Signature
b = c + (c_high - c) / N; E_c1 = c_low + (c_high - c_low) / (N + 1); E_b = E_c1 + (c_high - E_c1) / N
| Inputs | Definition | Unit |
|---|---|---|
c | The bidding supplier's own unit cost, known only to it, at least c_low and at most c_high | currency per unit, for example GBP per pack |
c_high | Upper end of the common uniform cost range | currency per unit |
N | Number of suppliers that submit bids, the bidder itself included, at least 2 | count |
c_low | Lower end of the common uniform cost range, at most c_high | currency per unit |
b | Price per unit the supplier with cost c offers when every supplier uses the same equilibrium bidding rule | currency per unit |
|---|---|---|
E_c1 | Expected value of the lowest of the N unit costs, from HE-FM-AUCT-001 | currency per unit |
E_b | Expected price per unit the purchaser pays under the first-price rule, the bid of the expected lowest-cost supplier | currency per unit |
Function
Expected price and equilibrium bids of a single-winner reverse auction with independent uniform costs
Maps the lowest and highest possible unit costs c_low and c_high, the number of bidding suppliers N and, for a bid, one supplier's own cost c to the expected lowest cost E_c1, the expected price E_p the purchaser pays, the winner's expected margin M_w, the symmetric equilibrium bid b of a first-price (pay-as-bid) tender and the expected winning bid E_b, for a single-winner procurement auction in which each supplier knows only its own cost. HE-FM-AUCT-001 gives the expected costs, price and margin under the second-price rule, which by revenue equivalence is also the expected price under the first-price rule, and HE-FM-AUCT-002 gives the first-price bid and checks that its expected winning bid equals HE-FM-AUCT-001's price. HE-CF-AUCT-001 runs one tender under both rules and checks the closed forms by a seeded simulation. Both closed forms take the affine form of the Nash bargaining price HE-FM-BARG-001, a lower limit plus a weight times a range, but the weights 2 / (N + 1) and 1 / N come from order statistics of the cost draws, not from bargaining power (derived). The article's supply-risk comparison of one winner against two is an expected cost over delivery branches, HE-FM-CHN-001, with branch probabilities such as 0.81 that both suppliers deliver given by path products, HE-FM-CHN-002, so it is not restated here. The two-type screening of one supplier with a hidden cost is HE-FN-MECH-001, and the Medicare DMEPOS single payment amount, a median, maximum or 75th percentile of winning bids, is a programme rule and has no record here. Notation follows the Auction Theory article, with c_low and c_high for its lowest and highest possible unit costs.
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Implementations
Excel
Excel first-price equilibrium bid and expected winning bid with uniform costs
With the supplier's cost in a cell named BidderCost, the cost range in CostLowestFP and CostHighestFP and the number of bidders in BiddersFP, the formulas return the bid in a cell named EquilibriumBid, the expected lowest cost in ExpLowestCostFP and the expected winning bid in ExpWinningBid, in that order. For several suppliers, put their costs in a column and copy the first formula down.
=BidderCost+(CostHighestFP-BidderCost)/BiddersFP; =CostLowestFP+(CostHighestFP-CostLowestFP)/(BiddersFP+1); =ExpLowestCostFP+(CostHighestFP-ExpLowestCostFP)/BiddersFP
Assumptions
Symmetric risk-neutral equilibrium in a first-price tender with uniform costs
All suppliers are risk neutral, draw their costs independently from the same uniform range, know only their own cost and use the same increasing bidding rule, the setting of Klemperer's Appendix D. Klemperer's section 5 (Risk Aversion) states that risk aversion makes bidders bid more aggressively in first-price auctions, which in a tender means bids nearer to cost (the mirror, derived), and suppliers whose costs come from different ranges are not symmetric, so the bid formula then no longer holds.
Single winner paid its own bid with at least two bidders and no binding ceiling
One supplier wins the whole contract at its bid, N is at least 2 and no price ceiling below c_high binds. With continuous costs two suppliers tie with probability zero, so tie-breaking does not change the expected values. At c = c_high the bid equals the cost and the margin is zero (derived).
Procurement bid derived from the printed selling bid by mirroring the cost range
The procurement bid is derived here from Klemperer's selling bid by mapping each cost c to the value c_low + c_high minus c and each bid back the same way, which turns v_low + ((n minus 1) / n) (v minus v_low) into c + (c_high minus c) / N. The identity held in 11,514 grid cases, a best-response grid search with step 0.0005 matched the bid to within 0.0002 for N of 2, 3, 5 and 8, and a seeded simulation matched E_b (Python checks here).
Worked examples
Lowest-cost supplier at GBP 2.60 bids GBP 3.40 in the article's draw
The article's one draw, illustrative. With three suppliers and costs uniform on the interval [2.00, 5.00] GBP per pack, the supplier with cost 2.60 adds 2.40 / 3 and bids 3.40, which wins. The expected winning bid is 3.50, equal to the second-price expected price of HE-EX-AUCT-001.
c = 2.6; c_high = 5; c_low = 2; N = 3; b = 3.4; E_c1 = 2.75; E_b = 3.5
Middle-cost supplier at GBP 3.20 bids GBP 3.80 in the article's draw
The same draw, illustrative. The supplier with cost 3.20 adds 1.80 / 3 and bids 3.80, which loses to 3.40.
c = 3.2; c_high = 5; c_low = 2; N = 3; b = 3.8; E_c1 = 2.75; E_b = 3.5
Highest-cost supplier at GBP 4.40 bids GBP 4.60 in the article's draw
The same draw, illustrative. The supplier with cost 4.40 adds 0.60 / 3 and bids 4.60. A supplier closer to c_high shades its bid less, since fewer rival costs can lie above its own.
c = 4.4; c_high = 5; c_low = 2; N = 3; b = 4.6; E_c1 = 2.75; E_b = 3.5
Supplier at the highest possible cost bids its cost
Limiting case with the article's range. A supplier whose cost equals c_high wins with probability zero, since every rival's cost lies below its own almost surely, and its equilibrium bid is its cost of 5.00 with no margin (derived).
c = 5; c_high = 5; c_low = 2; N = 3; b = 5; E_c1 = 2.75; E_b = 3.5
Common errors
First-price bids assumed equal to suppliers' costs
Bidding cost is dominant only under the second-price rule. Assuming that a pay-as-bid tender attracts bids at cost puts the expected price at the expected lowest cost, 2.75 per pack in the article's example, in place of 3.50, the same understatement of GBP 750,000 on 1,000,000 packs as HE-ER-AUCT-002 (computed here).
Bid shading spread over the whole cost range
The margin a supplier adds depends on how far its own cost lies below c_high, (c_high minus c) / N, not on the full range. Adding (c_high minus c_low) / N gives the 2.60 supplier a bid of 3.60 in place of 3.40 and an expected winning bid of 3.75 in place of 3.50, breaking revenue equivalence (computed here).
One tender's winning bid read as a test of revenue equivalence
Revenue equivalence concerns expected prices. In the article's draw the first-price rule pays 3.40 and the second-price rule 3.20, a gap of 0.20 in one tender, while both expected prices are 3.50, so a single tender, or a few, cannot show which rule is cheaper (computed here).
Sources
Klemperer on the first-price equilibrium bid with uniform values
Klemperer P. Auction theory: a guide to the literature. Author's copy dated May 1999, posted on the author's Nuffield College page (full text read), which states that it is published in Journal of Economic Surveys 1999;13(3):227-286, doi:10.1111/1467-6419.00083 (journal version not seen; page numbers below are the author's copy's). Appendix D (Examples Using the Uniform Distribution), page 57, states that in a first-price auction bidder i bids her expected payment conditional on winning an ascending auction, the expected highest of the other n minus 1 values given that they lie below v_i, which is v_low + ((n minus 1) / n) (v_i minus v_low). Footnote 139, pages 57 and 58, derives the same bid from the differential equation for the first-price strategy and takes its expectation at the highest value to confirm revenue equivalence with the second-price auction. Section 5 (Risk Aversion) states that risk aversion makes bidders bid more aggressively in first-price auctions. The procurement bid is derived here from the selling form.
Nobel 2020 background on first-price bidding without a dominant strategy
Royal Swedish Academy of Sciences. Improvements to auction theory and inventions of new auction formats. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2020. 12 October 2020 (full text read). Section 2.2 (Private Values), under the heading Main Findings and Insights, states that the first-price and Dutch auctions are strategically equivalent and have no dominant strategies, since lowering one's bid lowers the price if one wins but also lowers the probability of winning, and that optimal bids balance the two; it also reports that all four formats yield the same expected revenue in Vickrey's model.
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