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An example of multi-unit auctions with atypically many equilibria
Economics Letters 65 (1999) 67–69
An example of multi-unit auctions with atypically many equilibria Richard Engelbrecht-Wiggans* Department of Business Administration, University of Illinois, 1206 South Sixth Street, Champaign, IL 61820, USA Received 8 February 1999; accepted 4 May 1999
Abstract The appealing simple example of multi-unit uniform-price auctions with constant, uniformly distributed marginal values has atypically many equilibria. 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Auctions; Nash equilibrium JEL classification: D44
Examples play an important role in the theory of multi-unit auctions. The equilibrium analysis of auctions in which each bidder may win various subsets of the good(s) being offered has proven difficult. Relatively little is understood about the form that equilibria may take, necessary and / or sufficient conditions for existence, or the properties of different equilibria. Examples help illustrate the possibilities and suggest directions for additional research. Finding tractable examples has also proven difficult. Assuming that random variables have a uniform distribution often simplifies the calculations. But this may also result in anomalous examples. The purpose of this note is to explore one such example. Consider a uniform-price (price 5 high losing bid) auction of two units to n expected profit maximizing bidders with privately-known, uniformly distributed constant marginal values. Assuming uniformly distributed values results in an appealing simple example. Indeed, Ausubel and Cramton (1996) use this example to illustrate the possibility of multiple equilibria; they present two, one where each bidder bids (v,v), and a second in which each bidder bids (v,0), where v denotes the bidder’s value. I will argue that this example has atypically many equilibria. Many other examples do have the two *Tel.: 11-217-333-4240; fax: 11-217-244-7969. E-mail address: [email protected] (R. Engelbrecht-Wiggans) 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00117-2
1999 Published by Elsevier Science S.A. All rights reserved.
R. Engelbrecht-Wiggans / Economics Letters 65 (1999) 67 – 69
68
equilibria mentioned above. But the example with constant, uniformly distributed marginal values also has a continuum of other equilibria; in particular, an equilibrium results whenever each bidder bids (v, b(v)), where b(v) is any non-decreasing (not necessarily continuous) function of v with 0 # b(v) # v for all v. And no other example with non-increasing marginal values has the same richness of equilibria. To see that the two equilibria presented above are also equilibria in many other examples, consider non-uniform distributions for the constant marginal values. Clearly, bidding (v,v) is still a best response against others doing likewise. And if the density exists and is a non-decreasing function, then Corollary 4.4 of Engelbrecht-Wiggans and Kahn (1998) assures that everyone bidding (v,0) is an equilibrium. There are many such examples. To examine the multiplicity of equilibria, I draw on the work of Engelbrecht-Wiggans and Kahn. They consider auctions with non-negative, non-increasing, bounded marginal values. Let G1 and G2 denote the marginal distributions of each bidder’s two values. Define the function G(c,v) 5 e0#x #c (n 2 1)G1 (x)(n22 ) [G1 (x) 2 G2 (v) 1 g1 (x)(v 2 x)] dx, and the set C(v) 5 argmax 0#c G(c,v). Then, for example, by their Corollary 5.4, each bidder bidding (v, b(v)) is an equilibrium whenever C(v) is a non-decreasing correspondence and b(v) is a selection from C(v). Now let us see what happens in the case of constant, uniformly distributed marginal values. Here, G1 (v) 5 G2 (v) 5 v. Therefore G(c,v) 5 0 everywhere and any b(v) is a selection from C(v). C(v) is not a non-decreasing correspondence; Corollary 5.4 itself does not apply. However, the analysis leading up to Corollary 5.4 implies that an equilibrium results whenever each bidder bids (v, b(v)), where b(v) is any non-decreasing function with 0 # b(v) # v for all v. In the case of n52 bidders, this multiplicity of equilibria may be verified quite easily, directly. In particular, imagine that the bidder 2 has a value y and bids ( y, b( y)), where b( y) is any increasing function of y, and 0 # b( y) # y for all y. Let y(b) denote the inverse of b( y); define y(b) 5 1 for all b . b(1). Imagine that bidder 1 has a value x and bids B, where B # x. This gives bidder 1 an expected profit of
E 0#y #B
2(x 2 y) dy 1
E B #y #y(B )
(x 2 B) dy 1
E
(x 2 b( y)) dy
y(B )#y #y(x)
Differentiating this with respect to B gives the first order condition x 2 y(B) 5 0, which together with the monotonicity of y(B) implies that bidder 1 should set B 5 b(x). So, each bidder bidding (v, b(v)) gives an equilibrium. Such a multiplicity of equilibria is atypical. In particular, as before, consider the case of non-increasing, marginal values. With no loss of generality, let G1 and G2 have support [0,1]. For expository ease, assume that the density g1 exists. Imagine that G(c,v) 5 0 everywhere and see what this implies about G1 and G2 . In particular, it implies that dG(c,v) / dc 5 0 everywhere. Therefore G1 (c) 2 G2 (v) 1 g1 (c)(v 2 c) 5 0 for all c.0 and all v. Setting v 5 1 gives the differential equation G1 (c) 2 1 1 G 91 (c)(1 2 c) 5 0 which has as its solution G1 (x) 5 c(1 2 x) 1 1; the boundary condition G1 (0) 5 0 implies that G1 (x) 5 x. Substituting this back into dG(c,v) / dc 5 0 implies that c 2 G2 (v) 1 (v 2 c) 5 0 for all c.0 and all v. This simplifies to G2 (v) 5 v; both values must be uniformly distributed. Since we assumed non-increasing marginal values, the marginal values must be equal with probability one. In short, the only example with G(c,v) everywhere is that with constant, uniformly distributed marginal values.
R. Engelbrecht-Wiggans / Economics Letters 65 (1999) 67 – 69
69
References Ausubel, L.M., Cramton, P.C., 1996. Demand reduction and inefficiency in multi-unit auctions, Department of Economics, University of Maryland, Working paper no. 96-07. Engelbrecht-Wiggans, R., Kahn, C.M., 1998. Multi-unit auctions with uniform prices. Econ. Theor. 12, 227–258.
An example of multi-unit auctions with atypically many equilibria Richard Engelbrecht-Wiggans* Department of Business Administration, University of Illinois, 1206 South Sixth Street, Champaign, IL 61820, USA Received 8 February 1999; accepted 4 May 1999
Abstract The appealing simple example of multi-unit uniform-price auctions with constant, uniformly distributed marginal values has atypically many equilibria. 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Auctions; Nash equilibrium JEL classification: D44
Examples play an important role in the theory of multi-unit auctions. The equilibrium analysis of auctions in which each bidder may win various subsets of the good(s) being offered has proven difficult. Relatively little is understood about the form that equilibria may take, necessary and / or sufficient conditions for existence, or the properties of different equilibria. Examples help illustrate the possibilities and suggest directions for additional research. Finding tractable examples has also proven difficult. Assuming that random variables have a uniform distribution often simplifies the calculations. But this may also result in anomalous examples. The purpose of this note is to explore one such example. Consider a uniform-price (price 5 high losing bid) auction of two units to n expected profit maximizing bidders with privately-known, uniformly distributed constant marginal values. Assuming uniformly distributed values results in an appealing simple example. Indeed, Ausubel and Cramton (1996) use this example to illustrate the possibility of multiple equilibria; they present two, one where each bidder bids (v,v), and a second in which each bidder bids (v,0), where v denotes the bidder’s value. I will argue that this example has atypically many equilibria. Many other examples do have the two *Tel.: 11-217-333-4240; fax: 11-217-244-7969. E-mail address: [email protected] (R. Engelbrecht-Wiggans) 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00117-2
1999 Published by Elsevier Science S.A. All rights reserved.
R. Engelbrecht-Wiggans / Economics Letters 65 (1999) 67 – 69
68
equilibria mentioned above. But the example with constant, uniformly distributed marginal values also has a continuum of other equilibria; in particular, an equilibrium results whenever each bidder bids (v, b(v)), where b(v) is any non-decreasing (not necessarily continuous) function of v with 0 # b(v) # v for all v. And no other example with non-increasing marginal values has the same richness of equilibria. To see that the two equilibria presented above are also equilibria in many other examples, consider non-uniform distributions for the constant marginal values. Clearly, bidding (v,v) is still a best response against others doing likewise. And if the density exists and is a non-decreasing function, then Corollary 4.4 of Engelbrecht-Wiggans and Kahn (1998) assures that everyone bidding (v,0) is an equilibrium. There are many such examples. To examine the multiplicity of equilibria, I draw on the work of Engelbrecht-Wiggans and Kahn. They consider auctions with non-negative, non-increasing, bounded marginal values. Let G1 and G2 denote the marginal distributions of each bidder’s two values. Define the function G(c,v) 5 e0#x #c (n 2 1)G1 (x)(n22 ) [G1 (x) 2 G2 (v) 1 g1 (x)(v 2 x)] dx, and the set C(v) 5 argmax 0#c G(c,v). Then, for example, by their Corollary 5.4, each bidder bidding (v, b(v)) is an equilibrium whenever C(v) is a non-decreasing correspondence and b(v) is a selection from C(v). Now let us see what happens in the case of constant, uniformly distributed marginal values. Here, G1 (v) 5 G2 (v) 5 v. Therefore G(c,v) 5 0 everywhere and any b(v) is a selection from C(v). C(v) is not a non-decreasing correspondence; Corollary 5.4 itself does not apply. However, the analysis leading up to Corollary 5.4 implies that an equilibrium results whenever each bidder bids (v, b(v)), where b(v) is any non-decreasing function with 0 # b(v) # v for all v. In the case of n52 bidders, this multiplicity of equilibria may be verified quite easily, directly. In particular, imagine that the bidder 2 has a value y and bids ( y, b( y)), where b( y) is any increasing function of y, and 0 # b( y) # y for all y. Let y(b) denote the inverse of b( y); define y(b) 5 1 for all b . b(1). Imagine that bidder 1 has a value x and bids B, where B # x. This gives bidder 1 an expected profit of
E 0#y #B
2(x 2 y) dy 1
E B #y #y(B )
(x 2 B) dy 1
E
(x 2 b( y)) dy
y(B )#y #y(x)
Differentiating this with respect to B gives the first order condition x 2 y(B) 5 0, which together with the monotonicity of y(B) implies that bidder 1 should set B 5 b(x). So, each bidder bidding (v, b(v)) gives an equilibrium. Such a multiplicity of equilibria is atypical. In particular, as before, consider the case of non-increasing, marginal values. With no loss of generality, let G1 and G2 have support [0,1]. For expository ease, assume that the density g1 exists. Imagine that G(c,v) 5 0 everywhere and see what this implies about G1 and G2 . In particular, it implies that dG(c,v) / dc 5 0 everywhere. Therefore G1 (c) 2 G2 (v) 1 g1 (c)(v 2 c) 5 0 for all c.0 and all v. Setting v 5 1 gives the differential equation G1 (c) 2 1 1 G 91 (c)(1 2 c) 5 0 which has as its solution G1 (x) 5 c(1 2 x) 1 1; the boundary condition G1 (0) 5 0 implies that G1 (x) 5 x. Substituting this back into dG(c,v) / dc 5 0 implies that c 2 G2 (v) 1 (v 2 c) 5 0 for all c.0 and all v. This simplifies to G2 (v) 5 v; both values must be uniformly distributed. Since we assumed non-increasing marginal values, the marginal values must be equal with probability one. In short, the only example with G(c,v) everywhere is that with constant, uniformly distributed marginal values.
R. Engelbrecht-Wiggans / Economics Letters 65 (1999) 67 – 69
69
References Ausubel, L.M., Cramton, P.C., 1996. Demand reduction and inefficiency in multi-unit auctions, Department of Economics, University of Maryland, Working paper no. 96-07. Engelbrecht-Wiggans, R., Kahn, C.M., 1998. Multi-unit auctions with uniform prices. Econ. Theor. 12, 227–258.