Comparing the effectiveness of collusion devices in first-price procurement: an auction experiment

Abstract

Collusion in procurement auctions is illegal, but often observed. We compare experimentally three coordination mechanisms in how effectively they promote collusion in first-price procurement auctions. One mechanism aims at excluding competitive bids via bidding restrictions. The second one allows for promises on sharing the gains from collusion as in mutual shareholding. The third mechanism relies on unrestricted pre-play communication. Agreements made under the three mechanisms are non-binding. In the experiment, bidders interact with the same group of competitors only once as it is quite common in globalized (online) markets. We find that first-price procurement is quite collusion-proof regarding the first two mechanisms whereas pre-play communication, on average, increases profits. The communication protocols provide valuable insights about how to coordinate and implement non-binding collusion agreements in competitive one-shot interactions with private information.

Appendices

Appendix 1: Linear benchmark solution in case of mutual and symmetric shareholding

For i, j = 1, 2 with i ≠ j the payoff expectation for the risk neutral bidder i with cost value c _i  ∈ [0, 1] is

$$E_{i} \left( {b_{i} \left| {c_{i} } \right.} \right) = \left( {1 - s} \right)\int\limits_{{b_{i} < f\left( {c_{j} } \right)}} {\left[ {b_{i} - c_{i} } \right]{\text{d}}c_{j} + s} \int\limits_{{b_{i} \ge f\left( {c_{j} } \right)}} {\left[ {f\left( {c_{j} } \right) - c_{j} } \right]} {\text{d}}c_{j} ,$$

where f(.) is the linear symmetric and monotonic equilibrium bid function.

Let us rewrite E _i (b _i |c _i ) as

$$\begin{gathered} E_{i} \left( {b_{i} \left| {c_{i} } \right.} \right) = \left( {1 - s} \right)\left( {b_{i} - c_{i} } \right)\int\limits_{{f^{{ - 1}} \left( {b_{i} } \right) < c_{j} \le 1}} {{\text{d}}c_{j} + s} \int\limits_{{f^{{ - 1}} \left( {b_{i} } \right) \ge c_{j} \ge 0}} {\left[ {f\left( {c_{j} } \right) - c_{j} } \right]} {\text{d}}c_{j} \hfill \\ \quad = \left( {1 - s} \right)\left( {b_{i} - c_{i} } \right)\left[ {1 - f^{{ - 1}} \left( {b_{i} } \right)} \right] + s\left[ {F\left( {f^{{ - 1}} \left( {b_{i} } \right)} \right)} \right]\int\limits_{{f^{{ - 1}} \left( {b_{i} } \right) < c_{j} \le 1}} {{\text{d}}c_{j} } \hfill \\ \quad \quad + s\int\limits_{{f^{{ - 1}} \left( {b_{i} } \right) \ge c_{j} \ge 0}} {\left[ {f\left( {c_{j} } \right) - c_{j} } \right]} {\text{d}}c_{j} , \hfill \\ \end{gathered}$$

where F’(.) = f(.). For an interior best reply b _i to f(.) the first order condition is

$$\begin{aligned} s\left[ {b_{i} \frac{d}{{{\text{d}}b_{i} }}f^{{ - 1}} \left( {b_{i} } \right) - f^{{ - 1}} \left( {b_{i} } \right)\frac{d}{{{\text{d}}b_{i} }}f^{{ - 1}} \left( {b_{i} } \right)} \right] & = \left( {1 - s} \right)\left( {b_{i} - c_{i} } \right)\frac{d}{{{\text{d}}b_{i} }}f^{{ - 1}} \left( {b_{i} } \right) \\ & \quad - (1 - s)\left[ {1 - f^{{ - 1}} \left( {b_{i} } \right)} \right] \\ \end{aligned}$$

which can be simplified as follows:

$$\begin{aligned} &\frac{d}{{{\text{d}}b_{i} }}f^{ - 1} \left( {b_{i} } \right)\left[ {b_{i} - f^{ - 1} \left( {b_{i} } \right) - \frac{1 - s}{s}\left( {b_{i} - c_{i} } \right)} \right] = - \frac{1 - s}{s}\left[ {1 - f^{ - 1} \left( {b_{i} } \right)} \right] \hfill \\ &\quad \Leftrightarrow \frac{d}{{{\text{d}}b_{i} }}f^{ - 1} \left( {b_{i} } \right) = - \frac{1 - s}{s}\frac{{1 - f^{ - 1} \left( {b_{i} } \right)}}{{b_{i} - f^{ - 1} \left( {b_{i} } \right) - \frac{1 - s}{s}\left( {b_{i} - c_{i} } \right)}} \hfill \\ &\quad \Leftrightarrow \frac{{{\text{d}}b_{i} }}{{{\text{d}}f^{ - 1} \left( {b_{i} } \right)}} = - \frac{s}{1 - s}\frac{{b_{i} - f^{ - 1} \left( {b_{i} } \right) - \frac{1 - s}{s}\left( {b_{i} - c_{i} } \right)}}{{1 - f^{ - 1} \left( {b_{i} } \right)}}. \hfill \\ \end{aligned}$$

Now substituting b _i  = f(c _i ) and c _i  = f ⁻¹(b _i ) into the latter equation yields the ordinary differential equation

$$f'\left( {c_{i} } \right) = \frac{1 - 2s}{1 - s}\frac{{f\left( {c_{i} } \right) - c_{i} }}{{1 - c_{i} }}.$$

For the linear and monotonic solution f(c _i ) = α + βc _i with β > 0 we thus obtain

$$\left( {1 - s} \right)\beta - \left( {1 - s} \right)\beta c_{i} = \left( {1 - 2 s} \right)\alpha + \left( {1 - 2 s} \right)\left( {\beta - 1} \right)c_{i} .$$

Since the left and the right hand-side above have to coincide for all c _i  ∈ [0,1] this requires

$$\left( { 1 - s} \right)\beta = \left( { 1 - 2 s} \right)\alpha \quad {\text{or}}\quad \beta = \frac{1 - 2s}{1 - s}\alpha$$

and

$$- \left( {1 - s} \right)\beta = \left( {1 - 2s} \right)\left( {\beta - 1} \right)$$

or, after substituting for β,

$$- \left( {1 - 2s} \right)\alpha = \left( {1 - 2s} \right)\left[ {\frac{{\left( {1 - 2s} \right)\alpha }}{1 - s} - 1} \right]$$

and, thus,

$$\alpha = \frac{1 - s}{2 - 3s}\quad {\text{and}}\quad \beta = \frac{1 - 2s}{2 - 3s}.$$

Appendix 2: Instructions (translated from German)

2.1 Welcome to this experiment!

2.1.1 Preliminary remarks

In the following, you will take part in an experimental study in the field of economics in which the decision behavior of individuals is investigated. During the experiment, you will participate in a series of auction games in which you can earn money. How much you eventually earn depends on your own and others’ decisions (possible losses will be deducted from the show-up bonus of 2.50 Euro which you receive for participating in this experiment). At the end of the experiment, your accrued earnings will be converted into Euro at the rate of 1 ECU: 0.07 EURO and disbursed to you in cash.

Please read the subsequent instructions carefully. About 5 min after you have received these instructions, we will come to your place to answer any remaining questions. Afterwards, you will receive a questionnaire which is used to ensure that you have fully understood the rules of this experiment. We will not start with the experiment until all participants have correctly answered all the listed questions.

In case that you have further questions in the course of the experiment, please indicate this by raising your hand. We will then come to your place and answer your questions.

2.1.2 Description of the auction

In every period of the experiment, a generic “project” is auctioned off. The project is awarded to the bidder who states the lowest bid.

Bidders In each auction there are exactly two bidders, i.e., you and another bidder. In each period, the other bidder with whom you will interact is randomly assigned to you from a group of participants. It is ensured that you will not interact with the same participant in two consecutive periods.

Costs For every auction period and for every bidder, a cost value is independently and randomly assigned from the interval from 50 LD to 150 LD whereby each value in this range is equally likely. Before the start of an auction, you will be informed about your own cost value. Apart from this, you will not receive any further information.

Decision In each auction period you have to decide on the bid that you want to submit for the project.

If your bid for the project is less than the bid of the other bidder, you are awarded the project and your auction profit is the difference between your bid and your cost. It is possible to realize a loss if your bid is less than your cost.

If your bid for the project is greater than the bid of the other bidder, you do not win the auction. In this case your profit equals zero, since you were not awarded the project and therefore did not incur any cost.

If your bid is equal to the bid of the other bidder, you are awarded the project with a probability of 50 %.

2.1.3 Proposal stage

In every auction and before determining his/her bid, each of the two bidders has the possibility to make the other bidder a suggestion concerning the distribution of the bids that are to be submitted. For this purpose, both bidders independently select an integer value from the range of 0 to 100. After each bidder has decided on a particular value, both bidders are informed about the larger of the two stated values. In the following, this value shall be denoted as N.

Given N, each bidder is free to set his/her own bid according to the following rule:

$${\text{Own bid}} = 1 50{-}N + \left( {N/ 100} \right) \, * \, \left( {{\text{own cost}}{-} 50} \right)$$

This means that if your cost amounts to 50, your bid would be 150 − N. If you were assigned the maximal cost of 150, you would always bid 150, irrespective of the value of N. This shows that it is possible to constrict the bidding interval by agreeing on a small value of N. The smaller is N, the larger is the least “accepted” bid and the larger is the potential profit of the bidders.

Please notice that every bidder is free to decide whether (s)he sets his/her bid according to the above-mentioned formula or not.

2.1.4 Proposal stage

In every auction and before determining his/her bid, each of the two bidders has the possibility to make the other bidder a suggestion concerning the distribution of the yet unrealized auction profit between the two. For this purpose, both bidders independently select an integer value from the range of 0 to 100. After each bidder has decided on a particular value, both bidders are informed about the smaller of the two stated values. In the following, this value shall be denoted as N.

Given N, the winning bidder is free to divide the realized auction profit according to the following rule:

$${\text{The\, winner\, of\, the\, auction\, obtains}}:\;(200 - N/200{\text{ }}*{\text{ }}({\text{winner's\, bid}} - {\text{winner's\, cost}}).$$

$${\text{The losing bidder obtains:}}\;N/200{\text{ }}*{\text{ }}({\text{winner's bid}} - {\text{winner's cost}})$$

This means that the larger is N, the smaller is the difference between the payoff of the winning and the losing bidder.

Please bear in mind that every bidder is free to decide whether to split the realized auction profit according to the above-mentioned rule or not, after (s)he is informed that (s)he has won the auction.

2.1.5 Communication stage

Before an auction is conducted the two bidders have the possibility to communicate with each other via electronic (chat) messages before they then independently decide on their bid.

Generally, the content of your communication is totally up to you. You are, however, not allowed to:

• provide personal information about yourself such as your age, address, gender [please always use gender-neutral terms, e.g., “bidder A”, “bidder B”], field of studies [this also includes mentioning the names of professors, lectures or similar contents which allow to identify the other’s field of studies] and the like, or to

• negotiate any form of side payments.

In case that you do not respect these rules we will unfortunately have to exclude you from the experiment which means that you will not receive any payment at all in this experiment. The duration of the communication stage is limited to 5 min. You may, however, finish your conversation earlier as well.

2.1.6 Practice periods

Before the actual experiment starts you will have the possibility to familiarize yourself with the decision problem and the use of the software in the course of two practice periods. Note that in both periods, the other bidders’ decisions are simulated by the computer and are identical for all participants. All decisions that are made during the two practice periods are for training purposes only and will not affect your eventual payoff in the experiment.

2.1.7 Payment

After you have finished the two practice periods, you will participate in a series of auctions of which five auctions will be randomly selected to determine your payoff in this experiment. Once all auctions have been finished, your earnings in the respective five periods will be summed up, converted according to the exchange rate of 1 ECU: 0.07 EURO, and disbursed to you in cash.

2.1.8 Please note

All participants in this experiment have received the identical set of instructions. None of the participants will receive any information concerning the identity of any other participant.

Appendix 3

See Tables 6, 7, and 8.

Table 6 Submitted bids—summary statistics

Table 7 Period profits—summary statistics

Table 8 Linear mixed-effects model of period profits