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Revenue in contests with many participants
Operations Research Letters 42 (2014) 119–122
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Revenue in contests with many participants Arieh Gavious a,b,∗ , Yizhaq Minchuk c a
Faculty of Business Administration, Ono Academic College, Israel
b
Department of Industrial Engineering and Management, Ben-Gurion University, Israel
c
Department of Industrial Engineering and Management, Shamoon College of Engineering, Israel
article
info
Article history: Received 10 February 2013 Received in revised form 23 December 2013 Accepted 23 December 2013 Available online 3 January 2014
abstract We show that in a contest with a single prize, the expected effort made by the kth highest valuation participant bounds the sum of the expected efforts made by all of the participants with valuations less than the kth highest valuations. We also show that in the limit case of a contest with m prizes, the expected effort made by the kth highest valuation participant when the bidders are risk-neutral is greater than the expected effort in the risk-averse case. © 2013 Elsevier B.V. All rights reserved.
Keywords: Contest All-pay auction Revenue Risk aversion
1. Introduction We study a contest with m identical prizes where each participant’s valuation of the prize is private information drawn independently according to the same common knowledge distribution function. In such a contest, all participants make unrecoverable efforts such as money or resources. According to the revenue equivalence theorem, if the participants are risk-neutral, the sum of the expected efforts is identical to the expected revenue in any auction mechanism such as first- or second-price auctions (see Myerson [8], Riley and Samuelson [9]). However, in first-price auctions, the winner pays the bid he or she offered, whereas in contests, the bids made by the participants are usually efforts or other nonrecoverable resources. If the contest’s organizer cannot recover all of the efforts made by the participants, what portion of the total efforts invested by all of the participants is recoverable? The current study shows that when the number of participants is finite and there is a single prize, the expected payment made by the kth highest (i.e., highest with respect to effort) participant is at least twice the expected effort made by the k + 1th participant. Thus, it follows that if the organizer can recover only k of the highest efforts, the sum of the expected efforts that cannot be recovered is bounded by the kth participant’s expected effort.
∗ Corresponding author at: Faculty of Business Administration, Ono Academic College, Israel. E-mail addresses: [email protected], [email protected] (A. Gavious), [email protected] (Y. Minchuk). 0167-6377/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.orl.2013.12.008
In other words, even if only a few of the highest efforts can be recovered, the losses are minor. This result generalizes Archak and Sundararajan [1] who show a limit result when the number of participants is infinite. In other words, when there is a single prize in the contest, in the limit case, the participant with the greatest effort generates 1/2 of the total expected efforts (Moldovanu and Sela [7] proved the result for the participants with the greatest effort), the second generates 1/4, the third 1/8, etc. Note that by the revenue equivalence theorem the sum of all of the expected efforts approaches 1 when the number of participants approaches infinity. In addition, we propose a simpler proof for the limit case than the one presented by Archak and Sundararajan [1]. Archak and Sundararajan [1] also demonstrate the existence and uniqueness of the equilibrium bid function based on mechanism design considerations. Moreover, they show that when risk aversion increases, the optimal number of prizes is more than one. We use this simpler proof to prove a generalized result for the limit case with m ≥ 1 prizes and risk-averse participants. We show that in a contest with risk-averse participants, the expected effort made by the kth highest participant is bounded by the expected effort made by the kth participants in the risk-neutral case. Although the results in this paper are new to the literature, it is worth noting two other studies with a different setting that is similar to the setting we present in the current study. Chawla, Hartline and Sivan [3] consider an all-pay auction mechanism where the prize is divisible. They determine that in all possible (prize divisible) all-pay auctions mechanisms, the revenue maximizing auction is the one where the participant with the greatest effort wins the entire prize. Chawla, Hartline and Sivan [3] also show that the sum
120
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
of the expected efforts made by the participants (i.e., the seller’s expected revenue) is bounded by twice the expected highest effort. In the current study we will extend this result and give a bound to the ratio between the kth highest effort and the k + 1th highest effort. Cavallo and Jain [2] consider a different setting where the value for the contest organizer (seller) depends on the winner’s personal skill, his or her level of effort and the organizer’s personal quality parameters. They look for a mechanism that maximizes the social payoff, namely, the maximum expected quality for the organizer less the total efforts made by the participants. They also offer an efficient and incentive-compatible selling mechanism for their problem. 2. The model
u(ˆv ; v) = vi G(ˆv ) − bi (ˆv )
n − 1 j
F n−j−1 (v)(1 − F (v))j .
(n − 1)! F n−m−1 (v)(1 − F (v))m−1 f (v). (2) (m − 1)!(n − m − 1)!
Following standard arguments, the equilibrium effort function is found by solving ∂∂vˆ u(ˆv ; v)vˆ =v = 0, which yields v
G(s)ds.
(3)
0
Let Yk,n denote the distribution of the kth highest value of n participants (i.e., the kth order statistics). The distribution of Yk,n is given by FYk,n (v) =
k−1 n i=0
i
F n−i (v)(1 − F (v))i .
(4)
The organizer’s expected revenue generated by the kth highest valuation participant is given by 1
0
Proof. Integrating by parts (5) and rearranging gives Rk+1 = b(1) −
b(v)dFYk,n (v).
1
b′ (v)
k n
0
i=0
b′ (v)
n
0 1
b′ (v)
−
i=0
n k
0 1
b′ (v)
= Rk −
F n−i (v)(1 − F (v))i dv
k−1
1
i
F n−i (v)(1 − F (v))i dv
F n−k (v)(1 − F (v))k dv
n k
0
i
F n−k (v)(1 − F (v))k dv
1
(n − 1)v f (v)F n−2 (v)
= Rk − 0
×
n k
F n−k (v)(1 − F (v))k dv 1
= Rk −
n−k
0
×F
n−k−1 1
n−1
0
v F n−1 (v)f (v)
n! k!(n − k − 1)!
(v)(1 − F (v))k dv n−1 n−k
v F n−1 (v)dFYk+1,n (v).
Observe that the quantity v F n−1 (v) is the ex-ante willingness of a bidder with type v to pay. Obviously, it is higher than the bidder’s bid b(v). Thus, we have
≤ Rk −
n−1 n−k n−1 n−k
1
v F n−1 (v)dFYk+1,n (v)
0 1
0
b(v)dFYk+1,n (v) = Rk −
Rearranging completes the proof.
n−1 n−k
Rk+1 .
In the following proposition we show that the tail of the lowest n − k expected efforts is bounded by the kth expected effort. Proposition 2. Let m = 1. Then, the revenue generated by the kth highest effort is greater than the revenue generated by the sum of the n successive efforts, namely Rk ≥ i=k+1 Ri . Proof. By Proposition 1, Rk ≥
2n−1−k Rk+1 n−k
≥ 2Rk+1 . Thus,
Rk ≥ 2Rk+1 ≥ Rk+1 + 2Rk+2 ≥ Rk+1 + Rk+2 + 2Rk+3
≥ ··· ≥
n
Ri .
i=k+1
3. A single prize and a finite number of participants
Rk =
Rk+1 .
n−k
(1)
Thus, G is a distribution function of the mth highest valuation and its density is given by
b(v) = v G(v) −
2n − 1 − k
Rk+1 = Rk −
m−1
G′ (v) =
Rk ≥
= Rk −
where G(v) is the probability that in equilibrium, participant i will win one of the m prizes if his or her valuation of the prize is v . Given that the equilibrium effort function is monotonic with respect to v , G(v) is the probability that the value v is one of the m highest valuations among the n participants and is given by
j=0
Proposition 1. Let m = 1. Then,
= b(1) −
We initially consider a contest with m identical prizes and n risk-neutral participants, each one of whom has a unit demand. Later on, we will consider risk-averse participants. Each participant has a private valuation v for a prize that has been drawn independently from a continuously differentiable distribution function F (v) over the support [0, 1] with a strictly positive density F ′ = f > 0. Moreover, the valuations are the private information of the participants. Participant i makes an effort bi (e.g., resources, effort, etc.) independent of other participants. The m participants with the highest effort win a single prize, but all of the n participants pay their effort. To find the symmetric Bayesian equilibrium effort function, we follow the standard arguments (see, for example, Krishna [6]). Assume that there exists a symmetric and monotonic equilibrium effort function b(v). The expected payoff for a participant with value v when he or she is playing vˆ as his or her valuation, and all other n − 1 participants are playing according to the equilibrium effort strategy b(v), is given by
G(v) =
The following proposition gives the bounds on Rk for finite n.
(5)
The last result is based on the bound Rk ≥ 2Rk+1 , which is a weaker version of Proposition 1. However, the decline of Rk is sharper, particularly when k is increasing. In the extreme case, when k = n − 1, Rn−1 ≥ nRn . We conclude that if the organizer can recover just a few of the highest efforts, the sum of the efforts that are unrecoverable is minor.
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
Appendix. Proof of Lemma 1
4. Many prizes and many risk-averse participants In this section we consider the case with m prizes and many participants. We first provide a simpler proof for Archak and Sundararajan’s [1] result that we then use in the following result. Later on, we will use the method in the proof for Proposition 3. First, assume that the participants are risk-neutral. Lemma 1. Let R∞ k = limn→∞ Rk . Then, m−1
R∞ k =
k+i−1
1 2k+i
i
i=0
k = 1, 2, . . . .
(6)
ra,∞
.
Rk =
Observe that V (v) ≥ 0, because otherwise the participant would make zero efforts in order to avoid losses. By the concavity of U we have
V (v) ≤ U G(v)(v − b (v)) − (1 − G(v))b (v) = U G(v)v − bra (v) .
ra
ra
It follows that G(v)v ≥ bra (v). Thus, 1 0
0
0
v G(v)dFYk,n (v) ≥
0
1
bra (v)dFYk,n (v) = Rra k
,∞ v G(v)dFYk,n (v) ≥ Rra . From Lemma 1 k 1 ∞ (see Appendix A) we get Rk = limn→∞ 0 v G(v)dFYk,n (v).
and in the limit limn→∞
1
G(s)ds dFYk,n (v).
Define 1
v G(v)dFYk,n (v), 1
k
v
B = 0
(7)
G(s)ds dFYk,n (v).
(8)
First we will prove that limn→∞ Bk = 0 Integrating by parts (8) yields Bk =
1
G(s)ds −
1
G(v)FYk,n (v)dv
0
0 1
G(v)(1 − FYk,n (v))dv ≤
= 0
Since limn→∞
1 0
1
G(v)dv.
0
G(v)dv = 0, we find that limn→∞ Bk = 0.
(n)!
Now we will focus on Ak . Substituting dFYk,n (v) = (k−1)!(n−k)! F n−k (v)(1 − F (v))k−1 f (v)dv and (1) in (7) yields.
1
k
v
A = 0
m−1
n − 1 j
j =0
F
n −j −1
(v)(1 − F (v))
j
(n)! F n−k (v)(1 − F (v))k−1 f (v)dv (k − 1)!(n − k)! 1 m −1 (n)! n−1 = v f (v) j (k − 1)!(n − k)! 0 j =0 ×
× F 2n−j−k−1 (v)(1 − F (v))k−1+j dv m −1 (n)! n−1 = β(2n − k − j, k + j) (k − 1)!(n − k)! j=0 j 1 v f (v)F 2n−j−k−1 (v)(1 − F (v))k−1+j dv 0 × β(2n − k − j, k + j) m −1 (n)! n−1 = (k − 1)!(n − k)! j=0 j × β(2n − k − j, k + j)E (Yj+k,2n−1 ) dv
Proof. The expected utility for a participant with valuation v is V (v) = G(v)U (v − bra (v)) + (1 − G(v))U (−bra (v)).
v
v G(v) −
0
Proposition 3. Let m ≥ 1. Then,
1
Ak =
Proof. See Appendix A. Observe that in particular, if m = 1, then R∞ = 21k , k = k 1, 2, . . . . It follows that the contribution of the k highest participant to the organizer’s expected revenue is equal to the total contributions of all of the ranked from k + 1 to infinity. In participants ∞ ∞ other words, R∞ k = j=k+1 Rj . An additional interesting observation is that Rk is not necessarily monotonically increasing with n. For example, consider the case where F (v) = v 6 . Then, for n = 2 calculations show that R1 > 1/2. 1 , it follows that R1 is not increasing with n. However, since R∞ 1 = 2 n Notice, however, that j=1 Rnj , which is the sum of all of the efforts, is monotonically increasing with n (e.g., Krishna [6]). Assume that every participant has an identical utility function U (x) where U (0) = 0, U ′ > 0 and U ′′ < 0. Let Rra k be the kth highest ra,∞ = participant’s expected effort in the risk-averse case and let Rk limn→∞ Rra . Fibich and Gavious [4] show that when m = 1, R∞ > k Rra,∞ where R∞ , Rra,∞ are the total expected efforts made by all participants in the case of risk-averse participants risk-neutral and ∞ ∞ ra,∞ . Thus, in a contest respectively. Therefore, k=0 R∞ k > k=0 Rk with many participants, the contest’s organizer prefers riskneutral participants. We will show that this result is much stronger ra,∞ for every k. This result is not straightforward and that R∞ k ≥ Rk because according to Fibich, Gavious and Sela [5], a low type risk-averse participant bids less aggressively than a risk-neutral participant. However, a high type risk-averse participant bids more aggressively than a risk-neutral participant. Thus, revenue ranking is not immediate when comparing risk-neutral and risk-averse ra,∞ ∞ with Fibich settings. Observe that the result ∞Rk ∞≥ Rk∞, together ra,∞ , implies that and Gavious’ [4] result that k=0 Rk > k=0 Rk ra,∞ for some k’s, a strict inequality holds. In other words, R∞ . k > Rk
R∞ k ≥ Rk
Substituting (3) into (5) gives
0
,
121
where β(2n − k − j, k + j) is the beta function with parameters 2n − k − j and k + j given by
β(2n − k − j, k + j) =
(2n − k − j − 1)!(k + j − 1)! (2n − 1)!
(9)
and E (Yi+k,2n−1 ) is the expectation of the random variable Yi+k,2n−1 defined above in Eq. (4) as the i + kth order statistic of 2n − 1 i.i.d. random variables with the distribution F (v). Then, for fixed i + k, when n → ∞ the random variable Yi+k,2n−1 is the 100% percentile. It follows that, limn→∞ E (Yi+k,2n−1 ) = 1. Thus, lim Ak = lim
n→∞
n→∞
0
×
n!
(k − 1)!(n − k)!
m−1
n − 1 j=0
j
(2n − k − j − 1)!(k + j − 1)! (2n − 1)!
122
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
(k + j − 1)! (n − k + 1) · · · n · (n − j)(n − j + 1) · · · (n − 1) lim n→∞ ( k − 1 )! j ! (2n − k − j)(2n − k − j + 1) · · · (2n − 1) j=0
m−1
=
m−1
=
k + j − 1 j=0
j
yielding the result.
lim
n→∞
nk · nj
(2n)k+j
m−1
=
k + j − 1 1 j=0
j
2k+j
References [1] N. Archak, A. Sundararajan, Optimal design of crowdsourcing contests, in: International Conference of Information Systems Proceedings, paper 200, 2009.
[2] R. Cavallo, S. Jain, Efficient crowdsourcing contests, in: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, vol. 2, 2012, pp. 677–686. [3] S. Chawla, D.J. Hartline, B. Sivan, Optimal crowdsourcing contest, in: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 2011, pp. 856–868. [4] G. Fibich, A. Gavious, Large auctions with risk-averse bidders, Int. J. Game Theory 39 (2010) 359–390. [5] G. Fibich, A. Gavious, A. Sela, All-pay auction with risk-averse bidders, Int. J. Game Theory 34 (2006) 583–599. [6] V. Krishna, Auction Theory, Academic Press, New York, 2002. [7] B. Moldovanu, A. Sela, Contest architecture, J. Econom. Theory 126 (2006) 70–96. [8] R.B. Myerson, Optimal auction design, Math. Oper. Res. 6 (1981) 58–73. [9] J.G. Riley, W.F. Samuelson, Optimal auctions, Amer. Econ. Rev. 71 (1981) 381–392.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Revenue in contests with many participants Arieh Gavious a,b,∗ , Yizhaq Minchuk c a
Faculty of Business Administration, Ono Academic College, Israel
b
Department of Industrial Engineering and Management, Ben-Gurion University, Israel
c
Department of Industrial Engineering and Management, Shamoon College of Engineering, Israel
article
info
Article history: Received 10 February 2013 Received in revised form 23 December 2013 Accepted 23 December 2013 Available online 3 January 2014
abstract We show that in a contest with a single prize, the expected effort made by the kth highest valuation participant bounds the sum of the expected efforts made by all of the participants with valuations less than the kth highest valuations. We also show that in the limit case of a contest with m prizes, the expected effort made by the kth highest valuation participant when the bidders are risk-neutral is greater than the expected effort in the risk-averse case. © 2013 Elsevier B.V. All rights reserved.
Keywords: Contest All-pay auction Revenue Risk aversion
1. Introduction We study a contest with m identical prizes where each participant’s valuation of the prize is private information drawn independently according to the same common knowledge distribution function. In such a contest, all participants make unrecoverable efforts such as money or resources. According to the revenue equivalence theorem, if the participants are risk-neutral, the sum of the expected efforts is identical to the expected revenue in any auction mechanism such as first- or second-price auctions (see Myerson [8], Riley and Samuelson [9]). However, in first-price auctions, the winner pays the bid he or she offered, whereas in contests, the bids made by the participants are usually efforts or other nonrecoverable resources. If the contest’s organizer cannot recover all of the efforts made by the participants, what portion of the total efforts invested by all of the participants is recoverable? The current study shows that when the number of participants is finite and there is a single prize, the expected payment made by the kth highest (i.e., highest with respect to effort) participant is at least twice the expected effort made by the k + 1th participant. Thus, it follows that if the organizer can recover only k of the highest efforts, the sum of the expected efforts that cannot be recovered is bounded by the kth participant’s expected effort.
∗ Corresponding author at: Faculty of Business Administration, Ono Academic College, Israel. E-mail addresses: [email protected], [email protected] (A. Gavious), [email protected] (Y. Minchuk). 0167-6377/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.orl.2013.12.008
In other words, even if only a few of the highest efforts can be recovered, the losses are minor. This result generalizes Archak and Sundararajan [1] who show a limit result when the number of participants is infinite. In other words, when there is a single prize in the contest, in the limit case, the participant with the greatest effort generates 1/2 of the total expected efforts (Moldovanu and Sela [7] proved the result for the participants with the greatest effort), the second generates 1/4, the third 1/8, etc. Note that by the revenue equivalence theorem the sum of all of the expected efforts approaches 1 when the number of participants approaches infinity. In addition, we propose a simpler proof for the limit case than the one presented by Archak and Sundararajan [1]. Archak and Sundararajan [1] also demonstrate the existence and uniqueness of the equilibrium bid function based on mechanism design considerations. Moreover, they show that when risk aversion increases, the optimal number of prizes is more than one. We use this simpler proof to prove a generalized result for the limit case with m ≥ 1 prizes and risk-averse participants. We show that in a contest with risk-averse participants, the expected effort made by the kth highest participant is bounded by the expected effort made by the kth participants in the risk-neutral case. Although the results in this paper are new to the literature, it is worth noting two other studies with a different setting that is similar to the setting we present in the current study. Chawla, Hartline and Sivan [3] consider an all-pay auction mechanism where the prize is divisible. They determine that in all possible (prize divisible) all-pay auctions mechanisms, the revenue maximizing auction is the one where the participant with the greatest effort wins the entire prize. Chawla, Hartline and Sivan [3] also show that the sum
120
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
of the expected efforts made by the participants (i.e., the seller’s expected revenue) is bounded by twice the expected highest effort. In the current study we will extend this result and give a bound to the ratio between the kth highest effort and the k + 1th highest effort. Cavallo and Jain [2] consider a different setting where the value for the contest organizer (seller) depends on the winner’s personal skill, his or her level of effort and the organizer’s personal quality parameters. They look for a mechanism that maximizes the social payoff, namely, the maximum expected quality for the organizer less the total efforts made by the participants. They also offer an efficient and incentive-compatible selling mechanism for their problem. 2. The model
u(ˆv ; v) = vi G(ˆv ) − bi (ˆv )
n − 1 j
F n−j−1 (v)(1 − F (v))j .
(n − 1)! F n−m−1 (v)(1 − F (v))m−1 f (v). (2) (m − 1)!(n − m − 1)!
Following standard arguments, the equilibrium effort function is found by solving ∂∂vˆ u(ˆv ; v)vˆ =v = 0, which yields v
G(s)ds.
(3)
0
Let Yk,n denote the distribution of the kth highest value of n participants (i.e., the kth order statistics). The distribution of Yk,n is given by FYk,n (v) =
k−1 n i=0
i
F n−i (v)(1 − F (v))i .
(4)
The organizer’s expected revenue generated by the kth highest valuation participant is given by 1
0
Proof. Integrating by parts (5) and rearranging gives Rk+1 = b(1) −
b(v)dFYk,n (v).
1
b′ (v)
k n
0
i=0
b′ (v)
n
0 1
b′ (v)
−
i=0
n k
0 1
b′ (v)
= Rk −
F n−i (v)(1 − F (v))i dv
k−1
1
i
F n−i (v)(1 − F (v))i dv
F n−k (v)(1 − F (v))k dv
n k
0
i
F n−k (v)(1 − F (v))k dv
1
(n − 1)v f (v)F n−2 (v)
= Rk − 0
×
n k
F n−k (v)(1 − F (v))k dv 1
= Rk −
n−k
0
×F
n−k−1 1
n−1
0
v F n−1 (v)f (v)
n! k!(n − k − 1)!
(v)(1 − F (v))k dv n−1 n−k
v F n−1 (v)dFYk+1,n (v).
Observe that the quantity v F n−1 (v) is the ex-ante willingness of a bidder with type v to pay. Obviously, it is higher than the bidder’s bid b(v). Thus, we have
≤ Rk −
n−1 n−k n−1 n−k
1
v F n−1 (v)dFYk+1,n (v)
0 1
0
b(v)dFYk+1,n (v) = Rk −
Rearranging completes the proof.
n−1 n−k
Rk+1 .
In the following proposition we show that the tail of the lowest n − k expected efforts is bounded by the kth expected effort. Proposition 2. Let m = 1. Then, the revenue generated by the kth highest effort is greater than the revenue generated by the sum of the n successive efforts, namely Rk ≥ i=k+1 Ri . Proof. By Proposition 1, Rk ≥
2n−1−k Rk+1 n−k
≥ 2Rk+1 . Thus,
Rk ≥ 2Rk+1 ≥ Rk+1 + 2Rk+2 ≥ Rk+1 + Rk+2 + 2Rk+3
≥ ··· ≥
n
Ri .
i=k+1
3. A single prize and a finite number of participants
Rk =
Rk+1 .
n−k
(1)
Thus, G is a distribution function of the mth highest valuation and its density is given by
b(v) = v G(v) −
2n − 1 − k
Rk+1 = Rk −
m−1
G′ (v) =
Rk ≥
= Rk −
where G(v) is the probability that in equilibrium, participant i will win one of the m prizes if his or her valuation of the prize is v . Given that the equilibrium effort function is monotonic with respect to v , G(v) is the probability that the value v is one of the m highest valuations among the n participants and is given by
j=0
Proposition 1. Let m = 1. Then,
= b(1) −
We initially consider a contest with m identical prizes and n risk-neutral participants, each one of whom has a unit demand. Later on, we will consider risk-averse participants. Each participant has a private valuation v for a prize that has been drawn independently from a continuously differentiable distribution function F (v) over the support [0, 1] with a strictly positive density F ′ = f > 0. Moreover, the valuations are the private information of the participants. Participant i makes an effort bi (e.g., resources, effort, etc.) independent of other participants. The m participants with the highest effort win a single prize, but all of the n participants pay their effort. To find the symmetric Bayesian equilibrium effort function, we follow the standard arguments (see, for example, Krishna [6]). Assume that there exists a symmetric and monotonic equilibrium effort function b(v). The expected payoff for a participant with value v when he or she is playing vˆ as his or her valuation, and all other n − 1 participants are playing according to the equilibrium effort strategy b(v), is given by
G(v) =
The following proposition gives the bounds on Rk for finite n.
(5)
The last result is based on the bound Rk ≥ 2Rk+1 , which is a weaker version of Proposition 1. However, the decline of Rk is sharper, particularly when k is increasing. In the extreme case, when k = n − 1, Rn−1 ≥ nRn . We conclude that if the organizer can recover just a few of the highest efforts, the sum of the efforts that are unrecoverable is minor.
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
Appendix. Proof of Lemma 1
4. Many prizes and many risk-averse participants In this section we consider the case with m prizes and many participants. We first provide a simpler proof for Archak and Sundararajan’s [1] result that we then use in the following result. Later on, we will use the method in the proof for Proposition 3. First, assume that the participants are risk-neutral. Lemma 1. Let R∞ k = limn→∞ Rk . Then, m−1
R∞ k =
k+i−1
1 2k+i
i
i=0
k = 1, 2, . . . .
(6)
ra,∞
.
Rk =
Observe that V (v) ≥ 0, because otherwise the participant would make zero efforts in order to avoid losses. By the concavity of U we have
V (v) ≤ U G(v)(v − b (v)) − (1 − G(v))b (v) = U G(v)v − bra (v) .
ra
ra
It follows that G(v)v ≥ bra (v). Thus, 1 0
0
0
v G(v)dFYk,n (v) ≥
0
1
bra (v)dFYk,n (v) = Rra k
,∞ v G(v)dFYk,n (v) ≥ Rra . From Lemma 1 k 1 ∞ (see Appendix A) we get Rk = limn→∞ 0 v G(v)dFYk,n (v).
and in the limit limn→∞
1
G(s)ds dFYk,n (v).
Define 1
v G(v)dFYk,n (v), 1
k
v
B = 0
(7)
G(s)ds dFYk,n (v).
(8)
First we will prove that limn→∞ Bk = 0 Integrating by parts (8) yields Bk =
1
G(s)ds −
1
G(v)FYk,n (v)dv
0
0 1
G(v)(1 − FYk,n (v))dv ≤
= 0
Since limn→∞
1 0
1
G(v)dv.
0
G(v)dv = 0, we find that limn→∞ Bk = 0.
(n)!
Now we will focus on Ak . Substituting dFYk,n (v) = (k−1)!(n−k)! F n−k (v)(1 − F (v))k−1 f (v)dv and (1) in (7) yields.
1
k
v
A = 0
m−1
n − 1 j
j =0
F
n −j −1
(v)(1 − F (v))
j
(n)! F n−k (v)(1 − F (v))k−1 f (v)dv (k − 1)!(n − k)! 1 m −1 (n)! n−1 = v f (v) j (k − 1)!(n − k)! 0 j =0 ×
× F 2n−j−k−1 (v)(1 − F (v))k−1+j dv m −1 (n)! n−1 = β(2n − k − j, k + j) (k − 1)!(n − k)! j=0 j 1 v f (v)F 2n−j−k−1 (v)(1 − F (v))k−1+j dv 0 × β(2n − k − j, k + j) m −1 (n)! n−1 = (k − 1)!(n − k)! j=0 j × β(2n − k − j, k + j)E (Yj+k,2n−1 ) dv
Proof. The expected utility for a participant with valuation v is V (v) = G(v)U (v − bra (v)) + (1 − G(v))U (−bra (v)).
v
v G(v) −
0
Proposition 3. Let m ≥ 1. Then,
1
Ak =
Proof. See Appendix A. Observe that in particular, if m = 1, then R∞ = 21k , k = k 1, 2, . . . . It follows that the contribution of the k highest participant to the organizer’s expected revenue is equal to the total contributions of all of the ranked from k + 1 to infinity. In participants ∞ ∞ other words, R∞ k = j=k+1 Rj . An additional interesting observation is that Rk is not necessarily monotonically increasing with n. For example, consider the case where F (v) = v 6 . Then, for n = 2 calculations show that R1 > 1/2. 1 , it follows that R1 is not increasing with n. However, since R∞ 1 = 2 n Notice, however, that j=1 Rnj , which is the sum of all of the efforts, is monotonically increasing with n (e.g., Krishna [6]). Assume that every participant has an identical utility function U (x) where U (0) = 0, U ′ > 0 and U ′′ < 0. Let Rra k be the kth highest ra,∞ = participant’s expected effort in the risk-averse case and let Rk limn→∞ Rra . Fibich and Gavious [4] show that when m = 1, R∞ > k Rra,∞ where R∞ , Rra,∞ are the total expected efforts made by all participants in the case of risk-averse participants risk-neutral and ∞ ∞ ra,∞ . Thus, in a contest respectively. Therefore, k=0 R∞ k > k=0 Rk with many participants, the contest’s organizer prefers riskneutral participants. We will show that this result is much stronger ra,∞ for every k. This result is not straightforward and that R∞ k ≥ Rk because according to Fibich, Gavious and Sela [5], a low type risk-averse participant bids less aggressively than a risk-neutral participant. However, a high type risk-averse participant bids more aggressively than a risk-neutral participant. Thus, revenue ranking is not immediate when comparing risk-neutral and risk-averse ra,∞ ∞ with Fibich settings. Observe that the result ∞Rk ∞≥ Rk∞, together ra,∞ , implies that and Gavious’ [4] result that k=0 Rk > k=0 Rk ra,∞ for some k’s, a strict inequality holds. In other words, R∞ . k > Rk
R∞ k ≥ Rk
Substituting (3) into (5) gives
0
,
121
where β(2n − k − j, k + j) is the beta function with parameters 2n − k − j and k + j given by
β(2n − k − j, k + j) =
(2n − k − j − 1)!(k + j − 1)! (2n − 1)!
(9)
and E (Yi+k,2n−1 ) is the expectation of the random variable Yi+k,2n−1 defined above in Eq. (4) as the i + kth order statistic of 2n − 1 i.i.d. random variables with the distribution F (v). Then, for fixed i + k, when n → ∞ the random variable Yi+k,2n−1 is the 100% percentile. It follows that, limn→∞ E (Yi+k,2n−1 ) = 1. Thus, lim Ak = lim
n→∞
n→∞
0
×
n!
(k − 1)!(n − k)!
m−1
n − 1 j=0
j
(2n − k − j − 1)!(k + j − 1)! (2n − 1)!
122
A. Gavious, Y. Minchuk / Operations Research Letters 42 (2014) 119–122
(k + j − 1)! (n − k + 1) · · · n · (n − j)(n − j + 1) · · · (n − 1) lim n→∞ ( k − 1 )! j ! (2n − k − j)(2n − k − j + 1) · · · (2n − 1) j=0
m−1
=
m−1
=
k + j − 1 j=0
j
yielding the result.
lim
n→∞
nk · nj
(2n)k+j
m−1
=
k + j − 1 1 j=0
j
2k+j
References [1] N. Archak, A. Sundararajan, Optimal design of crowdsourcing contests, in: International Conference of Information Systems Proceedings, paper 200, 2009.
[2] R. Cavallo, S. Jain, Efficient crowdsourcing contests, in: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, vol. 2, 2012, pp. 677–686. [3] S. Chawla, D.J. Hartline, B. Sivan, Optimal crowdsourcing contest, in: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 2011, pp. 856–868. [4] G. Fibich, A. Gavious, Large auctions with risk-averse bidders, Int. J. Game Theory 39 (2010) 359–390. [5] G. Fibich, A. Gavious, A. Sela, All-pay auction with risk-averse bidders, Int. J. Game Theory 34 (2006) 583–599. [6] V. Krishna, Auction Theory, Academic Press, New York, 2002. [7] B. Moldovanu, A. Sela, Contest architecture, J. Econom. Theory 126 (2006) 70–96. [8] R.B. Myerson, Optimal auction design, Math. Oper. Res. 6 (1981) 58–73. [9] J.G. Riley, W.F. Samuelson, Optimal auctions, Amer. Econ. Rev. 71 (1981) 381–392.