Theoretical Economics Letters, 2013, 3, 197-201
http://dx.doi.org/10.4236/tel.[phone] 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
Received April 12, 2013; revised May 12, 2013; accepted June 12, 2013
Copyright © 2013 Chen Cohen, Moshe Schwartz. This is an open access article distributed under the Creative Commons Attribution
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
opyright © 2013 SciRes. TEL

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,.
i
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:
if
1 if0
3
if
ii
iii ij
iii j
xxx
uvx xx
vx xx




j







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:
if
1 if0
3
0 if 0
if
iij
iii j
i
ij
iii j
xxx
vx xx
u
xx
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:
if
1 if0,1
3
0 if 0 and 1
if
iij
iii j
i
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

1
21
2
i
i
Vi i
PV



, while his probability
of investing no effort is

1
01
21
23
V
V
V
PV



.
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
ii
Vxk
where k is the expected payoff
An additional equation shows that the sum of prob-
abilities is equal to 1.
1
0
1
V
i
i
p




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:

1
11
1
1
21
2
i
i
VV
ii
i
i
pi i
V






  



 ,
while this expectancy, according to the original CS mo-
del is:
1
1
11
2
V
i
V
i
V




,
The CSI model always yields a larger expectancy than
the original one:


11
1
11
1
11
1
1
1
1
21 1
2
10
2
VV
i
ii
i
iV
V
i ii
i
V
i
i
pi i
V
ii
V V
V








 



 










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:
if
1 if0,1
3
0 if 0and 1
if
iij
iii j
i
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

1
21
2
i
i
Vi i
PV



, and choose to
invest effort in level 1 with the probability

1
2
11
21
1
12
i
i
V
i
i
PVV








 
 






 and invest no effort in
the probability 01
PV

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
 1


prob' of winning
ii
Vxk


With k as the expected payoff
An additional equation shows that the sum of prob-
abilities is equal to 1.
1
01
V
i
i
p




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 1 5 1 5 1 5 2
CSI (Cohen and
Schwartz
Improvement)
11
80 15 80 18 80 12 80 24 80 2.28
OCSI
overextended CSI
model
16
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.
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C. COHEN, M. SCHWARTZ 201
Appendix 1
Numerical examples according to CSI for various V values.
6
P
5
P
4
P
3
P
2
P
1
P
0
P
V
1 1
3
4
1
4
2
2
4
1
4
1
4
3
12
32
6
32
9
32
5
32
4
24
80
12
80
18
80
15
80
11
80
5
16
64
8
64
12
64
10
64
11
64
7
64
6
96
448
48
448
72
448
60
448
66
448
63
448
43
448
7