Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Expected price and equilibrium bids of a single-winner reverse auction with independent uniform costs

a(c_low, c_high, N, c) = (E_c1, E_p, M_w, b, E_b)

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.

  • Expected lowest cost, expected price and winner's margin in a second-price reverse auction with uniform costs

    E_c1 = c_low + (c_high - c_low) / (N + 1); E_p = c_low + 2 * (c_high - c_low) / (N + 1); M_w = E_p - E_c1

    A purchaser buys from the lowest of N bidding suppliers whose unit costs are independent draws from a uniform distribution on the interval [c_low, c_high]. Under the second-price rule each supplier bids its cost and the winner is paid the second-lowest bid, so the expected price is the expected second-lowest cost. Klemperer prints the expected k-th highest of n uniform draws as v_low + ((n + 1 minus k) / (n + 1)) (v_high minus v_low); the lowest cost is the N-th highest and the second-lowest the (N minus 1)-th highest, which give E_c1 and E_p. By revenue equivalence the first-price rule has the same expected price (HE-FM-AUCT-002). The winner's expected margin is the gap between the two, (c_high minus c_low) / (N + 1) (derived), and both the price and the margin fall towards c_low and zero as N rises.

  • Symmetric first-price equilibrium bid and expected winning bid in a reverse auction with uniform costs

    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

    Under the first-price rule the lowest bidder wins and is paid its own bid, so a supplier bids above its cost, trading a higher price on winning against a lower chance of winning. With N suppliers whose costs are independent and uniform on the interval [c_low, c_high], the symmetric equilibrium bid adds to the supplier's cost c the expected gap to the lowest of the other N minus 1 costs, given that they all exceed c, which is (c_high minus c) / N. Klemperer prints the selling form, a bid of v_low + ((n minus 1) / n) (v minus v_low); the procurement form is derived by mirroring each cost to the value c_low + c_high minus c on the same range and checked here by a best-response search and by simulation. Because the bid is linear in cost, the expected winning bid is the bid at the expected lowest cost E_c1 of HE-FM-AUCT-001 (derived), and it equals that formula's expected price E_p, revenue equivalence for this case.

Auction Theory — Functions & Formulae | HealthEconomics.wiki