Abstract
With an effort to allocate divisible resources among suppliers and consumers, a double-sided auction model is designed to decide strategies for individual players in this chapter. Under the auction mechanism with the VCG-type payment, the incentive compatibility holds, and the efficient bid profile is a Nash equilibrium (NE). Different from the single-sided auction in the previous chapter, there exists an infinite number of NEs in the underlying double-sided auction game, which brings difficulties for players to implement the efficient solution. To overcome this challenge, we formulate the double-sided auction game as a pair of single-sided auction games which are coupled via a joint potential quantity of the resource. A decentralized iteration procedure is then designed to achieve efficient solution, where a single player, a buyer or a seller, implements his best strategy with respect to a given potential quantity and a constraint on his bid strategy. Accordingly, the potential quantity is updated with respect to iteration steps as well. It is verified that the system converges to the efficient NE within finite iteration steps in the order of \(\mathscr {O}(\ln (1/\varepsilon ))\) with \(\varepsilon \) representing the termination criterion of the algorithm.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R. Jain, J. Walrand, An efficient Nash-implementation mechanism for network resource allocation. Automatica 46, 1276–1283 (2010)
G. Iosifidis, I. Koutsopoulos, Double auction mechanisms for resource allocation in autonomous networks. IEEE J. Sel. Areas Commun. 28(1), 95–102 (2010)
S.K. Garg, S. Venugopal, J. Broberg, R. Buyya, Double auction-inspired meta-scheduling of parallel applications on global grids. J. Parallel Distrib. Comput. 73(4), 450–464 (2013)
A.R. Kian, J.B. Cruz, R.J. Thomas, Bidding strategies in oligopolistic dynamic electricity double-sided auctions. IEEE Trans. Power Syst. 20(1), 50–58 (2005)
P. Samimi, Y. Teimouri, M. Mukhtar, A combinatorial double auction resource allocation model in cloud computing. Inf. Sci. 357, 201–216 (2016)
A. Mohsenian-Rad, V.W.S. Wong, J. Jatskevich, R. Schober, A. Leon-Garcia, Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid. IEEE Trans. Smart Grid 1(3), 320–331 (2010)
R. Johari, J.N. Tsitsiklis, Communication requirements of VCG-like mechanisms in convex environments, in Proceedings of the Allerton Conference on Control, Communications and Computing, Princeton, pp. 1391–1396
S. Yang, B. Hajek, VCG-Kelly mechanisms for allocation of divisible goods: adapting VCG mechanisms to one-dimensional signals. IEEE J. Sel. Areas Commun. 25(6), 1237–1243 (2007)
P. Samadi, H. Mohsenian-Rad, R. Schober, V.W.S. Wong, Advanced demand side management for the future smart grid using mechanism design. IEEE Trans. Smart Grid 3(3), 1170–1180 (2012)
D. Fang, W. Jingfang, D. Tang, A double auction model for competitive generators and large consumers considering power transmission cost. Int. J. Electr. Power Energy Syst. 43(1), 880–888 (2012)
P.K. Tiwari, Y.R. Sood, An efficient approach for optimal allocation and parameters determination of TCSC with investment cost recovery under competitive power market. IEEE Trans. Power Syst. 28(3), 2475–2484 (2013)
X. Zou, Double-sided auction mechanism design in electricity based on maximizing social welfare. Energy Policy 37, 4231–4239 (2009)
R.T. Maheswaran, T. Basar, Social welfare of selfish agents: motivating efficiency for divisible resources, in IEEE 43rd Annual Conference on Decision and Control, vol. 2 (2004), pp. 1550–1555
A. Lazar, N. Semret, Design and analysis of the progressive second price auction for network bandwidth sharing. Telecommun. Syst. 13 (2001)
N. Semret, Market Mechanisms for network resource sharing. Ph.D. thesis, Columbia University (1999)
B. Tuffin, Revisited progressive second price auction for charging telecommunication networks. Telecommun. Syst. 20(3–4), 255–263 (2002)
P. Maillé, B. Tuffin, The progressive second price mechanism in a stochastic environment. Netnomics 5(2), 119–147 (2003)
P. Maillé, Market clearing price and equilibria of the progressive second price mechanism. RAIRO-Oper. Res. 41(4), 465–478 (2007)
S. Zou, Z. Ma, X. Liu, Auction-based distributed efficient economic operations of microgrid systems. Int. J. Control 87(12), 2446–2462 (2014)
X. Shi, Z. Ma, An efficient game for vehicle-to-grid coordination problems in smart grid. Int. J. Syst. Sci. 46(15), 2686–2701 (2015)
J. Zou, X. Hongwan, Auction-based power allocation for multiuser two-way relaying networks. IEEE Trans. Wirel. Commun. 12(1), 31–39 (2013)
L. Cao, W. Xu, J. Lin, K. Niu, Z. He, An auction approach to resource allocation in OFDM-based cognitive radio networks, in 75th IEEE Vehicular Technology Conference (VTC Spring), Yokohama (2012), pp. 1–5
D. Wu, Y. Cai, M. Guizani, Auction-based relay power allocation: Pareto optimality, fairness, and convergence. IEEE Trans. Commun. 62(7), 2249–2259 (2014)
D.C. Parkes, L.H. Ungar, Iterative combinatorial auctions: theory and practice, in 17th National Conference on Artificial Intelligence (AAAI-00) (2000), pp. 74–81
L.M. Ausubel, P. Milgrom, Ascending auctions with package bidding. Front. Theor. Econ. 1, 1–42 (2002)
P. Maillé, B. Tuffin, Multibid auctions for bandwidth allocation in communication networks, in 23rd Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 1 (2004), pp. 54–65
P. Maillé, B. Tuffin, Pricing the internet with multibid auctions. IEEE/ACM Trans. Netw. 14(5), 992–1004 (2006)
P. Jia, C.W. Qu, P.E. Caines, On the rapid convergence of a class of decentralized decision processes: quantized progressive second-price auctions. IMA J. Math. Control. Inf. 26(3), 325–355 (2009)
P. Jia, P. Caines, Analysis of quantized double auctions with application to competitive electricity markets. INFOR: Inf. Syst. Oper. Res. 48(4), 239–250 (2010)
P. Jia, P.E. Caines, Analysis of decentralized quantized auctions on cooperative networks. IEEE Trans. Autom. Control 58(2), 529–534 (2013)
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)
D.S. Damianov, J.G. Becker, Auctions with variable supply: Uniform price versus discriminatory. Eur. Econ. Rev. 54(4), 571–593 (2010)
H. Haghighat, H. Seifi, A.R. Kian, Pay-as-bid versus marginal pricing: the role of suppliers strategic behavior. Electr. Power Energy Syst. 42(1), 350–358 (2012)
D.E. Aliabadi, M. Kaya, G. Sahin, An agent-based simulation of power generation company behavior in electricity markets under different market-clearing mechanisms. Energy Policy 100, 191–205 (2017)
S. Zhou, Z. Shu, K. Tan, H.B. Gooi, S. Chen, Y. Gao, Study of market clearing model for Singapore’s wholesale real-time electricity market, in IEEE International Conference on Power System Technology (2016), pp. 1–7
M. Nazif Faqiry and Sanjoy Das, Double-sided energy auction in microgrid: equilibrium under price anticipation. IEEE J. Mag. 4, 3794–3805 (2016)
Y. Wang, W. Saad, Z. Han, H. Vincent Poor, T. Basar, A game-theoretic approach to energy trading in the smart grid. IEEE Trans. Smart Grid 5(3), 1439–1450 (2014)
A. Jin, W. Song, W. Zhuang, Auction-based resource allocation for sharing cloudlets in mobile cloud computing. IEEE Trans. Emerg. Top. Comput. PP(99), 1 (2017)
H. Zhou, J. Jiang, W. Zeng, An agent-based finance market model with the continuous double auction mechanism. Second WRI Glob. Congr. Intell. Syst. 2, 316–319 (2010)
H. Kebriaei, B. Maham, D. Niyato, Double-sided bandwidth-auction game for cognitive device-to-device communication in cellular networks. IEEE Trans. Veh. Technol. 65(9), 7476–7487 (2016)
P. Li, S. Guo, I. Stojmenovic, A truthful double auction for device-to-device communications in cellular networks. IEEE J. Sel. Areas Commun. 34(1), 71–81 (2016)
W. Dong, S. Rallapalli, L. Qiu, K.K. Ramakrishnan, Y. Zhang, Double auctions for dynamic spectrum allocation. IEEE/ACM Trans. Netw. 24(4), 2485–2497 (2016)
E. Bompard, Y. Ma, R. Napoli, G. Abrate, The demand elasticity impacts on the strategic bidding behavior of the electricity producers. IEEE Trans. Power Syst. 22(1), 188–197 (2007)
F.S. Wen, A.K. David, Strategic bidding for electricity supply in a day-ahead energy market. Electr. Power Syst. Res. 59, 197–206 (2001)
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendices
3.1.1 Proof of Lemma 3.1
It is equivalent to show the incentive compatibility for any buyer \(n \in \mathscr {N}\) by verifying that, for any bid \(b_n \equiv (\beta _n, d_n) \in \mathscr {B}_n\), there exists a truth-telling bid \(b_n^t \equiv (\beta _n^t, d_n^t) \in \mathscr {B}^t_n\), say \(\beta _n^t = v_n'(d_n^t)\), such that
for any given bid profile \(\varvec{r}_{-n}\) of other players.
Denote by \(x_n\) and \(x_n^t\) the allocations of buyer n with respect to \((b_n, \varvec{r}_{-n})\) and \((b_n^t, \varvec{r}_{-n})\), respectively. We will show (3.30) in the following:
-
(i)
In case \(\beta _n < v_n'(d_n)\).
Consider a bid \(b_n^t\) such that \(d_n^t = x_n \le d_n\). By Assumption 3.2, we have \(\beta _n^t \ge v_n'(d_n) > \beta _n\). Then by (3.5), we have \(x_n^t \ge x_n\). Also by \(x_n^t \le d_n^t = x_n\), we have \(x_n^t = x_n\). Hence, by the specification of the payoff functions of players, we have \(f_n(b_n^t, \varvec{r}_{-n}) = f_n(b_n, \varvec{r}_{-n})\).
-
(ii)
In case \(\beta _n > v_n'(d_n)\).
Consider a bid \(b_n^t\) such that \(d_n^t = d_n\); then \(\beta _n > v_n'(d_n) = v_n'(d_n^t) = \beta _n^t\). By (3.5), we have \(x_n^t \le x_n\). When \(x_n^t = x_n\), the payoffs \(f_n(b_n^t, \varvec{r}_{-n}) = f_n(b_n, \varvec{r}_{-n})\). Next we consider the case \(x_n^t < x_n\). The following holds:
$$\begin{aligned} f_n&(b_n, \varvec{r}_{-n}) - f_n(b_n^t, \varvec{r}_{-n}) \\&= v_n(x_n) - v_n(x_n^t) + \tau _n(b_n^t, \varvec{r}_{-n}) - \tau _n(b_n, \varvec{r}_{-n}) \\&\le \beta _n^t (x_n - x_n^t) + U(\varvec{z}) - \beta _n x_n - U(\varvec{z}^t) + \beta _n^t x_n^t \\&\le \beta _n^t (x_n - x_n^t) - \beta _n^t (x_n - x_n^t) = 0. \end{aligned}$$
In conclusion, by (i, ii) given above, we obtain that (3.30) holds.
Following the same technique applied for buyers, we can also verify the incentive compatibility for any seller \(m \in \mathscr {M}\).
3.1.2 Proof of \(Q^{k} = \varGamma ^{k}\) in Theorem 3.1
By adopting Algorithm 3.1, we have \(\varGamma ^{0} < \sum _{i \in \mathscr {N}} d_i^{0}\) and \(\varGamma ^{0} < \sum _{j \in \mathscr {M}} h_j^{0}\); by which together with (3.13a) and (3.14a), we have \(\varGamma ^{0} = \sum _{i \in \mathscr {N}} x_i^{0} = \sum _{j \in \mathscr {M}} y_j^{0}\); then \(Q^{0} \equiv \min \Big \{ \sum _{i \in \mathscr {N}} x_i^{0}, \sum _{j \in \mathscr {M}} y_j^{0} \Big \} = \varGamma ^{0}\).
At iteration step \(k \ge 1\), a buyer n asks to implement his best response in the buyer-sided auction. Denote by \(\varvec{b}^{k} = (b_n^{k}, \varvec{b}_{-n}^{k})\) the updated bid profile at step \(k-1\) where \(\varvec{b}_{-n}^{k} = \varvec{b}_{-n}^{k-1}\), and \(b_n^{k} = (\beta _n^{k}, d_n^{k})\) represents the best response of buyer n. Denote by \(\varvec{x}^{k} = \varvec{x}(\varvec{b}^{k},\varGamma ^{k})\), i.e., \(\varvec{x}^{k}\) represents the allocation of buyers.
We will show \(Q^{k} = \varGamma ^{k}\) by verifying that \(\sum _{i \in \mathscr {N}} x_i^{k} = \varGamma ^{k}\) and \(\sum _{j \in \mathscr {M}} y_j^{k} = \varGamma ^{k}\) in (I) and (II), respectively.
(I). To show \(\sum _{i \in \mathscr {N}} x_i^{k} = \varGamma ^{k}\) by proof of contradiction below.
Firstly since \(\varGamma ^{k}\) is the total allocated quantity, we have \(\sum _{i \in \mathscr {N}} x_i^{k} \le \varGamma ^{k}\); then suppose that \(\sum _{i \in \mathscr {N}} x_i^{k} < \varGamma ^{k}\), by (3.13a), we have \(x_i^{k} = d_i^{k}\) for all \(i \in \mathscr {N}\), i.e., all the buyers are fully allocated. Hence \(\sum _{i \in \mathscr {N}} x_i^{k} = \sum _{i \in \mathscr {N}} d_i^{k} < \varGamma ^{k}\).
By (3.18a), we have
We will show that \(d_n^{k} < D_n^{k-1}\) in (i)–(ii) below.
-
(i)
In case \(\varGamma ^{k} - \sum _{i \in \mathscr {N}} x_i^{k-1} < 0\). The following inequalities hold:
$$\begin{aligned}&\varGamma ^{k} < \sum _{i \in \mathscr {N}} x_i^{k-1} \le x_n^{k-1} + \sum _{i \in \mathscr {N}/\{n\}} d_i^{k-1} \\& = x_n^{k-1} + \sum _{i \in \mathscr {N}/\{n\}} d_i^{k}, \\&\varGamma ^{k} > \sum _{i \in \mathscr {N}} d_i^{k} = d_n^{k} + \sum _{i \in \mathscr {N}/\{n\}} d_i^{k}; \end{aligned}$$then it implies that \(d_n^{k} < x_n^{k-1}\). Also by (3.31), we have \(D_n^{k-1} = x_n^{k-1}\) in case \(\varGamma ^{k} - \sum _{i \in \mathscr {N}} x_i^{k-1} < 0\); then we have \(d_n^{k} < D_n^{k-1}\).
-
(ii)
In case \(\varGamma ^{k} - \sum _{i \in \mathscr {N}} x_i^{k-1} \ge 0\). We have the following:
$$\begin{aligned} \varGamma ^{k} > \sum _{i \in \mathscr {N}} d_i^{k} = d_n^{k} + \sum _{i \in \mathscr {N}/\{n\}} d_i^{k-1} \ge d_n^{k} + \sum _{i \in \mathscr {N}/\{n\}} x_i^{k-1}, \end{aligned}$$which implies that \(d_n^{k} < \varGamma ^{k} - \sum _{i \in \mathscr {N}/\{n\}} x_i^{k-1}\). Also by (3.31), we have
$$\begin{aligned} D_n^{k-1} = x_n^{k-1} + \varGamma ^{k} - \sum _{i \in \mathscr {N}} x_i^{k-1} = \varGamma ^{k} - \sum _{i \in \mathscr {N}/\{n\}} x_i^{k-1}, \end{aligned}$$(3.32)in case \(\varGamma ^{k} - \sum _{i \in \mathscr {N}} x_i^{k-1} \ge 0\); then we have \(d_n^{k} < D_n^{k-1}\).
By (i)–(ii) above, we have \(d_n^{k} < D_n^{k-1}\). Also by \(\sum _{i \in \mathscr {N}} d_i^{k} < \varGamma ^{k}\), we can define another bid profile for player n, denoted by \(\widehat{b}_n \equiv \left( \widehat{\beta }_n, \widehat{d}_n \right) \), such that
By \(\varepsilon < \varGamma ^{k} - \sum _{i \in \mathscr {N}} d_i^{k}\), we have \(\sum _{i \in \mathscr {N}} d_i^{k} + \varepsilon = \sum _{i \in \mathscr {N}/\{n\}} d_i^{k} + \widehat{d}_n < \varGamma ^{k}\); then by (3.13a), we have \(\widehat{x}_n \equiv x_n \left( (\widehat{b}_n,\varvec{b}_{-n}^{k}),\varGamma ^{k} \right) = \widehat{d}_n\), and \(x_i \left( (\widehat{b}_n,\varvec{b}_{-n}^{k}),\varGamma ^{k}\right) = d_i^{k} = x_i^{k}\) for all \(i \in \mathscr {N}/\{n\}\).
Hence by the payment of buyer n specified in (3.13b), we have \(\tau _n(\varvec{b}^{k},\varGamma ^{k}) = \tau _n \big ((\widehat{b}_n,\varvec{b}_{-n}^{k}),\varGamma ^{k}\big )\). Thus we have
where the last inequality holds by Assumption 3.2, which is contradicted with the consideration that \(b_n^{k}\) is the best response of buyer n. It implies that \(\sum _{i \in \mathscr {N}} x_i^{k} < \varGamma ^{k}\) cannot be held. Hence \(\sum _{i \in \mathscr {N}} x_i^{k} = \varGamma ^{k}\).
(II). By the same technique applied in the proof of \(\sum _{i \in \mathscr {N}} x_i^{k} = \varGamma ^{k}\) given in (I) above, we can show \(\sum _{j \in \mathscr {M}} y_j^{k} = \varGamma ^{k}\) as well.
In summary, by the specification of \(Q^{k}\) given in (3.17), we have
3.1.3 Proof of \(p_b^{k} \ge p_s^{k}\) for all \(k \ge 0\) in Theorem 3.1
Under Algorithm 3.1, the matched prices \(p_b^{0}\) and \(p_s^{0}\) subject to the initial bid profile are supposed to satisfy the inequality of \(p_b^{0} \ge p_s^{0}\); then to show \(p_b^{k} \ge p_s^{k}\) holds for all \(k \ge 0\), it is equivalent to verify that \(p_b^{k+1} \ge p_s^{k+1}\) holds in case \(p_b^{k} \ge p_s^{k}\) with \(k \ge 0\).
By (3.23) and the assumed \(p_b^{k} \ge p_s^{k}\) for some \(k \ge 0\), it can be shown that
Also suppose that at iteration step k, in the buyer-sided auction, buyer n updates his best response, denoted by \(b_n^{k+1}\).
By Appendix 3.5, we can have \(\varGamma ^{k} = \sum _{i \in \mathscr {N}} x_i^{k}\), by which together with (3.18a) and (3.33), the following holds:
By the concavity of \(v_n\) under Assumption 3.2 and (3.15a), we have
Moreover, by (3.20a), the best response of buyer n satisfies \(d_n^{k+1} \le D_n^{k}\); then by which together with Assumption 3.2, we have \(\beta _n^{k+1} \ge v_n'(D_n^{k})\).
In the following, we will verify that
in (I)–(III) below.
(I) In case \(x_n^{k} \in (0, d_n^{k})\).
By (3.13a) and (3.16a), we have \(p_b^{k} = \beta _n^{k}\). Also by (3.13a) we have
for all \(i \in \mathscr {N}/\{n\}\), and by which, together with \(\beta _n^{k+1} \ge v_n'(D_n^{k})\), we can obtain that those buyers \(i \in \mathscr {N}/\{n\}\), such that \(x_i^{k} = 0\) and \(\beta _i^{k} \ge v_n'(D_n^{k})\), can increase their own allocations, respectively. Thus by (3.16a), we have \(p_b^{k+1} \ge v_n'(D_n^{k})\). Hence by (3.35), we get that
which implies that the inequality (3.36) holds by \(p_b^{k+1} \ge v_n'(D_n^{k})\), and where the inequality holds by \(x_n^{k} < d_n^{k}\), and the 2nd equality holds by \(p_b^{k} = \beta _n^{k}\).
(II) In case \(x_n^{k} = 0 < d_n^{k}\). By (3.23), we have \(\varGamma ^{k+1} = \varGamma ^{k}\).
By (3.34) and \(\varGamma ^{k+1} = \varGamma ^{k}\), we have \(D_n^{k} = 0\) by which together with \(d_n^{k+1} \le D_n^{k}\), we have \(x_n^{k+1} = d_n^{k+1} = 0 = x_n^{k}\). Hence we have \(p_b^{k+1} = p_b^{k}\) which implies (3.36).
(III) Other cases besides those considered in (I) and (II).
By adopting Algorithm 3.1, we have \(x_i^{k} = d_i^{k}\) for all \(i \in \mathscr {N}\), i.e., all buyers are fully allocated, and \(\beta _n^{k} = \max _{i \in \mathscr {N}} \{ \beta _i^{k} \}\); then \(p_b^{k} = \min _{i \in \mathscr {N}} \{ \beta _i^{k} \} \le \beta _n^{k}\).
By \(\beta _n^{k+1} \ge v_n'(D_n^{k})\), if \(v_n'(D_n^{k}) > p_b^{k}\), we have \(p_b^{k+1} = p_b^{k}\); then \(p_b^{k+1} \ge p_b^{k} - \bar{\rho } (\varGamma ^{k+1} - \varGamma ^{k})\) by \(\varGamma ^{k+1} \ge \varGamma ^{k}\); else, we have \(p_b^{k+1} \ge v_n'(D_n^{k})\). Hence by (3.35) and \(p_b^{k} \le \beta _n^{k}\), the inequality (3.36) holds.
By adopting the same technique to verify (3.36) for the matched price on the buyer-sided auction game in (I)–(III) above, we can show the following inequality property for the matched price on the seller-sided auction game as well:
By (3.23), \(\varGamma ^{k+1} = \varGamma ^{k}\) or \(\varGamma ^{k+1} = \varGamma ^{k} + \frac{p_b^{k} - p_s^{k}}{\bar{\rho } + \bar{\sigma }}\); then we have the following analysis:
-
In case \(\varGamma ^{k+1} = \varGamma ^{k}\). By (3.36), (3.37) and \(p_b^{k} \ge p_s^{k}\), we have \(p_b^{k+1} \ge p_s^{k+1}\).
-
In case \(\varGamma ^{k+1} = \varGamma ^{k} + \frac{p_b^{k} - p_s^{k}}{\bar{\rho } + \bar{\sigma }}\). We have \(p_b^{k} - \bar{\rho } (\varGamma ^{k+1} - \varGamma ^{k}) = p_s^{k} + \bar{\sigma } (\varGamma ^{k+1} - \varGamma ^{k})\); then by (3.36) and (3.37), we have \(p_b^{k+1} \ge p_s^{k+1}\).
In summary, we have \(p_b^{k+1} \ge p_s^{k+1}\) on the condition that \(p_b^{k} \ge p_s^{k}\), i.e., the conclusion holds.
3.1.4 Proof of (3.25) in Theorem 3.1
We will verify (3.25) in (I)–(II) below.
(I). To show \(x_i^{k+1} \ge x_i^{k}\) for all \(i \in \mathscr {N}\).
Suppose that \(x_n^{k+1} < x_n^{k}\). Consider another bid \(\widehat{b}_n = (\widehat{\beta }_n, \widehat{d}_n)\) with \(\widehat{\beta }_n = v_n'(\widehat{d}_n)\) and \(\widehat{d}_n = x_n^{k}\), and denote by \(\widehat{\varvec{x}}\) the allocation with respect to the bid profile \(\varvec{\widehat{b}} \equiv (\widehat{b}_n, \varvec{b}_{-n}^{k+1})\).
By (3.24), (3.13a) and \(x_n^{k+1} < x_n^{k} = x_n^*(\varvec{b}^{k},\varGamma ^{k})\), buyer n does not grab other buyers’ allocation under both \(\varvec{b}^{k+1}\) and \(\varvec{\widehat{b}}\). Since buyer n is assigned to implement his best response, he must satisfy one of the three cases specified in (3.21a) in sequence. Then by (3.13a), we obtain that those buyers whose bid prices satisfy \(\beta _i^{k} > \beta _n^{k}\), for some \(i \in \mathscr {N}/\{n\}\), must be fully allocated. In other words, their allocation cannot be increased. Since for all \(i \in \mathscr {N}/\{n\}\), \(\beta _i^{k} = \beta _i^{k+1}\) holds, we obtain that only the buyers whose bid prices satisfy \(\beta _i^{k+1} \le \beta _n^{k}\) can increase the allocation. That is,
then by the definition of the payoff function \(f_n\), the following holds:
which is contradicted with the fact that \(b_n^{k+1}\) is the best response of buyer n, and where the 1st inequality holds by Assumption 3.2, the 2nd inequality holds by (3.38), and the last one holds by \(x_n^{k+1} < x_n^{k} \le d_n^{k}\). Hence \(x_n^{k+1} \ge x_n^{k}\) holds.
By (3.24), we have \(\varGamma ^{k+1} \ge \varGamma ^{k} = \sum _{i \in \mathscr {N}} x_i^{k}\), by which together with (3.18a), we have \(D_n^{k} = x_n^{k} + \varGamma ^{k+1} - \varGamma ^{k}\). By (3.20a), the best response of buyer n satisfies \(d_n^{k+1} \le D_n^{k}\); then by \(x_n^{k+1} = d_n^{k+1}\) specified in (3.22a) in Lemma 3.4, we have
by which we can show that \(x_i^{k+1} \ge x_i^{k}\) for all \(i \in \mathscr {N}/\{n\}\), i.e., the allocations of other buyers will not decrease.
(II). By the same technique applied in (I), we can show \(y_j^{k+1} \ge y_j^{k}\) for all \(j \in \mathscr {M}\).
3.1.5 Proof of Theorem 3.2
By Theorem 3.1 together with (3.22), we can obtain that after certain iteration steps \(\widehat{k} \le \max \{N,M\}\), the following holds:
-
(i)
The potential quantity of resource is completely distributed among players, i.e., \(\varGamma ^{k} = \sum _{i \in \mathscr {N}} x_i^{k} = \sum _{j \in \mathscr {M}} y_j^{k}\) for any \(k \ge \widehat{k}\).
-
(ii)
All the players are fully allocated, i.e., \(x_i^{k} = d_i^{k}, \forall i\) and \(y_j^{k} = h_j^{k}, \forall j\) for any \(k \ge \widehat{k}\).
Suppose that buyer n and seller m are the players who are assigned to update their best responses, respectively, at iteration step k with \(k > \widehat{k}\); then \(\Vert \varvec{d}^{k+1} - \varvec{d}^{k}\Vert _1 = |d_n^{k+1} - d_n^{k}|\), and \(\Vert \varvec{h}^{k+1} - \varvec{h}^{k}\Vert _1 = |h_m^{k+1} - h_m^{k}|\).
First in the buyer-sided auction, by (3.19a) and (i&ii), it gives
By the definition of \(p_b\) and \(p_s\) specified in (3.16), and (ii), we have
where i, j represent the buyer and the seller who update strategies at iteration step \(k - 1\), respectively.
By (3.39), we have
which implies that \(\Vert \varvec{d}^{k+1} - \varvec{d}^{k}\Vert _1\) is bounded by \(\frac{\beta _i^k - \alpha _j^k}{\bar{\rho } + \bar{\sigma }}\) from below.
By (3.25) and (ii) in earlier part of this section, we have \(\varvec{d}^{k+1} \ge \varvec{d}^{k}\) for any k with \(k \ge \widehat{k}\); then by which together with (3.40), at step k, we adopt
to specify an upper bound for the convergence steps of the algorithm.
By the specifications of \(\underline{\rho }\) and \(\underline{\sigma }\) given in (3.26), and (3.39), we can obtain that
by which, together with (3.39), we have
Also by (3.16), we have \(\beta _i^{k-1} \ge p_b^{k-1}\) and \(\alpha _j^{k-1} \le p_s^{k-1}\) which and (3.39) imply that
Then by (3.41) and (3.42), we have
by which together with (3.43), we have
By the same method used above, the following holds for the seller-sided auction:
By \(\beta _l = v'_l(d_l), l \in \mathscr {N}\) and \(\alpha _l = c'_l(h_l), l \in \mathscr {M}\), together with (3.15a) and (3.15b),
By the termination condition in Algorithm 3.1, we can verify that the system converges within \(K = \widehat{k} + \widehat{K}\) iteration steps, with \(\widehat{K}\) given in (3.27).
3.1.6 Verification of (3.28) and (3.29) in Theorem 3.3
Suppose that, under Algorithm 3.1, the system converges at step k, and at the equilibrium as specified in Algorithm 3.1, we have \(\varvec{r}^{k+1} = \varvec{r}^{k}\) and \(\varGamma ^{k+1} = \varGamma ^{k}\).
First we analyze the properties of \(\varvec{b}^{k+1}\) in the buyer-sided auction. Suppose that the system assigns buyer n to update his best response with respect to the allocation \(\varvec{x}^{k}\) and \(\varvec{b}^{k}\), and \(b_n^{k+1}\) is the best response updated by buyer n at step k. By (3.22), we have \(x_n^{k+1} = d_n^{k+1}\).
By \(\varvec{b}^{k+1} = \varvec{b}^{k}\), \(\varGamma ^{k+1} = \varGamma ^{k}\) and (3.13a), we have \(\varvec{x}^{k+1} = \varvec{x}^{k}\); then by \(d_n^{k} = d_n^{k+1}\), we have \(x_n^{k} = d_n^{k}\), i.e., buyer n assigned by the system is fully allocated before his best response is implemented. By the rule to assign buyer n in Algorithm 3.1, we have
Suppose \(\beta _n^{k} = \lambda \), for some positive valued \(\lambda \); then by (3.45), \(\varvec{x}^{k+1} = \varvec{x}^{k}\) and \(\varvec{r}^{k+1} = \varvec{r}^{k}\), the following properties hold, for all \(i \in \mathscr {N}\),
From \(\varGamma ^{k+1} = \varGamma ^{k}\) and Appendix 3.5, we have \(\varGamma ^{k+1} = \sum _{i \in \mathscr {N}} x_i^{k}\); then \(D_n^{k} \equiv D_n(\varvec{b}^{k},\varGamma ^{k},\varGamma ^{k+1}) = d_n^{k+1}\) by (3.18a) and (3.46). By \(\beta _n = v_n'(d_n)\), we have \(v_n'(D_n^{k}) = \beta _n^{k+1}\); then by (3.46), (3.47) and (3.16a), we can verify that \(p_b^{k+1} = \lambda \).
Hence by (3.16a), we have \(\beta _i^{k+1} \ge \lambda \) in case with \(x_i^{k+1} > 0\) for all \(i \in \mathscr {N}\), by which together with (3.46) and (3.47), we have \(\beta _i^{k+1} = \lambda \) in case \(d_i^{k+1} > 0\) for all \(i \in \mathscr {N}\).
In summary, we obtain that the updated bid profile \(\varvec{b}^{k+1}\) at step k satisfies (3.28) and \(\sum _{i \in \mathscr {N}} d_i^{k+1} = \varGamma ^{k+1}\), and the matched price of buyer is specified as \(p_b^{k+1} = \lambda \).
Since \(\varGamma ^{k+1} = \varGamma ^{k}\), we have \(p_b^{k} = p_s^{k}\); then by the same technique of the analysis of \(\varvec{b}^{k+1}\), we can verify (3.29) and \(\sum _{j \in \mathscr {M}} s_j^{k+1} = \varGamma ^{k+1}\) as well.
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Ma, Z., Zou, S. (2020). Double-Sided Auction Games for Efficient Resource Allocation. In: Efficient Auction Games. Springer, Singapore. https://doi.org/10.1007/978-981-15-2639-8_3
Download citation
DOI: https://doi.org/10.1007/978-981-15-2639-8_3
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-2638-1
Online ISBN: 978-981-15-2639-8
eBook Packages: EnergyEnergy (R0)