The demand curve type determines how to calculate supply and demand. See calcDemand for more details. Where applicable, the Int field gives the number of points to use when calculating demand by repeated runs of the LP.

Data for plotting a supply and demand curve graph.

Determine which graphs to generate for this supply by default. For a horizontal supply we generate UnadjustedDemand (and UnadjustedDemandWithTQSS if there is a constraint-based TQSS), which do not require the number of points to be specified. For a vertical supply we generate AdjustedDemand with a small number of points. Do not generate demand curves for more complex supply orderings.

Calculate a list of supply and demand curves for each good.

The number of units (horizontal axis) gives the amount allocated to the good and all successive goods. The price (vertical axis) gives the demand price of the good relative to the demand price of the previous good.

For UnadjustedDemand and UnadjustedDemandWithTQSS, the demand of all the bids on this good is aggregated, ignoring bidding on other goods (so paired bids are counted for multiple goods). This only really makes sense for horizontal auctions. The latter adds the TQSS vertically to the supply curve.

For other demand curve types, this uses calcDemandCurveLP to run the auction repeatedly over a range of quantities. The supply of the current good is fixed to be the desired quantity. The supply of other goods is adjusted depending upon the DemandCurveType:

• For VariableTotalAllocation, the base goods keep their original supply curve, and for other goods, we fix the supply based on the allocations resulting from the original run of the auction. Fixed supply curves have a single step of height zero.
• For FixedTotalAllocation, the supply curves of all goods are fixed based on the allocations. Fixed supply curves have a single step of height zero.
• For AdjustedDemand, all goods other than the current good keep their original supply curves. The fixed supply curve for the current good has a single step with a large negative height.

Calculate the normalised total demand as a decreasing step function from quantities to prices, represented as a list of points.

This works essentially as follows:

• Assign each bid's quantity demanded to one or more goods, giving an absolute demand curve for each good.
• Normalise each absolute demand curve by subtracting the supply curve for the corresponding good.
• Add the normalised demand curves horizontally.

For the first step, we must convert a bid into the demand on one or more goods. If the bid is definitely fully allocated (its surplus is strictly positive), its demand corresponds to what it is actually allocated. If the bid has a zero surplus, but is allocated something, its demand includes what it is actually allocated and uses an arbitrary allocated good for the remainder. Otherwise, if the bid is not allocated anything, use an arbitrary good that maximises its (possibly negative) surplus.

Note that for bids whose surplus is not strictly positive this is an approximation: making different arbitrary choices would change the demand graph (but only for quantities beyond the total allocated).

Convert a list of steps into a list of points. This includes two points for each step (giving the left and right ends for each given price), plus an extra point indicating the line going off to infinity.

Given the auction prices and a bid, calculate:

• the maximal surplus m achieved by the bid (i.e. the value to the bid of its preferred goods at the given prices);
• the set of goods gs that achieve surplus m;
• the maximal surplus m' achieved on goods that may be marginal for generalised bids (i.e. goods with a surplus of zero or a non-unique positive value);
• the set of goods gs' that achieve surplus m'.