Documentation

Return all the candidate auction rates for the given bids.

Given all the (i, p_i) pairs of good number and price, compute the resulting hyperplanes (both hods and flanges) and find all the intersections of all choices of n of them.

Return all the hyperplanes generated by the given list of (i, p_i) pairs, both hods and flanges.

The resulting list will then be used to pick dim hyperplanes at a time and consider their intersection by solving the corresponding linear system.

Generate all the linear systems resulting from the intersection of the given number of hyperplanes taken from the given list.

map '(<>)' over the given systems to get solutions

For standard auctions, all intersections should be non-negative integers, so round the points appropriately and remove any that are negative.

For budget-constrained auctions, remove points that are too close to each other, and round extremely small values to zero.

Replace any extremely small value with a genuine zero.

Rounding error may result in multiple points that are very close together, so we remove all but one point in each case.

select n xs returns all the possible combinations of n elements of xs, modulo permutations (it doesn't return [1,2] and [2, 1] but rather just one of them, as in our case we don't care in which order we put the hyperplane in the matrix, the system is equivalent whatever the "order of the equations" is).

H as b corresponds to the hyperplane with equation a_0*x_0 + ... + a_(n-1)*x_(n-1) = b.

Return (A, b) where Ax = b is the linear system corresponding to the intersection of all the given hyperplanes.

(i, p_i) pair

Hod i p corresponds to the hyperplane with equation x_i = p.

Flange i p_i j p_j corresponds to the hyperplane with equation x_ip_i = x_jp_j (for budget-constrained auctions), or x_i-p_i = x_j-p_j (for standard auctions).

Build a Hyperplane of the given type in a space of the given dimension.

A very small floating point. < epsilon will be used instead of == 0.