Minimizing Shirking in Auctions and Tournaments

doi:10.4236/tel.2013.34033

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Theoretical Economics Letters, 2013, 3, 197-201

http://dx.doi.org/10.4236/tel.2013.34033 Published Online August 2013 (http://www.scirp.org/journal/tel)

Minimizing Shirking in Auctions and Tournaments

Chen Cohen1, Moshe Schwartz2

1Department of Economics, Ashkelon Academic College, Ashkelon, Israel

2Department of Sociology and Anthropo logy, Ashkelon Academic College, Ashkelon, Israel

Email: [email protected], [email protected]

Received April 12, 2013; revised May 12, 2013; accepted June 12, 2013

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The present paper deals with two-player all pay contests in which a tie is due to slacking, showing that to reduce the

likelihood of such an occurrence, slackers should be denied any reward, or even punished. The denial of reward, or the

punishment, inflicted on substandard performers, may spur some players to bigger efforts, or induce others to avoid

contests in which they are unable to meet standards. However denying any reward to those making small but not sub-

standard efforts, would not raise the proportion of those making the maximum effort, while more competitors would

abstain from any effort at all, detracting from overall performance. The point allocation rule su ggested by this paper is

thus shown to improve on its alternatives. The paper proposes changing the rules of point allocation in soccer, to reduce

the incidence of non -scoring draws, of ten the outcome of bad playing or of extr eme risk avoidance, expressed in purely

defensive game strategies. Under the new rules, a win would award a team with three points, a scoring draw would enti-

tle the teams to one point each, but neither team would receive any points for non-scoring draws. We show that this

change would reduce the numbers of games ending in 0:0, while raising the numbers of goals in other games, thus

boosting spectator enjoyment.

Keywords: Tournaments; Auctions; Nash Equilibrium; Soccer; Draws; All-Pay-Contests

1. Introduction

In winner-takes-all contests, the winner is the one who

invested the most in the game and the sole prize receiver,

even though all contestants have incurred costs. Some

winner-take-all contests are adjudicated through objec-

tive valuations, such as tests of skill and ability, or by

votes, others—through more subjective valuations. If the

best contestants perform the same, the contest ends

without either side winning, i.e. with a tie. Ties occur in

sports competitions, but also in other contests, such as

elections, if a runoff is required between recipients of the

largest numbers o f vo t e s.

Winner-takes-all formal game theory has been applied

to rent-seeking by Hillman and Samet [1] and Hillman

and Riley [2]; lobbying by Becker [3] and Che and Gale

[4]; R&D races by Dasgupta [5]; political contests by

Snyder [6]; and wars of attrition by O’Neil [7]. Cohen

and Sela (CS) [8] modeled the points system in soccer

(European football) matches using competitions of the

“All Pay” type (evaluating the optimal prize p olicy in the

case of a draw, as against a win).Like CS, we look at the

contest output (rather than at the inputs) and deny any

prize to low performing teams, even if the teams they

have played against have done no better. CS has shown

that the players’ efforts in equilibrium do not depend on

the expected prize in case of a tie, as long as it is one

third or less than the prize for winning.

Minimax strategies are compatible with defens ive soc-

cer games, ending up in draws, and often frustrating

spectators. To reduce the likelihood of draws, it has been

suggested to lessen their relative reward, by increasing

pay-offs for wins. This has led to the 3-1-0 (the three

point victory) system. Moreover, it has been demon-

strated that a 4-1-0 or 5-1-0 rule, though intuitively at-

tractive, would not improve results [8]. However, Broca

and Carillo [9] have also shown that under some condi-

tions, the 3-1-0 rule is counterproductive, inducing play-

ers to play more defensively than under 2 PV (to avoid

being led early on in the match). Another discussed im-

provement is the GG (Golden Goal) rule, whereby when

two teams are tied at the end of the regular time, the first

to score within the 30 minutes of overtime, wins the

match. Thus the GG rule reduces play time, driving up

the likelihood o f offens ive strateg ies in th e seco nd half of

the game. Broca and Carillo [9] have also shown that

combining the GG rule and the 3-1-0 system leads to

C. COHEN, M. SCHWARTZ

198

better results than 3-1-0 alone.

Statistical analysis vindicates the decision of the Eng-

lish Football Leagu e, in 1982, to award three points for a

win, instead of the previous two points (thus reducing the

prize for a draw to one third of that for a win). Indeed,

following the League’s decision, the average number of

goals scored in its’ matches increased dramatically Dob-

son and Goddard [10]. Moschini [11] has used a data set

spanning 30 years and 35 countries, showing that the

system has achieved its purpose (even without being

augmented by the GG r ul e).

The original contribution of the present paper, hence-

forth CSI (Cohen and Schwartz Improvement), is reduc-

ing the likelihood of ties due to lack of a serious effort to

win on the part of the contestants. We show that to

achieve this purpose, slacking teams and those who have

played a purely defensive game, failing to take any risks,

should be denied any reward whatsoever. This denial of

reward inflicted on substandard performers, may spur

some teams to bigger efforts, while inducing others to

avoid contests if they are unwilling or unable to meet

minimal standards.

However if we denied any reward to those make small

but not substandard efforts (Overextended CSI rule,

henceforth OCSI), more competitors would abstain from

any effort at all, while the numbers of those making the

maximum effort would not increase. Thus the CSI rule,

suggested by this paper, is shown to improve on both CS

and OCSI.

In the present paper, we show that the CSI model,

which adds to the CS model the denial of any points to

the teams involved in scoreless draws, will reduce their

frequency. We also show that the proposed change in-

creases the average number of goals per match.

Part 2 of this paper presents the model and its Nash

equilibrium point. Part 3 analyzes the results, offers ex-

amples and presents conclusions.

2. The Model and the Nash Equilibrium

Point

Let us assume that two players are vying for one prize in

a one-stage all-pay contest. Each player invests

. The investment includes money, time,

human capital and physical efforts (depending on com-

petition type—as listed above), as well as the energy

expended in the actual competition. Every player shoul-

ders such costs, irrespective of whether he ends up by

winning or losing: in such a game, the one investing the

most, wins. We assume complete information and a sym-

metrical model, in which winning the competition, car-

ries the same value V for both players. In case of a draw

where both players invested equal efforts, they will each

have a 1/3 probability of winning the prize.



1, 2, 3,.

x

To allow positive probability for a draw, we assume

that the number of possible strategies is finite. Thus we

have to use a finite game theory model. The competition

planner must determine the prize, not only for a win, but

also for a draw, with a view to ensuring that in the latter

case the probable prize is less than half th e prize in a win.

2.1. The CS Model

According to this model, a draw gains one poin t for each

of the participating teams, whereas a win gains three

points for the victorious team.

The payment functions per team in this model are:

1 if0

iii ij

iii j

xxx

uvx xx

vx xx





















2.2. The CSI Model

According to this model, a draw gains one poin t for each

of the participating teams, except if it is a 0:0 draw; in

this later case no team gets any points. A win gains

three points for the victorious team.

The payment function per team in this model are:

1 if0

0 if 0

iij

iii j

xxx

vx xx























Considering the payment function for teams, one may

observe that any team invests an effort to win (all pay),

which implies that if it loses the game or ends up with

0:0, it has incurred a negative gain, as its investment has

yielded no fruit.

2.3. The OCSI Model

This model fails to achieve an additional improv ement in

point allocation. To the contrary, it worsens results.



The model awards one point for draws, except for 0:0

and 1:1 draws. In the event of a win, the winning team

gets three points and the losing one-none.

The payment function in this model is:

1 if0,1

0 if 0 and 1

iij

iii j

ij ij

iii j

xxx

vx xx

uxx xx

vx xx



























C. COHEN, M. SCHWARTZ 199

2.4. Equilibrium

In the CSI model, an equal equ ilibrium of pure strategies

cannot exist, since, when player i invests output x1 and V

> x1, there is an incentive for player j to invest one unit

mor e tha n x1. Similarly, for player i there will now be an

incentive to invest more than player j and so on, up to

value V, at which point an additional investment will be

not be worthwhile for either player. However, in such a

game, a Nash equilibrium point may exist for mixed

strategies, which means that one can invest effort with a

given probability in th e original game, while maintaining

the likelihood of equal gain for any strategy employed.

Proposition 1: Consider two players with the same

prize valuation V, who compete in an all-pay contest for

a unique prize. If the rules are those of the CSI model,

there is a symmetric equilibrium in which any given

player chooses any given effort with



1, ,1xv



the probability



Vi i







, while his probability

of investing no effort is









.

Equilibrium in a mixed strategy is derived from as-

sumption that every effort assigned a positive probability

provides the player (in our case the individual team) with

the same expected gain. Thus the probabilities below are

the derivative of V.

Equations showing the equal probability of winning

with any strategy: the equations have the form:





prob' of winning

Vxk

where k is the expected payoff

An additional equation shows that the sum of prob-

abilities is equal to 1.







Examples of equilibrium for values other than V can

be found in Appendix 1.

Proposition 2: Consider two players with the same

prize valuation V who compete in all-pay contest for a

unique prize. The expectancy of goals in the CSI model

is larger then the expectancy in the original CS model.

Proof. In the CSI model the expectancy of goals per

player per match is:



pi i











 









 ,

while this expectancy, according to the original CS mo-

del is:









,

The CSI model always yields a larger expectancy than

the original one:



21 1

pi i

















 







 



















Thus, the expected output is likely to rise, due to the

proposed change.

However, denying any reward to those attaining 1:1

ties as well as to those attaining 0:0 ties, would not im-

prove the results but rather make them worse (OCSI

Model). In such a case, the payment function for the

player is:

1 if0,1

0 if 0and 1

iij

iii j

ij ij

iii j

xxx

vx xx

uxx xx

vx xx



























Proposition 3: Consider two players with the same

prize valuation V who compete in all-pay contest for a

unique prize. If the rules are those of the overextended

ISC model, then, there is symmetric equilibrium in which

any given player chooses every effort



2, ,1xv

with the probability



Vi i







, and choose to

invest effort in level 1 with the probability



PVV





















 













 and invest no effort in

the probability 01



for every V



i > 0, when i is a

positive integer.

The above equilibrium can also be derived from the

equation system representing the equal expectancy of

gain for any investment strategy with a positive probabil-

ity between very small and V







prob' of winning

Vxk





With k as the expected payoff

An additional equation shows that the sum of prob-

abilities is equal to 1.







From the above equation systems we derive the given

probabilities at equilibrium:

C. COHEN, M. SCHWARTZ

200

2.5. Comparing CS, CSI and OCSI

V = 5 0

P 1

P 2

P 3

P 4

P 5

PT.E

CS 1

5 1

5 2

CSI (Cohen and

Schwartz

Improvement)

80 15

80 18

80 12

80 24

80 2.28

OCSI

overextended CSI

model

80 10

80 18

80 12

80 24

80 2.22

Allocating 0 poin ts for a 1 :1 draw (in add ition to 0 po ints

for a 0:0 draw) does not spur the players to make addi-

tional efforts but rather discourages them. Those who

made no efforts in the CSI case, will keep abstaining

from them, while those who made a small effort, will

desist from it. Meanwhile, the proportion of those mak-

ing a serious effort will not change.

3. Analysis and Conclusions

Requiring a minimum entrance price (in an auction or in

other contests) under conditions of incomplete informa-

tion, or imposing a substantial fine on contestants who

fail to comply with what they committed to do (deliver

goods, do a job, or stick to the tender if they win, rather

than “coping out” at the last minute), is only effective if

matters are clear cut (either the contestant stuck to his

commitment or he did not). However, sometimes the

inputs required in the contest are multiple, complex, and

not directly observable. Thus, it is not immediately clear

whether enough effort has been invested, which requires

a more sophisticated tool, to induce the players to invest

at least the minimum. The tool we propose looks at the

output of the contest (rather than at the inputs) and denies

any prize to players who have demonstrated low per-

formance, even if others have done no better.

Like the original CS model, our CSI model (Cohen

and Schwartz Improvement) adds realism without fos-

tering complication. It adds realism, by using the all pay

assump tion, which implies that players make efforts,

whether they end up by winning, losing, or with a draw,

and thus these efforts are to be viewed as sunken costs.

Still, our model allows for the possibility of investing

less then the minimum acceptable effort. However, such

an occurrence becomes less likely. In addition, we show

that with the new change in decision rule, the likelihood

of a player’s investing less than the minimal acceptable

effort, is below the likelihood of his producing any other

output and thus less than the average probability, 1/V.

We show that the amended decision rule reduces the

frequency of cases in which both players invest less than

the acceptable minimum. Thus, the planner wishing to

maximize players’ investment in the competition, will

achieve better results with the proposed decision rule, as

it causes a participant investing below the minimal level

to lose the potential prize, irrespective of how much his

competitor has invested. Our proposal lessens the likeli-

hood of situations in which two competitors, both of

whom failed to invest the minimum, still manag e to share

the spoils between them.

We also show that the amended decision rule improves

over both CS and OCSI, as r aising beyond that minimu m

the effort required for a prize (the overextended ISC

model) lowers overall game performance, causing those

willing to make a small effort to desist from it, without

raising the share of those making a serious effort.

REFERENCES

[1] G. Becker, “A Theory of Competition among Pressure

Groups for Political Influence,” Quarterly Journal of

Economics, Vol. 98, No. 3, 1983, pp. 371-400.

doi:10.2307/1886017

[2] I. Broca and J. D. Carrillo, “Do the Three-Point Victory

and Golden Goal Rules Make Soccer More Exciting?”

Journal of Sports Economics, Vol. 5, No. 2, 2004, pp.

169-185. doi:10.1177/1527002503257207

[3] Y. K. Che and I. Gale, “Caps on Political Lobbying,”

American Economic Review, Vol. 88, No. 3, 1998, pp.

643-651.

[4] C. Cohen and A. Sela, “Contests with Ties,” The B.E.

Journal of Theoretical Economics, Vol. 7, No. 1, 2007,

Article 43.

[5] P. Dasgupta, “The Theory of Technological Competi-

tion,” In: J. Stiglitz and G. F. Mathewson, Eds., New De-

velopments in the Analysis of Market Structure, MIT

Press, Cambridge, 1986, pp. 519-545.

[6] S. Dobson and J. Goddard, “The Economics of Football,”

Cambridge University Press, Cambridge, 2001.

doi:10.1017/CBO9780511493225

[7] A. Hillman and J. G. Riley, “Politically Contestable Rents

and Transfers,” Economics and Politics, Vol. 1, No. 1,

1989, pp. 17-39. doi:10.1111/j.1468-0343.1989.tb00003.x

[8] A. Hillman and D. Samet, “Dissipation of Contestable

Rents by Small Numbers of Contenders,” Public Choice,

Vol. 54, No. 1, 1987, pp. 63-82.

doi:10.1007/BF00123805

[9] G. Moschini, “Incentives and Outcomes in a Strategic

Setting: The 3-Points-for-a-Win System in Soccer,” Work-

ing Paper # 08021, Department of Economics Working

Paper Series, Iowa State University, 2008.

[10] B. O’Neal, “International Escalation and the Dollar Auc-

tion,” Journal of Conflict Resolution, Vol. 30, No. 1,

1968, pp. 33-50. doi:10.1177/0022002786030001003

[11] J. Snyder, “Elections Goals and the Allocation of Cam-

paign Resources,” Econometrica, Vol. 57, No. 3, 1989,

pp. 637-660. doi:10.2307/1911056

C. COHEN, M. SCHWARTZ 201

Appendix 1

Numerical examples according to CSI for various V values.

1 1

448