A Numerical Study with Experimental Data on
Risk-Averse Subcontractors in Procurement
Auctions with Subcontract Bids
Naoki Watanabe†† Keio Business School, JapanIEEE BigData 2021 (CDEC),
Dec 15-18, 2021

Introduction
Theoretical Study
Numerical Study
Conclusion

My student and I were preparing for a presentation on the
incentive schemes for information exchange among employees
of a leading company in Japan, which may be called internal
information exchange market.
Unfortunately, it is still incomplete.
This time, I will propose a methodological issue for theexperiment which is supposed to be conducted in the near
future.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

1. Introduction: Subcontract Auctions

Prime contractors (PCs) solicit bids from subcontractors(SCs), before their procurement (”reverse”) auctions,◦ to estimate the costs of subcontractable works◦ to lower the total project costs

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

1. Introduction: Subcontract Auctions

Prime contractors (PCs) solicit bids from subcontractors(SCs), before their procurement (”reverse”) auctions,◦ to estimate the costs of subcontractable works◦ to lower the total project costs
In a subcontract (”reverse”) auction, there are◦ a PC as auctioneer◦ SCs as bidders

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

A Simple Model
SCij , j = 1, 2,

SC11

SC12

SC21

SC22

t11

t12

t21

t22

C



C
s21 C



s11 C
C
CW

 s12



 s22



C
CW

PC1

PC2

c1 = p1c2 = p2C
b1 C◦ cost: tij , iid,
◦ bid: sij to PCi◦ payment:
pij1st = min{si1 , si2 }pij2nd = max{si1 , si2 }if sij is lowest;
pij = 0 otherwise.


b2 



C
CCW
Procurement Buyer

2 PCs, i = 1, 2
◦ payment to SC:
pi = pij1st or pij2nd◦ cost: ci = pi
◦ bid: bi

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

A Simple Model
$
'

'

$

SCij , j = 1, 2,

SC11

SC12

SC21

SC22

t11

t12

t21

t22

C



C
s21 C



GU1 s11 C
'
C
CW

 s12


PC1

◦ payment:
 s22 GU2
pij1st = min{si1 , si2 }$pij2nd = max{si1 , si2 }if sij is lowest;
pij = 0 otherwise.

c2 = p2

%
&
C

%


b1 C

2 PCs, i = 1, 2
◦ payment to SC:
pi = pij1st or pij2nd

b2 



C
CCW
Procurement Buyer&◦ bid: sij to PCi

PC2

c1 = p1

&




C
CW

◦ cost: tij , iid,

◦ cost: ci = pi

GD%◦ bid: bi

4 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Theory and Experiment
Assume that subcontractors are equally risk-averse.
Theoretical Predictions

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Theory and Experiment
Assume that subcontractors are equally risk-averse.
Theoretical Predictions
1(i) SCs submit higher bids in FPA (1st price reverse auction)
than in SPA (2nd price reverse auction), regardless of theirdegree of risk-aversion. (ii) In FPA, SCs lower their bids as
their degree of risk-aversion increases.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Theory and Experiment
Assume that subcontractors are equally risk-averse.
Theoretical Predictions
12(i) SCs submit higher bids in FPA (1st price reverse auction)
than in SPA (2nd price reverse auction), regardless of theirdegree of risk-aversion. (ii) In FPA, SCs lower their bids as
their degree of risk-aversion increases.
SPA renders higher expected profits to PCs than FPA.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Theory and Experiment
Assume that subcontractors are equally risk-averse.
Theoretical Predictions
123(i) SCs submit higher bids in FPA (1st price reverse auction)
than in SPA (2nd price reverse auction), regardless of theirdegree of risk-aversion. (ii) In FPA, SCs lower their bids as
their degree of risk-aversion increases.
SPA renders higher expected profits to PCs than FPA.
The ex post efficient allocation of subcontract work is achieved
when the subcontract auctions are FPAs, whereas it is notnecessarily achieved when the subcontract auctions are SPAs.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Theory and Experiment
Assume that subcontractors are equally risk-averse.
Theoretical Predictions
123(i) SCs submit higher bids in FPA (1st price reverse auction)
than in SPA (2nd price reverse auction), regardless of theirdegree of risk-aversion. (ii) In FPA, SCs lower their bids as
their degree of risk-aversion increases.
SPA renders higher expected profits to PCs than FPA.
The ex post efficient allocation of subcontract work is achieved
when the subcontract auctions are FPAs, whereas it is notnecessarily achieved when the subcontract auctions are SPAs.

Experimental Results
All predictions were statistically observed except Prediction 2.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Numerical Study and its Suggestions
Remove the assumption that subcontractors are equally risk-averse.
Numerical Study
The unclear experimental result might be due to the presenceof extreme patterns in the distributions of risk aversion
coefficients among the subjects.

6 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Numerical Study and its Suggestions
Remove the assumption that subcontractors are equally risk-averse.
Numerical Study
The unclear experimental result might be due to the presenceof extreme patterns in the distributions of risk aversion
coefficients among the subjects.
Suggestions
In experiments for data exchange or trades of information
which will be conducted, construct first a subject pool inwhich the coefficients of subjects’ risk attitudes are measured
in advance.

6 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Numerical Study and its Suggestions
Remove the assumption that subcontractors are equally risk-averse.
Numerical Study
The unclear experimental result might be due to the presenceof extreme patterns in the distributions of risk aversion
coefficients among the subjects.
Suggestions
In experiments for data exchange or trades of information
which will be conducted, construct first a subject pool inwhich the coefficients of subjects’ risk attitudes are measured
in advance.
It is difficult to assume that subjects’ risk attitude are kept
intact during a session.

6 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontract Auctions
A Simple Model
Main Result

Main Result: Numerical Study and its Suggestions
Remove the assumption that subcontractors are equally risk-averse.
Numerical Study
The unclear experimental result might be due to the presenceof extreme patterns in the distributions of risk aversion
coefficients among the subjects.
Suggestions
In experiments for data exchange or trades of information
which will be conducted, construct first a subject pool inwhich the coefficients of subjects’ risk attitudes are measured
in advance.
It is difficult to assume that subjects’ risk attitude are kept
intact during a session.
Controlling subjects’ (moderate or extreme) risk attitudes⇒ Robustness check of the performance of (internal or formal)
Data Exchange Markets.
6 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

2. Theoretical Study: Subcontractors’ Bidding Functionsik = s(tik ); bidding function of subcontractors.
tik is independently and identically distributed over [t, t̄].
Subcontractors are risk-averse. Subcontractor SCik obtains
utility u(y ) = y rik , when it receives income y ;(1 − rik ) ∈ (0, 1) is the risk aversion coefficient (constantrelative risk aversion, CRRA). For any SCik , let rik = r for
simplicity (for the theoretical part).
A dominant strategy (equilibrium bidding function of
subcontractors) in SPA for each SCik iss ∗∗ (tik ) = tik ,(1)regardless of any risk aversion coefficients
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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

In FPA, assuming that PCs use an identical bidding function,
each SCik determines his or her bid sik so as to maximize hisor her expected utility(sik − tik )r Prob(sik < min(sik 0 , sj1 , sj2 ))= (sik − tik )r Prob(s −1 (sik ) < tik 0 )3= (sik − tik )r [1 − Prob(s −1 (sik ) ≥ tik 0 )]3
s −1 (sik ) − t 3] ,= (sik − tik )r [1 −t̄ − t
The first order condition (hereafter, FOC) iss 0 (tik )r (sik − tik )r −1 [t̄ − tik ]3 = 3(sik − tik )r [t̄ − tik 0 ]2 .
The symmetric equilibrium bidding function of subcontractors
in FPA ist̄ − tiks ∗ (tik ) = tik +
r.
(2)
3+r
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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Prime Contractors’ Bidding Functionbi = b(ci ); bidding function of prime contractors.
Prime contractors are risk-neutral.
If SPA is used in subcontract auctions, the cost ci of PCi is
independently and identically distributed over [c, c̄] = [t, t̄],∗∗ (t ) = t .
because si,ki,ki,kEach PCi determines its bid bi so as to maximize theexpected payoff(bi − ci )Prob(bi < bj ) = (bi − ci )Prob(b −1 (bi ) < cj )= (bi − ci )Prob(b −1 (bi ) < max(tj1 , tj2 ))= (bi − ci )[1 − Prob(b −1 (bi ) ≥ max(tj1 , tj2 ))]= (bi − ci )[1 − Prob(b −1 (bi ) ≥ tj1 )Prob(b −1 (bi ) ≥ tj2 )]b −1 (bi ) − c 2) ].
= (bi − ci )[1 − (c̄ − c9 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

The FOC isb 0 (ci )[(c̄ − c)2 − (ci − c)2 ] − 2bi (ci − c) = −2ci (ci − c).
The symmetric equilibrium bidding function of the prime
contractors in SPA isb ∗∗ (ci ) =(2/3)(c̄ 3 − ci3 ) − c(c̄ 2 − ci2 ),(c̄ − c)2 − (ci − c)2(3)

where ci = max{si1 , si2 } is NOT a function of r . (See SCs’
bidding function (1).)

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

When FPAs are used as subcontract auctions, the cost
cj = pj = min(s ∗ (tj1 ), s ∗ (tj2 )) of PCi is independently andidentically distributed over [c, c̄], where c = (3t + t̄r )/(3 + r )and c̄ = t̄, given the SCs’ bidding function (2).
Each PCi determines its bid bi so as to maximize theexpected payoff(bi − ci )Prob(bi < bj ) = (bi − ci )Prob(b −1 (bi ) < cj )= (bi − ci )Prob(b −1 (bi ) < min(s ∗ (tj1 ), s ∗ (tj2 )))= (bi − ci )Prob(b −1 (bi ) < s ∗ (tj1 ))×Prob(b −1 (bi ) < s ∗ (tj2 ))3ti1 + t̄r)= (bi − ci )Prob(b −1 (bi ) <3+r
3ti2 + t̄r
×Prob(b −1 (bi ) <)3+r((3 + r )b −1 (bi ) − c̄r )/3 − c 2= (bi − ci )[1 −)] .
c̄ − c11 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Let β(ci ) = ((3 + r )ci − c̄r )/3 (< c̄).
The FOC is
3b 0 (ci )(c̄ − β(ci ))2 − 2(bi − c)(c̄ − β(ci ))(3 + r ) = 0, i.e.,b 0 (ci )(c̄ − ci ) − 2(bi − c) = 0.
The symmetric equilibrium bidding function of prime
contractors in FPA isb ∗ (ci ) =c̄ 3 − ci2 (3c̄ − 2ci ),3(c̄ − ci )2(4)where ci = min{si1 , si2 } is a function of r . (See SCs’ bidding
function (2).)

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Theoretical Predictions

Suppose that subcontractors are equally risk-averse.
Prediction 1. (i) Subcontractors submit higher bids in FPA thanin SPA, regardless of their degree of risk-aversion. (ii) In FPA,
subcontractors lower their bids as their risk-aversion increases.
Prediction 2. SPA renders higher expected profits to prime
contractors than FPA.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Let t ∗ be the cost of the subcontractor who is awarded thesubcontract work.
We say that an allocation of subcontract work is ex post
efficient if the social surplus V − t ∗ is maximized.
In ex post efficient allocations, subcontractor SCik is awarded
the subcontract work if and only if sik < min(sik 0 , sj1 , sj2 ),where k 0 6= k and j 6= i.

Prediction 3. If the subcontractors are equally risk-averse, thenthe ex post efficient allocation of subcontract work is achieved
when the subcontract auctions are FPAs, whereas it is notnecessarily achieved when the subcontract auctions are SPAs.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Allocative efficiency in SPA: An example
SC11

SC12

SC21

SC22

1000

1900

1600

1800

C
1000 C 1900



C
CW


 1800C
1600 C



C
CW

PC1

PC2

c1 = p1c2 = p2

C


b2 



b C
1

C
CW
Procurement Buyer

DWL generated by SPA is 600 (risk-neutral PCs and SCs).
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Introduction
Theoretical Study
Numerical Study
Conclusion

Subcontractors’ Bidding Function
Prime Contractors’ Bidding Function
Theoretical Predictions

Allocative efficiency in SPA: An example
SC11

SC12

SC21

SC22

1000

1900

1600

1800

C
1000 C 1900



C
CW


 1800C
1600 C



C
CW

PC1

PC2

c1 = 1900

c2 = 1800

C


b2 



b C
1

C
CW
Procurement Buyer

DWL generated by SPA is 600 (risk-neutral PCs and SCs).
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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

3. Numerical Study: The Experiment
In total, 16 experimental sessions were conducted fromJanuary 2010 to March 2014.
Subjects were recruited from all over the campuses atUniversity of Tsukuba and Osaka University.
SC’s cost ∼ U[1, 000, 2, 000], procurement buyer’s
reserve=2,000.

Subsession 1
Subsession 2
Subsession 3

no. of PCs

PCs playedby machine

no. SCsfor each PC

no. subcont.
auctions

no. ofperiods

no. subjects
per period

1
2
2

—
Yes
No

2
2
2

20
20
40

10
10
20

4
4
6

Subjects were randomly matched and assigned to SCs or PCs
at the beginning of every period.
Machine bidders were assumed to be risk-neutral.
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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Subcontractors’ Bids
Bids
2000

Bids
2000

1800

1800

1600

1600

First-price auctions

1400

Second-price auctions

1400

Session 1: 1 PC
Session 2: 2 machine PCs

Session 1: 1 PC

1200

1200
Session 3: 2 human PCs

Session 2: 2 machine PCs

Theoretical Bid Function (Session 1)

Session 3: 2 human PCs
Theoretical Bid Function

Theoretical Bid Function (Session 2,3）
1000
1000

1200

1400

Costs

1600

1800

2000

1000
1000

1200

1400

1600

1800

2000

Costs

Figure: Typical Bids: Risk Aversion (Left)
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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Experimental Results

PC’s profit: Prediction 2 was not observed at the 5% level ofsignificance; p = 0.1155 for the permutation test in subsession
2 where machine bidders played as prime contractors.
FPA: the average was 215.4 with standard deviation 23.4.
SPA: the average was 251.6 with standard deviation 142.5.

DWL: Prediction 3 was statistically confirmed in subsession 2;
p = 0.040 for the permutation test.

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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Given sik , tik , and t̄ = 2000, the value of rik can be computedfrom the experimental data using formula (2); the averagevalue is the estimate of the common value of r .
The average value of rik was 0.4596 with standard deviation
0.4124 in a session, while it was 0.5870 with standard
deviation 0.4653 in another session. The average is about 0.5.
If subjects were equally risk-averse, then Prediction 2 would
be surely observed at the 1% level of significance, because thep-values for r = 0.5. (SCs’ costs were taken from the values
realized in the experiment.)

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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Table: Prime contractors’ profits simulated in the FPA with the value of CRRA
coefficients and those simulated in the SPA.
r = 1.0

r = 0.9

r = 0.8

r = 0.7

r = 0.6

mean 200.0062 205.1346 210.5329 216.223 222.2292
std dev 38.39252 39.37695 40.41318 41.50543 42.65836
p-value <0.00001 <0.00001 0.0000129 0.0000712 0.000377
r = 0.5

SPA

mean
228.5786 271.2249
std dev
43.87717 141.0522
p-value0.001889–a
The p-values for the permutation test: FPA vs. SPA

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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Numerical Inference

Assumption A: Examine two extreme patterns in distributionsof CRRA coefficients (the average is 0.5):(a) r11 = 0.4, r12 = 0.4, r21 = 0.8, and r22 = 0.8.
(b) r11 = 0.8, r12 = 0.4, r21 = 0.8, and r22 = 0.4.

Assumption B: Each subcontractor believes that the othersubcontractors were as risk-averse as he or she is.
Given the data of subcontractors’ costs, the profits of prime
contractors can be generated using the bidding functions (1),(2), (3), and (4).

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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inferencecase (a): p = 0.1219 for the permutation test: No significant
difference in PC’s profit between the FPA and SPA.
the average profit of the winning prime contractors was 218.0
with standard deviation 37.5 when the subcontract auctions
were FPAs, whereas it was 255.6 with standard deviation 122.1
when the subcontract auctions were SPAs.

case (b): similar results.
This numerical computation thus replicated the experimental
results, which were
FPA: the average was 215.4 with standard deviation 23.4.
SPA: the average was 251.6 with standard deviation 142.5.

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Introduction
Theoretical Study
Numerical Study
Conclusion

The Experiment
Numerical Inference

Conclusion. It is plausible that unclear results observed in theprevious experimental sessions was due to the presence of extreme
patterns in the distributions of risk aversion coefficients among the
subjects.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

4. Conclusion: Summary

For verifying the theoretical predictions in the experiment,
the difficulty did not lie in the bidding behavior of risk-averse
subjects, but it was controlling the distribution of subjects’risk attitudes.

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Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

Remarks: Discussion

Truly, it is difficult to assume that subjects’ risk attitude are
kept intact during a session. ⇒ Cox et al. (1985) proposed
the ex post adjustments of subjects’ risk attitudes.

25 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

Remarks: Discussion

Truly, it is difficult to assume that subjects’ risk attitude are
kept intact during a session. ⇒ Cox et al. (1985) proposed
the ex post adjustments of subjects’ risk attitudes.
It is, however, never difficult to construct a subject pool inwhich subjects’ risk aversion coefficients are measured inadvance.

25 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

Remarks: Discussion

Truly, it is difficult to assume that subjects’ risk attitude are
kept intact during a session. ⇒ Cox et al. (1985) proposed
the ex post adjustments of subjects’ risk attitudes.
It is, however, never difficult to construct a subject pool inwhich subjects’ risk aversion coefficients are measured inadvance.
moderate risk aversion coefficients⇒ verification of the validity of theoretical predictions in(internal or formal) Data Exchange Markets

25 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

Remarks: Discussion

Truly, it is difficult to assume that subjects’ risk attitude are
kept intact during a session. ⇒ Cox et al. (1985) proposed
the ex post adjustments of subjects’ risk attitudes.
It is, however, never difficult to construct a subject pool inwhich subjects’ risk aversion coefficients are measured inadvance.
moderate risk aversion coefficients⇒ verification of the validity of theoretical predictions in(internal or formal) Data Exchange Markets
extreme risk aversion coefficients⇒ robustness check of the performance of those markets

25 / 26

Introduction
Theoretical Study
Numerical Study
Conclusion

Summary
Discussion

Thank you for listening.
In the next opportunity, I will report the experimental results of“internal information exchange market” in a Japanese company.

26 / 26