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VCG mechanism for multi-unit auctions and appearance of information: a subject experiment


This paper investigates whether, in multi-unit auctions, different types of appearance of information associated with bidding generate different levels of allocative efficiency and sellers’ revenue when the VCG mechanism is applied to human subject experiments of those auctions. We examine two types of appearance of information about bidders’ valuations of the item given to them and the bids they are asked to submit: One type is unit valuations and the unit bids themselves and the other type is unit valuations and the unit bids multiplied by the number of units. We observed that there was no significant difference on average in either allocative efficiency or the seller’s revenue between these two types of appearance of information. Rather, for each appearance of information, there was a significant difference in subjects’ bidding behavior between different display types of draws of unit valuations. This behavioral difference, however, did not significantly affect allocative efficiency. The performance of the VCG mechanism is robust against display types of those draws as well as against types of appearance of information.

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  1. Takahashi and Shigeno (2011) and Takahashi et al. (2018) also showed in their numerical experiments that as the number of units of the item increases, the computation time in VCG rapidly increases, whereas the increase in computation time is suppressed in GBA.

  2. \( ({\text{AP}})_{{\text{B}}} \) is known to be \({\mathcal{NP}}\)-hard.

  3. A more detailed summary is available upon request.

  4. Engelmann and Grimm (2009) conducted sessions in which an auction rule was used in the first 10 rounds and another auction rule was used in the second 10 rounds, and analyze the data taken from the first 10 rounds to compare the auction results with the Mann–Whitney U test. In our analysis, we assumed that possible learning effect on subjects’ bidding behavior was excluded in the first 5 rounds out of 10 rounds in each display type of draws, according to a convention.

  5. Engelmann and Grimm (2009) compared the performances of a uniform-price sealed-bid auction, a uniform-price clock auction, a discriminatory auction, a static Vickrey auction, and a dynamic Vickrey auction in the same experimental environment, including a brief but nice literature review on laboratory experiments of multi-unit auctions. Many experimental studies in multi-unit auctions were integrated in the paper. Among those studies, Kagel and Levin (2001) was a seminal paper to study the demand reduction in uniform-price auctions.


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The authors wish to thank Toru Suzuki and Hiroshi Tanaka for their excellent research assistance.


This research was supported by JSPS Grant-in-Aid for Young Scientists (B) 26870200 and Grant-in-Aid for Scientific Research (B) 15H02972 (Takahashi), and the Japan Center for Economic Research, ORA-Plus research project “BEAM”, and JSPS Grant-in-Aid for Challenging Research (Pioneering) 17H06190 (Watanabe).

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Correspondence to Naoki Watanabe.

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Appendix: Instruction

This is the instruction for the participants in sessions for Appearance 1 when draws of unit valuations are displayed as they are in the first 10 rounds.

Appendix: Instruction

Welcome to this experiment!

Thank you very much for taking the time to participate in our auction experiment. The experiment lasts for about 100 min, including the payment session.

1.1 At first

  • Please follow the instructions given by the experiment administrators.

  • Please remain silent, and do not talk to or exchange notes with other participants.

  • Please do not look at what other participants are doing.

  • Please do not change your position. Please do not lean in your chair.

  • Please do NOT do anything other than what you are instructed to do.

  • Please turn off and refrain from using your cell phones.

  • Please quietly raise your hand if you have questions or need help.

In this experiment, a total of 20 auctions will be held and 5 units of a virtual item are auctioned off to 3 bidders in each auction.

1.2 Compensation

After all the 20 auctions end, a computer will randomly choose 3 auctions each from the 10 auctions in the first and second halves, that is, a total of 6 auctions. You will be compensated on the basis of the total points you earned in the 6 auctions. The final compensation will be the amount based on those points in addition to a compensation of 1500 JPY for participation.

1.3 Group selection

At the beginning, you will be assigned an ID. Your ID will remain the same throughout the session you participate in, and it will be displayed on your computer screen. A computer will randomly make groups of 3 participants with different IDs before each auction. The participants in your group will be different in each auction and you will not be able to know who are in your group.

Please raise your hand if you have questions on the above contents.

1.4 Auction

In each auction, 5 units of an identical item are auctioned off to 3 bidders. Please bid for all units within 120 s. If no one in the same group bids within this time limit, all bidders in the group obtain zero points. The outcome will not be shown until the remaining time is up, even if everyone bids within the time limit.

At the beginning of each auction, each bidder is given unit valuations of the item for each unit. When the unit valuation is, e.g., 15 for 3 units, the total valuation is \(15\times 3=45\). You are asked to submit your unit bids for each unit. Please press the “bid” button after you fill in your unit bids on your screen. Then, a pop-up window appears and shows your total bids for each unit. If you click on the “OK” button in the pop-up window, your bids will then be sent to the server computer to compute the outcome of the auction. If you click on “cancel” button there, you can then go back to the screen to fill in your unit bids.

For each bidder, unit valuations are drawn as integers independently of those for the other bidders with equal probability between 1 and 200. Please bid in non-negative integers. The remaining time is displayed on the right upper corner of your screen. When the auction ends, the outcome is shown on your screen. The next auction will start after 5 s. The rule of the auction is explained next.

Please raise your hand if you have questions on the above content.

1.5 Item allocation in the auction

Below is an example of the auction in which 3 units of an item auctioned off to 2 bidders; In Table 8, valuations (or bids) are displayed as unit valuations (or unit bids) multiplied by the number of units. Note that the unit valuations and unit bids shown in this example do not suggest any bidding strategy in the auctions you participate in.

Table 8 Example

The item will be allocated to bidders such that the total amount of bids is maximized as follows. Find an allocation that maximizes the total amount of bids among all possible allocations; In the example, (0, 0): 0, (1, 1): \(70{\times }1+40 {\times }1=110, (1, 0): 70{\times }1=70\), \((2, 0): 55{\times }2=110, (3, 0): 50{\times }3=150\), \((0, 1): 40{\times }1=40, (0, 2): 60{\times }2=120\), \((0, 3): 65{\times }3=195, (1, 2): 70{\times }1+60{\times }2=190\), \((2,1): 55{\times }2+40{\times }1 =150\). Thus, this auction allocates 3 units to bidder 2. The total amount of bids is 195. When there are two or more allocation in each of which the total amount of bids is maximized, one of those allocations is chosen at random.

1.6 Payment determination in the auction

The payments of bidders are determined as follows.

$$\begin{aligned}&\hbox {payment of bidder }i & = \hbox {(total amount of bids in the auction that excludes bidder }i\hbox {)} \\&\quad - \hbox {(total amount of bids in the auction)} \\&\quad + \hbox {(bidder } i'\hbox {s bid for the unit assigned to } i\hbox {)}. \\ \end{aligned}$$

In the example,

  • payment of bidder 1 \(=(65{\times }3)-195+0=0\),

  • payment of bidder 2 \(=(50{\times }3)-195+(65{\times }3)=150\).

1.7 Point

The amount of points each bidder earn is calculated as follows.

$$\begin{aligned} \hbox {bidder }i'\hbox {s points }= & {} \hbox { (total valuation for the units bidder }i\hbox { is allocated)} \\&- \hbox { (payment of bidder }i\hbox {)}. \end{aligned}$$

In the example,

  • bidder 1’s points \(=0-0=0\),

  • bidder 2’s points \(=(65\times 3)-150=45\).

You will be compensated on the basis of the points you earned. The exchange rate is 1 point \(=\) 1 JPY. As mentioned, you will be compensated on the basis of the total points you earned in the 6 auctions, 3 out of the first 10 auctions and 3 out of the second 10 auctions.

1.8 Configuration

In the first 10 auctions, unit valuations are given on your computer screen and you are asked to submit unit bids there, as shown in the Example (Table 8). In the second 10 auctions, total valuations are given on your screen and you are asked to submit total bids there, as shown in Table 9.

Table 9 Another display

1.9 Practice

At the beginning of each sequence of 10 auctions, an auction is held as a practice so that you can familiarize yourself with how to do with the computer. The points you earn in the practice auctions are not counted as those for the compensation.

Please raise your hand if you have questions.

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Takahashi, S., Izunaga, Y. & Watanabe, N. VCG mechanism for multi-unit auctions and appearance of information: a subject experiment. Evolut Inst Econ Rev 16, 357–374 (2019).

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