# Bidding with Securities: Profit-Share (Equity) vs. Cash Auctions

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1 Elmar G. Wolfstetter 1/13 Bidding with Securities: Profit-Share (Equity) vs. Cash Auctions Elmar G. Wolfstetter

2 Elmar G. Wolfstetter 2/13 Outline 1 Introduction 2 Equilibrium 3 (Almost) full surplus extraction 4 Example/Illustrations

3 Elmar G. Wolfstetter 3/13 Problem Suppose the seller can verify the winner s valuation for the item, after the auction. In that case it becomes possible to set the price to be a share of that valuation. We call such an auction a profit share auction and refer here to standard auctions as cash auctions. We shall 1 establish the equilibrium of profit share auctions, 2 show that profit share auctions are more profitable, 3 show how the seller can achieve full surplus extraction.

4 Elmar G. Wolfstetter 4/13 Specification of the share auction 1 The bidder who offers the highest share, s, wins and pays his share; losers pay nothing. 2 The sellers throws in a small cash-reward c to all bidders winner and losers alike. 3 The reward paid to the winner increases the winner s valuation from say x to x + c; therefore, the winner pays the price s(x)(x + c) and losers earn c.

5 Elmar G. Wolfstetter 5/13 Further assumptions/notation 1 n 2 bidders 2 independent private values drawn from the continuous c.d.f. F : (0, 1) (0, 1). 3 bids are shares of the private value, s [0, 1]. 4 s(x): equilibrium share strategy; β(x): equilibrium strategy of the standard first-price cash auction.

6 Elmar G. Wolfstetter 6/13 Outline 1 Introduction 2 Equilibrium 3 (Almost) full surplus extraction 4 Example/Illustrations

7 Elmar G. Wolfstetter 7/13 Equilibrium Working hypothesis: game has symmetric equilibrium in strictly monotone increasing strategy s, which will be confirmed below (for c > 0). x = arg max G(y)(x + c)(1 s(y)) + (1 G(y))c, y Therefore, s has to solve the differential equation (after rearranging) (G(x)s(x)) = G x (x) x + c, s(0) = 0(initial condition). Proposition x. s(x) = s(x)= x 0 x 0 y y + c y G (y) y + c G(x) dy s (x) > 0 G (y) G(x) dy < x 0 G (y) G(x) dy = 1

8 Elmar G. Wolfstetter 8/13 Outline 1 Introduction 2 Equilibrium 3 (Almost) full surplus extraction 4 Example/Illustrations

9 Elmar G. Wolfstetter 9/13 lim s(x) = 1. c 0 Surplus extraction Proposition Choosing c > 0 arbitrarily small, the profit-share auction is an optimal selling mechanism since it achieves (almost) full surplus extraction by the seller, unlike standard (cash) auctions. Remark 1 The game has no strictly monotone equilibrium for c = 0; in that case the differential equation yields s(x) 1, which contradicts the assumed strict monotonicity. Therefore, the cash prize is essential! 2 The superiority of the profit-share auction is an example of the linkage principle, according to which linking the price to a variable that is correlated with bidders private information lowers bidders information rent (Milgrom, 1987).

10 Elmar G. Wolfstetter 10/13 Outline 1 Introduction 2 Equilibrium 3 (Almost) full surplus extraction 4 Example/Illustrations

11 Elmar G. Wolfstetter 11/13 Example Let n = 2 and F (x) x (uniform distribution). Then, s(x) = 1 c ( ) c + x x ln. c

12 Elmar G. Wolfstetter 12/13 Illustration 1 Figure 1 plots s(x) for c {0.1, 0.01, 0.001}. This nicely illustrates how the equilibrium bid function increases uniformly as c becomes smaller, and approaches s = 1 as c goes to zero x x x v

13 Elmar G. Wolfstetter 13/13 Illustration Figure: Expected profit of the share auction (solid) and of the cash auction (dashed)

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