Analysis of Decentralized Quantized Auctions on ... - Semantic Scholar
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Analysis of Decentralized Quantized Auctions on Cooperative Networks Peng Jia and Peter E. Caines
Abstract This paper considers a decentralized quantity allocation problem over networks. By employing the so-called UQ-PSP auction algorithm [1], distributed auctions on a two-level network are developed so as to achieve efficient resource allocations (in the sense of maximization of social welfare). In the formulation in this paper, each vertex in the higher level network is regarded as a supplier for a uniquely associated lower level network, and each lower level network consists of a set of agents which represent buyers. Each lower level network with its associated supplier is assumed to constitute a local UQ-PSP auction, generically denoted as Al . The adjustment of the quantities supplied to any Al is via a consensusbased dynamical system which exchanges quantities depending upon the limit prices of the recursive bidding processes of the local auctions in the neighborhood of Al in the higher level network. Such a consensus auction system is formulated as a discrete-time weighted-average consensus problem with an associated family of time-varying and asymmetric Perron matrices. Subject to continuous valued pricing, the corresponding dynamical system converges to a global limit price, which is independent of the initial data and corresponds to an efficient quantity allocation. Exponential convergence is established by using the passivity property of UQ-PSP auctions, and using the primitive and SIA (stochastic, indecomposable and aperiodic) properties of the family of Perron matrices.
I. Introduction Game theoretic methods have been applied for the design of market pricing and resource allocation in several practical areas such as social foraging problem [2], and network flow, shortest path, assignment and transportation problems [3]. Each agent in such scenarios will typically This work was supported by NSERC Discovery and Strategic Grants. P. Jia and P. E. Caines are with the Centre for Intelligent Machines (CIM) and the Department of Electrical and Computer Engineering, McGill University, Montreal, Canada. {pjia; peterc}@cim.mcgill.ca
2
have its own demand and utility functions, where the latter depends upon both individual and mass behaviours. In [4], [5], a so-called Progressive Second Price (PSP) (auction) algorithm was proposed for dynamical market-pricing and variable-size resource allocation in communication networks. This auction algorithm efficiently and dynamically allocates a divisible resource to agents capable of exerting their market power to generate successive price and quantity bids. However, the rate of convergence of (continuous valued) PSP auction algorithms is inversely proportional to a bid fee defined in the specification of the algorithm. P. Maille and B. Tuffin presented a one-step bid version of the PSP algorithm in [6] to avoid the slow convergence of PSP algorithms, but this is achieved by the communication of (approximations of) all agents’ demand functions to all agents, and hence is not a decentralized algorithm in the sense used in this paper. Quantized PSP algorithms were subsequently developed by P. Jia, C. W. Qu and P. Caines in [1], [7] to deal with the slow convergence and signal overhead problems of PSP algorithms. In this paper we study a potential application of the Unique-limit Quantized PSP (UQPSP) (auction) algorithm to resource allocation problems on distributed networks. The UQ-PSP algorithm was first introduced in [1], [8]; it has the following desirable properties: (a) The assumption of the quantized prices in the UQ-PSP algorithm is a practically meaningful and well motivated constraint (for example, in practical auctions the bid increment amount is usually predetermined and bounded from below and therefore the pricing is not continuous, see e.g., [9], [10]). (b) A UQ-PSP algorithm is a VCG (Vickrey-Clarke-Groves Mechanism)-type dynamical auction: at each bidding iteration, agents’ UQ-PSP strategies are optimal in terms of individual utility up to a tolerance γ and truth-telling (i.e., at each iteration their bids accurately reflect their demand functions), where γ depends on the initial price quantization. (c) If a virtual auctioneer is introduced to bid the lowest quantized market clearing price, the dynamical system associated with the UQ-PSP algorithm converges to a unique limit price such that the aggregate demand of all agents at this price is just less than (i.e., equal to, or one quantization induced unit step below) the total quantity C, and hence the limit price is independent of the initial bid data. (d) The limit prices of UQ-PSP auction algorithms are γ-Nash equilibria (i.e., no agent can gain more than O(γ) in its own utility by unilaterally deviating from its strategy) and the
3
limit allocations are efficient up to a quantized level. (e) The number of iterations for convergence of a UQ-PSP algorithm is bounded by a constant which is a function of the auction parameters and is bounded from above by the cardinality of the quantized price set plus one. The work in this paper is motivated by the fact that (i) agents in communication networks, power grid and social networks are often intrinsically unable to access and process all the information communicated over such networks but can communicate locally, and (ii) in principle, UQ-PSP based local auctions can be implemented with the properties (a)-(e) above. Cooperative distributed decision-making via consensus algorithms has been extensively studied in computer science [11], decision theory [12], and systems and control theory [13], [14], [15]. In the present paper, we introduce two-level network structures where vertices in a higher level network (with an arbitrary topology) are regarded as cooperative suppliers and vertices in lower level (clique) networks are considered as noncooperative buyers, where each lower level network is uniquely associated with a vertex in the higher level network. In this framework, a socalled consensus UQ-PSP auction algorithm is formulated where the dynamics occur in two nested iterations: (i) each agent participates in a local UQ-PSP auction for a given quantity of resource; (ii) the quantities associated with local auctions are recursively adjusted via a consensus algorithm, and the cycle is repeated until convergence occurs up to a tolerance level. Realistic examples of the cooperative suppliers considered in this context are provided by retail branches of a company or independently owned subsidiary power grids on a national network. The key motivation for the cooperation assumption in such a case would lie in the increase of social welfare of the whole network (market), which would coincide with the efficiency of the PSP and UQ-PSP auction algorithms (see [5], [1]). It is next assumed that the pricing quantization in the local auctions is sufficiently fine that the dynamical system is adequately represented by a model where the prices take values in the real number continuum R1 . It will then be shown that distributed network based consensus UQ-PSP auction algorithms converge to a global limit price (which corresponds to the global market clearing price), where, moreover, global efficiency is achieved. The high level network consensus dynamics are analyzed as a discrete-time weighted-average consensus system with a family of time-varying and asymmetric Perron matrices. In this paper, the Perron matrices are of the form M(k) = I −αβ(k)L, where α > 0, β(·) is a diagonal matrix with positive diagonal entries,
4
and L is the higher level network Laplacian; the properties of (i) the uniform boundedness of the weightings β(k) and (ii) the SIA (stochastic, indecomposable, aperiodic) and primitive features of the Perron matrices M(k), are sufficient to permit the proof of a general (geometric convergence rate) consensus in Theorem 3.3. The paper is organized as follows. •
Previous results of PSP auctions are reviewed in Section II and are applied in Section III as a starting point. We begin Section II-A with an introduction of the PSP mechanism, and then in Section II-B the dynamical UQ-PSP auction algorithm is formulated and its convergence analysis is briefly presented.
•
In Section III-A, a two-level network based auction framework is formulated subject to a cooperation assumption among local auctions and local suppliers. The dynamical behaviours between local suppliers are then formulated as a consensus problem and convergence is analyzed in Section III-B. Numerical examples and further discussion for two-level network based auctions are given in Section III-C.
•
Finally, the results are summarized in Section IV. II. Quantized PSP Auction Algorithms
For self-contained analysis, in this section we will briefly review the previous work in the Progressive Second Price (PSP) auction and the Unique-limit Quantized PSP (UQ-PSP) algorithm. The convergence results in this part will be immediately applied in Section III as a starting point. A. Problem Formulation [5] In a non-cooperative game, N agents compete for a divisible good of quantity C. Each agent i, 1 ≤ i ≤ N, makes a two-dimensional bid si = (pi , qi ) to a seller, where qi is the quantity the agent desires and pi is the unit-price the agent would like to pay for qi . The bidding profile is defined as s B [si ]1≤i≤N , and s−i B [s1 , · · · , si−1 , si+1 , · · · , sN ] as the profile of Agent i’s opponents. It is noted here that s is public information and is received by all agents in the system. All agents in such a game update their bids by observing the present bidding profile s. The updated bidding profile is then announced publicly and agents make new bids iteratively. The bidding process repeats until some kind of equilibrium is achieved. Here we call such a bidding
5
process a progressive (dynamical) auction. Specifically, in each bidding iteration, we define the following notation. The market price function (MPF) of Agent i is defined as: X Pi (q, s−i ) = inf p ≥ 0 : C − q ≥ q , k pk >p,k,i
(1)
which is interpreted as the minimum price an agent bids in order to obtain the quantity q given the opponents’ profile s−i . Clearly the function is only reasonable when q > 0. Its inverse function Qi is defined as follows: + X Qi (p, s−i ) = C − qk , pk >p,k,i
which means the maximum available quantity at a bid price of p given s−i . The enveloping market price function is as follows: X E P (q, s) = inf p ≥ 0 : C ≥ q + q , k
0 ≤ q ≤ C,
(2)
pk >p
which denotes the market price function without an agent being omitted; this gives an upper contour to the set of market price curves. Each agent is associated with a private demand function δi : R+ → R+ , 1 ≤ i ≤ N, whose value δi (q) is interpreted to be the marginal price of a certain amount of resource q for an agent Rq i. Define Di = δ−1 i , as the inverse demand function and Zi (q) = 0 δi (z)dz, as the valuation (or reward) functions of agents. It is assumed that Di (p) ≥ 0 for all p ≥ 0 (i.e., set Di (p) = 0 for all p ≥ δi (0)). Assumption 2.1: For i = 1, 2, · · · , N, δi satisfies the elasticity assumption ([5], [8]), that is to say, (i) Zi (0) = 0; (ii) δi ≥ 0 is decreasing and continuous; (iii) for any C ≥ q > q0 ≥ 0, there exists γ > 0 such that δi (q) > 0 implies δi (q) − δi (q0 ) < −γ(q − q0 ). Remark 2.2: The third condition of Assumption 2.1 is also referred as a strongly monotone mapping over [0, C] for −δi (see [16]), that is to say, for each −δi : R → R, (q − q0 )(−δi (q) + δi (q0 )) ≥ ||q − q0 ||2 , for all p, p0 ∈ [0, C]. In general, the utility of Agent i’s is defined to be ui (s) = Zi (ai (s)) − ci (s),
(3)
6
where ai denotes the quantity Agent i obtains by a bid si (when the opponents bid s−i ) and the charge to Agent i by the seller is denoted ci . Since Zi is a private valuation function, each agent’s utility ui is hence unknown to the other agents and the auctioneer. The PSP allocation rule presented in [17] is formulated as follows: ( ) qi ai (s) = ai (si , s−i ) = min qi , P Qi (pi , s−i ) , k:pk =pi qk X ci (s) = ci (si , s−i ) = p j · a j (0; s−i ) − a j (si ; s−i ) .
(4) (5)
j,i
Here ai corresponds to the minimum of Agent i’s bid quantity qi and the available market quantity at the bid price pi . For the PSP allocation rule, the charge ci is defined as the (opportunity) cost shown in (5), which is based upon the exclusion compensation principle [5] and the VickreyClarke-Groves (VCG) mechanism [18]; it represents the potential difference in revenue between that contributed by all the other agents distinct from Agent i when (i) Agent i is absent from the auction and (ii) Agent i participates in the auction. The Progressive Second Price auction is named after this feature [5]. B. Quantized Algorithm with a Unique Iteration Limit In this paper we assume that all agents are myopic and egoistic, that is to say, they maximize only their own utilities by reacting to the current bidding profile without considering the previous or the anticipated market bidding behaviours. Given s−i , the bid si = (wi , vi ) is an -best reply for agent i (see [5] and Appendix of [19]) uniquely determined by ( vi = sup q ≥ 0 : δi (q) > Pi (q, s−i );
q
Z
Pi (η, s−i )dη ≤ bi − 0
wi = δi (vi ),
)
, δi (0) (6)
where > 0 is the bid fee, bi is Agent i’s budget, and δi satisfies the elasticity assumption above. It is shown in [5] that vi above is indeed the best quantity reply in the sense that ui (s) with respect to qi is supremised up to at the unique solution to (6). The strategy (6) is also truthful [5], that is to say, the bid price wi is calculated truthfully based upon the private demand function δi and the best quantity reply vi . Moreover, given s−i , the bid price wi guarantees that ai ((wi , vi ), s−i ) = vi , and hence the utility ui is maximized up to , which is called -best reply.
7
The convergence and efficiency of the dynamical PSP algorithm have been studied by Lazar and Semret in [5], [20]. Under the assumption that all agents apply -best strategies, a PSP auction algorithm converges to a 2 Nash Equilibrium in time of O(1/) where the bid fee is a design parameter and used to trade off between social efficiency and convergence rate. Quantized price PSP auction algorithms were introduced in [7], [1] to accelerate the convergence speed of the PSP algorithms; in this formulation we assume all (myopic and egoistic) agents in the auction only choose bid prices from a quantized price set B0p at each bidding iteration k to maximize its own utility, which is a realistic assumption [9]. Assumption 2.3: Given a bidding profile sk−1 at iteration (k − 1), k ≥ 1, all agents in a Uniquelimit Quantized PSP (UQ-PSP) auction synchronously apply the quantized strategies (UQ-PSP strategies) ski = (pki , qki ), 1 ≤ i ≤ N, at iteration k as follows: pki = T vki , sk−1 , B0p k k−1 if vki = pmax ; Pi (vi , s−i ), B n o min p j ∈ B0p ; p j > Pi (vki , sk−1 ) , otherwise, −i qki = Di (pki ),
(7)
0 where (wki , vki ), derived from (6) by setting = 0, is the best reply w.r.t. sk−1 −i , B p is defined as
the initial quantized bid price set and pmax B max B0p . k k k Lemma 2.4: [19] Given sk−1 −i , the quantized strategy si = (pi , qi ) specified in (7) is a γ-best
reply, where γ depends solely upon the initial quantized price set B0p . Here the γ-best reply is with respect to all feasible strategies without necessarily quantized pricing constraint. The relationship between Agent i’s demand function, its market price function, its best strategy given in (6), and its UQ-PSP strategy given in (7) is shown in Figure 1, where purely for simplicity of portrayal, it is assumed that δi is linear. By the definition of the function T in (7), each agent in a UQ-PSP auction bids a quantized price pi ∈ B0p , immediately higher than the best price reply wi in (6), or wi itself when wi corresponds to the maximum quantized price. For the sake of simplicity, we assume that all agents in the auction do not have budget constraints. Because the values of all agents’ cost functions vanish in the limit, that is to say, ci (s∗ ) = 0, for all i, 1 ≤ i ≤ N, at the equilibrium s∗ (see Theorem 2.7), the budget {bi > 0, 1 ≤ i ≤ N} will not influence the limit quantity allocation.
8
& (! #%$ )
Analysis of Decentralized Quantized Auctions on Cooperative Networks Peng Jia and Peter E. Caines
Abstract This paper considers a decentralized quantity allocation problem over networks. By employing the so-called UQ-PSP auction algorithm [1], distributed auctions on a two-level network are developed so as to achieve efficient resource allocations (in the sense of maximization of social welfare). In the formulation in this paper, each vertex in the higher level network is regarded as a supplier for a uniquely associated lower level network, and each lower level network consists of a set of agents which represent buyers. Each lower level network with its associated supplier is assumed to constitute a local UQ-PSP auction, generically denoted as Al . The adjustment of the quantities supplied to any Al is via a consensusbased dynamical system which exchanges quantities depending upon the limit prices of the recursive bidding processes of the local auctions in the neighborhood of Al in the higher level network. Such a consensus auction system is formulated as a discrete-time weighted-average consensus problem with an associated family of time-varying and asymmetric Perron matrices. Subject to continuous valued pricing, the corresponding dynamical system converges to a global limit price, which is independent of the initial data and corresponds to an efficient quantity allocation. Exponential convergence is established by using the passivity property of UQ-PSP auctions, and using the primitive and SIA (stochastic, indecomposable and aperiodic) properties of the family of Perron matrices.
I. Introduction Game theoretic methods have been applied for the design of market pricing and resource allocation in several practical areas such as social foraging problem [2], and network flow, shortest path, assignment and transportation problems [3]. Each agent in such scenarios will typically This work was supported by NSERC Discovery and Strategic Grants. P. Jia and P. E. Caines are with the Centre for Intelligent Machines (CIM) and the Department of Electrical and Computer Engineering, McGill University, Montreal, Canada. {pjia; peterc}@cim.mcgill.ca
2
have its own demand and utility functions, where the latter depends upon both individual and mass behaviours. In [4], [5], a so-called Progressive Second Price (PSP) (auction) algorithm was proposed for dynamical market-pricing and variable-size resource allocation in communication networks. This auction algorithm efficiently and dynamically allocates a divisible resource to agents capable of exerting their market power to generate successive price and quantity bids. However, the rate of convergence of (continuous valued) PSP auction algorithms is inversely proportional to a bid fee defined in the specification of the algorithm. P. Maille and B. Tuffin presented a one-step bid version of the PSP algorithm in [6] to avoid the slow convergence of PSP algorithms, but this is achieved by the communication of (approximations of) all agents’ demand functions to all agents, and hence is not a decentralized algorithm in the sense used in this paper. Quantized PSP algorithms were subsequently developed by P. Jia, C. W. Qu and P. Caines in [1], [7] to deal with the slow convergence and signal overhead problems of PSP algorithms. In this paper we study a potential application of the Unique-limit Quantized PSP (UQPSP) (auction) algorithm to resource allocation problems on distributed networks. The UQ-PSP algorithm was first introduced in [1], [8]; it has the following desirable properties: (a) The assumption of the quantized prices in the UQ-PSP algorithm is a practically meaningful and well motivated constraint (for example, in practical auctions the bid increment amount is usually predetermined and bounded from below and therefore the pricing is not continuous, see e.g., [9], [10]). (b) A UQ-PSP algorithm is a VCG (Vickrey-Clarke-Groves Mechanism)-type dynamical auction: at each bidding iteration, agents’ UQ-PSP strategies are optimal in terms of individual utility up to a tolerance γ and truth-telling (i.e., at each iteration their bids accurately reflect their demand functions), where γ depends on the initial price quantization. (c) If a virtual auctioneer is introduced to bid the lowest quantized market clearing price, the dynamical system associated with the UQ-PSP algorithm converges to a unique limit price such that the aggregate demand of all agents at this price is just less than (i.e., equal to, or one quantization induced unit step below) the total quantity C, and hence the limit price is independent of the initial bid data. (d) The limit prices of UQ-PSP auction algorithms are γ-Nash equilibria (i.e., no agent can gain more than O(γ) in its own utility by unilaterally deviating from its strategy) and the
3
limit allocations are efficient up to a quantized level. (e) The number of iterations for convergence of a UQ-PSP algorithm is bounded by a constant which is a function of the auction parameters and is bounded from above by the cardinality of the quantized price set plus one. The work in this paper is motivated by the fact that (i) agents in communication networks, power grid and social networks are often intrinsically unable to access and process all the information communicated over such networks but can communicate locally, and (ii) in principle, UQ-PSP based local auctions can be implemented with the properties (a)-(e) above. Cooperative distributed decision-making via consensus algorithms has been extensively studied in computer science [11], decision theory [12], and systems and control theory [13], [14], [15]. In the present paper, we introduce two-level network structures where vertices in a higher level network (with an arbitrary topology) are regarded as cooperative suppliers and vertices in lower level (clique) networks are considered as noncooperative buyers, where each lower level network is uniquely associated with a vertex in the higher level network. In this framework, a socalled consensus UQ-PSP auction algorithm is formulated where the dynamics occur in two nested iterations: (i) each agent participates in a local UQ-PSP auction for a given quantity of resource; (ii) the quantities associated with local auctions are recursively adjusted via a consensus algorithm, and the cycle is repeated until convergence occurs up to a tolerance level. Realistic examples of the cooperative suppliers considered in this context are provided by retail branches of a company or independently owned subsidiary power grids on a national network. The key motivation for the cooperation assumption in such a case would lie in the increase of social welfare of the whole network (market), which would coincide with the efficiency of the PSP and UQ-PSP auction algorithms (see [5], [1]). It is next assumed that the pricing quantization in the local auctions is sufficiently fine that the dynamical system is adequately represented by a model where the prices take values in the real number continuum R1 . It will then be shown that distributed network based consensus UQ-PSP auction algorithms converge to a global limit price (which corresponds to the global market clearing price), where, moreover, global efficiency is achieved. The high level network consensus dynamics are analyzed as a discrete-time weighted-average consensus system with a family of time-varying and asymmetric Perron matrices. In this paper, the Perron matrices are of the form M(k) = I −αβ(k)L, where α > 0, β(·) is a diagonal matrix with positive diagonal entries,
4
and L is the higher level network Laplacian; the properties of (i) the uniform boundedness of the weightings β(k) and (ii) the SIA (stochastic, indecomposable, aperiodic) and primitive features of the Perron matrices M(k), are sufficient to permit the proof of a general (geometric convergence rate) consensus in Theorem 3.3. The paper is organized as follows. •
Previous results of PSP auctions are reviewed in Section II and are applied in Section III as a starting point. We begin Section II-A with an introduction of the PSP mechanism, and then in Section II-B the dynamical UQ-PSP auction algorithm is formulated and its convergence analysis is briefly presented.
•
In Section III-A, a two-level network based auction framework is formulated subject to a cooperation assumption among local auctions and local suppliers. The dynamical behaviours between local suppliers are then formulated as a consensus problem and convergence is analyzed in Section III-B. Numerical examples and further discussion for two-level network based auctions are given in Section III-C.
•
Finally, the results are summarized in Section IV. II. Quantized PSP Auction Algorithms
For self-contained analysis, in this section we will briefly review the previous work in the Progressive Second Price (PSP) auction and the Unique-limit Quantized PSP (UQ-PSP) algorithm. The convergence results in this part will be immediately applied in Section III as a starting point. A. Problem Formulation [5] In a non-cooperative game, N agents compete for a divisible good of quantity C. Each agent i, 1 ≤ i ≤ N, makes a two-dimensional bid si = (pi , qi ) to a seller, where qi is the quantity the agent desires and pi is the unit-price the agent would like to pay for qi . The bidding profile is defined as s B [si ]1≤i≤N , and s−i B [s1 , · · · , si−1 , si+1 , · · · , sN ] as the profile of Agent i’s opponents. It is noted here that s is public information and is received by all agents in the system. All agents in such a game update their bids by observing the present bidding profile s. The updated bidding profile is then announced publicly and agents make new bids iteratively. The bidding process repeats until some kind of equilibrium is achieved. Here we call such a bidding
5
process a progressive (dynamical) auction. Specifically, in each bidding iteration, we define the following notation. The market price function (MPF) of Agent i is defined as: X Pi (q, s−i ) = inf p ≥ 0 : C − q ≥ q , k pk >p,k,i
(1)
which is interpreted as the minimum price an agent bids in order to obtain the quantity q given the opponents’ profile s−i . Clearly the function is only reasonable when q > 0. Its inverse function Qi is defined as follows: + X Qi (p, s−i ) = C − qk , pk >p,k,i
which means the maximum available quantity at a bid price of p given s−i . The enveloping market price function is as follows: X E P (q, s) = inf p ≥ 0 : C ≥ q + q , k
0 ≤ q ≤ C,
(2)
pk >p
which denotes the market price function without an agent being omitted; this gives an upper contour to the set of market price curves. Each agent is associated with a private demand function δi : R+ → R+ , 1 ≤ i ≤ N, whose value δi (q) is interpreted to be the marginal price of a certain amount of resource q for an agent Rq i. Define Di = δ−1 i , as the inverse demand function and Zi (q) = 0 δi (z)dz, as the valuation (or reward) functions of agents. It is assumed that Di (p) ≥ 0 for all p ≥ 0 (i.e., set Di (p) = 0 for all p ≥ δi (0)). Assumption 2.1: For i = 1, 2, · · · , N, δi satisfies the elasticity assumption ([5], [8]), that is to say, (i) Zi (0) = 0; (ii) δi ≥ 0 is decreasing and continuous; (iii) for any C ≥ q > q0 ≥ 0, there exists γ > 0 such that δi (q) > 0 implies δi (q) − δi (q0 ) < −γ(q − q0 ). Remark 2.2: The third condition of Assumption 2.1 is also referred as a strongly monotone mapping over [0, C] for −δi (see [16]), that is to say, for each −δi : R → R, (q − q0 )(−δi (q) + δi (q0 )) ≥ ||q − q0 ||2 , for all p, p0 ∈ [0, C]. In general, the utility of Agent i’s is defined to be ui (s) = Zi (ai (s)) − ci (s),
(3)
6
where ai denotes the quantity Agent i obtains by a bid si (when the opponents bid s−i ) and the charge to Agent i by the seller is denoted ci . Since Zi is a private valuation function, each agent’s utility ui is hence unknown to the other agents and the auctioneer. The PSP allocation rule presented in [17] is formulated as follows: ( ) qi ai (s) = ai (si , s−i ) = min qi , P Qi (pi , s−i ) , k:pk =pi qk X ci (s) = ci (si , s−i ) = p j · a j (0; s−i ) − a j (si ; s−i ) .
(4) (5)
j,i
Here ai corresponds to the minimum of Agent i’s bid quantity qi and the available market quantity at the bid price pi . For the PSP allocation rule, the charge ci is defined as the (opportunity) cost shown in (5), which is based upon the exclusion compensation principle [5] and the VickreyClarke-Groves (VCG) mechanism [18]; it represents the potential difference in revenue between that contributed by all the other agents distinct from Agent i when (i) Agent i is absent from the auction and (ii) Agent i participates in the auction. The Progressive Second Price auction is named after this feature [5]. B. Quantized Algorithm with a Unique Iteration Limit In this paper we assume that all agents are myopic and egoistic, that is to say, they maximize only their own utilities by reacting to the current bidding profile without considering the previous or the anticipated market bidding behaviours. Given s−i , the bid si = (wi , vi ) is an -best reply for agent i (see [5] and Appendix of [19]) uniquely determined by ( vi = sup q ≥ 0 : δi (q) > Pi (q, s−i );
q
Z
Pi (η, s−i )dη ≤ bi − 0
wi = δi (vi ),
)
, δi (0) (6)
where > 0 is the bid fee, bi is Agent i’s budget, and δi satisfies the elasticity assumption above. It is shown in [5] that vi above is indeed the best quantity reply in the sense that ui (s) with respect to qi is supremised up to at the unique solution to (6). The strategy (6) is also truthful [5], that is to say, the bid price wi is calculated truthfully based upon the private demand function δi and the best quantity reply vi . Moreover, given s−i , the bid price wi guarantees that ai ((wi , vi ), s−i ) = vi , and hence the utility ui is maximized up to , which is called -best reply.
7
The convergence and efficiency of the dynamical PSP algorithm have been studied by Lazar and Semret in [5], [20]. Under the assumption that all agents apply -best strategies, a PSP auction algorithm converges to a 2 Nash Equilibrium in time of O(1/) where the bid fee is a design parameter and used to trade off between social efficiency and convergence rate. Quantized price PSP auction algorithms were introduced in [7], [1] to accelerate the convergence speed of the PSP algorithms; in this formulation we assume all (myopic and egoistic) agents in the auction only choose bid prices from a quantized price set B0p at each bidding iteration k to maximize its own utility, which is a realistic assumption [9]. Assumption 2.3: Given a bidding profile sk−1 at iteration (k − 1), k ≥ 1, all agents in a Uniquelimit Quantized PSP (UQ-PSP) auction synchronously apply the quantized strategies (UQ-PSP strategies) ski = (pki , qki ), 1 ≤ i ≤ N, at iteration k as follows: pki = T vki , sk−1 , B0p k k−1 if vki = pmax ; Pi (vi , s−i ), B n o min p j ∈ B0p ; p j > Pi (vki , sk−1 ) , otherwise, −i qki = Di (pki ),
(7)
0 where (wki , vki ), derived from (6) by setting = 0, is the best reply w.r.t. sk−1 −i , B p is defined as
the initial quantized bid price set and pmax B max B0p . k k k Lemma 2.4: [19] Given sk−1 −i , the quantized strategy si = (pi , qi ) specified in (7) is a γ-best
reply, where γ depends solely upon the initial quantized price set B0p . Here the γ-best reply is with respect to all feasible strategies without necessarily quantized pricing constraint. The relationship between Agent i’s demand function, its market price function, its best strategy given in (6), and its UQ-PSP strategy given in (7) is shown in Figure 1, where purely for simplicity of portrayal, it is assumed that δi is linear. By the definition of the function T in (7), each agent in a UQ-PSP auction bids a quantized price pi ∈ B0p , immediately higher than the best price reply wi in (6), or wi itself when wi corresponds to the maximum quantized price. For the sake of simplicity, we assume that all agents in the auction do not have budget constraints. Because the values of all agents’ cost functions vanish in the limit, that is to say, ci (s∗ ) = 0, for all i, 1 ≤ i ≤ N, at the equilibrium s∗ (see Theorem 2.7), the budget {bi > 0, 1 ≤ i ≤ N} will not influence the limit quantity allocation.
8
& (! #%$ )
