Scaling and Equity Auctions: The Types-as-Marginal-Costs Case

The experiment will implement a mixed between- and within-subject design: participants will participate in 15 rounds of cash auctions and 15 rounds of either equity-bid or scaling auctions.

2026-03-02

2026-05-31

The focus of our study will be the outcomes of the auctions, both in terms of the expected amount the auctioneer would need to pay the winning bidder (the equivalent of the auctioneer's earnings in a standard auction) and the expected costs that would need to be spent on the project (equivalent to efficiency).

This study compares cash, scaling (unit-price), and equity-bid auctions under uncertainty about marginal costs. In each round, participants bid for the right to complete a project that requires two costly inputs and a fixed setup cost. In cash rounds, bidders specify their required payment (in points) and the lowest bidder wins. In the scaling treatment, bidders specify per-unit prices for each input, and the bidder with the lowest score (sum of the per-unit prices) wins. In the equity treatment, bidders request a percentage reimbursement of their variable costs, and the lowest requested share wins. After winning, payoffs depend on realized input quantities, which vary randomly to capture cost uncertainty. All subjects participate in cash auctions in odd rounds, and either scaling or equity-bid auction in even rounds. Moreover, subjects are randomized into environments where input quantities are either positively or negatively correlated. The design allows us to compare how the different mechanisms allocate risk and select efficient providers when project requirements are uncertain.

Our experiment is designed to compare how cash, scaling, and equity-bid auctions perform in a procurement setting with input uncertainty. We will perform comparisons in two environments: one in which the requirements of the two inputs are positively correlated, and another in which they are negatively correlated. Participants face a reverse auction for the rights to complete a project. To complete the project, the participant must pay a non-contractible fixed cost and also provide two types of costly inputs that are potentially contractible.

In cash auctions, participants make bids in terms of the number of points they must be paid to complete the project. The winner, who is the lowest bidder, is paid their bid and must pay the costs associated with the project. In scaling auctions, participants must state how much they wish to be paid for each unit of each type of input. The auctioneer then multiplies these prices by fixed weights (which are both equal to one in this experiment) to compute a "score," and the winner is the bidder with the lowest score. The winner then pays the costs associated with the project and receives payments for each type of input equal to their bid. In equity-bid auctions, participants must make bids in terms of what percentage of their variable costs they request reimbursement of. The lowest bidder wins the auction, pays the costs associated with the project, and receives both an up-front payment of 4400 points and the percentage reimbursement they requested. The experiment will implement a mixed between- and within-subject design: participants will participate in 15 rounds of cash auctions and 15 rounds of either equity-bid or scaling auctions.

In the experiment, the auctions are explicitly framed such that the government is the auctioneer that needs a project to be completed and participants are bidding to determine who will complete the project and how much they will be paid. Participants are informed that payments will be made in terms of points and that in the round that counts for payment, 200 points will be worth $1. Bidders are informed that in order to complete the project, the winner must pay a fixed cost of 2400 points and provide two types of inputs, referred to as Input 1 and Input 2. The quantity of each input is random in each round and unknown when the subject is making their bid. In the "Positive Correlation" treatments, the required inputs are (2,2) or (6,6) with equal probability, while in the "Negative Correlation" treatments, the inputs are (1,5) or (5,1) with equal probability. Marginal costs are randomly and independently drawn in each round, unique to each bidder, and private. In the positive correlation environment, the marginal cost of each unit of input 1 is uniformly drawn from {320,324,...,396,400} and the marginal cost of each unit of input 2 is one half the cost of input 1. In the negative correlation environment, the marginal cost of each unit of input 1 is uniformly drawn from {560,567,...,693,700} and the marginal cost of each unit of input 2 is one seventh the cost of input 1.

In all treatments, subjects participate in 30 rounds of auctions, with groups rematched each round. Odd rounds are cash auctions, while even rounds are scaling or equity auctions, depending on the treatment. Bids are made using sliders. Bids in the cash treatment are multiples of 20 between 0 and 10,000. The bid represents the payment, in points, that the government will pay if the bidder wins the auction. Bids in the equity treatment are integers between 0 and 50, and represent the percent of variable costs that will be paid by the government if the bidder wins the auction. Bids in the scaling treatment are multiples of 10 between 0 and 1600, and represent the payments per unit of input used if the bidder wins the auction. Subjects are informed that these bids will be added together to form a score, with the bidder with the lowest score winning the auction. This computation is completed for them below the sliders.

The experimental software will randomize automatically. Groups and types are re-randomized each round.

Groups and types are re-randomized each round.

No

Our final sample is expected to be 135-155 subjects.

With 144 subjects participating in 30 rounds, the total number of observations is expected to be 4320 bids and 2160 auctions.

We expect to have 36 subjects in each of the four between-subject treatments.