An experimental study of VCG mechanism for multi-unit auctions: competing with machine bidders

Abstract

This paper complements the main experimental result reported in Takahashi et al. (Evolut Inst Econ Rev 16:357–374, 2019) to a deeper understanding of subjects’ bidding behavior under the VCG mechanism. In the experiment, there are two types of appearance of information about bidders’ valuations of the item given to them and the bids they are asked to submit: one is unit valuations and the unit bids themselves (Appearance 1) and the other is unit valuations and the unit bids multiplied by the number of units (Appearance 2). For subjects who compete with truth-telling machine bidders in multi-unit auctions, we confirmed that in Appearance 1, they choose truth-telling bids more frequently, and efficient allocations are observed more frequently, as compared to the situation where they compete with human bidders. This result suggests a possibility that in Appearance 1, subjects learn their dominant strategy not by practicing with other subjects but by practicing with machine bidders in experiments for multi-unit auctions, although the item allocation and payment determination under the VCG mechanism is never intuitively understandable to the subjects.

Appendices

Appendix 1: Bid plots and regression results

Figures Figures 1, 2, 3, 4, 5, 6, 7, 8 amd Tables 10, 11.

Fig. 1

Appearance 2, session 1, at random

Fig. 2

Appearance 2, session 1, descending

Fig. 3

Appearance 2, session 2, at random

Fig. 4

Appearance 2, session 2, descending

Fig. 5

Appearance 1, session 3, at random

Fig. 6

Appearance 1, session 3, descending

Fig. 7

Appearance 1, session 4, at random

Fig. 8

Appearance 1, session 4, descending

Table 8 Regression results for Appearance 2: session 1

Table 9 Regression results for Appearance 2: session 2

Table 10 Regression results for Appearance 1: session 3

Table 11 Regression results for Appearance 1: session 4

Appendix 2: Instruction

³ 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. You will be matched with 2 machine bidders. During the experiment, this matching will be fixed.

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; one is you and the other two are machine bidders. The bidding strategy of each machine bidder is programmed in advance. 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 the “cancel” button there, you can then go back to the screen for filling 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 in 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 an auction in which 3 units of an item are auctioned off to 2 bidders. In Table 12, valuations (or bids) are displayed as unit valuations (or unit bids) multiplied by the number of units.⁴ Note that the unit valuations and unit bids shown in this example do not suggest any bidding strategy in the auctions you will participate in.

Table 12 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): $6 \times 2=120$, (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 allocations that maximize the total amount of bids, one of those allocations is chosen at random.

1.6 Payment determination in the auction

The payments of bidders are determined as follows.

Payment of bidder i $=$ (total amount of bids in the auction that excludes bidder i):

• (total amount of bids in the auction) $+$ (bidder i’s bid for the unit assigned to i).

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 earns is calculated as follows.

Bidder i’s points $=$ (total valuation for the units bidder i is allocated)

• (payment of bidder i).

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 12). 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 13.

Table 13 Another display type

1.9 Practice

At the beginning of each sequence of 10 auctions, an additional 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 will not be among those counted for the compensation.

Please raise your hand if you have questions.