A Fast Distributed Algorithm for Uniform Price Auction with Bidding Information Protection

Abstract

In this paper, a fast distributed algorithm is proposed for solving the winners and price determination problems in a uniform price auction in which each bidder bids for multiple units out of a lot of k identical items with a per-unit price. In a conventional setting, all bidders disclose their bidding information to an auctioneer and let the auctioneer allocate the items and determine the uniform price, i.e., the least winning price. In our setting, all bidders do not need to disclose their bidding information to the auctioneer. The bidders and the auctioneer collaboratively compute by the distributed algorithm to determine in a small number of steps the units allocated and the uniform price. The number of steps is independent of the number of bidders. At the end of the computing process, each bidder can only know the units allocated to him/her and the uniform price. The auctioneer can only know the units being allocated to the bidders and the uniform price. Therefore, neither the bidders nor the auctioneer are able to know the per-unit bidding prices of the bidders except the uniform price. Moreover, the auctioneer is not able to know the bidding units of the losing bidders. Bidders’ per-unit bidding prices are protected, and the bidding units of the losing bidders are protected. Bidding information privacy is preserved.

1. Introduction

In this paper, a fast distributed algorithm is proposed for solving the unit allocation and uniform price determination for a uniform price auction. Uniform price auction is a multiunit auction [1,2], in which k identical items (equivalently, k units) are auctioned off to m bidders, each bidding for multiple units with a per-unit bidding price If each bidder can only bid for one unit; the winner determination problem is identical to the classical kWTA problem [3,4].

1.1. Uniform Price Auction (UPA)

In convention, a uniform price auction is conducted in a centralized manner. Each bidder submits his/her bid, which includes the number of units to bid and the per-unit bidding price, in sealed to an auctioneer. Then, the auctioneer allocates the units to the bidders in accordance with the descending order of the per-unit bidding prices. Once the unit allocation has been performed, all winners pay for each unit the least winning per-unit bidding price. While the conventional method for solving a uniform price auction is efficient as the auctioneer is in charge of the unit allocation and price determination, bidding information (including the number of units to bid and the per-unit bidding price) privacy is not protected, and thus it is not applicable to the situation when some bidders do not want to disclose their bidding information. In this regard, a new distributed algorithm has to be developed to solve these problems with bidding information privacy protection. However, algorithms developed along this direction are scarce. To the best of our knowledge, all these algorithms are developed for sealed-bid auctions in which each bidder can only bid for one unit of items.

1.2. Distributed/Decentralized Algorithms

For the first price auctions, a few distributed or decentralized algorithms have been developed in recent years to deal with sealed-bid auctions with bidding price protection and auctioneer ripping off. Some algorithms are developed based on encryption technique [5,6] for the bidding information together with a specialized algorithm for the auctioneer to rank the encrypted bidding prices. For multiunit auctions, where each bidder can only bid for one unit, kWTA-based algorithms have been developed to ensure partial bidding prices protection [7,8] and support full bidding prices protection [9].

It should be stressed that decentralization has long been investigated in auction implementations [10,11,12]. Some decentralized implementations focused on the algorithm for winner determination auctioneer rips them out only [10] Bidding prices protection is not considered. If the bidding information is not protected, auctioneer could learn from the bidding information collected from the historical auctions so as to maximize the revenue of the future auctions [13,14,15,16]. On the other hand, many distributed kWTA algorithms introduced in recent years [17,18,19,20] can be applied in solving the winner determination problem in an uniform price auction if each bidder can only bid for one unit and if the bidding price is set to be the input. However, these algorithms are unable to determine the uniform price.

1.3. Multiunit UPA with Bidding Price Protection

On top of the above issues on winner determination and uniform price determination, some decentralized algorithms have been developed and implemented with consideration on the issue on bidding prices protection [11,12]. Inspired by a recent developed fast distributed kWTA algorithm for uniform price auctions in which each bidder can bid for one unit [9], we propose in this paper a fast distributed algorithm for the multiunit uniform price auctions in which each bidder can bid for more than one unit. An exemplar market of this kind is the electricity auction market [21]. Our proposed algorithm is able to complete the units allocation and uniform price determination in a very few number of steps. Moreover, this algorithm is able to protect the bidding information, including the units to bid and the per-unit bidding price. The idea behind the algorithm development is simply based on the bisection method which has been applied in a fast distributed algorithm as presented in [9].

1.4. Organization of the Paper

In the next section, the uniform price auction to be solved in this paper is formally defined. Existing solutions for solving the problems (unit allocation and uniform price determination) in a uniform price auction are presented. Then, the proposed fast distributed algorithm is presented in Section 3. Convergence analysis, stopping criteria and sufficient steps for convergence are analyzed in the section. Benefits of our fast distributed algorithm are re-iterated in Section 4. Simulation results with comparisons to the existing methods are presented in Section 5. Discussions on the applicability of the proposed algorithm in other auction settings are commented onin Section 6. Finally, the conclusion of the paper is presented in Section 7.

2. Uniform Price Auction (UPA)

Depending on the settings of (a) a bidder’s bidding price vector; (b) the allocation rules; and (c) the payment determination rules, various mechanisms have been introduced for uniform price auction [2].

In this paper, we consider a sealed-bid auction with k identical units to be auction-off. There are m ($m>k$) bidders who are willing to bid. Each bidder might bid for multiple units. Let $(q_i,u_i)$ be the bid vector of the $i^{th}$ bidder. Here, $q_i$ is the number of units to bid and $u_i$ be the per-unit bidding price. As a convention, we let $u_{{\pi }_1}<u_{{\pi }_2}<\cdots <u_{{\pi }_m}$. To proceed, we need to make the following assumptions.

Assumption 1.  (a) The total bidding quantities is larger than the number of units to be auctioned off. That is to say, $n={\sum }_{i=1}^m{q}_i>k$. (b) All per-unit bidding prices are distinct positive integers. That is to say, $u_i\ne u_{i^{\prime }}$ if $i\ne i^{\prime }$. Moreover, $0<u_i\le M$, where M is the maximum per-unit bidding price. (c) The maximum number of units to be bid by a bidder is limited to Q, i.e., $q_i\le Q$ for $i=1,\cdots ,m$. (d) Bidders do not want to disclose their bidding prices. □

It should be noted that per-unit price, per-unit bidding price and bidding price are used interchangeably in this paper. Moreover, other settings like those for sealed-bid combinatorial multiunit auctions are not treated in this paper. For a combinatorial multiunit auction, the $i^{th}$ bidder bids for all $q_i$ units by $u_i$. A winner must be allocated with all the units being bid. This units allocation problem is equivalent to a knapsack problem which is a $\mathcal{NP}$ hard problem [22]. One should not confuse that our proposed algorithm is applicable to this combinatorial auction problem.

2.1. Unit Allocation and Price Determination

To allocate the units, it starts from the highest per-unit bidding price, $u_{{\pi }_m}$. Allocate the bidder with the highest per-unit bidding price the units being bid. If there are units remaining, allocate the bidder with the second highest per-unit bidding price, i.e., $u_{{\pi }_{m-1}}$, for the units being bid. The process repeats until all the units are allocated. For the payment, it is determined by the least winning per-unit bidding price, i.e., the uniform price. All winning bidders pay for this uniform price.

2.2. Illustrative Examples

For illustration, Table 1 shows two examples. The bidders’ bids are $(50,2)$, $(100,1)$, $(75,2)$, $(40,3)$ and $(80,1)$. The difference between these two examples is in the total number of units for bid, (1) $k=4$ and (2) $k=5$.

Table 1. Units allocation and uniform price determination.

(1) $m=5$ and $k=4$.
	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$
Per-Unit Price $u_i$	50	100	75	40	80
Biding Quantity $q_i$	2	1	2	3	1
Units Allocated	–	1	2	–	1
Uniform Price	–	75	75	–	75
(2) $m=5$ and $k=5$.
	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$
Per-Unit Price $u_i$	50	100	75	40	80
Bidding Quantity $q_i$	2	1	2	3	1
Units Allocated	1	1	2	–	1
Uniform Price	50	50	50	–	50

Figure 1 shows the idea behind unit allocation and uniform price determination. For $k=4$, the units are allocated to the bidders $B_2$ with one unit, $B_3$ with two units and $B_5$ with one unit. The uniform price is 75. For $k=5$, the units are allocated to the bidders $B_1$ with one unit, $B_2$ with one unit, $B_3$ with two units and $B_5$ with one unit. The uniform price is 50.

Figure 1. The idea behind unit allocation and uniform price determination. (Left): For $k=4$, the uniform price is 75. (Right): For $k=5$, the uniform price is 50.

2.3. Solutions for UPA and Their Limitations

Here, we introduce two solutions for solving the unit allocation and uniform price determination for a uniform price auction. The first one is based on the conventional sealed-bid auction mechanism [1]. All bidders submit their bids to an auctioneer and let the auctioneer allocate the units and determine the uniform price. The second one is inspired by the Wang kWTA neural network [23,24].

2.3.1. Centralized Algorithm

To solve the unit allocation and uniform price determination problems in a uniform price auction, the classical solution is based on a centralized method in which an auctioneer is needed. All bidders submit their bidding information to the auctioneer, as shown in Figure 2a. The auctioneer will then perform the unit allocation and determine the uniform price.

Figure 2. Solutions for the uniform price auction. (a) The classical solution is with an auctioneer to collect all bidding information, i.e., $(q_i,u_i)$ for $i=1,\cdots ,m$. Then, the auctioneer allocates the units to the bidders and determines the uniform price. (b) A recurrent neural network-based distributed solution in which the bidders only submit their ${\widehat{q}}_i\left(t\right)$ to the auctioneer and the auctioneer updates $y\left(t\right)$ upon the ${\widehat{q}}_i\left(t\right)$s having been received. (c) The fast algorithm proposed in this paper. Here, each rectangle box corresponds to a software running in a computing device, such as a computer or a cell phone. Data communication is conducted via a wireless communication channel.

First, the per-unit bidding prices are sorted in ascending order, i.e., $u_{{\pi }_1}<\cdots <u_{{\pi }_m}$. Starting from $u_{{\pi }_m}$, $min\{k,q_{{\pi }_m}\}$ is allocated to the ${{\pi }_m}^{th}$ bidder. If $k-min\{k,q_{{\pi }_m}\}>0$, allocate $min\{k-q_{{\pi }_m},q_{{\pi }_{m-1}}\}$ to the ${{\pi }_{m-1}}^{th}$ bidder. Repeat the allocation process until all k units have been allocated. The uniform price is defined as the least winning per-unit price. Once the results have been obtained, the auctioneer broadcasts the results to the bidders. Algorithm 1 lists the detailed steps for this centralized algorithm.

Clearly, this centralized method is very efficient, as the time span (equivalently, the number of steps) for unit allocation and uniform price determination is just one. However, the bidding price privacy is definitely not protected.

Algorithm 1 Centralized Algorithm

Bidders: Set $0<u_i<M$ and $1\le q_i\le Q$. Send $(q_i,u_i)$ to the auctioneer. Auctioneer: (Unit Allocation) Rank $u_i$s in ascending order, i.e., $u_{{\pi }_1}<\cdots <u_{{\pi }_m}$. Set $j=m$ and $q_r=k$ (remaining units). while {$q_r>0$} do       Allocate $q_{{\pi }_j}^{\ast }=min\{q_{{\pi }_j},q_r\}$ to ${{\pi }_j}^{th}$ bidder.       Set $y=u_{{\pi }_j}$ and $q_r=q_r-min\{q_{{\pi }_j},q_r\}$.       $j=j-1$. end while for $j^{\prime }=1,\cdots ,j$, do       $q_{{\pi }_j^{\prime }}^{\ast }=0$. end for Auctioneer: (Bidding Results Broadcast) for $j=1,\dots ,m$, do     Send $(q_j^{\ast },y)$ to the $j^{th}$ bidder. end for

2.3.2. Recurrent Neural Network-Based Distributed Algorithm

Another solution is based on a distributed method as presented in Appendix A and shown in Figure 2b. Note that this recurrent neural network-based distributed algorithm has not been presented in the literature for solving the unit allocation and uniform price determination problems in a uniform price auction. Bidders do not need to be sent to the auctioneer agent any bidding information.

Setting $y\left(0\right)=M$. In each round t, the bidders simply send their ${\widehat{q}}_i\left(t\right)=q_i\varphi (u_i-y\left(t\right))$ to the auctioneer. $\varphi (·)$ is a piecewise linear function defined as follows:

$$\varphi \left(s\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if} s\ge \mathsf{\Delta },\hfill \\ s\hfill & \mathrm{if} 0<s<1/\mathsf{\Delta },\hfill \\ 0\hfill & \mathrm{if} s\le 0,\hfill \end{array}\right.$$

(1)

where $\mathsf{\Delta }<1/2$. Then, the auctioneer updates $y\left(t\right)$ upon the receiving of ${\widehat{q}}_i\left(t\right)$ for $i=1,\cdots ,m$.

$$y(t+1)=y\left(t\right)+\mu \left\{\sum _{i=1}^m\widehat{q}\left(t\right)-k\right\},$$

(2)

where $0<\mu \ll 1$ is the step size. Finally, the updated $y\left(t\right)$ is broadcasted to all the bidders for their updating of ${\widehat{q}}_i$.

However, this distributed method suffers from two problems. First, this method can only protect the losing bidding price. All bidders and the auctioneer are able to recover the winners’ bidding prices. Second, the time span for unit allocation and uniform price determination can be long, as the number of steps for completing the task is in the order of ${10}^3$ to ${10}^4$. If the communication delay in each iteration is in the order of ${10}^{-2}$, the time span will be in the order of 10 to ${10}^2$. This time span will be scaled up if either the number of bidders or the number of units to be auctioned off scales up.

In the sequel, it is needed to develop an efficient distributed algorithm that is able to protect the bidding price privacy.

3. Fast Distributed Algorithm

In view of the limitations of the algorithms presented in the last section regarding the bidding price privacy and the long completion time, a fast distributed algorithm is developed to simultaneously allocate the units to the bidders and determine the uniform price in a small number of steps. Moreover, the fast distributed algorithm is able to protect the bidding price privacy.

The key for the algorithm development is based on the property of $f_I\left(y\right)$, where

$$f_I\left(y\right)=\sum _{i=1}^m\underset{{\widehat{q}}_i}{\underbrace{q_i\varphi (u_i-y)}}-k.$$

(3)

As $f_I\left(y\right)$ is a non-increasing function, we can then state without proof the following theorem regarding the property of $f_I\left(y\right)$:

Theorem 1. 

$f_I\left(y\right)$ as stated in Equation (3) is a non-increasing function of y, i.e., $f_I\left(y\right)\ge f_I\left(y^{\prime }\right)$ if $y\le y^{\prime }$, with $f_I\left(0\right)>0$ and $f_I\left(M\right)<0$. In addition, there exists $y^{\ast }$ such that $f_I\left(y^{\ast }\right)=0$. □

If the uniform price is set to be the least winning price, $y^{\ast }={max}_y\left\{y\right|f_I\left(y\right)=0\}$. As $f_I\left(y\right)$ is a non-increasing function of a single variable y, a distributed algorithm can easily be developed by the method of bisection.

3.1. Bisection Method-Based Distributed Algorithm

The distributed algorithm is depicted in Algorithm 2 and its implementation is shown in Figure 2c. By the fact that $f_I\left(y\right)$ is a single variable function and $f_I\left(y\right)$ is a non-increasing function, as stated in Theorem 1, method of bisection can be applied. We define three variables, namely $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$.

Let ${\pi }_i$ be the index of the winner with the least winning price $u_{{\pi }_i}$. We also let $y^{\ast }$ be the convergent value of $y\left(t\right)$ and $q_{{\pi }_i}^{\ast }$ be the units allocated to the ${{\pi }_i}^{th}$ bidder. Either one of the following two cases will happen:

(a)

$1\le q_{{\pi }_i}^{\ast }<q_{{\pi }_i}$ and hence $u_{{\pi }_i}-\mathsf{\Delta }<y^{\ast }<u_{{\pi }_i}$.

$$f_I\left(y\right)\left\{\begin{array}{cc}>0\hfill & \mathrm{if} y<y^{\ast }\hfill \\ =0\hfill & \mathrm{if} y=y^{\ast },\hfill \\ <0\hfill & \mathrm{if} y>y^{\ast }.\hfill \end{array}\right.$$

(b)

$q_{{\pi }_i}^{\ast }=q_{{\pi }_i}$ and hence $y^{\ast }=u_{{\pi }_i}-\mathsf{\Delta }$.

$$f_I\left(y\right)\left\{\begin{array}{cc}>0\hfill & \mathrm{if} y<u_{{\pi }_{i-1}},\hfill \\ =0\hfill & \mathrm{if} u_{{\pi }_{i-1}}\le y\le u_{{\pi }_i}-\mathsf{\Delta },\hfill \\ <0\hfill & \mathrm{if} y>u_{{\pi }_i}-\mathsf{\Delta }.\hfill \end{array}\right.$$

Figure 3 shows the shapes of $f_I\left(y\right)$ for these cases.

Figure 3. Shapes of $f\left(y\right)$. Let ${\pi }_i$ be the winner with the least winning price. (a) $1\le q_{{\pi }_i}^{\ast }<q_{{\pi }_i}$. $f_I\left(y^{\ast }\right)=0$, $f_I\left(y\right)>0$ if $y<y^{\ast }$ and $f_I\left(y\right)<0$ if $y>y^{\ast }$. (b) $q_{{\pi }_i}^{\ast }=q_{{\pi }_i}$. $f_I\left(y\right)=0$ if $u_{{\pi }_{i-1}}<y\le u_{{\pi }_i}-\mathsf{\Delta }$. $f_I\left(y\right)>0$ if $y<u_{{\pi }_{i-1}}$ and $f_I\left(y\right)<0$ if $y>u_{{\pi }_i}-\mathsf{\Delta }$.

For $t=0$, the auctioneer sets that $\underline{y}\left(0\right)=0$, $y\left(0\right)=M/2$ and $\overline{y}\left(0\right)=M$. Then, $y\left(0\right)$ is broadcast to all bidders. Once the bidders have received $y\left(0\right)$, they calculate their ${\widehat{q}}_i\left(0\right)=q_i\varphi (u_i-y\left(0\right))$ and then send their ${\widehat{q}}_i\left(0\right)$ to the auctioneer. After that, the auctioneer calculates $\underline{y}\left(1\right)$, $y\left(1\right)$ and $\overline{y}\left(1\right)$ based on the values ${\widehat{q}}_i\left(0\right)$s received. Again, $y\left(1\right)$ is broadcasted to all bidders.

For $t\ge 1$, all bidders calculate their ${\widehat{q}}_i\left(t\right)$ based on the $y\left(t\right)$ received and then send them to the auctioneer. The auctioneer then updates $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$ by the following Equation (4).

$$\left[\begin{array}{c}\underline{y}(t+1)\\ y(t+1)\\ \overline{y}(t+1)\end{array}\right]=\left\{\begin{array}{cc}\left[\begin{array}{c}y\left(t\right)\\ (y\left(t\right)+\overline{y}\left(t\right))/2\\ \overline{y}\left(t\right)\end{array}\right]\hfill & \mathrm{if} f_I\left(y\left(t\right)\right)\ge 0,\hfill \\ \left[\begin{array}{c}\underline{y}\left(t\right)\\ (y\left(t\right)+\underline{y}\left(t\right))/2\\ y\left(t\right)\end{array}\right]\hfill & \mathrm{if} f_I\left(y\left(t\right)\right)<0,\hfill \end{array}\right.$$

(4)

where $f_I\left(y\left(t\right)\right)={\sum }_{i=1}^m{\widehat{q}}_i\left(t\right)-k$.

Algorithm 2 Bisection Method-Based Distributed Algorithm

Initialization: Set M, Q, $ϵ$ and $\mathsf{\Delta }$. Bidders: for  $i=1,\cdots ,m$  do in parallel       Set $0<u_i<M$ and $1\le q_i\le Q$. end for Auctioneer: Set k, $\underline{y}=0$, $y=M/2$, $\overline{y}=M$. Broadcast y to all bidders. while {$(\overline{y}-\underline{y})\ge 2ϵ\mathsf{\Delta }/Q$} do       Bidders:       for $i=1,\cdots ,m$ do in parallel             Calculate ${\widehat{q}}_i=q_i\varphi (u_i-y)$             Send ${\widehat{q}}_i$ to the auctioneer.       end for       Auctioneer:       Calculate $f_I={\sum }_{i=1}^m{\widehat{q}}_i-k$.       if $f_I\ge 0$ then             $\underline{y}=y$ and $y=(\underline{y}+\overline{y})/2$.       else             $\overline{y}=y$ and $y=(\underline{y}+\overline{y})/2$.       end if        Broadcast y to all bidders. end while Units allocated are $\left[{\widehat{q}}_1\right],\cdots ,\left[{\widehat{q}}_m\right]$. Uniform price is $\left[y\right]$.

3.2. Convergence Analysis

Let the ${\pi }_i$ be the last winner, $q_{{\pi }_i}^{\ast }$ be the units being allocated and $y^{\ast }$ be the convergent of $y\left(t\right)$. It can be shown that $y\left(t\right)\to y^{\ast }$.

Theorem 2.

By setting $\underline{y}\left(0\right)=0$ and $\overline{y}\left(0\right)=M$, we can obtain (i) $y^{\ast }\in [\underline{y}\left(t\right),\overline{y}\left(t\right)]$ for all $t\ge 0$ and (ii) ${lim}_{t\to \infty }y\left(t\right)=y^{\ast }$. Moreover, $\left[y^{\ast }\right]=u_{{\pi }_i}$.

Proof. 

See Appendix B.    □

3.3. Stopping Criteria

To ensure that $\left[q_{{\pi }_i}\varphi (u_{{\pi }_i}-y\left(t\right))\right]=q_{{\pi }_i}^{\ast }$, the following stopping criteria must hold:

$$|q_{{\pi }_i}\varphi (u_{{\pi }_i}-y\left(t\right))-q_{{\pi }_i}\varphi (u_{{\pi }_i}-y^{\ast })|=ϵ<1/2.$$

As $q_{{\pi }_i}\le Q$, the above condition holds if the following condition holds:

$$|\varphi (u_{{\pi }_i}-y\left(t\right))-\varphi (u_{{\pi }_i}-y^{\ast })|=ϵ/Q.$$

To ensure $|y\left(t\right)-y^{\ast }|=ϵ\mathsf{\Delta }/Q$, we need the condition that $(\overline{y}\left(t\right)-\underline{y}\left(t\right))=2ϵ\mathsf{\Delta }/Q$. As $y^{\ast }\in [u_{{\pi }_i}-\mathsf{\Delta },u_{{\pi }_i}]$, $\underline{y}\left(t\right)\ge u_{{\pi }_i}-\mathsf{\Delta }-2ϵ/Q$. To ensure $\left[y\left(t\right)\right]=u_{{\pi }_i}$, we need another condition that $(1+2ϵ/Q)\mathsf{\Delta }<1/2$. As a result, we can state without proof the following stopping criteria.

Theorem 3.

The sufficient conditions to ensure that $\left[y\left(t\right)\right]=u_{{\pi }_i}$ and $\left[q_{{\pi }_i}\varphi (u_{{\pi }_i}-y\left(t\right))\right]=q_{{\pi }_i}^{\ast }$ are given as follows:

$$\begin{array}{cc}\hfill \left(a\right)& 0<ϵ,\mathsf{\Delta }<1/2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(b\right)& 0<(\overline{y}\left(t\right)-\underline{y}\left(t\right))<{\displaystyle \frac{2ϵ\mathsf{\Delta }}{Q}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \hspace{1em} \hspace{1em}\left(c\right)& 0<\left(1+{\displaystyle \frac{2ϵ}{Q}}\right)\mathsf{\Delta }<{\displaystyle \frac{1}{2}},\hfill \end{array}$$

(7)

where Q is given.

3.4. Number of Steps $\mathcal{T}$

Similarly to the case of distinct bid prices, we can state the following theorem on the number of steps $\mathcal{T}$ for the convergence of $y\left(t\right)$ such that   

$$\left[q_{{\pi }_i}\varphi (u_{{\pi }_i}-y\left(\mathcal{T}\right))\right]-k=0.$$

Theorem 4.

A sufficient condition on the number of steps $\mathcal{T}$ for the algorithm, as stated in Equation (4), to allocate the units and determine the uniform price is given by

$$\mathcal{T}=⌈{log}_2\left(M\right)+{log}_2\left(Q\right)-{log}_2\left(ϵ\mathsf{\Delta }\right)-1⌉.$$

(8)

Proof. 

By Equations (5)–(7), $(\overline{y}\left(t\right)-\underline{y}\left(t\right))<2ϵ\mathsf{\Delta }/Q$. On the other hand, the initial condition $(\overline{y}\left(0\right)-\underline{y}\left(0\right))=M$ implies that $(\overline{y}\left(t\right)-\underline{y}\left(t\right))=2^{-t}M$. To ensure that $(\overline{y}\left(\mathcal{T}\right)-\underline{y}\left(\mathcal{T}\right))=2ϵ\mathsf{\Delta }/Q$, $2^{-\mathcal{T}}M=2ϵ\mathsf{\Delta }/Q$. Equivalently,

$$\mathcal{T}=⌈{log}_2\left(M\right)+{log}_2\left(Q\right)-{log}_2\left(ϵ\mathsf{\Delta }\right)-1⌉.$$

The proof is completed.    □

3.5. Optimal Setting

From Equation (8), it is clear that the smallest number of steps occurs when $ϵ\mathsf{\Delta }$ is maximum. By Equation (7), we get

$$\mathsf{\Delta }<{\displaystyle \frac{Q}{2(Q+2ϵ)}}.$$

We let $F\left(ϵ\right)={\displaystyle \frac{ϵQ}{2(Q+2ϵ)}}$. It can be shown that $dF\left(ϵ\right)/dϵ>0$ for $0<ϵ<1/2$. $F\left(ϵ\right)$ is maximum at $ϵ={(1/2)}^{-}$. That is to say, $F\left(ϵ\right)\le Q/\left(4\right(Q+1\left)\right)$. Equivalently,

$$max\left\{ϵ\mathsf{\Delta }\right\}={\displaystyle \frac{Q}{4(Q+1)}}.$$

(9)

For illustration, the curves for $\mathsf{\Delta }=Q/\left(2\right(Q+2ϵ\left)\right)$ and $ϵ\mathsf{\Delta }=Q/\left(4\right(Q+1\left)\right)$ are shown in Figure 4.

Figure 4. Plots of the curves for $\mathsf{\Delta }=Q/\left(2\right(Q+2ϵ\left)\right)$ (the solid lines) and for $ϵ\mathsf{\Delta }=Q/\left(2\right(Q+1\left)\right)$ (the dot-dash lines). It is shown that $ϵ\mathsf{\Delta }$ is maximum if $ϵ=1/2$.

We can now state without proof the following theorem for the setting of $ϵ$, $\mathsf{\Delta }$ and $\mathcal{T}$ for the algorithm as stated in Equation (4).

Theorem 5.

Given M and Q, ϵ, Δ and the stopping criteria are, respectively, set as follows:

$$ϵ={\displaystyle \frac{1}{2}},\mathsf{\Delta }={\displaystyle \frac{Q}{4(Q+1)}},(\overline{y}\left(t\right)-\underline{y}\left(t\right))<{\displaystyle \frac{1}{4(Q+1)}}.$$

(10)

The number of steps for unit allocation and uniform price determination is given by

$$\mathcal{T}=⌈{log}_2\left(M\right)+{log}_2(Q+1)+1⌉.$$

(11)

It should be noted that the number of steps $\mathcal{T}$, as stated in Equation (11) is independent of m once M and Q are fixed. Therefore, the time span for completing the unit allocation and uniform price determination depends on the total communication delay in the data exchange in a step times the total number of steps $\mathcal{T}$. It will be discussed in the next section; this time span is still short as compared with the recurrent neural network solution.

4. Benefits of the Fast Algorithm

In summary, the proposed fast distributed algorithm as stated in this section has many advantages as compared with the algorithms presented in Section 2.3. A comparison among the algorithms presented is depicted in Table 2.

Table 2. Comparisons of the bidding information privacy among algorithms.

Algorithm	On $q_{{\pi }_j}(j=1,\cdots ,m)$	On $u_{{\pi }_j}(j=1,\cdots ,m)$
Centralized	Auctioneer knows $q_{{\pi }_m},\cdots ,q_{{\pi }_1}$.	Auctioneer knows $u_{{\pi }_m},\cdots ,u_{{\pi }_1}$.
	Bidder ${\pi }_j$ only knows $q_{{\pi }_j}$.	Bidder ${\pi }_i$ only knows $u_{{\pi }_j}$.
RNN-Based	Auctioneer knows $q_{{\pi }_m},\cdots ,q_{{\pi }_{i+1}}$.	Auctioneer knows $u_{{\pi }_m},\cdots ,u_{{\pi }_i}$.
	Bidder ${\pi }_j$ only knows $q_{{\pi }_j}$.	Bidders know $u_{{\pi }_m},\cdots ,u_{{\pi }_i}$.
Ours	Auctioneer knows $q_{{\pi }_m},\cdots ,q_{{\pi }_{i+1}}$.	Auctioneer only knows $u_{{\pi }_i}$.
	Bidder ${\pi }_j$ only knows $q_{{\pi }_j}$.	Bidders only know $u_{{\pi }_i}$.
Algorithm	Steps	Winning Bid Price	Losing Bid Price
Centralized	One.	Not protected.	Not protected.
RNN-Based	$\mathcal{O}\left(M\right)$.	Not protected.	Protected.
Ours	$\mathcal{O}(log(M\left)\right)$.	Protected.	Protected.

It is assumed that ${\pi }_m,\cdots ,{\pi }_i$ are the winners.

4.1. Bidding Information Privacy Protection

In each implementation, the bidding prices ($u_i$s or $u_{ij}$s) and bidding quantities ($q_i$s) are not communicated during the process. Therefore, the above implementations are able to protect both winners’ and losers’ bidding information privacy. A centralized algorithm does not protect any bidding information. The RNN-based algorithm only partially protects some bidding information.

4.2. Low Communication Delay

In this implementation, sending data ${\widehat{q}}_i\left(t\right)$ and $y\left(t\right)$ in between bidders and the auctioneer is accomplished by a wireless communication protocol [Chapter 2] [25]. Equivalently, broadcasting y to all bidder agents is accomplished by the same wireless communication protocol. Suppose that there are m bidders. The time span for unit allocation and uniform price determination is $(m+1)d_{comm}\mathcal{T}$, where $d_{comm}$ is the communication delay of sending a message from one device to another.

It should be noted that the number $(m+1)$ is based on the fact that m bidders need to send their $z_i\left(t\right)$s to the auctioneer and the auctioneer needs to broadcast the $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$ to all the bidders. For WiFi connection, the medium access control (MAC) is based on the protocol carrier sense medium access with collision avoidance (CSMA/CA). For m bidders successfully send their data to the auctioneer, the time complexity is in the order of $\mathcal{O}(m\times d_{comm})$. Practically, the time span is hardly more than $10\times m\times d_{comm}$. Owing not to distract the readers on the elegant of our algorithm, we simply state that the time span is $m\times d_{comm}$. For a more realistic bound, we simply have to change 10 (resp. 1) seconds to 100 (resp. 10) seconds for Bluetooth (resp. WiFi).

By Equation (11), we can get that the time span is equal to $(m+1)\lceil {log}_2\left(M\right)+{log}_2(Q+1)+1\rceil d_{comm}$. If (i) we set the total number of bidders $m\le 99$ and $M=1024$; and (ii) consider Bluetooth communication protocol (i.e., $d_{comm}=0.01$ second), the time span is around 10 s. For WiFi communication, the communication delay is $0.001$ s. The time span is around one second.

In our algorithm, M has to be pre-set and it has to be larger than ${max}_j\left\{u_j\right\}$. Clearly, we can only guess an upper bound for this value. As the time span is proportional to ${log}_2\left(M\right)$, the time span is still short even if we guess the ${max}_j\left\{u_j\right\}$ from M to $2^{10}M$. This time span scales as the number of bidders increases. So, our algorithm is not suitable to be applied in those auctions requiring real-time decision.

5. Simulations

Simulation results are shown in this section with a focus on (i) the small number of steps required for convergence and (ii) the number of steps for convergence as compared with the RNN-based solution.

5.1. Finite-Time Convergence

With reference to the bidding information depicted in Table 1, we set that $M=256$ and $Q=4$. Thus, we can obtain $\mathsf{\Delta }=1/5$ and $\mathcal{T}=12$. By that, two simulations are conducted. One is for $k=4$ and the other is for $k=5$. Their results on $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$ are shown in Figure 5.

Figure 5. The changes in $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$. Here, $M=256$ and $Q=4$. Hence, $\underline{y}\left(1\right)=0$, $\overline{y}\left(1\right)=256$ and $\mathsf{\Delta }=1/5$ (from Theorem 5).

For the case of $m=5,k=4$ in Table 1, the uniform price is given by $\left[y\right(\mathcal{T}\left)\right]=75$ (as shown in Figure 5a) and the units being allocated are given by $[0,1,2,0,1]$. For the case of $m=5,k=5$ in Table 1, the uniform price is given by $\left[y\right(\mathcal{T}\left)\right]=50$ (as shown in Figure 5b) and the units being allocated are given by $[1,1,2,0,1]$.

5.2. Fast Convergence

To compare the fast convergence of the proposed Equation (4) as compared with the RNN-based algorithm, extensive simulations are conducted for that

$$M=2^5,2^6,2^7,\cdots ,2^{15}.$$

For all simulations, $m=5$, $k=5$ and $Q=4$.

5.2.1. Our Algorithm

For our proposed algorithm, the stopping criteria is given by

$$\overline{y}\left(t\right)-\underline{y}\left(t\right)\le 1/20,$$

which is from Theorem 5. Hence, the number of steps required for our Equation (4) to determine the unit allocation and the uniform price is given by

$$\mathcal{T}=\lceil {log}_2\left(M\right)+{log}_2\left(5\right)+1\rceil .$$

(12)

This number is independent of m, k, the per-unit bidding prices $u_1,\cdots ,u_m$ and the number of units to bid $q_1,\cdots ,q_m$.

5.2.2. RNN-Based

For the RNN-based algorithm, the number of steps to determine the uniform price and the unit allocation depends on the per-unit bidding prices and the number of units to bid. To have a fair comparison, a set of 1000 samples are generated for each M and $Q=4$. In each sample, a set of five per-unit bidding prices are randomly generated in between 0 and M. In addition, a set of five bidding units are randomly generated in the range from 1 to 4. Then, the number of steps for an RNN-based algorithm to determine the uniform price and the unit allocation for this sample is counted. The average number of steps over 1000 samples for a specific M is recorded. The step size and the stopping criteria are set as follows:

$$\mu =0.01\mathrm{and}\left|y\right(t+1)-y(t\left)\right|\le 0.0001.$$

5.2.3. Comparison Results for Different M

The comparison results are shown in Figure 6 and Figure 7. Here, the line with circles is the result from the RNN-based algorithm. The line with triangles is the result from our proposed algorithm. Let $T_{RNN}\left(M\right)$ be the average number of steps for the RNN-based algorithm, with price range $[0,M]$, to determine the unit allocation and uniform price. It is clear from Figure 6 that

$$log\left(T_{RNN}\left(M\right)\right)={\kappa }_0+{\kappa }_{1}log\left(M\right)$$

for the RNN-based algorithm. Equivalently,

$$T_{RNN}\left(M\right)=exp\left({\kappa }_0\right)M^{{\kappa }_1}.$$

Figure 6. Comparison between the number of steps obtained by the RNN-based algorithm and our proposed algorithm. The line with triangle marks is the result based on Equation (12). Here, the horizontal axis corresponds to M and the vertical axis corresponds to the number of steps.

Figure 7. Comparison between the number of steps obtained by the RNN-based algorithm and our proposed algorithm. Here, $Q=4$, $k=5$, the horizontal axis corresponds to the number of bidders m and the vertical axis corresponds to the number of steps. Note that the number of steps of our proposed algorithm is independent of the number of bidders m.

The number of steps obtained by the RNN-based algorithm is much larger than our proposed algorithm, especially when M is large.

5.2.4. Comparison Results for Different m

For $M=4096$, the results for different numbers of bidders are shown in Figure 7. Let $T_{RNN}\left(M\right)$ be the average number of steps for the RNN-based algorithm with $k=5$ and $Q=4$. Again, the average number of steps obtained by the RNN-based algorithm is much larger than our proposed algorithm, especially when m is small. It is clear from Figure 7 that

$$log\left(T_{RNN}\left(m\right)\right)={\kappa }_0^{\prime }-{\kappa }_1^{\prime }log\left(m\right).$$

Equivalently,

$$T_{RNN}\left(m\right)=exp\left({\kappa }_0^{\prime }\right)m^{-{\kappa }_1^{\prime }}.$$

For a small number of m, the difference is in the order of magnitude $\mathcal{O}\left({10}^4\right)$. For practical consideration, our algorithm is applied to a small number of bidders. Therefore, our proposed algorithm is preferred for solving the unit allocation and uniform price determination problems.

5.3. On Bidding Price Privacy

For Case (1) in Table 1, the proposed Equation (4) is applied with $\underline{y}\left(0\right)=0$ and $\overline{y}\left(0\right)=128$. During the process of unit allocation and uniform price determination, the auctioneer can obtain information ${\widehat{q}}_1\left(t\right),\cdots ,{\widehat{q}}_5\left(t\right)$ from the bidders as well as $\underline{y}\left(t\right)$, $y\left(t\right)$ and $\overline{y}\left(t\right)$. All this information is depicted in Table 3 for illustration.

Table 3. Numerical results for the Case (1) in Table 1.

t	${\widehat{\mathit{q}}}_1\left(\mathit{t}\right)$	${\widehat{\mathit{q}}}_2\left(\mathit{t}\right)$	${\widehat{\mathit{q}}}_3\left(\mathit{t}\right)$	${\widehat{\mathit{q}}}_4\left(\mathit{t}\right)$	${\widehat{\mathit{q}}}_5\left(\mathit{t}\right)$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	$\underline{\mathit{y}}\left(\mathit{t}\right)$	$\mathit{y}\left(\mathit{t}\right)$	$\overline{\mathit{y}}\left(\mathit{t}\right)$
0	–	–	–	–	–	–	–	–	–	–	0	64.0000	128.0000
1	0	1	2	0	1	$[0,64]$	$[64,128]$	$[64,128]$	$[0,64]$	$[64,128]$	64.0000	96.0000	128.0000
2	0	1	0	0	0	$[0,64]$	$[96,128]$	$[64,96]$	$[0,64]$	$[64,96]$	64.0000	80.0000	96.0000
3	0	1	0	0	0	$[0,64]$	$[96,128]$	$[64,80]$	$[0,64]$	$[64,80]$	64.0000	72.0000	80.0000
4	0	1	2	0	1	$[0,64]$	$[96,128]$	$[72,80]$	$[0,64]$	$[72,80]$	72.0000	76.0000	80.0000
5	0	1	0	0	1	$[0,64]$	$[96,128]$	$[72,76]$	$[0,64]$	$[76,80]$	72.0000	74.0000	76.0000
6	0	1	2	0	1	$[0,64]$	$[96,128]$	$[74,76]$	$[0,64]$	$[76,80]$	74.0000	75.0000	76.0000
7	0	1	0	0	1	$[0,64]$	$[96,128]$	$[74,75]$	$[0,64]$	$[76,80]$	74.0000	74.5000	75.0000
8	0	1	2	0	1	$[0,64]$	$[96,128]$	$[74.5,75]$	$[0,64]$	$[76,80]$	74.5000	74.7500	75.0000
9	0	1	2	0	1	$[0,64]$	$[96,128]$	$[75.75,75]$	$[0,64]$	$[76,80]$	74.7500	74.8750	75.0000
10	0	1	1.250	0	1	$[0,64]$	$[96,128]$	$[74.875,75]$	$[0,64]$	$[76,80]$	74.7500	74.8125	74.8750
11	0	1	1.875	0	1	$[0,64]$	$[96,128]$	$[74.8125,74.875]$	$[0,64]$	$[76,80]$	74.7500	74.7812	74.8125

Here, $\underline{y}\left(0\right)=0$ and $\overline{y}\left(0\right)=128$. From the above information, the auctioneer can best guess that $q_2=1$, $q_2=2$ and $q_4=1$ but not $q_1$ and $q_2$. For the per-unit bidding prices, except $u_3$, the auctioneer can only guess the ranges for $u_1$, $u_2$, $u_4$ and $u_5$.

From the above results, the auctioneer is able to determine from $\left[{\widehat{q}}_1\left(t\right)\right],\cdots ,\left[{\widehat{q}}_5\left(t\right)\right]$ the number of units to be allocated and $\left[y\right(t\left)\right]$ the uniform price for the payment of the units. For other bidding information, the auctioneer can best guess that $q_2=1$, $q_2=2$ and $q_4=1$. The auctioneer is unable to guess $q_1$ and $q_2$. For the per-unit bidding prices, except $u_3$, the auctioneer can only know the ranges for $u_1$, $u_2$, $u_4$ and $u_5$. The exact values of $u_1$, $u_2$, $u_4$ and $u_5$ are unknown. Per-unit bidding prices are protected.

6. Discussions

In this section, two important issues are discussed. The first one is on the identical per-unit price bidding problem. The other is on the case that the uniform price is defined as the highest losing bidding price. The third one is on the per-unit bidding price protection.

6.1. Identical Per-Unit Bidding Price

It should be noted that our proposed algorithm is not applicable to the situation that multiple bidders have identical per-unit bidding price. To deal with this problem, one solution is to add a time-dependent markup on a bid. The earlier bidder will be allocated with units first.

Let T be the period for the bidders to set their bid. Let us say the bid setting starts from 1:00 p.m. and ends at 1:15 p.m. So, the bidding period T is 900 s. Further, let $t_i$ be the time when the $i^{th}$ bidder has just completed his/her setting. Then, the input $u_i$ is replaced by ${\tilde{u}}_i$ as follows:

$${\tilde{u}}_i=\alpha u_i+\underset{Markup}{\underbrace{\beta \left({\displaystyle \frac{T-t_i}{T}}\right)}}.$$

(13)

For the settings of $\alpha$ and $\beta$, we need to solve the condition that $|{\tilde{u}}_i-{\tilde{u}}_j|\ge 1$. For this, we need to consider two conditions: (i) $u_i=u_j=\overline{u}$ and (ii) $u_i=\overline{u}$ and $u_j=\overline{u}+1$.

For Case (i), we can get $|{\tilde{u}}_i-{\tilde{u}}_j|=\beta |t_j-t_i|/T$. To ensure that $|{\tilde{u}}_i-{\tilde{u}}_j|\ge 1$, we have to set

$$\beta \ge {\displaystyle \frac{T}{min\left\{\right|t_j-t_i\left|\right\}}}.$$

(14)

For Case (ii), we can get ${\tilde{u}}_i=\alpha \overline{u}+\beta (T-t_i)/T$ and ${\tilde{u}}_j=\alpha (\overline{u}+1)+\beta (T-t_j)/T$. Thus, ${\tilde{u}}_j-{\tilde{u}}_i=\alpha +\beta (t_i-t_j)/T$ and $min\left\{{\tilde{u}}_j-{\tilde{u}}_i\right\}=\alpha -\beta$. To ensure that $|{\tilde{u}}_i-{\tilde{u}}_j|\ge 1$,

$$\alpha \ge \beta +1.$$

(15)

If we assume that $min\left\{\right|t_j-t_i\left|\right\}$ is larger than ${\delta }_t$, the settings of $\alpha$ and $\beta$ are given by $\beta =T/{\delta }_t$ and $\alpha =(T/{\delta }_t)+1$. The value ${\delta }_t$ is set to assume that no two or more bids are submitted within ${\delta }_t$ time. By that, the additional number of iterations is simply ${log}_2\left(\alpha \right)$ (resp. ${log}_2((T/{\delta }_t)+1)$). The total number of steps $\mathcal{T}$ is given by

$$\mathcal{T}=\lceil {log}_2((T/{\delta }_t)+1)+{log}_2\left(M\right)+{log}_2(Q+1)+1\rceil .$$

(16)

The modified algorithm for solving the winner determination and price determination is depicted in Algorithm 3.

Algorithm 3 Modified Bisection Method-Based Distributed Algorithm

Initialization: Set M, Q, $ϵ$, $\mathsf{\Delta }$, $\alpha$, $\beta$ and ${\delta }_t$. Bidders: for  $i=1,\cdots ,m$ do in parallel       Set $0<u_i<M$ and $1\le q_i\le Q$.       Set ${\tilde{u}}_i=\alpha u_i+\beta (T-t_i)/T$. end for Auctioneer: Set k, $\underline{y}=0$, $y=M/2$, $\overline{y}=\alpha M$. Broadcast y to all bidders. while {$(\overline{y}-\underline{y})\ge 2ϵ\mathsf{\Delta }/Q$} do       Bidders:       for $i=1,\cdots ,m$ do in parallel             Calculate ${\widehat{q}}_i=q_i\varphi ({\tilde{u}}_i-y)$             Send ${\widehat{q}}_i$ to the auctioneer.       end for       Auctioneer:       Calculate $f_I={\sum }_{i=1}^m{\widehat{q}}_i-k$.       if $f_I\ge 0$ then             $\underline{y}=y$ and $y=(\underline{y}+\overline{y})/2$.     else             $\overline{y}=y$ and $y=(\underline{y}+\overline{y})/2$.       end if       Broadcast y to all bidders. end while Units allocated are $\left[{\widehat{q}}_1\right],\cdots ,\left[{\widehat{q}}_m\right]$. Uniform price is $[y/\alpha ]$.

6.2. Highest Losing Bidding Price as the Uniform Price

If the uniform price is set to be the highest losing per-unit bidding price, our proposed distributed algorithm as stated in Equation (4) can be modified by changing the condition in Equation (4) as follows:

$$\left[\begin{array}{c}\underline{y}(t+1)\\ y(t+1)\\ \overline{y}(t+1)\end{array}\right]=\left\{\begin{array}{cc}\left[\begin{array}{c}\underline{y}\left(t\right)\\ (\underline{y}\left(t\right)+y\left(t\right))/2\\ y\left(t\right)\end{array}\right]\hfill & \mathrm{if} f_I\left(y\left(t\right)\right)\ge 0,\hfill \\ \left[\begin{array}{c}y\left(t\right)\\ (y\left(t\right)+\overline{y}\left(t\right))/2\\ \overline{y}\left(t\right)\end{array}\right]\hfill & \mathrm{if} f_I\left(y\left(t\right)\right)<0,\hfill \end{array}\right.$$

(17)

The benefits as presented in Section 4 carry over to this algorithm.

Moreover, it can readily be shown that Theorem 5 holds true for this case. ${lim}_{t\to \infty y\left(t\right)}=y^{\ast }$ and ${lim}_{t\to \infty }\widehat{q}\left(t\right)=q^{\ast }$. If the stopping criteria is set to be that $\overline{y}\left(t\right)-\underline{y}\left(t\right)\le 2ϵ\mathsf{\Delta }/Q$, the number of steps for this algorithm to determine the unit allocation and the uniform price (highest losing price) is identical to the number of steps for Algorithm 2. That is to say,

$$\mathcal{T}=⌈{log}_2\left(M\right)+{log}_2(Q+1)+1⌉$$

if $ϵ=1/2$ and hence $\mathsf{\Delta }=1/\left(4\right(Q+1\left)\right)$.

6.3. Bidding Information Privacy

For the per-unit bidding price (resp. bidding unit) privacy, we can only demonstrate and present in Section 5.3 for Case (1) in Table 1 that the auctioneer is unable to determine the per-unit bidding prices and the units being bid from the data obtained during the process of unit allocation and uniform price determination by our proposed distributed algorithm as stated in Equation (4). Once the unit allocation and uniform price have been determined, the auctioneer can obtain from Table 1 the following results.

Bidder	1	2	3	4	5
$u_i$	$[0,64]$	$[96,128]$	75	$[0,64]$	$[76,80]$
$q_i$	0	1	2	0	1

Except the lowest winning per-unit price, the auctioneer cannot know other per-unit bidding prices. For the bidding units, the auctioneer can know that $q_2=1$, $q_3\ge 2$ and $q_5=1$. The bidding quantities of other bidders are unknown.

Therefore, our algorithm could protect the per-unit bidding prices and partially protect the bidding units. Theoretical analysis on bidding price privacy protection has yet to be accomplished. We leave it as a work for future investigation, if it is possible. We can only conjecture that our proposed Equation (4) is able to protect the per-unit bidding prices $u_i$ for $i=1,\cdots ,m$.

6.4. Limitations on Encryption-Based Privacy Protection

Protecting bidding price has recently attracted cryptologists developing techniques to tackle such problem [6,11,12,26,27,28,29,30]. Consider a sealed-bid first price auction. One approach is to develop an encryption method for the bidding price together with a comparison algorithm for ranking two encrypted bidding prices [5]. By repeatedly applying the comparison algorithm among the n encrypted bidding prices, the centralized agent is able to determine the encrypted bidding price corresponding to the highest bid (resp. the winner) and announce the result to all the bidders. Finally, the winner announces his/her bidding price to the bidders.

These methods can protect the bidding prices. However, it is not suitable for a uniform price auction in which each bidder can bid for multiple units with a single per-unit price. While the auctioneer is able to obtain the ranking order of the per-unit bidding prices, the auctioneer needs to know the bidding quantities from the bidders to complete the unit allocation. Bidding information privacy is not all protected.

6.5. Other Multi-Unit Auction Mechanisms

It is clear that our proposed algorithm is limited to uniform price auction. For auctioning off multi-unit, other mechanisms like discriminatory price auction and combinatorial auction [31] can be applied. It is very true. However, extended modification is needed if our algorithm is applied in a discriminatory price auction to protect the losing bidding information. For combinatorial auctions, our algorithm is not able to apply, as solving the winner determination problem itself requires all bidding information. Privacy cannot be protected.

6.5.1. Discriminatory Price Auction

For a discriminatory auction, each bidder has a not-to-disclose bid $(q_i,u_i)$. The auctioneer allocates the units to the bidders starting from the bidder with the highest per-unit bidding price until the units have all been allocated. Then, each winning bidder pays for each unit based on his/her per-unit bidding price.

For this auction mechanism, our fast algorithm can be applied. The lowest per-unit winning price can be immediately obtained. Let $(q_{{\pi }_j},u_{{\pi }_j})$ be the bid of the ${{\pi }_j}^{th}$ winner with the lowest per-unit bidding price. After the winner determination process has been completed, the auctioneer is able to know that

$$k-\sum _{i=j-1}^n{q}_{{\pi }_i}>0\mathrm{and}k-\sum _{i=j}^n{q}_{{\pi }_i}\le 0.$$

Moreover, the auctioneer knows the per-unit bidding price $u_{{\pi }_j}$. After that, the auctioneer can set $k^{\prime }={\sum }_{i=j-1}^n{q}_{{\pi }_i}$ and then re-run the winner determination process to obtain the second lowest per-unit bidding price. This process repeats until all per-unit winning prices have been obtained.

Therefore, our fast algorithm can be extended and applied in discriminatory price auctions with losing per-unit prices protection. The complexity of this extended algorithm is just in the order of $\mathcal{O}(k\times log(M)$. Losing bid information is protected.

6.5.2. Combinatorial Auction

For a combinatorial auction, each bidder has a not-to-disclose bid $(q_i,U_i)$. Here, $U_i$ is the amount to be paid for all $q_i$ units if the bidder has successfully won the bid. As mentioned earlier in the paper, the winner determination problem of auctioning off multi-units via a combinatorial auction mechanism is identical to a knapsack problem, which is a well-known $\mathcal{NP}$-hard problem [22]. Just for solving the winner determination problem, the auctioneer needs to know all the bidding information. It is unable to protect any bidding information.

7. Conclusions

In this paper, a fast distributed algorithm has been presented for solving the unit allocation and uniform price determination problems for a uniform price auction, in which the bidders can bid for multiple units with a per-unit bidding price. The key idea is based on applying the idea of the bisection method. In the sequel, the proposed distributed algorithm is able to allocate the units to the bidders and determine the uniform price in a very small number of steps as compared with the recurrent neural network-based algorithm. Furthermore, the distributed algorithm is able to protect the bidding information, including the bidders’ per-unit bidding price and their units to bid. Theoretical analyses on the properties of the proposed algorithm, including its convergence property and the stopping criteria, are presented. Finally, the advantages of the proposed algorithm are elucidated and demonstrated by simulations. To the best of our knowledge, applying the method of bisection for solving the unit allocation and uniform price determination problems in a distributed manner with bidding price protection has not been presented in the literature.

Appendix A.1. Algorithm

For the case that each bidder bids for multiple units, the allocation and the uniform price can be solved by the following recurrent neural network solution which is in essence a modification of Wang kWTA [23] which consists of a single recurrent state.

$$\begin{array}{ccc}\hfill y(t+1)& =& y\left(t\right)+\mu \underset{f_I\left(y\left(t\right)\right)}{\underbrace{\left\{\sum _{i=1}^m{q}_i{z}_i\left(t\right)-k\right\}}},\hfill \end{array}$$

(A1)

$$\begin{array}{ccc}\hfill z_i\left(t\right)& =& \varphi (u_i-y\left(t\right))\hfill \end{array}$$

(A2)

where $y\left(0\right)=M$, $0<\mu \ll 1$ is the step size.

Once Equations (A1) and (A2) converge, the units allocated are $\left[q_1{z}_1\left(t\right)\right],\cdots ,\left[q_m{z}_m\left(t\right)\right]$ and the uniform price is $\left[y\right(t\left)\right]$. Here, $[·]$ is the rounding operator Here, we assume that the algorithm terminates in finite steps, i.e., $1\ll t<\infty$. For $t\to \infty$, the rounding operator can be removed. This property can be confirmed by the following theorem.

Theorem A1.

Given that $y\left(0\right)=M$, the algorithm as stated in Equations (A1) and (A2) is able to converge and determine the units to be allocated to the bidders and the uniform price.

Proof. 

Consider the following scalar function of y.

$$V_I\left(y\right)=\sum _{i=1}^m{q}_i\int \varphi (u_i-y)dy+ky.$$

(A3)

Differentiating $V_I\left(y\right)$ with respect to y, we can get $f_I\left(y\right)=V_I\left(y\right)/dy$, and hence Equation (A1) is a gradient descent system. Equivalently, $y(t+1)=y\left(t\right)-\mu d{V}_I\left(y\left(t\right)\right)/dy$. Therefore, ${lim}_{t\to \infty }y\left(t\right)$ exists and ${lim}_{t\to \infty }{f}_I\left(y\left(t\right)\right)=0$. As $y\left(0\right)=M$ and $f_I\left(y\right)$ is a non-increasing function, $y\left(t\right)$ decreases from M to the rightmost value of y in $\left\{y\right|f_I\left(y\right)=0\}$. ${lim}_{t\to \infty }y\left(t\right)$ is the least winning per-unit price. The uniform price is determined.

Let $y^{\ast }$ be this value and ${\pi }_i$ be the corresponding winner. As $\mathsf{\Delta }<1/2$, it is clear that $\left[y^{\ast }\right]=u_{{\pi }_i}$. In addition, we can get $\left[q_{{\pi }_j}\varphi (u_{{\pi }_j}-y^{\ast })\right]=q_{{\pi }_j}$ for $j>i$, $\left[q_{{\pi }_j}\varphi (u_{{\pi }_j}-y^{\ast })\right]=0$ for $j<i$ and $\left[q_{{\pi }_i}\varphi (u_{{\pi }_i}-y^{\ast })\right]=k-{\sum }_{j=i+1}^m{q}_{{\pi }_j}$. As a result, the algorithm as stated in Equations (A1) and (A2) is able to determine the units to be allocated to the bidders. The proof is completed. □

Appendix A.2. Winners’ Bidding Price Not Protected

While the algorithm as stated in Equations (A1) and (A2) is able to solve the problems in a uniform price auction, the algorithm is unable to protect the winning per-unit bidding price information. For the examples as presented in Table 1, Figure A1 shows the changes in $y\left(t\right)$ and $dy/dt$ obtained. From the change in $dy/dt$, i.e., Figure A1b,d, the bidders and the auctioneer are able to reveal the winning per-unit bidding prices.

Let $t_1,t_2$ and so on be the time instances when $dy/dt$ jumps abruptly. The winning per-unit bidding prices are $y\left(t_1\right),y\left(t_s\right)$ and so on. When does $dy/dt$ change abruptly, as shown in Figure A1b,d? The auctioneer is able to reveal the winning per-unit prices. The winners’ per-unit bidding prices are not protected.

Appendix A.3. Communication Delay

From Figure A1, one can see that the number of steps for the unit allocation and uniform price determination is in the order of ${10}^3$. Consider that the bidders and the auctioneer are agents. The algorithm as stated in Equations (A1) and (A2) is implemented in a distributed manner. Sending ${\widehat{q}}_i\left(t\right)=q_i{z}_i\left(t\right)$, for $i=1,\cdots ,m$, from the bidder agents to the auctioneer agent will cause communication delay. Similarly, sending $y\left(t\right)$ from the auctioneer agent to the bidder agents will cause another communication delay. Therefore, a large number of steps will definitely hamper the application of Equations (A1) and (A2) for a uniform price auction.

Figure A1. Simulation results for the illustrative examples as presented in Table 1. Here, $\mu =0.01$ and $M=200$. (a,c): Change in $y\left(t\right)$ against t. (b,d): Change in $dy/dt$ against t. By observing when $dy/dt$ changes abruptly, as shown in (b,d), the auctioneer and the bidders are able to reveal the winners’ bidding prices. From (a,c), one can see that the number of steps for the RNN-based algorithm to determine the unit allocation and the uniform price is in the order of $\mathcal{O}\left({10}^3\right)$.

Figure A1. Simulation results for the illustrative examples as presented in Table 1. Here, $\mu =0.01$ and $M=200$. (a,c): Change in $y\left(t\right)$ against t. (b,d): Change in $dy/dt$ against t. By observing when $dy/dt$ changes abruptly, as shown in (b,d), the auctioneer and the bidders are able to reveal the winners’ bidding prices. From (a,c), one can see that the number of steps for the RNN-based algorithm to determine the unit allocation and the uniform price is in the order of $\mathcal{O}\left({10}^3\right)$.

Appendix B.1. Case (a)

For the case that $q_{{\pi }_i}^{\ast }=q_{{\pi }_i}$, it is clear that $y^{\ast }=u_{{\pi }_i}-\mathsf{\Delta }$. From that, we can get $f_I\left(y\right)\ge 0$ for all $y\le y^{\ast }$ and $f_I\left(y\right)<0$ for all $y>y^{\ast }$.

For $t=0$, it is clear that $y^{\ast }\in [\underline{y}\left(0\right),\overline{y}\left(0\right)]$. It is assumed that the argument is true for $t=t^{\prime }$, i.e., $y^{\ast }\in [\underline{y}\left(t^{\prime }\right),\overline{y}\left(t^{\prime }\right)]$. For $t=t^{\prime }$, we can get $y\left(t^{\prime }\right)=(\underline{y}\left(t^{\prime }\right)+\overline{y}\left(t^{\prime }\right))/2$.

If $f_I\left(y\left(t^{\prime }\right)\right)\ge 0$, we can obtain by Equation (4) that $\underline{y}(t^{\prime }+1)=y\left(t\right)$, $\overline{y}(t^{\prime }+1)=\overline{y}\left(t^{\prime }\right)$. As $f_I\left(y\left(t^{\prime }\right)\right)\ge 0$, $y\left(t^{\prime }\right)\le y^{\ast }$. As a result, $y^{\ast }\in [\underline{y}(t^{\prime }+1),\overline{y}(t^{\prime }+1)]$.

If $f_I\left(y\left(t^{\prime }\right)\right)<0$, we can obtain by Equation (4) that $\underline{y}(t^{\prime }+1)=\underline{y}\left(t\right)$, $\overline{y}(t^{\prime }+1)=y\left(t\right)$. As $f_I\left(y\left(t^{\prime }\right)\right)<0$, $y\left(t^{\prime }\right)>y^{\ast }$. As a result, $y^{\ast }\in [\underline{y}(t^{\prime }+1),\overline{y}(t^{\prime }+1)]$.

Therefore, the argument is true for $t=t^{\prime }+1$.

Appendix B.2. Case (b)

For the case that $1\le q_{{\pi }_i}^{\ast }<q_{{\pi }_i}$ and hence $u_{{\pi }_i}-\mathsf{\Delta }<y^{\ast }<u_{{\pi }_i}$, we can get $f_I\left(y\right)>0$ for all $y<y^{\ast }$ and $f_I\left(y\right)<0$ for all $y>y^{\ast }$.

If $f_I\left(y\left(t^{\prime }\right)\right)>0$, we can obtain by Equation (4) that $\underline{y}(t^{\prime }+1)=y\left(t\right)$, $\overline{y}(t^{\prime }+1)=\overline{y}\left(t^{\prime }\right)$. As $f_I\left(y\left(t^{\prime }\right)\right)>0$, $y\left(t^{\prime }\right)<y^{\ast }$. As a result, $y^{\ast }\in [\underline{y}(t^{\prime }+1),\overline{y}(t^{\prime }+1)]$.

If $f_I\left(y\left(t^{\prime }\right)\right)<0$, we can obtain by Equation (4) that $\underline{y}(t^{\prime }+1)=\underline{y}\left(t\right)$, $\overline{y}(t^{\prime }+1)=y\left(t\right)$. As $f_I\left(y\left(t^{\prime }\right)\right)<0$, $y\left(t^{\prime }\right)>y^{\ast }$. As a result, $y^{\ast }\in [\underline{y}(t^{\prime }+1),\overline{y}(t^{\prime }+1)]$.

Therefore, the argument is true for $t=t^{\prime }+1$.

Based on the analysis in Case (a) and Case (b); and by the principle of mathematical induction, we can conclude that $y^{\ast }\in [\underline{y}\left(t\right),\overline{y}\left(t\right)]$ for $t\ge 0$.

Appendix B.3. Convergence

Note that $\overline{y}\left(t\right)-\underline{y}\left(t\right)=2^{-t}M$, ${lim}_{t\to \infty }(\overline{y}\left(t\right)-\underline{y}\left(t\right))=0$. Together with the fact that $y\left(t\right)\in [\underline{y}\left(t\right),\overline{y}\left(t\right)]$ for all $t\ge 0$, we conclude that ${lim}_{t\to \infty }y\left(t\right)=y^{\ast }$. As $u_{{\pi }_i}-\mathsf{\Delta }\le y^{\ast }\le u_{{\pi }_i}$ and $\mathsf{\Delta }<1/2$, $\left[y^{\ast }\right]=u_{{\pi }_i}$. The proof is completed.