parameter ρ adaptively [30,31], though this comes at the sacrifice of theoretical convergence guarantees. To implement the integrated procedure of bidding and control, we propose Algorithm 3, which does inexact optimization for bidding and feedback control in each iteration. In practice, the algorithm converges rapidly to efficient equilibrium allocation and supply although the bids may be far from equilibrium at the beginning. Given a proper step-size δ , we observe that Algorithm 3 converges almost at the same rate as Algorithm 2, which is a considerable acceleration. We also introduce a normalization step in Algorithm 3 which significantly improves stability of the numerical solutions. Take the BID–CONTROL–DEMAND procedure for example, we always normalize the L1 norm of the price weights to 1 (line 4, 16). The rationale is as follows. If we examine the price weights update in Algorithm 1, it is easy to check that ∥pk+1 ∥1 = ∥pk ∥1 since dk ∈ D. Nevertheless, since we are doing inexact optimization in algorithm 3, each dk may no longer be feasible, i.e., dk ∈ / D, which leads to possible oscillation of pk . Normalization can stabilize the updates while not incurring side effects according to Corollary 1. Without the normalization, we might need to choose a very small step-size δ and suffer much slower convergence.

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Algorithm 3 Integrated Distributed Bidding and Feedback Control 1: 2: 3: 4: 5: 6: 7: 8: 9:

procedure BID-CONTROL-DEMAND(p, ρ, δ, ϵ ) initialize z1 , µ1 ∈ RM randomly mediator broadcast ρ to agents p1 ← p/∥p∥1 k←1 repeat mediator sends pki , zik , µki to agent i, ∀i for i ← 1, M do in parallel dki ← arg max vˆi (d, pki ) − µki d − ( 12 ρ )(d − zik )2

▷ normalize price weights

0≤d≤X

10: 11: 12: 13: 14: 15:

16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28:

bid bki ← µki dki to the mediator end for mediator updates dk : dki ← bki /µki , ∀i [ ] zk+1 ← M1 X − 1T (dk + µ k /ρ ) 1 + dk + µ k /ρ µ k+1 ← µ k + ρ((dk − zk+1 ) ) pk+1 ← pk + δ

X −dk (N −1)X

−

▷ shadow price

pk ∥ pk ∥ 1

▷ price weights ▷ normalize

pk+1 ← pk+1 /∥pk+1 ∥1 k←k+1 until ∥pk+1 − pk ∥1 < ϵ end procedure procedure BID-CONTROL-SUPPLY(q, ρ, δ, ϵ ) initialize z1 , ν 1 ∈ RN randomly mediator broadcast ρ to agents q1 ← q/∥q∥1 k←1 repeat mediator sends qkj , zjk , νjk to agent j, ∀j for j ← 1, N do in parallel skj ← arg max νjk s − cˆj (s, qkj ) − 12 ρ (s − zjk )2

▷ normalize price weights

0≤s≤X

30:

bid wjk ← νjk (X − skj ) to the mediator end for

31:

mediator updates sk : skj ← X − ∑

29:

32: 33: 34: 35: 36: 37: 38:

wjk n

wnk

(N − 1)X

← X − 1 (s − ν /ρ ) 1 + s − ν k /ρ k k k+1 ν ← ν − ρ (s ( − z k) ) qk (N −2)X +s qk+1 ← qk + δ (N −1)2 X − ∥qk ∥ z

k+1

k+1

1 N

[

T

k

qk+1 ← qk+1 /∥qk+1 ∥1 k←k+1 until ∥qk+1 − qk ∥1 < ϵ end procedure

k

]

k

▷ shadow price

1

▷ normalize

5. Achieving market efficiency In this section, we concentrate on the global properties of the auction mechanism. We first study how to determine the optimal resource level X ∗ , if not known a priori, to achieve global efficiency. Then we discuss the profit aspects of the auction mechanism and show that at the optimal resource level X ∗ , the mediator needs to subsidize the agents so as to enable the transactions. 5.1. Optimal resource level Up till now, we have focused on the scenario where the market participants have consensus on the resource level X . In this subsection, we shift our attention to the scenario where the mediator needs to determine the optimal resource level X ∗ and solve the global social welfare problem MAX-WELFARE. We first analyze its structural properties, then design an algorithm to determine X ∗ .

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225

Since social welfare is the aggregate values less the aggregate costs, it will be helpful to use MAX-VALUE and MIN-COST to represent MAX-WELFARE. For notational convenience, we refer to the optimization problem (1) and (2) as MAX-VALUE(X ) and MIN-COST(X ) respectively to emphasize that they are parameterized by the resource level X . We further denote the optimal objective value of optimization problems MAX-VALUE(X ) and MIN-COST(X ) as value(X) and cost(X) respectively. Thus, the optimization problem MAX-WELFARE is equivalent to

OPT-RESOURCE maximize X ≥0

welfare(X ) = value(X ) − cost(X ).

(19)

The function welfare(X ) represents the maximum welfare that can be obtained given a fixed resource level X . The following analysis indicates that welfare(X ) is a concave function of X , and characterizes its derivative by the agents’ marginal utility and cost when both the demand-side and supply-side efficiencies are achieved. Definition 3. Suppose d solves MAX-VALUE(X ) and s solves MIN-COST(X ). Define the marginal utility mv(X ) ≜ vi′ (di ) for any di > 0 and the marginal cost mc(X ) ≜ ci′ (si ) for any sj > 0. From the KKT conditions of MAX-VALUE(X ) and MIN-COST(X ), we know that at efficient outcomes all active agents have equal marginal utility (or cost), thus mv(X ) and mc(X ) are well-defined. Lemma 2. The function welfare(X ) is concave in X , with derivative d welfare(X ) = mv(X ) − mc(X ). dX Moreover, X ∗ solves OPT-RESOURCE if and only if mv (X ∗ ) = mc(X ∗ ). Lemma 2 provides the concavity of the function welfare(X ), which inspires us to use gradient ascent to update X as follows: X k+1 = X k + δ[mv(X ) − mc(X )]. However, the mediator cannot directly observe agents’ marginal utility and cost. Fortunately, the following lemma states that if both the demand-side and supply-side efficiencies are achieved, mv(X ) and mc(X ) can be revealed at the Nash equilibrium. Lemma 3. If the demand-side game (M, p) and supply-side game (N , q) induce efficient outcome, i.e., d(p) solves MAX-VALUE(X ) and s(q) solves MIN-COST(X ), then for any active agent i, j at equilibrium, i.e., di (p) > 0 and sj (q) > 0, we have mv(X ) =

pi µ(p) 1 − di (p)/X

and

mc(X ) =

qj ν (q) 1 + sj (q)/[(N − 2)X ]

,

where µ(p) and ν (q) are the baseline market-clearing price at the unique Nash equilibrium. Lemma 3 allows us to design a gradient ascent algorithm (Algorithm 4)3 to determine the optimal resource level X ∗ as follows. Algorithm 4 Determine X ∗ to Achieve Global Efficiency 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

procedure OPTIMAL-RESOURCE(X 1 , δ, ϵ ) k←1 repeat broadcast X k to all agents (dk , pk , µk ) ← BID-CONTROL-DEMAND (sk , qk , ν k ) ← BID-CONTROL-SUPPLY pick active agents i, j such that dki > 0, skj > 0 X k+1 ← X k + δ

[

pki µk 1−dki /X k

−

qkj ν k

]

1+skj /[(N −2)X k ]

k←k+1 until |X k+1 − X k |/X k < ϵ end procedure

5.2. Payment transfer of the mechanism Finally, we discuss the payment transfer of the auction mechanism when both demand-side and supply-side efficiencies are achieved. We characterize the payment transfer by the Herfindahl index [32] achieved at the efficient equilibrium outcomes. Our results indicate that at global efficiency, the mediator needs to subsidize the agents so as to enable the transactions. 3 We omit in Algorithm 4 the parameters passed to BID–CONTROL–DEMAND and BID–CONTROL–SUPPLY for notational convenience.

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Theorem 6. If the demand- and supply-side games (M, p) and (N , q) achieve efficient outcomes, i.e., equilibrium allocation d(p) solves MAX-VALUE(X ) and s(q) solves MIN-COST(X ), the mediator receives payment transfer X · mv(X )(1 − Hd ) from the demand-side agents and make payment transfer

( X · mc(X ) 1 +

)

1 N −2

Hs

to the supply-side agents, where the Herfindahl index Hd and Hs are defined as Hd ≜

∑ [ di (p) ]2 i∈M

X

and Hs ≜

∑ [ sj (q) ]2 j∈N

X

.

The Herfindahl index Hd and Hs measure the degree of competition when efficiency is achieved: a higher index indicates lower degree of competition and higher market power. Furthermore, from Lemma 2 we see that mv(X ∗ ) = mc(X ∗ ) for the optimal resource level X ∗ , which immediately implies the following corollary. Corollary 2. At the optimal resource level X ∗ , if demand and supply side games (M, p) and (N , q) achieve efficient outcomes, then the mediator makes positive net payment transfer to agents. The corollary indicates that the mediator needs to subsidize the agents so as to achieve global efficiency. This result is in line with the impossibility result in classic mechanism design [33,13], where budget balance does not come for free when efficiency and incentive-compatibility are required. 6. Numerical evaluations In this section, we conduct numerical simulations to complement the analyses carried out in previous sections. Since both our feedback control algorithm and distributed bidding algorithm have theoretical convergence guarantees, we can safely focus on more complex market scenarios. 6.1. Achieve global market efficiency First of all, we are interested in the market scenario where the resource level X is not given a priori. We consider a market where there are M = 3 agents on the demand-side and N = 3 agents on the supply-side. The demand-side agents are

√

associated with concave value functions v1 (d) = 0.5d, v2 (d) = d + 1/2, v3 (d) = log (d + 1). The supply-side agents have convex cost functions c1 (s) = 0.1s2 , c2 (s) = 0.08(s2 + 2s), c3 (s) = e0.2s − 1. The coefficients in the cost functions ensure that the aggregate values of demand side agents and the aggregate costs of supply side agents have the same order of magnitude, which is common in real world markets. We use Algorithm 4 to determine the optimal resource level X ∗ and set the initial resource level X 1 = 1, step size δ = 3, and precision ϵ = 1e−4. Fig. 4 plots the dynamics of resource level and equilibrium allocation and supply (on the left) as well as the dynamics of social welfare (on the right). Starting from a low resource level X 1 = 1, Algorithm 4 keeps increasing the resource level until social welfare is maximized. During the process, we observe that the overall resource level and individual’s allocation or supply grow in concave curves. Moreover, at the beginning of the iterations when resource is scarce, agent 3 on the supply side is excluded from competition due to her high marginal cost; when resource becomes abundant, she becomes active in the supply game. The numerical experiments suggest that Algorithm 4 converges stably towards global market efficiency. 6.2. Adaptivity with arrival and departure We consider a dynamical market where the arrivals and departures of agents may happen. We assume the amount of resource is X = 10. In addition to the six agents introduced above, we further simulate two agents with v4 (d) = d and c4 (s) = e0.1s − 1. Since the equilibrium allocation and supply is invariant under scaling of price weights and our market algorithm are robust to initializations, we handle the arrival and departure of an agent as follows. Take the demand side for 1 on her and scale example, when an agent arrives at time t and the number of agents become N(t), we place price weight N(t) N(t)−1

the other agents’ weights by N(t) . When an agent departs from the system, we also scale the other agents’ weights such that their sum remains 1. In this way we are keeping the relative values of weights among old agents and the L1 norm of weight vector unchanged. From now on, we refer to an agent by her value or cost function for notational convenience. We assume agent v4 arrives at t = 26 and leaves at t = 55; agent c4 arrives at t = 46 and stays until the end of the simulation. Each time an agent joins or leaves the market, we adjust the price weights and drive the market to maximal social welfare by the market algorithms. Fig. 5 plots the dynamics of social welfare. We use the scenario where there is no price differentiation, i.e., p = 1 and q = 1, as baseline. As the figure suggests, the efficiency loss in this scenario can be around

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227

Fig. 4. Determine the optimal resource level.

Fig. 5. Social welfare dynamics under agents’ arrivals and departures.

Fig. 6. Equilibrium allocation dynamics under agents’ arrivals and departures.

20% to 40%. From Fig. 5 we observe that each time an agent joins the market, the maximal social welfare increases, and our market mechanism and algorithms are very efficient in driving the market back to efficiency. Figs. 6 and 7 depict the dynamics of equilibrium allocation and supply, respectively. For the demand side game, the arrival of agent v4 drives v1 , v2 out of competition eventually. This should be especially clear for v1 since v4 will always dominate v1 . When agent v4 leaves the market at t = 56, the demand side market gradually returns to the previous state. As for the supply game, the arrival of agent c4 significantly reduces the amount of resource other agents supply at equilibrium.

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Fig. 7. Equilibrium supply dynamics under agents’ arrivals and departures.

To conclude, the numerical experiments demonstrates that our market mechanism and algorithms are very effective and robust in dynamic and complex market scenarios. 7. Conclusion In this work, we consider a market of divisible resource consisting of agents on both sides of demand and supply. We design an adaptive auction framework for a market mediator to achieve efficient resource allocation and acquisition. Our novel design generalizes demand/supply function bidding mechanisms by introducing price differentiation via tunable parameters. We design algorithms that enable the mediator and agents to jointly run the market in an adaptive fashion: the mediator sends market signals to agents; each agent submits her bid based on the signals in a distributed manner; the mediator adjusts tunable parameters based on bids and update market signals. We also design an adaptive algorithm to dynamically determine the optimal amount of resource that needs to be transacted so as to maximize social welfare, if not known a priori. By utilizing our market mechanisms, the market mediator will be able to reach an efficient market outcome under Nash equilibrium. We believe that our design can also foster further research: a simple price differentiation scheme can achieve efficiency while incurring only marginal computation cost. Acknowledgment This work was supported by National Nature Science Foundation of China under Grant No. 61772486. Appendix A. Proofs We first present a very useful lemma. ∑ b Lemma 4. For the demand game (M, p), suppose b is a bidding profile, µ = Xi i is the corresponding baseline market-clearing price and d is the induced allocation, then b is a Nash equilibrium if and only if for all i ∈ M, 1 pi

vi′ (di )(1 −

di X

{ )

= µ, ≤ µ,

if

di > 0 ;

if

di = 0 . ∑

wj

For the supply game (N , q), if w is a bidding profile, ν = (N −j 1)X is the corresponding baseline market-clearing price and s is the induced supply, then w is a Nash equilibrium if and only if for each j ∈ N ,

⎧ ] ⎨≥ ν, c ′ (sj ) 1 + = ν, qj j (N − 2)X ⎩ ≤ ν, 1

[

sj

if

sj = 0;

if

0 < sj < X ;

if

sj = X .

Proof of Lemma 4. Demand Game. The ‘only if’ part. Suppose d is a Nash equilibrium. Then b−i ̸ = 0 for all i ∈ M. Otherwise, if b−i = 0 for some agent i, then her utility is ui (bi , b−i ) =

{

vi (X ) − pi bi , vi (0),

if bi > 0; if bi = 0.

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229

Clearly no equilibrium exists due to the discontinuity at 0. Therefore, we can express each agent’s utility by ui (bi , b−i ) = vi

(

bi B

) X

− pi bi ,

∀i ∈ M,

which is continuous over bi ∈ [0, +∞) since b−i ̸ = 0. We take its first order partial derivative w.r.t. bi :

∂ ui (bi , b−i ) = vi′ ∂ bi =

(

bi B

pi X

[

B

) X

B − bi B2

X − pi

(

1

vi′ (di ) 1 −

pi

di

)

X

] −µ .

In fact, one can easily check it is strictly concave over bi ∈ [0, +∞) by checking ∂ 2 ui (bi , b−i )/∂ b2i < 0. Therefore, by applying the first order optimality conditions for concave functions, we immediately have the results in the lemma. The ‘if’ part. We first show that 0 ≤ di < D for all i ∈ M. Otherwise, suppose di = D, then dk = 0, ∀k ̸ = 1, k ∈ M. According to the condition for di in the lemma, we have µ = 0; as for k ̸ = q, we have µ = p1 vk′ (0) > 0, a contradiction. k Hence b−i ̸ = 0 for all i ∈ M, and each agent’s utility is ui (bi , b−i ) = vi

(

bi B

) X

− pi bi ,

∀i ∈ M,

which is continuous over bi ∈ [0, +∞) and strictly concave in di . Reverse the steps in the ‘only if’ parts and we have that b is indeed a Nash equilibrium. Supply Game. In principle, the proof of the supply game follows the same flow and techniques as those of the demand game. We only highlight the differences here. We first show that if w is an equilibrium, then w−j ̸ = 0 for all j ∈ N . Otherwise, suppose w−j = 0, then the utility of agent j is uj (wj , w−j ) =

{ −cj (X ), if wj = 0; −2 − NN − w , if wj > 0. 1 j

No equilibrium exists due to the discontinuity at 0. Therefore, the utility for each agent j ∈ N is uj (wj , w−j ) = qj

(

W N −1

) ( ) wj − wj − cj X − (N − 1)X , W

a continuous and concave function. Next we show that when N = 2, no equilibrium exists. Suppose (w1 , w2 ) is an equilibrium, then we have w1 > 0, w2 > 0 by the above discussion. The utility of agent 1 is u1 (w1 , w2 ) = q1 w2 − c1

( X−

) w1 X , w1 + w2

which is increasing in w1 , a contradiction. Finally, we point out that when analyzing the case of N > 3, we can restrict our attention to 0 ≤ wj ≤ ∑ wk

k̸ =j wk , N −2

∑

a closed

interval. This is because when wj > N −2 , the payment qj − wj she receives is negative, clearly not something in her best interest. Given the above ingredients, the rest part is only a mirror of the demand side counterpart, which we avoid here for simplicity. □ k̸ =j

(

W N −1

)

Proof of Theorems 1 and 2. The general idea is to reverse-engineer the conditions in Lemma 4 into global optimization problems NASH-DEMAND and NASH-SUPPLY, and show that the Lagrangian multiplier of each problem is exactly the baseline market-clearing price. Existence. Since NASH-DEMAND(p) and NASH-SUPPLY(q) are optimization problems with strictly concave (convex) objective function and compact feasible region, they admit unique solution d(p) and s(q) respectively. Since they meet the Slater’s Condition [28], they each have a unique Lagrangian multiplier µ(p) > 0 and ν (q) > 0 that satisfy the KKT conditions:

vˆi ′ (di )

′

cˆi (si )

{ = µ(p), di > 0; ≤ µ(p), di = 0,

{

= ν (q), si > 0; ≥ ν (q), si = 0,

which satisfy the conditions of Lemma 4. Thus b = µ(p)d(p) is a Nash equilibrium of the demand game, and w = ν (X − s(q)) is a Nash equilibrium of the supply game.

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M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234

Uniqueness. Suppose b is a Nash equilibrium of the demand game, then by Lemma 4, 1 pi

vi′ (di )(1 −

di X

{ )

= µ, ≤ µ,

if

di > 0;

if

di = 0,

where d is the induced allocation. These are exactly the KKT conditions of NASH-DEMAND(p), which immediately imply that µ = µ(p) and d = d(p). Therefore b is unique. As for the supply game, suppose the induced supply s of a Nash equilibrium w has at least two positive components, then the above arguments still apply and w is unique. The only slightly more complicated case is when the induced supply s is dictatorial. For example, s = (X , 0, . . . , 0), then there are infinitely many ν that satisfy the conditions in Lemma 4:

⎧ ] ⎨≥ ν, c (sj ) 1 + = ν, qj j (N − 2)X ⎩ ≤ ν, 1

[

′

sj

if

sj = 0;

if

0 < sj < X ;

if

sj = X .

The smallest feasible ν is ν (q) = c1 (X ). In this scenario, the equilibrium of the supply game is of the form w = ν (X − s), where ′

c1′ (X ) ≤ ν ≤

1 qj

[

cj′ (sj ) 1 +

]

sj (N − 2)X

,

∀j ∈ N , j ̸ = 1 ,

and they all induce the same supply vector s.

□

Proof of Corollary 1. Since scaling price weights by k > 0 is equivalent to scaling the objective of NASH-DEMAND or NASHSUPPLY by 1/k, the equilibrium allocation or supply will remain unchanged. □ Proof of Theorem 5. As for the demand game, the epigraphs of the objective mapping and nonempty convex set. Then we examine the unaugmented Lagrangian L0 (d, z, µ ) = −g(z) +

∑

∑

i∈M

vˆi (di ) and g(z) are both closed

vˆi (di , pi ) − µ T (d − z).

i∈M

Let µ∗ = µ(p) · 1, and d∗ = z∗ = d(p), then (d∗ , z∗ , µ∗ ) is a saddle point of L0 (d, z, µ): L0 (d, z, µ∗ ) ≤ L0 (d∗ , z∗ , µ∗ ) ≤ L0 (d∗ , z∗ , µ).

By Section 3.2.1 of [30] and the uniqueness of the optimal solution to NASH-DEMAND(p), we have our convergence guarantees. The supply game part is just a mirror of the above discussion. □ Proof of Theorem 3. We first analyze the demand game. For any two agents i, j at equilibrium, the KKT conditions [28] of MAX-VALUE imply that vi′ (di ) = vj′ (dj ). Moreover, the KKT conditions of NASH-DEMAND(p) imply 1 pi

(

vi (di ) 1 − ′

)

di

=

X

1

(

vj (dj ) 1 − ′

pj

dj

)

X

.

Combining the two equations, we have X − di (p) pi

=

X − dj (p) pj

∑ =

j

X − dj (p)

∑

j pj

=

(N − 1)X

∑

j

pj

.

Conversely, if the conditions X − di (p) (N − 1)X

pi

= ∑

k

pk

,

∀i ∈ M

hold, then the conditions together with the KKT conditions of NASH-DEMAND(p) imply that vi′ (di ) = vj′ (dj ) for all agents i, j. Therefore, d(p) also solves MAX-VALUE and the proof is completed. Mirroring the above arguments we can also prove the theorem for the supply game. □ Proof of Lemma 1. The most important property of the variational inequalities VI(D, F ) and VI(S , G) is that the mappings F (·) and G(·) are separable in each dimension. We next use VI(D, F ) to demonstrate how we can employ theorem 3.1 in [34] to derive the sensitivity. For any p ∈ RM first focus on the scenario where d > 0 ++ , let d = d(p) be the equilibrium allocation. We ∑M to show the main procedures. When d > 0, the only binding constraint is g(d) = i=1 di − X . Let G = ∇d g(d) = [1, . . . , 1], where Q is the projection matrix onto the row spaces of G, then we have Q = GT (GGT )−1 G = [ M1 ]M ×M . According to

M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234

231

theorem 3.1 in [34], we have Q ∇p g(p) = 0,

(A.1)

(I − Q )(−∇d F )(I − Q )∇p g(p) = (I − Q )∇p F ,

(A.2)

where I is the identity matrix. By definition of αi and βi , we have ∇d F = diag(α1 , . . . , αM ), and ∇p F = diag(β1 , . . . , βM ). Combining (A.1) and (A.2), we have −1

−1

− (I − Q )∇d F ∇p g(p) = (I − Q )∇p F .

(A.3) 2

Note that the projection matrix Q satisfies Q = Q , thus rank(Q ) + rank(I − Q ) = M holds, which indicates that (A.1) and (A.3) admits a unique ∇p g(p). By partitioning of matrix, we can obtain the results of the theorem when d > 0. The complete results can be obtained by induction on the number of zero components of d. The above discussion can be directly applied to the sensitivity of VI(S , G) and we avoid the duplication here. □ Proof of Theorem 4. We first prove that the demand side dynamical system converges to optimality. We have the first order condition: d

∑ ∂ di X − dj ∑ ∂ di Xpj ∑ − . ∂ pj N − 1 ∂ pj k pk

di (t) =

dt

j∈M

j∈M

Since d(p) is a homogeneous function of degree 0, by Euler’s equation [35], the second item on the RHS of the above equation is 0. Then we have d dt

∑ ∂ di X − dj ∂ pj N − 1 j∈M ) ( 1 ∑ α ∑ i − 1j=i αj βj (X − dj ). = N −1 k αk

di (t) =

(A.4)

j∈M

From Lemma 4, we have 1 pj

vj′ (dj )(1 −

dj X

) = µ,

∀j ∈ M,

where µ is the baseline market-clearing price. Thus,

βj (X − dj ) =

1

X µ2

pj

vj′ (dj )

v ′ (d )(1 − dj /X ) = 2 j j

.

By incorporating this into (A.4), we have

⎡ ⎤ ∑ αj 1 1 ⎣ ⎦. ∑ di (t) = − ′ ′ dt N −1 vi (di ) k αk vj (dj ) X αi µ2

d

(A.5)

j

1 }. Take any It is worth noticing that the summation part on the RHS of the above equation is a weighted average of { v ′ (d ) i

i

snapshot of the system at time t, if p(t) ̸ = p∗ , from the KKT conditions we know that the agents cannot have equal marginal ′ values vi′ (di ). Therefore, we can always find agents m, n ∈ M such that vm (dm ) ≥ vi′ (di ), vn′ (dn ) ≤ vi′ (di ) for all i ∈ M and d d at least one of each group of inequalities is strict. Thus, we have dt dm (t) > 0 and dt dn (t) < 0. After an infinitesimal time ′ period δ t, dm (t + δ t) increases and dn (t + δ t) decreases, which implies that the marginal value vm (dm (t + δ t)) decreases ′ and vn (dn (t + δ t)) increases. We choose a series of timestamps that go to infinity and construct the according marginal ′ ′ value series {vm (dm )} and {vn′ (dn )}.4 Then we can conclude that {vm (dm )} is decreasing and bounded from below, while ′ {vn (dn )} is increasing and bounded from above. Therefore, they are both convergent series and must converge to the same limit. Otherwise, if they converge to distinct limits, then according to (A.5), the equilibrium allocation will still change at convergence, which is a contradiction. Thus far, we have proven that the equilibrium allocation will converge and agents have equal marginal values at convergence, which implies that demand side efficiency is achieved. As for the supply side dynamical system, the first order condition of equilibrium supply is: d dt

si (t) =

∑ ∂ si (N − 2)X + sj (t) ∂ si qj (t) ∑ − X. ∂ qj (N − 1)2 ∂ qj k qk (t) j∈N

4 Technically speaking the indexes m and n may vary with time, but this does not affect the correctness of our discussion.

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M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234

Since s(q) is a homogeneous function of degree 0, by Euler’s equation [35], the second item on the RHS of the above equation is 0. Then we have

∑ ∂ si (N − 2)X + sj (t) ∂ qj (N − 1)2 j∈N ) [ ( ] (N − 2)X ∑ αi sj (t) ∑ = − 1 α β 1 + . j=i j j (N − 1)2 (N − 2)X k αk

d

si (t) =

dt

(A.6)

j∈N

From Lemma 4, we have 1 qj

[

′

]

sj

cj (sj ) 1 +

(N − 2)X

= ν,

∀j ∈ N ,

where ν is the baseline market-clearing price. Thus,

[ βj 1 +

ν2

]

sj (N − 2)X

=−

′

cj (sj )

.

By incorporating this into (A.6), we have d dt

si (t) = −

⎡ ⎤ ∑ αj 1 1 ⎣ ∑ − ′ ⎦, ′ ci (si ) k αk cj (sj )

(N − 2)X αi ν 2 (N − 1)2

(A.7)

j

which is the counterpart of Eq. (A.5) on the supply side. Thus, we can slightly modify the analysis of (A.5) to prove that the equilibrium supply will converge to efficiency. □ Proof of Lemma 2. By the envelop theorem [35], we have d dX

value(X ) = mv(X ),

d dX

cost(X ) = mc(X ).

And the lemma immediately follows. □ Proof of Lemma 3. By definition, mv (X ) = vi′ (di ). And the KKT conditions of NASH-DEMAND(p) give that 1 pi

( ) di vi′ (di ) 1 − = µ(p). X

As a result, we have mv(X ) =

Xpi µ(p) X − di (p)

.

Likewise, by the KKT conditions of NASH-SUPPLY(q), we also have mc(X ) =

(N − 2)Xqj ν (q) (N − 2)X + sj (q)

. □

Proof of Theorem 6. From Lemma 4, we have that 1

di

v ′ (di )(1 − ) = µ(p), ∀i ∈ M, pi i X [ ] 1 ′ sj cj (sj ) 1 + = ν (q), j ∈ N . qj (N − 2)X We also have by definition that mv(X ) = vi′ (di ) and mc(X ) = cj′ (sj ). Therefore, the demand side agents pay

∑

pi µ(p)di = X · mv(X )(1 − Hd ),

i∈M

and the supply side agents receive

∑ j∈ N

(

qj ν (q)sj = X · mc(X ) 1 +

1 N −2

) Hs

. □

M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234

233

Proof of Corollary 2. At the optimal resource level X ∗ , we have that mv(X ∗ ) = mc(X ∗ ). Then the net payment transfer from the mediator to the agents is

( X · mc(X ) 1 +

( = X · mc(X )

1

− X · mv(X )(1 − Hd ) ) Hs + Hd > 0. □

N −2 1

N −2

) Hs

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M. Zou et al. / Performance Evaluation 127–128 (2018) 212–234 Mao Zou received the B.Eng degree in computer science and technology and the Ph.D. degree in computer software and theory from University of Science and Technology of China (USTC), in 2013 and 2018, respectively. His current research interests include Internet economics and performance evaluation.

Richard T.B. Ma received the B.Sc. degree (Hons.) in computer science and the M.Phil. degree in computer science and engineering from the Chinese University of Hong Kong, in 2002 and 2004, respectively, and the Ph.D. degree in electrical engineering from Columbia University, in 2010. During the Ph.D. degree, he was a Research Intern with the IBM T.J. Watson Research Center, NY, USA, and the Telefonica Research, Barcelona. He is currently an Assistant Professor with the Department of Computer Science, National University of Singapore. His current research interests include distributed systems and network economics. He is a co-recipient of the Best Paper Award in the IEEE Workshop on Smart Data Pricing 2015, the IEEE ICNP 2014, and the IEEE IC2E 2013.

Yinlong Xu received the B.S. degree in mathematics from Peking University, Beijing, China, in 1983, and the M.S. and Ph.D. degrees in computer science from the University of Science and Technology of China (USTC), Hefei, China, in 1989 and 2004, respectively. He is currently a Professor with the School of Computer Science and Technology, USTC. He served the Department of Computer Science and Technology, USTC as an Assistant Professor, a Lecturer, and an Associate Professor. He is currently leading a group of research students in doing some networking, storage and high performance computing research. His current research interests include storage system, file system, social network, and high performance I/O. Prof. Xu was a recipient of the Excellent Ph.D. Advisor Award of the Chinese Academy of Sciences in 2006.