Sequential Two-Prize Contests
Aner Sela
September, 2009
Abstract
We study two-stage all-pay auctions with two identical prizes. In each stage, the players compete for
one prize. Each player may win either one or two prizes. We analyze the unique subgame-perfect equilibrium of our model with two players where each player’s marginal values for the prizes are decreasing,
constant or increasing.
Jel Classification: D44, D82, J31, J41.
Keywords: Multi-stage contests, All-pay auctions.
1
Introduction
In winner-take-all contests, all contestants including those who do not win the prize, incur costs as a result
of their e¤orts. However, only the winner receives the prize. Winner-take-all contests have been applied
to numerous settings including rent-seeking and lobbying in organizations, R&D races, political contests,
promotions in labor markets, trade wars, military con‡icts and biological wars of attrition. The most
common winner take-all contest is the all-pay auction. In the standard (one-stage) all-pay auction each
player submits a bid (e¤ort) for the prize and the player who submits the highest bid receives the prize,
but, independently of success, all players bear the cost of their bids. In the economic literature, all-pay
auctions are usually studied under complete information where each player’s value for the prize is common
Department of Economics, Ben-Gurion University of the Negev, Beer–Sheva 84105, Israel. Email: [email protected]
1
knowledge,1 or under incomplete information where each player’s value for the prize is private information
to that player and only the distribution of the players’values is common knowledge.2 Most of this literature
has focused on one-stage all-pay auctions with a single prize which is awarded to the player with the highest
bid3 or, alternatively, on one-stage all-pay auctions with several prizes which are awarded to the players with
the highest bids.4 In the real world, however, we can …nd numerous multi-stage contests with several prizes
where in each stage a single prize is awarded or several prizes are awarded simultaneously. For example,
the situation in which employees seek to climb the organizational hierarchy often consists of several types
of well-de…ned positions, and political contests where any candidate, regardless of whether he wins or loses,
may compete again in the next round of the contest for the same job.
In this paper, we analyze the equilibrium behavior in sequential two-prize all-pay auctions under complete
information where the prizes are identical. Each player may win more than one prize, and each player’s
marginal values for the …rst and the second prize are either decreasing, constant or increasing. The di¤erence
between the players’ marginal values in both stages accords well with environments in which there are
synergies between prizes. A synergy between prizes does not necessarily have to be the same for both
players. In fact, it could even be positive for one player and negative for the other. In other words, our
two-stage all-pay auction encompasses any variation between the players’ marginal values for prizes, and
therefore the model suits any dynamic environment. Because it is possible to win more than one prize, our
model is more complex than the standard one-stage all-pay auction with a single prize. In one-prize contests,
the players’strategies are quite simple since the players have only two options, either to win the prize or to
lose it, while in sequential multi-prize contests more intricate strategies are involved since each player may
win more than one prize and therefore players may face many options that depend on the identity of the
winner in each stage. Furthermore, each of these options may have a di¤erent e¤ect on the chance of each
1 See,
for example, Hillman and Riley (1989), Baye, Kovenock and de Vries (1993), Che and Gale (1998)), Kaplan, Luski and
Wettstein (2003) and Konrad (2006).
2 See,
for example, Amman and Leininger (1996), Krishna and Morgan (1998), Moldovanu and Sela (2001) and Cohen,
Kaplan and Sela (2008).
3 Baye,
Kovenock and de Vries (1996) provided a complete characterization of equilibrium behavior in the complete informa-
tion all-pay auction with one prize.
4 See,
for example, Barut and Kovenock (1998) and Moldovanu and Sela (2001, 2006).
2
player to win the other prizes in the later stages. In particular, in sequential multi-prize contests, each player
has to decide in which stages he will compete to win and in which stages he will quit and conserve his e¤ort
(resources) for the other rounds.
In our sequential two-prize all pay auction with two players, we …nd the unique sub-game perfect equilibrium where no player quits in the …rst stage such that both players compete to win both of the prizes.
The players use mixed strategies in both stages and therefore each has a chance to win both prizes as well
as none of them. The players’ behavior in the second stage is similar to that of the standard one-stage
all-pay auction, but their behavior in the …rst stage is completely unique. For example, in contrast to the
standard one-stage all-pay auction, both players may have positive expected payo¤s. Moreover, even if a
player is weaker than the other player (his marginal values are lower than those of his opponent) he may
have a positive expected payo¤. On the other hand, if both players’marginal values are increasing, then one
of the players necessarily has an expected payo¤ of zero (similar to the standard one-stage all-pay auction).
We also show that, in contrast to the standard one-stage all-pay auction, both players may have expected
payo¤s of zero even if they have di¤erent marginal values for the prizes awarded in the contest.
In our sequential two-prize all-pay auction with more than two players the equilibrium is not unique,
and a complete characterization of the equilibrium behavior is quite complicated. However, the extension of
the equilibrium of our model with two players to the case with more than two players is quite simple. We
illustrate this extension where the players’marginal values are decreasing. A similar extension holds for any
relation between the players’marginal values.
There is an extensive literature on multi-stage contests, particularly multi-stage all-pay auctions. For
example, Moldovanu and Sela (2006) studied two-stage all-pay auctions under incomplete information where
the winners of the sub-contests at the …rst stage compete among themselves in a second stage. Groh et al.
(2008) studied the same model of a two-stage all-pay auction under complete information for the purpose
of …nding the optimal seeding from the designer’s point of view. Konrad and Kovenock (2005) analyzed a
multi-stage contest in which two players compete against each other in a sequence of all-pay auctions, and
the player who is the …rst to accumulate a su¢ ciently larger number of wins than his rival is awarded the
single prize. Konrad and Kovenock (2009) studied best-of-k contests with two players where players are
3
matched in the all-pay auction and the …rst to win
k+1
2
games is the winner of the contest.5 In contrast
to our multi-prize model, in all of the above papers the players compete for one prize that is allocated to
the winner in the …nal stage. The paper most related to our work is Kovenock and Roberson (2009) who
studied a two-stage game in which players compete in a complete information all-pay auction in each stage,
where winning (or losing) in the …rst stage a¤ects the winner’s utility function and his opponent’s utility in
the second stage. One of the main di¤erences between their model (as well as those discussed above) and
ours is that while in their model winning in a stage provides an advantage for the winner in the next stage,
in our model winning in the …rst stage may yield either an advantage or disadvantage for the winner in the
second stage depending on the relation between the players’marginal values for the prizes in both stages.
The paper is organized as follows: In Section 2 we introduce our sequential two-prize all-pay auction, and
in Section 3 we analyze the unique equilibrium behavior in two-prize all-pay auctions with two players. In
Section 4 we show several conclusions about the players’expected payo¤s in the two-prize all-pay auction.
In Section 5 we illustrate an extension of our results to the case with more than two players. Section 6
concludes.
2
The model
We consider a sequential all-pay auction with two players which we denote by i = a; b; and two stages which
we denote by t = 1; 2. In each of the stages a single (identical) prize is awarded. Let vji denote player i0 s
marginal value for winning his j-th prize. That is, if player i wins only one prize his value is v1i and if he
wins two prizes his value is v1i + v2i : We assume that all players’marginal values are common knowledge.
Each player i submits a bid (e¤ort) x 2 [0; 1) in the …rst stage. The player with the highest bid wins
the …rst prize and all the players pay their bids. The players know the identity of the winner in the …rst
stage before the beginning of the second stage, such that the players’values in the second stage are common
knowledge. The player with the highest bid in the second stage wins the second prize and all the players
pay their bids.
5 Klumpp
and Polborn (2006) studied best-of-k contests where players are matched in the Tullock contest.
4
3
Equilibrium
In order to analyze the subgame-perfect equilibrium of a sequential two-prize all-pay auction we begin by
analyzing the second stage and go backwards to the …rst stage.
3.1
The second stage
Assume that player i’s value in the second stage is v i ; i = a; b, and, without loss of generality, assume that
the players’ values satisfy v a
v b : According to Baye, Kovenock and de Vries (1996), there is always a
unique mixed-strategy equilibrium in which players a and b randomize on the interval [0; v b ] according to
their e¤ort cumulative distribution functions, which are given by
v a F b (x)
x = va
v b F a (x)
x =
vb
0
Thus, player a’s equilibrium e¤ort is uniformly distributed, that is
F a (x) =
x
vb
while player b’s equilibrium e¤ort is distributed according to the cumulative distribution function
F b (x) =
va
vb + x
va
The respective expected payo¤s in the second stage are ua2 = v a
v b and ub2 = 0: Now, given the equilibrium
strategies in the second stage we can analyze the equilibrium strategies in the …rst stage.
3.2
The …rst stage
Assume that player i’s value in stage t = 1; 2 is vti : If player a wins in the …rst stage his payo¤ in this stage
is v1a and then, if v2a > v1b ; the winning in the …rst stage guarantees to player a an expected payo¤ of v2a
in the second stage, otherwise, if v2a
v1b
v1b , his expected payo¤ in the second stage is zero. Thus, if player a
wins in the …rst stage, his expected payo¤ is v1a + maxfv2a
v1b ; 0g:
If player a doesn’t win in the …rst stage then, if v1a > v2b ; his expected payo¤ in the second stage is
v1a
v2b , otherwise, if v1a
v2b , his expected payo¤ in the second stage is zero. Thus, if player a does not win
5
in the …rst stage his expected payo¤ is maxfv1a
v2b ; 0g: A similar argument holds for player b: The induced
marginal value of each player in the …rst stage is the di¤erence between his expected payo¤ in the contest if
he wins or not in the …rst stage. Thus, the induced marginal values of the players in the …rst stage are:
vb1a
vb1b
= v1a + maxfv2a
v1b ; 0g
maxfv1a
v2b ; 0g
= v1b + maxfv2b
v1a ; 0g
maxfv1b
v2a ; 0g
(1)
Note that since the players’marginal values are positive, the induced marginal values given by (1) are positive
as well. Then, using the induced marginal values, the players’equilibrium strategies in the …rst stage of the
sequential two-prize all-pay auction are stated in the following result.
Theorem 1 In the unique subgame-perfect equilibrium of the sequential two-prize all-pay auction, the players’ strategies in the …rst stage are as follows:
1. If vb1a
vb1b ; player a’s equilibrium e¤ ort is distributed according to
F1a (x) =
x
vb1b
while player b’s equilibrium e¤ ort is distributed according to
F1b (x) =
x + vb1a
vb1a
vb1b
The respective expected payo¤ s in the …rst stage are ua1 = vb1a
expected payo¤ s in the contest are
a
b
= vb1a
=
vb1b + maxfv1a
maxfv1b
vb1b and ub1 = 0: Then, the players’ total
v2b ; 0g
v2a ; 0g
2. If vb1a < vb1b ; player a’s equilibrium e¤ ort is distributed according to
F1a (x) =
x + vb1b
vb1b
while player b’s equilibrium e¤ ort is distributed according to
F1b (x) =
6
x
vb1a
vb1a
(2)
The respective expected payo¤ s in the …rst stage are ua1 = 0 and ub1 = vb1b
vb1a . Then, the players’ total
expected payo¤ s in the contest are
a
=
maxfv1a
b
= vb1b
v2b ; 0g
(3)
vb1a + maxfv1b
v2a ; 0g
Proof. Assume …rst that vb1a > vb1b . Then, the players randomize on the interval [0; vb1b ] according to their
e¤ort cumulative distribution functions, F1a (x) and F1b (x); which are given by the indi¤erence conditions:
(v1a + maxfv2a
v1b ; 0g)F1b (x) + (maxfv1a
v2b ; 0g)(1
F1b (x))
(v1b + maxfv2b
v1a ; 0g)F1a (x) + (maxfv1b
v2a ; 0g)(1
F1a (x))
x = vb1a
x =
vb1b + maxfv1a
maxfv1b
v2b ; 0g
v2a ; 0g
Using the induced values vb1a ; vb1b ; the above indi¤erence conditions can be written as
vb1a F1b (x)
x = vb1a
vb1b F1a (x)
x =
0
vb1b
(4)
The system of equations (4) describing the players’ mixed strategies F1a (x); F1b (x) in the …rst stage of
the sequential two-prize all-pay auction is identical to the system of equations describing the equilibrium
strategies of players in the standard one-stage all-pay auction where the players’ values are the induced
marginal values vb1a ; vb1b . Hence, according to Baye, Kovenock and de Vries (1996), there is a unique mixed
strategy equilibrium in the …rst stage of the sequential two-prize all-pay auction which is given by:
F1a (x)
=
F1b (x)
=
where the players’expected payo¤s are ua1 = vb1a
x
vb1b
(5)
x + vb1a
vb1a
vb1b
v1b and ub1 = 0:
Similarly, if vb1a < vb1b ; the players randomize on the interval [0; vb1a ] according to their e¤ort cumulative
distribution functions, F1a (x) and F1b (x); which are given by the indi¤erence conditions:
(v1a + maxfv2a
v1b ; 0g)F1b (x) + (maxfv1a
v2b ; 0g)(1
F1b (x))
x =
(v1b + maxfv2b
v1a ; 0g)F1a (x) + (maxfv1b
v2a ; 0g)(1
F1a (x))
x = vb1b
7
maxfv1a
v2b ; 0g
vb1a + maxfv1b
v2a ; 0g
or alternatively by:
vb1b F1a (x)
x = vb1b
v1a F1b (x)
x =
vb1a
0
(6)
Thus, the explicit distribution functions are:
F1a (x)
=
F1b (x)
=
x + vb1b
vb1b
x
vb1a
where the players’expected payo¤s are ua1 = 0 and ub1 = vb1b
4
vb1a
(7)
vb1a :
Results
According to Theorem 1, the players’ expected payo¤s in the …rst stage of the contest are determined by
their induced values. Even if player a; for example, has the larger marginal value in the …rst stage (v1a > v1b )
he may stay out of the contest with a positive probability. Moreover, he has an expected payo¤ of zero in
the …rst stage if he does not have the higher induced value (b
v1a < vb1b ). By (1), the di¤erence between the
players’induced values is given by
vb1a
vb1b = (v1a
v1b ) + v2a
v1b
v1a
v2b
(8)
Using equation (8) we can derive several results about the players’total expected payo¤s in the contest.
Proposition 1 In the sequential two-prize all-pay auction, if each player’s marginal values are decreasing,
the player with the highest marginal value has the highest total expected payo¤ .
Proof. We assume that v1a
v1b :6 Suppose …rst that vb1a
vb1b
0:
Then, since the marginal values are decreasing, by our assumption we obtain that v1a
by (2) and (8), player a0 s total expected payo¤ is
a
6A
= (v1a
v1b ) + v2a
similar argument holds if v1a < v1b :
8
v1b
v1b
v2b : Then,
and player b0 s total expected payo¤ is
b
= maxfv1b
v2a ; 0g
Thus, the di¤erence between the players’expected payo¤s satis…es
a
Suppose now that vb1a
b
(v1a
v1b )
0
vb1b < 0: Then, by (3) player a’s total expected payo¤ is
a
= v1a
v2b
and player b’s total expected payo¤ is
b
= (v1b
v1a ) + (v1a
v2b )
vb1
v2a + maxfv1b
v2a ; 0g:
Thus, the di¤erence between the players’expected payo¤s satis…es
a
b
(v1a
v1b )
0
In contrast to the standard one-stage all-pay auction with two players in which only the player with the
higher value has a positive expected payo¤, we have
Proposition 2 In the sequential two-prize all-pay auction, both players may have positive expected payo¤ s.
Furthermore, even if both marginal values of a player are lower than both marginal values of his opponent,
he still might have a positive expected payo¤ .
Proof. By (2) and (3), if v1b > v2a and v1a > v2b no player in the sequential two-prize all-pay auction has
an expected payo¤ of zero.
Now, suppose that minfv1a ; v2a g > maxfv1b ; v2b g: Then, by (8) we obtain that
Thus, by (3 ), if 2v1b
v2b
vb1b
vb1a = 2v1b
v2b
v2a
v2a > 0; player b has a positive expected payo¤ although both of his marginal
values are lower than player a’s marginal values.
Similar to the standard all-pay auction with two players in which at least one of the players has an
expected payo¤ of zero, we obtain
9
Proposition 3 In the sequential two-prize all-pay auction, if both players’ marginal values are increasing,
then one of the players necessarily has an expected payo¤ of zero.
Proof. By (2) and (3) we need to show that vb1a
vb1b
Assume that vb1a
v2a > 0: Then, since player a’s marginal values are
implies that v1a
v2b
0: Below we show that vb1a
vb1b
vb1b
0: Suppose now that v1b
0 implies that v1b
0 implies that v1b
v2a
v2a
0:7
0, and vb1a
vb1b
0
increasing, by (8) we obtain,
vb1a
vb1b = (v1a
v1b ) + (v1b
v2a )
v1a
v2b = (v1a
Thus we obtain a contradiction to our assumption vb1a
vb1b
positive.
v2a )
v1a
v2b < 0
0 and therefore the di¤erence v1b
v2a is not
In contrast to the standard all-pay auction with two players in which both players have expected payo¤s
of zero i¤ they have identical values, we obtain
Proposition 4 In the sequential two-prize all-pay auction, both players may have expected payo¤ s of zero
even if they have di¤ erent marginal values.
Proof. By (2) and (3), no player has a positive expected payo¤ in the contest i¤ the following three
conditions are satis…ed:
v2a
>
v1b
v2b
>
v1a
(9)
and given (9) and (8)
vb1a
vb1b = 2(v1a
v1b ) + (v2a
v2b ) = 0
(10)
For example, if v2b = 2v1a and v2a = 2v1b ; the conditions (9) and (10) are satis…ed.
5
Extensions
Up to this point we considered sequential contests with only two players. It is important to note that similar
to the standard one-stage all-pay auction with more than two players (n > 2), in our model with more
7 In
a similar way it can be shown that v
b1a
v
b1b
0 implies that v1a
10
v2b
0.
than two players, the equilibrium strategies are not necessarily unique. Below we show the extension of the
equilibrium of our model with two players to the case with more than two players. The players’strategies
in the second stage are similar to the standard all-pay auction (see Baye, Kovenock and de Vries, 1996) and
therefore we focus only on the equilibrium strategies in the …rst stage where the players’ marginal values
are decreasing.8 In this equilibrium, the two players with the highest induced marginal values are the only
active players in the …rst stage.
First, we denote by vmax h the h-highest marginal value in the set fv1i gi2N [ fv2i gi2N where N is the set
of players. That is, vmax 1 is the highest marginal value and vmax 2n is the lowest marginal value. Denote the
two active players by a and b. Now, we have four possible cases to deal with as follows:
1) v1a = vmax 1 and v1b = vmax 2 ; v2b = vmax 3 : In this case the induced marginal values of players a and b in
the …rst stage are:
vb1a
Since for every player i 2 N
vb1b
= vmax 1
(vmax 1
vmax 3 ) = vmax 3
= vmax 2
(vmax 2
vmax 4 ) = vmax 4
fa; bg; vb1i
vmax 4 ; and vb1a
vb1b ; by Theorem 1 we obtain that in the …rst stage
players a and b randomize on the interval [0; vb1b ] according to the e¤ort cumulative distribution functions
which are given by (5). Thus, the respective expected payo¤s in the …rst stage are ua1 = vmax 3 -vmax 4 , and
ui1 = 0 for all i 6= a:
2)v1a = vmax 1 ; v2a = vmax 3 and v1b = vmax 2 : In this case the induced marginal value of players a and b in
the …rst stage are:
vb1a
Since for every player i 2 N
vb1b
= vmax 1
(vmax 1
vmax 4 ) = vmax 4
= vmax 2
(vmax 2
vmax 3 ) = vmax 3
fa; bg; vb1i
vmax 4 ; and vb1b
vb1a ; by Theorem 1 we obtain that in the …rst stage
players a and b randomize on the interval [0; vb1a ] according to the e¤ort cumulative distribution functions
which are given by (7). Thus, the respective expected payo¤s in the …rst stage are ub1 = vmax 3 -vmax 4 ; and
ui1 = 0 for all i 6= b:
8 Similar
arguments hold for other relations among the players’marginal values.
11
3) v1a = vmax 1 ; v2a = vmax 2 and v1b = vmax 3 : In this case the induced marginal value of players a and b in
the …rst stage are:
vb1a
vb1b
=
(vmax 1 + vmax 2
vmax 3 )
(vmax 1
vmax 4 ) = vmax 2 + vmax 4
vmax 3
= vmax 3
Now, if vmax 2 + vmax 4
2vmax 3 ; then vb1a
vb1b : Thus by Theorem 1 we obtain that in the …rst stage players
a and b randomize on the interval [0; vb1b ] according to the e¤ort cumulative distribution functions which are
given by (5). Thus, the respective expected payo¤s in the …rst stage are ua1 = vmax 2 + vmax 4
2vmax 3 ,
and ui1 = 0 for all i 6= a. Otherwise, if vmax 2 + vmax 4 < 2vmax 3 , then vb1a < vb1b , and we obtain that in the
…rst stage players a and b randomize on the interval [0; vb1a ] according to the e¤ort cumulative distribution
functions which are given by (7). Thus, the respective expected payo¤s in the …rst stage are ub1 = 2vmax 3
vmax 2
vmax 4 ; and ui1 = 0 for all i 6= b:
4) v1a = vmax 1 ; v1b = vmax 2 and v1c = vmax 3 .9 In this case the induced marginal values of players a and b
in the …rst stage are:
vb1a
Since for every player i 2 N
vb1b
= vmax 1
(vmax 1
vmax 3 ) = vmax 3
= vmax 2
(vmax 2
vmax 3 ) = vmax 3
fa; bg; vb1i
vmax 3 ; and vb1a = vb1b ; by Theorem 1 we obtain that in the …rst stage
players a and b randomize on the interval [0; vb1a ] according to the e¤ort cumulative distribution functions
which are given by (5). Thus, the respective expected payo¤s in the …rst stage are ui1 = 0 for all i:
6
Concluding remarks
In the sequential two-prize all-pay auction we presented, if players have diminishing, constant or increasing
marginal returns with respect to winning additional auction, the players’ implicit valuations in the …rst
stage depend non-linearly on both of the marginal valuations of both of the players. Hence, the players’
behavior in the …rst stage is unexpected which suggest that we are dealing with an interesting and challenging
multi-stage model.
9 Player
c is not an active player.
12
Indeed, our results showed that the characteristics of the equilibria in the standard one-stage and sequential two-prize all-pay auctions are not the same. For example, in a one-stage all-pay auction the player with
the lowest value necessarily has an expected payo¤ of zero for any equilibrium, while in a sequential two-prize
all pay auction with two players, even if a player has two marginal values which are both smaller than those
of his opponent, he may have a positive expected payo¤ in the …rst stage. Furthermore, in contrast to the
standard one-stage all-pay auction, on the one hand, both players may have positive expected payo¤s, and
on the other hand, both players may have expected payo¤s of zero even if they have di¤erent marginal values
in the sequential two-prize all-pay auction.
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