**Introduction**

The bid functions often cannot be found analytically in a first price auction. Here we offer a program that numerically calculates the inverse bid functions for asymmetric independent private values first price auctions and procurements. The code was created by W.-R. Gayle and J.-F. Richard based on the algorithm suggested in their paper “Numerical Analysis of Asymmetric, First Price, Independent Private Values Auctions.” The program is written in FORTRAN-90 and uses the standard IMSL library.

**The setup**

Several bidders participate in a standard single object sealed-bid first-price auction. The bidders have independent private valuations of the object. The valuation distributions have the same support; however, they can differ across agents. Prior to the auction and the realization of the valuations, the bidders can form coalitions based on their knowledge of the valuation distributions. Each coalition submits only one bid. A coalition is treated as a single player whose private value is the highest one of the coalition members’ values.

**The Program**

In the menu of the program one can choose the following parameters:

- Number of the bidders;
- Distribution functions of the valuations for each of the bidders;
- Composition of the coalitions of bidders;
- Common support of the bidders’ valuation distribution functions;
- Specified reserve price or no reserve price.

The program produces an inverse bid function for each bidder/coalition and the following optional statistics and functions:

- Auctioneer’s expected revenue;
- Expected surplus of each bidder/coalition;
- Probability of winning for each bidder/coalition;
- Probability of retention;
- Optimal reserve price.

The program also compares bidders’ expected revenues under alternative collusive agreements.

**Related Information**

Numerical Solutions of Asymmetric, First-Price, Independent Private Values Auctions

Wayne-Roy Gayle

Jean-Francois Richard

**Distributions**

The program is currently limited to the following valuation distributions: two-parameter normal, two-parameter lognormal, two-parameter Weibull (including exponential distribution), and two-parameter Beta (including uniform and power distributions). Arbitrary user-defined distributions can be added in the form of FORTRAN subroutines.

**The Program**

The executable version of the program only allows the use of the standard built-in distributions of bidders’ valuations. It is self-sufficient and fully functional in all other aspects and calculates the aforementioned functions and statistics. One needs to use FORTRAN-90 and compile the program below, which requires access to the IMSL library.

**User Guide**

W.-R. Gayle and J.-F. Richard provide several illustrations of the capabilities of the program in their paper “Numerical Analysis of Asymmetric, First Price, Independent Private Values Auctions.” Here we give step-by-step instructions for two examples from the paper. In the first example several different bidders act individually. In the second example two high type bidders and one low type bidder collude in the auction against three non-collusive bidders of low type. Each command prompt is shown in italic, followed by the user input and then a brief explanation, which discusses the choice of the input and necessary changes for different possible scenarios.