Reverse Auctions - The Contractors Game

Transcription

1 Elmar G. Wolfstetter 1/12 Reverse Auctions - The Contractors Game June 24, 2014

2 Elmar G. Wolfstetter 2/12 Motivation Observers are often puzzled by the wide range of bids from similar consultants or contractors Wide ranging bids are sometimes viewed as evidence of collusion (Posner). However, dispersion is a consequence of the cost of drafting a bid, as we show here.

3 Elmar G. Wolfstetter 2/12 Motivation Observers are often puzzled by the wide range of bids from similar consultants or contractors Wide ranging bids are sometimes viewed as evidence of collusion (Posner). However, dispersion is a consequence of the cost of drafting a bid, as we show here.

4 Elmar G. Wolfstetter 2/12 Motivation Observers are often puzzled by the wide range of bids from similar consultants or contractors Wide ranging bids are sometimes viewed as evidence of collusion (Posner). However, dispersion is a consequence of the cost of drafting a bid, as we show here.

5 Elmar G. Wolfstetter 3/12 Model One object is procured from two potential contractors. The buyer values it at $1. Contractors have two kinds of cost and they know each others cost (which is their private information): cost of bidding d performance cost c, d + c < 1. The procurer employs a first-price auction, stipulating that bids are only valid if they do not exceed 1. The bidding game is a simultaneous moves game; actions are: make a bid [c + d, 1] or do not bid. Bids are denoted by b [d + c, 1]. The analysis is done for n 2 contractors (and can be extended to multiple objects).

6 Elmar G. Wolfstetter 3/12 Model One object is procured from two potential contractors. The buyer values it at $1. Contractors have two kinds of cost and they know each others cost (which is their private information): cost of bidding d performance cost c, d + c < 1. The procurer employs a first-price auction, stipulating that bids are only valid if they do not exceed 1. The bidding game is a simultaneous moves game; actions are: make a bid [c + d, 1] or do not bid. Bids are denoted by b [d + c, 1]. The analysis is done for n 2 contractors (and can be extended to multiple objects).

7 Elmar G. Wolfstetter 4/12 (Mixed) Strategies 1 B : [c + d, 1] [0, 1] (c.d.f. of bids) Case of two contractors 2 q := Pr{bid} [0, 1] (probability of bid resp. not bid) Proposition Unique symmetric equilibrium: q := Pr { participate and bid} = 1 c d 1 c B(b) := Pr { b b } = 1 b c d (1 c)(b c d) 1 1 c d = (b c)(1 c d)

8 Elmar G. Wolfstetter 4/12 (Mixed) Strategies 1 B : [c + d, 1] [0, 1] (c.d.f. of bids) Case of two contractors 2 q := Pr{bid} [0, 1] (probability of bid resp. not bid) Proposition Unique symmetric equilibrium: q := Pr { participate and bid} = 1 c d 1 c B(b) := Pr { b b } = 1 b c d (1 c)(b c d) 1 1 c d = (b c)(1 c d)

9 Elmar G. Wolfstetter 5/12 For c = 0.5, d = 0.1 and n = 2. Illustration of B(b) B b b

10 Elmar G. Wolfstetter 6/12 Proof Q: Why no pure strategy equilibrium with q = 1 and B(b) = 1 for some b? A: If one bidder would play this strategy, the other would undercut and bid lower (if b > c + d) or not bid (if b c + d); in either case at least one bidder regrets. Q: Why is (q, B) an equilibrium? Equilibrium expected profit: d + (b c)((1 q) + q(1 B(b))) }{{} bid b [c, d] d + (b c)(1 qb(b)) = 0 = }{{} 0, b [c, d] not bid

11 Elmar G. Wolfstetter 7/12 Extension to n 2 contractors A bidder wins iff he wins against each and every rival; therefore, payoff d + (b c)(1 q + q(1 B(b))) n 1, and ( d q = 1 1 c ( B(b) = 1 d b c 1 ( d 1 c ) 1 n 1 ) 1 n 1 ) 1 n 1 (prob. of bidding) (c.d.f. of bids) b [d + c, 1] (support of B)

12 Elmar G. Wolfstetter 8/12 Adverse Effect of Competition Compute the c.d.f. of the equilibrium price, denoted by H(p) and apply the convention that if no transaction takes place, this is equivalent to a transaction at price p = 1 (price=the buyer s valuation). Proposition H(p) is decreasing in n (first-order stochastic dominance shift); hence, more competition raises the random equilibrium price.

13 Elmar G. Wolfstetter 9/12 H(p, n) := Pr{lowest bid is p}, p < 1 = Pr{at least one bid is p} = Pr{not all bids are > p} =1 Pr{all bids > p} =1 (1 q + q(1 B(p)) n =1 (1 qb(p)) n =1 1 (1 ( ) 1 ( ) d n c )(1 d n 1 p c 1 ( d 1 c ) 1 n 1 ( ) n ( ) n d n 1 d n =1 1 > 1 p c p c n =: H(p, n ), n > n

14 Illustration of FSD shift The solid line is the cdf of the equilibrium price for n = 2; the dashed line is for n = 10. Pr P p Elmar G. Wolfstetter 10/12 p

15 Elmar G. Wolfstetter 11/12 H(1, n)=? Note, the cdf of equilibrium price, H(p, n), has a discontinuity at p = 1 which is due to our convention that no sale is treated as p = 1: { H(p, 1) if p < 1 H(1, n) = H(p, 1) + (1 q) n if p = 1. Also, for n + the c.d.f. approaches the limiting distribution H(p) = 1 d, for p < 1. p c

16 Elmar G. Wolfstetter 12/12 Corollary The auctioneer should restrict participation to two contractors. As it is typically the case in high tech procurements airplanes, weapons systems, etc. The performance of the auction can be improved under certain conditions by intelligent rebate policies.

17 Elmar G. Wolfstetter 12/12 Corollary The auctioneer should restrict participation to two contractors. As it is typically the case in high tech procurements airplanes, weapons systems, etc. The performance of the auction can be improved under certain conditions by intelligent rebate policies.

18 Elmar G. Wolfstetter 12/12 Corollary The auctioneer should restrict participation to two contractors. As it is typically the case in high tech procurements airplanes, weapons systems, etc. The performance of the auction can be improved under certain conditions by intelligent rebate policies.