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Information, stochastic dominance and bidding: The case of Treasury auctions
Accepted Manuscript Information, stochastic dominance and bidding: The case of Treasury auctions Patrick Leoni, Frederik Lundtofte PII: DOI: Reference:
S0165-1765(17)30049-6 http://dx.doi.org/10.1016/j.econlet.2017.02.004 ECOLET 7500
To appear in:
Economics Letters
Received date: 10 November 2016 Revised date: 25 January 2017 Accepted date: 3 February 2017 Please cite this article as: Leoni, P., Lundtofte, F., Information, stochastic dominance and bidding: The case of Treasury auctions. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.02.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights
We explore the link between signals, stochastic dominance and equilibrium bids. Informativeness is related to conditional first-order stochastic dominance. Our framework is relevant for discussing US Treasury auctions.
*Manuscript Click here to view linked References
Information, stochastic dominance and bidding: The case of Treasury auctions Patrick Leoni∗
Frederik Lundtofte†
January 2017
Abstract We explore the link between informativeness of signals, stochastic dominance and equilibrium bids in a multi-unit auction with risk averse bidders. We show that for a particular class of signal distributions, informativeness is related to conditional first-order stochastic dominance, so that a higher degree of informativeness in the signal-fundamental distribution induces higher bids and therefore higher revenues. Our framework is relevant for discussing total revenues and informativeness in US Treasury auctions.
Keywords:
Common value auctions, informativeness, stochastic dominance
JEL codes:
D44, G12
∗
Kedge Business School, Domaine de Luminy BP 921, 13 288 Marseille cedex 9, France.
†
Department of Economics and Knut Wicksell Centre for Financial Studies, Lund University,
P.O. Box 7082, S-220 07 Lund, Sweden. E-mail: [email protected]
Phone:
+46 46 222 8670.
Fax:
+46 46 222 4118.
1
Introduction
This paper analyzes the link between informativeness of signals, stochastic dominance and equilibrium bids in a uniform-price auction when bidders are risk averse. Using the informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014), we show that, for a particular class of signal-fundamental distributions, a higher degree of informativeness is associated with higher ranking in terms of first-order stochastic dominance and thus higher bids and revenues. The most closely related papers are Murto and V¨alim¨aki (2015) and Brandt, Drees, Eckwert, and V´ardy (2014). We contribute to the literature by extending Murto and V¨alim¨aki (2015) to multi-unit auctions and combining it with the informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014), thereby obtaining a framework that we think is particularly relevant for discussing total revenues and informativeness in US Treasury auctions.
2
Model
Building on the model by Murto and V¨alim¨aki (2015), we consider an auction with k identical objects (units) for sale, each with value V , which is unknown to the bidders. It is a (k + 1)th price auction, such that the kth highest bidders pay the (k + 1)th highest price. There are n bidders in total. Before participating in the bidding, each bidder i ∈ {1, 2, ..., n} observes a signal θi ∈ [0, 1], correlated with the true value V . The signals and the value are distributed according to a joint density function g(θi , v), and conditional 1
on V = v, θi is independent of θj for all i 6= j. The bidders share the same prior density on the true value, ρ(v), and the same utility function, u, and after having received their signals, they update in a Bayesian manner. It follows from Theorem 3.1 in Milgrom (1981) that there exists a symmetric equilibrium with bid functions that are strictly increasing in the signals. Now, let {θ1 , θ2 , ..., θn } (k+1)
be a random sample of realized signals. Let θn
be the signal of order (k + 1) in a
sample of size n. Then, the equilibrium price is given by the bid of the (k + 1)th highest (k+1)
bidder: Pnk+1 = bk+1 n (θn
(k)
(k)
). As it turns out, it is helpful to define Zn ≡ n(1 − θn ), be-
(1)
cause, e.g., Zn converges in distribution to an exponentially distributed random variable (Murto and V¨alim¨aki, 2015). By induction, one can thus show the following. (1)
(2)
(1)
(3)
(2)
(k+1)
Proposition 1. [Zn , Zn − Zn , Zn − Zn , ..., Zn
(k)
− Zn ] converges in distribution to
a vector of k + 1 independent exponential random variables with parameter γv ≡ g(1 | v). Proof. See Proposition 2 in Murto and V¨alim¨aki (2013). (k+1)
Since Zn
(1)
(2)
(1)
(3)
(2)
(k+1)
= Zn + (Zn − Zn ) + (Zn − Zn ) + ... + (Zn
(k)
− Zn ) converges to a (k+1)
sum of k + 1 independent exponentially distributed random variables, we have that Zn converges to a Gamma distribution: (k+1)
Corollary 1. Zn
converges to a Gamma distribution with parameters k + 1 and γv .
Proof. The result follows immediately from the above proposition. In the (k + 1)th price auction, the equilibrium price in the large-game limit as n tends to infinity, can be determined by the (k + 1)th highest bidder’s bid which equals 2
(k+1)
his willingness to pay. Conditional on Zn
= z, his willingness to pay, bk+1 (z), can be
found from EV u V − bk+1 (z) | Z (k+1) = z = u(0),
(1)
where Z (k+1) ∼ Gamma(k + 1, γv ) (Murto and V¨alim¨aki, 2015).
The following two propositions establish a link between stochastic dominance and bids. Proposition 2. Let ¯bk+1 (z) be the bid function associated with the (conditional) cdf F¯ (v|z), let ˆbk+1 (z) be the bid function associated with the (conditional) cdf Fˆ (v|z) and let Z be the support for the random variable Z (k+1) . If F¯ (v|z) first-order stochastically dominates Fˆ (v|z) for all z ∈ Z and u(c) is an increasing function, then ¯bk+1 (z) > ˆbk+1 (z) for all z ∈ Z. Proof. Suppose ¯bk+1 (z) ≤ ˆbk+1 (z) for some z ∈ Z. Then, we have that h i k+1 (k+1) k+1 (k+1) ¯ ˆ ¯ ¯ u(0) = EV u V − b (z) | Z = z ≥ EV u V − b (z) | Z =z
h i k+1 (k+1) ˆ ˆ > EV u V − b (z) | Z = z = u(0),
(2)
which is a contradiction, and so the supposition must be false. Proposition 3. With the same notation as in Proposition 2; if F¯ (v|z) second-order stochastically dominates Fˆ (v|z) for all z ∈ Z and u(c) is an increasing, strictly concave function, then ¯bk+1 (z) > ˆbk+1 (z) for all z ∈ Z. Proof. The proof is very similar to the proof of the previous proposition and is therefore omitted.
3
To discuss the degree of information contained in the signals, we will make use of the first informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014): ¯ v) ≥ G(θ, ˆ v) ∀ (θ, v) ∈ T × V, Definition 1. g¯ is weakly more informative than gˆ if G(θ, where T = [0, 1] and V = [0, +∞). With this definition in place, we can easily obtain the following result, which turns out to be helpful. Lemma 1. Given identical priors over V , ρ(v), we have that a distribution g¯ that satisfies ¯ | v) ≥ G(θ ˆ | v) ∀ (θ, v) ∈ T × V, where T = [0, 1] and V = [0, +∞), is weakly more G(θ informative. Proof. The result follows directly from the relation G(θ, v) =
2.1
Rv
−∞
ρ(v 0 )G(θ | v 0 )dv 0 .
A particular class of signal-fundamental distributions permitting further analysis
Now, we focus on a particular type of signal-fundamental distributions and priors where it turns out that increasing informativeness and first-order stochastic dominance go handin-hand. We assume that the bidders share a Gamma-distributed prior on V , with parameters α and β, and that the conditional signal distribution looks as follows: G(θ | v) = θλv , where λ > 0, θ ∈ [0, 1] and v ∈ [0, +∞). Thus, g(θ | v) = λvθλv−1 and g(1 | v) = λv. 4
(3)
We note that, by Lemma 1, G becomes weakly less informative as λ increases and that, given identical priors, λ is the only free parameter (with respect to Definition 1). Further, given the information structure stated above, the posterior distribution (V | Z (k+1) = z) is a Gamma distribution with parameters updated from (α, β) to (α + k + 1, β + λz). It then follows from, e.g., Ali (1975) that, if comparing two distributions, where one has a higher value on λ, the one with the lower λ-value first-order stochastically dominates the one with the higher λ-value. That is, in the case of this distribution, informativeness and first-order stochastic dominance are related in the following way: given identical priors, a weakly less informative signal-fundamental distribution (through a higher value on λ) implies a lower-ranked conditional distribution in terms of first-order stochastic dominance. From Proposition 2, we get that, in this case, given identical priors, a more informative signal-fundamental distribution results in higher bids for all bidders with an increasing utility function (i.e., risk neutral as well as risk averse bidders), which in turn results in higher revenues for the seller. If we further specialize to CARA utility, u(x) = −
e−ηx , η
(4)
where η is the coefficient of absolute risk aversion, the bid function found from the relation in (1) is given by b
k+1
log EV e−ηV | Z (k+1) = z . (z) = − η
Note that, abstracting from signals, the bid of agent i is bk+1 =− i
(5) log EiV [e−ηV ] . η
Since
e−ηV is a convex function, it follows that, for a risk averse agent (η > 0), a perceived 5
mean-preserving spread of V will decrease the his bid, whereas a risk neutral agent’s (η → 0) bid remains unaffected by mean-preserving spreads. Following the derivations in Murto and V¨alim¨aki (2015), we can solve explicitly for the bid in (5) as b
k+1
η α+k+1 log 1 + , (z) = η β + λz
(6)
where we see that the bid is decreasing in λ for each possible value on z. As we approach risk-neutrality (η → 0), the above expression for the bid function converges to (α + k + 1)/(β + λz), which is again decreasing in λ for each possible value on z. This is in line with the fact that a higher λ induces a lower-ranked conditional distribution in terms of first-order stochastic dominance.
3
Conclusion
We analyze the connection between informativeness of signals, stochastic dominance and equilibrium bids in a multi-unit auction with risk-averse bidders. For a particular class of signal distributions, there is a clear association between informativeness and conditional first-order stochastic dominance: given identical priors, a higher degree of informativeness in the signal-fundamental distribution induces higher bids and therefore higher revenues. We argue that our framework is particularly relevant for discussing total revenues and informativeness in US Treasury auctions.
6
References Ali, M. M. (1975): “Stochastic Dominance and Portfolio Analysis,” Journal of Financial Economics, 2, 205–229. ´ rdy (2014): “Information and the Brandt, N., B. Drees, B. Eckwert, and F. Va Dispersion of Posterior Expectations,” Journal of Economic Theory, 154, 604–611. Milgrom, P. R. (1981): “Rational Expectations, Information Acquisition and Competitive Bidding,” Econometrica, 49, 921–943. ¨ lima ¨ ki (2013): “Delay and Information Aggregation in Stopping Murto, P., and J. Va Games with Private Information,” Journal of Economic Theory, 148, 2404–2435. (2015): “Large Common Value Auctions with Risk Averse Bidders,” Games and Economic Behavior, 91, 60–74.
7
S0165-1765(17)30049-6 http://dx.doi.org/10.1016/j.econlet.2017.02.004 ECOLET 7500
To appear in:
Economics Letters
Received date: 10 November 2016 Revised date: 25 January 2017 Accepted date: 3 February 2017 Please cite this article as: Leoni, P., Lundtofte, F., Information, stochastic dominance and bidding: The case of Treasury auctions. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.02.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights
We explore the link between signals, stochastic dominance and equilibrium bids. Informativeness is related to conditional first-order stochastic dominance. Our framework is relevant for discussing US Treasury auctions.
*Manuscript Click here to view linked References
Information, stochastic dominance and bidding: The case of Treasury auctions Patrick Leoni∗
Frederik Lundtofte†
January 2017
Abstract We explore the link between informativeness of signals, stochastic dominance and equilibrium bids in a multi-unit auction with risk averse bidders. We show that for a particular class of signal distributions, informativeness is related to conditional first-order stochastic dominance, so that a higher degree of informativeness in the signal-fundamental distribution induces higher bids and therefore higher revenues. Our framework is relevant for discussing total revenues and informativeness in US Treasury auctions.
Keywords:
Common value auctions, informativeness, stochastic dominance
JEL codes:
D44, G12
∗
Kedge Business School, Domaine de Luminy BP 921, 13 288 Marseille cedex 9, France.
†
Department of Economics and Knut Wicksell Centre for Financial Studies, Lund University,
P.O. Box 7082, S-220 07 Lund, Sweden. E-mail: [email protected]
Phone:
+46 46 222 8670.
Fax:
+46 46 222 4118.
1
Introduction
This paper analyzes the link between informativeness of signals, stochastic dominance and equilibrium bids in a uniform-price auction when bidders are risk averse. Using the informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014), we show that, for a particular class of signal-fundamental distributions, a higher degree of informativeness is associated with higher ranking in terms of first-order stochastic dominance and thus higher bids and revenues. The most closely related papers are Murto and V¨alim¨aki (2015) and Brandt, Drees, Eckwert, and V´ardy (2014). We contribute to the literature by extending Murto and V¨alim¨aki (2015) to multi-unit auctions and combining it with the informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014), thereby obtaining a framework that we think is particularly relevant for discussing total revenues and informativeness in US Treasury auctions.
2
Model
Building on the model by Murto and V¨alim¨aki (2015), we consider an auction with k identical objects (units) for sale, each with value V , which is unknown to the bidders. It is a (k + 1)th price auction, such that the kth highest bidders pay the (k + 1)th highest price. There are n bidders in total. Before participating in the bidding, each bidder i ∈ {1, 2, ..., n} observes a signal θi ∈ [0, 1], correlated with the true value V . The signals and the value are distributed according to a joint density function g(θi , v), and conditional 1
on V = v, θi is independent of θj for all i 6= j. The bidders share the same prior density on the true value, ρ(v), and the same utility function, u, and after having received their signals, they update in a Bayesian manner. It follows from Theorem 3.1 in Milgrom (1981) that there exists a symmetric equilibrium with bid functions that are strictly increasing in the signals. Now, let {θ1 , θ2 , ..., θn } (k+1)
be a random sample of realized signals. Let θn
be the signal of order (k + 1) in a
sample of size n. Then, the equilibrium price is given by the bid of the (k + 1)th highest (k+1)
bidder: Pnk+1 = bk+1 n (θn
(k)
(k)
). As it turns out, it is helpful to define Zn ≡ n(1 − θn ), be-
(1)
cause, e.g., Zn converges in distribution to an exponentially distributed random variable (Murto and V¨alim¨aki, 2015). By induction, one can thus show the following. (1)
(2)
(1)
(3)
(2)
(k+1)
Proposition 1. [Zn , Zn − Zn , Zn − Zn , ..., Zn
(k)
− Zn ] converges in distribution to
a vector of k + 1 independent exponential random variables with parameter γv ≡ g(1 | v). Proof. See Proposition 2 in Murto and V¨alim¨aki (2013). (k+1)
Since Zn
(1)
(2)
(1)
(3)
(2)
(k+1)
= Zn + (Zn − Zn ) + (Zn − Zn ) + ... + (Zn
(k)
− Zn ) converges to a (k+1)
sum of k + 1 independent exponentially distributed random variables, we have that Zn converges to a Gamma distribution: (k+1)
Corollary 1. Zn
converges to a Gamma distribution with parameters k + 1 and γv .
Proof. The result follows immediately from the above proposition. In the (k + 1)th price auction, the equilibrium price in the large-game limit as n tends to infinity, can be determined by the (k + 1)th highest bidder’s bid which equals 2
(k+1)
his willingness to pay. Conditional on Zn
= z, his willingness to pay, bk+1 (z), can be
found from EV u V − bk+1 (z) | Z (k+1) = z = u(0),
(1)
where Z (k+1) ∼ Gamma(k + 1, γv ) (Murto and V¨alim¨aki, 2015).
The following two propositions establish a link between stochastic dominance and bids. Proposition 2. Let ¯bk+1 (z) be the bid function associated with the (conditional) cdf F¯ (v|z), let ˆbk+1 (z) be the bid function associated with the (conditional) cdf Fˆ (v|z) and let Z be the support for the random variable Z (k+1) . If F¯ (v|z) first-order stochastically dominates Fˆ (v|z) for all z ∈ Z and u(c) is an increasing function, then ¯bk+1 (z) > ˆbk+1 (z) for all z ∈ Z. Proof. Suppose ¯bk+1 (z) ≤ ˆbk+1 (z) for some z ∈ Z. Then, we have that h i k+1 (k+1) k+1 (k+1) ¯ ˆ ¯ ¯ u(0) = EV u V − b (z) | Z = z ≥ EV u V − b (z) | Z =z
h i k+1 (k+1) ˆ ˆ > EV u V − b (z) | Z = z = u(0),
(2)
which is a contradiction, and so the supposition must be false. Proposition 3. With the same notation as in Proposition 2; if F¯ (v|z) second-order stochastically dominates Fˆ (v|z) for all z ∈ Z and u(c) is an increasing, strictly concave function, then ¯bk+1 (z) > ˆbk+1 (z) for all z ∈ Z. Proof. The proof is very similar to the proof of the previous proposition and is therefore omitted.
3
To discuss the degree of information contained in the signals, we will make use of the first informativeness criterion in Brandt, Drees, Eckwert, and V´ardy (2014): ¯ v) ≥ G(θ, ˆ v) ∀ (θ, v) ∈ T × V, Definition 1. g¯ is weakly more informative than gˆ if G(θ, where T = [0, 1] and V = [0, +∞). With this definition in place, we can easily obtain the following result, which turns out to be helpful. Lemma 1. Given identical priors over V , ρ(v), we have that a distribution g¯ that satisfies ¯ | v) ≥ G(θ ˆ | v) ∀ (θ, v) ∈ T × V, where T = [0, 1] and V = [0, +∞), is weakly more G(θ informative. Proof. The result follows directly from the relation G(θ, v) =
2.1
Rv
−∞
ρ(v 0 )G(θ | v 0 )dv 0 .
A particular class of signal-fundamental distributions permitting further analysis
Now, we focus on a particular type of signal-fundamental distributions and priors where it turns out that increasing informativeness and first-order stochastic dominance go handin-hand. We assume that the bidders share a Gamma-distributed prior on V , with parameters α and β, and that the conditional signal distribution looks as follows: G(θ | v) = θλv , where λ > 0, θ ∈ [0, 1] and v ∈ [0, +∞). Thus, g(θ | v) = λvθλv−1 and g(1 | v) = λv. 4
(3)
We note that, by Lemma 1, G becomes weakly less informative as λ increases and that, given identical priors, λ is the only free parameter (with respect to Definition 1). Further, given the information structure stated above, the posterior distribution (V | Z (k+1) = z) is a Gamma distribution with parameters updated from (α, β) to (α + k + 1, β + λz). It then follows from, e.g., Ali (1975) that, if comparing two distributions, where one has a higher value on λ, the one with the lower λ-value first-order stochastically dominates the one with the higher λ-value. That is, in the case of this distribution, informativeness and first-order stochastic dominance are related in the following way: given identical priors, a weakly less informative signal-fundamental distribution (through a higher value on λ) implies a lower-ranked conditional distribution in terms of first-order stochastic dominance. From Proposition 2, we get that, in this case, given identical priors, a more informative signal-fundamental distribution results in higher bids for all bidders with an increasing utility function (i.e., risk neutral as well as risk averse bidders), which in turn results in higher revenues for the seller. If we further specialize to CARA utility, u(x) = −
e−ηx , η
(4)
where η is the coefficient of absolute risk aversion, the bid function found from the relation in (1) is given by b
k+1
log EV e−ηV | Z (k+1) = z . (z) = − η
Note that, abstracting from signals, the bid of agent i is bk+1 =− i
(5) log EiV [e−ηV ] . η
Since
e−ηV is a convex function, it follows that, for a risk averse agent (η > 0), a perceived 5
mean-preserving spread of V will decrease the his bid, whereas a risk neutral agent’s (η → 0) bid remains unaffected by mean-preserving spreads. Following the derivations in Murto and V¨alim¨aki (2015), we can solve explicitly for the bid in (5) as b
k+1
η α+k+1 log 1 + , (z) = η β + λz
(6)
where we see that the bid is decreasing in λ for each possible value on z. As we approach risk-neutrality (η → 0), the above expression for the bid function converges to (α + k + 1)/(β + λz), which is again decreasing in λ for each possible value on z. This is in line with the fact that a higher λ induces a lower-ranked conditional distribution in terms of first-order stochastic dominance.
3
Conclusion
We analyze the connection between informativeness of signals, stochastic dominance and equilibrium bids in a multi-unit auction with risk-averse bidders. For a particular class of signal distributions, there is a clear association between informativeness and conditional first-order stochastic dominance: given identical priors, a higher degree of informativeness in the signal-fundamental distribution induces higher bids and therefore higher revenues. We argue that our framework is particularly relevant for discussing total revenues and informativeness in US Treasury auctions.
6
References Ali, M. M. (1975): “Stochastic Dominance and Portfolio Analysis,” Journal of Financial Economics, 2, 205–229. ´ rdy (2014): “Information and the Brandt, N., B. Drees, B. Eckwert, and F. Va Dispersion of Posterior Expectations,” Journal of Economic Theory, 154, 604–611. Milgrom, P. R. (1981): “Rational Expectations, Information Acquisition and Competitive Bidding,” Econometrica, 49, 921–943. ¨ lima ¨ ki (2013): “Delay and Information Aggregation in Stopping Murto, P., and J. Va Games with Private Information,” Journal of Economic Theory, 148, 2404–2435. (2015): “Large Common Value Auctions with Risk Averse Bidders,” Games and Economic Behavior, 91, 60–74.
7