APPENDIX A-H

FOR

TIGER WOMEN: AN ALL-PAY AUCTION EXPERIMENT

ON GENDER SIGNALING OF DESIRE TO WIN

Appendix A: Proofs of Inference Rules

MMvsFF

Proposition 2: If males have higher valuation than females (V:M>F) and are less
risk averse, i.e., more risk prone (R:M>F), then MM>FF:

r ru^ _ ^(0) - U M (-b)

f-'MMK' 3 ) —

u M (y M - b) - u M (-b)

U M (0) - U M {-b)

<

u M (v F -b)+ mu m (v m - v F ) - u M (-b)

U M (0) - U M (-b)

U M (V F - b) - U M (-b)

^ I/ F (0) - U F {-b)

< u F (v F -b)-u F (-b) - GffW

MFvsFM

Proposition 3: If males have higher valuations than females (V: M>F), and are

more risk averse (R: M<F) , then MF>FM.

Proof:

r fh ._ u M (YM-VF)-u M (.-b)

fmW U m {V m - b) - U M (-b)
u m (o)+mu m (v m - v F ) - u M (-b)

~ u M (y F -b)+ mu m (v m - v F ) - u M (-b)

U M (0) - U M (-b)

>

U M (V F -b)-U M (-b)

t/ F (0) - u F {-b) _
> u F (v F -b)-u F (-b) - GmfW

Electronic copy available at: http://ssrn.com/abstract=2141181

MFvsMM

Proposition 4: If males have higher valuations than females (V: M>F) and are less
risk averse (R: M>F), then MF<MM.

Proof:

U P (0) - U p (-b)

G MF {b) =

>

u F (y F -b)- U F (-b)
f/ F (0) - U F C-b)

U F (Vp -b)+ MU F (V M - Vp) - Up{-b)
U F (0) - Up(-b)

u F (y M -b)- Up{-b)

t/ M (0) - U M (-b) _
Um(Ym -b) - U M {-b)

Proposition 5: If males have lower valuations than females (V: M<F) and are less
risk averse (R: M>F), then MF<MM

Proof:

U M (0) - U M (-b)

Gmm\P)

<

<

U M (V M - b) - U M (-b)
U M (0)+MU M (y F - V M ) - U M (-b)

U m (Vm ~b)+ MU M (V F - V M ) - U M (-b)
U M (yF-VM)-U M (-b)
U M (y F - b) - U M (-b)

^ U F (Vp - V M ) - Up(-b) _

< Up(v F -b)- Up(-b) - GmfW

Therefore, as long as male are less risk averse, MF should be lower than MM.

MFVS.FF

Proposition 6: If males have higher valuations than females (V: M>F), then
MF=FF

Proof:

_ t/ F (0) - Up(-b) _ Up(q) - u F i-b) _
GffW ~ Up{Vp-b)-Up{-b) ~ Up{Vp -b)- Upt-b) ~ GmfW

Electronic copy available at: http://ssrn.com/abstract=2141181

Proposition 6: If males have lower valuations than females (V: M<F), then MF<FF

Proof:

GmfW =

U F (V F -V m )-U F (-b)

U F (V F -b)- U F (-b)
E/ F (0) + MU F (V F - V m ) - U F (-b)

U F (V F -b)- U F (-b)

^ t/ F (o) - uA-b)

> t^tz - = G FF {b)

U F (V F -b)- U F {-b)

Therefore, whenever males have higher valuation, MF=FF; whenever males have
lower valuation, MF<FF

Appendix B: Studies on Gender Difference in
Children

Changing the culture, the task and the age of gender may mitigate the possible
demand effects of self-consciously competitive choices, as apparently changing the
task from math to verbal in adults. A number of studies show no gender difference in
competitive attitude among children. See Table A.l.

Table A. 1: Recent studies on gender difference in competitive attitude among children.

Authors

Tasks

Age

Country

Results

Booth &
Nolan (2011)

Solving mazes

(Gneezy, Niederle,

and Rustichini,

2003)

Mean
under

15

UK

No difference between single-sex

school girls and boys from co-ed

and single schools

Cotton,
Mclntyre &
Price, 2009

Simple math,
reading

9-12

US

No difference after 1 st round of

5 -round math contest. No

difference in non-contest or

reading treatments.

Cardenas et al.
(2011)

Running, skipping

rope, math and

word search

9-12

Colombia
and Sweden

No difference in Colombia;

Swedish girls were more

competitive than boys in some

tasks, and boys more likely to

choose to compete in general

Dreber, Von

Running;

7-10

Sweden

No difference

Essen, and
Ranehill
(2011):

skipping rope and
dancing

Gneezy &

Rustichini

(2004)

Running

9-10

Israel

Boys increase performance when
competing, but girls do not.

Khachatryan
(2011):

Running and

skipping/j umping

rope; Math or word

search task

8-16

Armenian

No difference for physical and

math task, but more competitive

in word search task.

Savikhin
(2011)

Electronic fishing
task

3-5

US

No difference

Sutter &
Rutzler (2010)

Younger: running;
older: math

3-18

Austria

Men are much more willing to

enter a competition than women

in any age.

Zhang (201 lb)

Simple math

(NV2007)

15,16

Rural Han
Chinese

No difference in competitive
attitude, control for ability,
confidence and risk attitude

Appendix C: Instructions (read out loud)

Welcome to our experiment! In this experiment, you will need to make a decision
which will affect your payoff. So please pay attention when I explain the rules of the
experiment. If you miss any details, please raise your hand. Please do not talk to other
subjects sitting around you and please shut off your cell phone. Any behavior that
violates the rules of the experiment is prohibited, and we will refuse to pay the subject.
The whole experiment will be finished within 15 minutes. Now please open your
envelope and take out the paper.

• Please write down your name in the left block.

• Your opponent is also a subject in this experiment, you can find the school or
school and gender of your opponent in the right block.

• We paired you and your opponent randomly before today.

• There will be an auction between the two of you.

• From now on, each of you is endowed with 10 CNY

• When the auction begins, you will use the 10 CNY we give you to bid in the
auction.

The prize of this auction is 10 CNY as well.

Both you and your opponent can only bid once in the auction. When you

decide your bid, please mark the corresponding circle on the right graph.

Please mark only one circle. Mark more than one will be treated as a mistake.

If you do not mark any circle in the graph, then you are bidding zero.

Please note that if your opponent chooses a lower bid than you do, you are the

winner in the auction and earn the extra 10 CNY, which is the prize of the

auction. Your opponent earns no extra money since he/she loses, but he/she

still has to pay his/her bid.

In the same reasoning, if your opponent chooses a higher bid than you do,

he/she will be the winner and earns the extra 10 CNY. But both of you have to

pay your own bid.

If both of you choose the same bid, you will split the 10 CNY prize and pay

your own bid.

If both of you bid zero, that is, both of you do not mark any circle, none of you

earns any additional payment.

Please note that your final payment in this experiment is exactly equal to the

payment after you bid using the 10 CNY we give you in this auction. We do

not provide any other payment.

Here are some examples to help you understand this auction

■ If you bid 3, and your opponent bids 7, you receive 7.

■ If you bid 5, and your opponent bids 4, you receive 15.

■ If you bid 10, and your opponent bids 9, you receive 10.

■ If you bid 6, and your opponent bids 6, you receive 9.

■ If you bid 8, and your opponent bids 10, you receive 2.

■ If you bid 0, and your opponent bids 0, you receive 10.

■ If you bid 2, and your opponent bids 1, you receive 18.

■ If you bid 1, and your opponent bids 8, you receive 9.

■ If you bid 7, and your opponent bids 2, you receive 13.

■ If you bid 10, and your opponent bids 10, you receive 5.

Please now decide your bid.

After you finish marking your bid, please write down your name and bank

account in the box, and then put this paper back to the envelop.

Appendix D: Tie Breaking and Discretization

Let inc=the discrete increment of the bid. If ties are broken by equal probability of
getting the prize (hereafter "flip", as flip of a coin), than in a mixed strategy equilibrium,
player 1 's expected utility equals the utility from the expected payoff from the auction:

U 1 {V 1 - b 1 )G 2 {b 1 - inc) + I/iC-fciXl - G 2 (^)) + (il/^Vi -

t>i) +\u 1 {-b 1 )\ (G 2 Oi) - G2O1 - inc)) = Unexpected payoff)

(1)
After rearrangement,

G 2 {b 1 inc) - Ui(Vi _ bi) _ Ui( _ bi) + 2 * PWJ

(2)
If ties are broken by splitting the prize (hereafter "split"), the mixed strategy
equilibrium relation becomes:

U 1 {V 1 - b 1 )G^b 1 - inc) + U^-b^l - G 2 *(^)) +
Ud\V- MCGIOi) - GKbi ~ inc)) = U^epected payoff*)

(3)
After rearrangement,

, . Unexpected payofD-Uit-b!) , ( t/ i(| y - ft i)- t/ i(- ft i)J , f .

G 7 ( D-i — inC ) = ; ; ; : ; — * pr ? ( Di )

(4)

Note that in our experiment, subjects received 5 CNY in ties: where V is the
monetary prize of the auction, which is equal to both players..

It is easy to see that for risk averse players,

1 (uA\v-b i )-Ui(.-b{))

2 t/i(Vi-bi)-Ui(-bi)

(Che and Gale, 1998) showed that when there was a cap m in an all-pay auction

and m G (— , V2), then the expected payoff of player 1 is V ± — V 2 , and that of player 2

is 0, (assuming V x > V 2 ), which is the same as the no cap case. Thus, the mixed
strategy equilibrium for both players in each case are:
Flip:

(6)

G 2 (^ inc) - Ui(Vi _ bi) _ Ui( _ bi) +~*pr 2 i.b J

Split:

_„. . , y 2 ( )-y 2 (-& 2 ) , (y 2 (^-6 2 )-y 2 (-b 2 ))

Gi O2 - inc) = / - + - — -, ^ ; — - 1 * pr-, *(b 2 )

1V 2 7 u 2 (v 2 -b 2 )-u 2 (-b 2 ) u 2 (y 2 -b 2 )-u 2 {-b 2 ) H 1 v lJ

G * (b inc) - "i^HiK) ■ Ki^O^t^) , pr * fZj -)

I 2V X ' l/iCVi-J>i)-tfi(-6i) l/iCVi-6i)-tfi(-6i) ^ 2 V ^

To see the difference between the flip and split CDFs in our experiment, we

simulated them for CRRA utility function:

X i-Y

U(x) = -

1-7

Based on the settings in the experiment, initial wealth w=bidding cap=10,

increment=0.5, then we can solved CDFs by the following systems of linear equations:

1
Flip: For each b 1 G (-, 1 10), we have an equation with one unknown

ffrfa - inc) = il0 X-Z^-W-Z*-r* + 2 * W ^ (8)

G 2 (10) = 1

with a total of 20 equations for 20 unknowns.

Split: For each b 1 G (-, 1 10), we have an equation with one unknown

(G (b inc-) = 1 ° 1 " Kl -( 1 °- b i) 1 " Kl I ((iQ+s-hQWi-do-ftQi-ri) t f .

j 2V ! ' (10+V 1 -fc 1 ) 1 -l / l-(10-fc 1 ) 1 -l / l (10+K 1 -fc 1 ) 1 -l / l-(10-fc 1 ) 1 -l / l V 2V ^

I G 2 (10) = 1

(9)

with a total of 20 equations for 20 unknowns.

The following examples show that these flip and split tie breaking rules have

similar effect on CDF of bids, given the CRRA risk aversion coefficient y equals to
0.00001 (nearly risk neutral), 0.5 (normally risk averse), and 0.99999 (extremely risk
averse). As can be seen in Figure I below, the CDFs of bids under different tie breaking
rules: flip a coin or split the prize are very similar.

■gamma=0.00001_flip
■gammal=0.00001_split

gammal=0.5_flip
•gammal=0.5_split
•gamma l=0.99999_flip

gammal=0.99999_split

0.5 1 1-5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Figure I: Shape of G2 when Vl=10, V2=10, cap=10

-gammal=0.00001_flip
■gamma 1=0. 00001_split

gammal=0.5_flip
-gammal=0.5_split
»gammal=0.99999_flip

gammal=0.99999_split

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 55 6 6.5 7 7.5 8 8.5 9 9.5 10

Bids

Figure II: Shape of G2 when Vl=15, V2=10, cap=10

-gammal=0.00001_flip
-gammal=0.00001_split

gammal=0.5_flip
»gammal=0.5_split
-gammal=0.99999_flip

gammal=0.99999_split

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Bids

Figure III: Shape of G2 when Vl=20, V2=10, cap=10

Appendix E: Cumulative distribution functions

FOR INITIAL STUDY

Only males' bids increased significantly after finding out the gender of opponent
(p=10%).

6{b)

— MC=4.0
— MF=5.4

— i — i i — i — i — i i — i — i — i — i — i i — i — i — i — i — i — i — i — t D— CIMY
0051 152253 3.5 4455556 6.5 775885995 10

G(b)

— FC=4.9

— FM=5.1

i t i — i — i— i— i— i — i — i— t— i— i— i— i — i— i — i — i— i b— CNY
0.5 1 1.5 2 2 5 3 3 5 4 4.5 5 5.5 6 6.5 7 7 5 8 8 5 9 9 5 10

Figure IV: CDF of initial study with initial sample of 92 subjects

G(b)

— MC=4.8
— MF=4.6

b=CNY

5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5 6 6 5 7 7 5 8 8 5 9 9 5 10

G(b)

— FC=5.0
— FM=5.4

b=CNY

5 1 1 5 2 2 5 3 3 5 i 4 5 5 5 5 6 6 5 7 7 5 8 8 5 9 9 5 10

Figure V: CDF of initial study with enlarged sample 156 subjects.

Appendix F: CDF of Bids in Treatments

G(b>

: : = . . =

LS 5 -3-5 1 t-5 5 5.5 6 &. S

:s 3 3.5 9 S.S lO

Figure VI: CDF of bids within SZ.

Figure VII: CDF of bids in UT

Figure VIII: CDF of SZ bids against UT.

Appendix G: An example of Bidding sheet

You are...
Name :

Please write down your name hi the left block
Your opponent is also a subject in this experiment. You
can find the school or school and gender of your
opponent in the right block.

We paired you and your opponent randomly before today.
There will be an auction between the two of you.
From now on, each of you is endowed with 10 CNY.
When the auction begins, you will use the 10 CNY we
gave you to bid in the auction.
The prize of this auction is 10 CNY as well.
Both you and your opponent can only bid once in the
auction. When you decide your bid. please mark the
corresponding circle on the right graph. Please mark only
one circle. Marking more than one will be treated as
mistake. If you do not mark any circle in the graph, you
are bidding zero.

-Y- Please note that if your opponent chooses a lower bid
than you do, you are the winner in the auction and
earn the extra 10 CNY which is the prize of the
auction. Your opponent earns no extra money since
he/she loses, but he/she still has to pay his bid.
In the same reasoning, if your opponent chooses a
higher bid than you do. he/she will be the winner and
earns the extra 10 CNY But both of you have to pay
your own bid.

If the two of you choose the same bid, you will split
the 1 CNY prize and pay your own bid.
If both of you bid zero, that is, both of you do not
mark any circle, then none of you earns any
additional payment.
Please note that your final payment in this experiment is
exactly equal to the payment after you bid using the 10
CNY we give you hi this auction. We do not provide any
other payment.
Please now decide your bid.

After you finish marking your bid, please write down
your name and bank account in the box, and then put this
paper back to the envelop.

>

A

♦

Your opponent is...

A Female student in UT

Name

Bank account No.

10

o

95

o

9

o

8.5

o

8

o

7.5

o

7

o

6,5

o

6

o

5.5

o

5

o

4.5

o

4

o

3.5

o

9
.1

o

2.5

o

2

o

1,5

o

1

o

0,5

o

Figure IX: Bidding sheet

As discussed in the main text, the only thing which changed in our bidding sheets for
different treatments was the gender or school of the opponent, or both. Subjects in the

incomplete information treatments did not know either.

Appendix H: Payment in treatments

SZ-M

SZ-F

SZ-C

UT-M

UT-F

>

UT-C

SZ-M

8.8

56%

8.8

7.5

9.7

74%

9.4

63%

73%
26%

87%

0.05%

83%

63%

SZ-F

9

48%

8.2

10.8

8.6

UT-M

11.3

10.3

29%

11.4

8.1

27%

7.5%

78%

2.7%
18%

0.68%

29%

UT-F

■

9.5

74%

8.8

9.8

8%

8.1

9.4

Figure X: Average payoffs of bidders in CNY and p-values. % between numbers are p-value of

differences.

Appendix I: Other evidence of women's greater

SELF-DISCIPLINE

We now discuss how our main result, that women have a higher DTW, aligns with
empirical evidence suggesting that women are more self -disciplined than men. DTW,
derived from willingness to pay to win in auctions should be predictive of willingness
to pay to win in other domains of competition. Perhaps the most important expression
of DTW is the willingness to forgo leisure in the preparation for real life competitions,
where leisure is a constant temptation. Assuming that studying was the cost of doing
well on exams, our results would be consistent with women being more competitive
than men for exams for which they can prepare/forgo leisure. In fact, Duckworth and
Seligman (2006), Duckworth, Quinn, and Tsukayama (2011) showed that girls do
better than boys in non-IQ type tests, e.g., spelling competitions, and that due to greater

self-discipline or "grit". The greater persistence of women was also suggested by
Cotton, Mclntyre, and Price (2009), which showed that the male advantage in math
competition against females in the US disappeared after the 1 st round. Desjarlais (2009)
showed that girls selected into math competitions (AMC 8 Contest) at virtually the
same rate as boys (183,857 males vs. 178,857 females) with no differences in measured
abilities. Girls are graduating high school, college and graduate schools at higher rates
in the US (Buchmann, DiPrete, and McDaniel 2008), though males do better than
females in most standardized tests including SAT, GRE, GMAT, AP (Coley 2001). The
pattern is similar for grade school education in less developed countries (Grant and
Berhman, 2010), including China (Lai, 2010). Even when fixing the course of study to
law, the only field for which we could find data, the almost equal number of women
LSAT test takers (Dalessandro et al., 2010), though women have lower measured
ability on the LSAT, suggests that women are in fact more competitive than men. Lower
ability with higher achievement implies that women are paying more in effort and
leisure than men. Chinese girls may be particularly willing to pay due to the traditional
Chinese cultural preference for boys, which has been recently exacerbated by the one
child policy. This is supported by anecdotal evidence that girls may have to "prove their
worth" to the family and the exceptionally high suicide rate of girls in China 2 .
Consistent with this, Zhang (2011b) found no gender differences in competitive attitude
among Han Chinese women, but did find it with their neighboring non-Han (minority)
Chinese women, who were less restricted by the one child policy.

The significant change in risk attitude from SZ and UT could be due to Chinese
universities reliance upon entrance exams with predictable content. For these exams,
success is mostly a matter of very tedious preparation. In contrast, US schools may
exert less gender specific selection by IQ like tests, grades, recommendations,
interviews, and extracurricular activities all of which may be difficult to prepare for.
Furthermore, a man's marriageability is often a matter of his income. Graduate school
is not known for it's lucrativeness. These factors may select out more daring men who

http://www.who.int/mental_health/prevention/suicide_rates/en/

might want to try their luck in the market.

The fact that women are apparently less competitive in their labor market
outcomes, given greater competitiveness in school, could be due to other factors like
marriage to even more competitive husbands and motherhood. Ancetol (2011) showed
that labor market participation of white women is non-monotonic on their level of
education. This could be due to their education being correlated with their husbands'
education and ambition. Shafer (201 1) showed that women's labor force participation is
decreasing on the income gap with their husband's income. Australian women's
reported life satisfaction increased if their partner worked full time but decreased if they
worked full time (Booth and Ours, 2009). Charles (2011) showed that women in richer
countries tended to adopt more traditional gender roles.

Finally, risk attitude may be more important than educational attainment for
becoming a top executive. Capelli and Harmoni (2005) found in 2001 that only 10% of
top executives at Fortune 500 companies had ivy-league educations. Selection by risk
attitude may be even more important for founding CEOs in high tech industries like Bill
Gates , Steve Jobs , or Larry Ellison , who were all conspicuous college dropouts.
Male lower self discipline could be an advantage for the most able men for such
positions. The susceptibility to the temptation for salient competitions, i.e., showing off,
could drive low ability men into street fights, basketball games, or dropping out of
highschool, at the same time that it drives high ability men to the "workaholism" that
might be required to become top executives. It may be to women's credit that they are
"under-represented" in risky winner takes all fields that require an all consuming
investment of effort to even have a chance of success. To make a valid assessment, we
would also need to find the proportion of women among the failed Gates, Jobs and
Ellisons, who subsequently regretted dropping out of college. To our knowledge, no
such data exists.

2 http://en.wikipedia.org/wiki/Bill Gates#Early life

3 http://en.wikipedia.org/wiki/Steve Jobs#Early life and education

4 http://en.wikipedia.org/wiki/Larry Ellison#Career

Women's greater risk aversion may be the key to reconciling the data prior to what
we presented in this paper, both empirical and experimental. Greater risk aversion
makes women put in less effort in competitions in which they cannot prepare, like those
in the laboratory setting. At the same time, it could make them prepare more if they
could prepare, like in academic competitions. Costly preparation allows women to
simultaneously raise their grades, while lowering the variance of their performance, but
at the cost of decreasing their total surplus from lost leisure. Women would be more
willing to make that trade-off than men if they are more risk averse. In such an
equilibrium, they would have a higher mean performance, but be under-represented in
the top tail of the distribution.