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Pre-Contest Communication Incentives
Mustafa Yildirim!
Department of Economics
Stockholm School of Economics
Sveav‰gen 65, Stockholm, SWEDEN
E-mail: [email protected]
April 8, 2015
Abstract
To demonstrate resolution and psychological strength, players often engage
in pre-contest communication by publicly stating their desire to win an upcoming contest. Existing explanations for this phenomenon revolve around
incomplete information and signaling. In this paper, I o§er a complementary
explanation that does not rely on signaling. Within a complete information
setup, I show that when communication involves not only an audience cost,
e.g., a reputation loss, in case of a false statement (as assumed in the literature) but also an audience reward, e.g., a credibility gain, in case of a true
statement, players may have an incentive for pre-contest communication.
1
Introduction
In many competitive and conáict settings, players publicly state their strong desire
to win an upcoming contest. Examples abound. In business, Wal-Mart Stores, Inc.ís
CEO Mike Duke reportedly said at a company ceremony that ìno matter what environment weíre inótoday, a year from now, or Öve years from nowówe are driven
to win. And weíre never satisÖed at Walmart until we do.î1 In politics, prior to the
!
This paper is based on my dissertation to the Stockholm School of Economics. I am indebted to
Karl W‰rneryd for his continuous guidance and encouragement. I thank Dan Kovenock and Huseyin
Yildirim for helpful discussions and comments. All errors are mine.
1
For more information see <http://www.theshelbyreport.com/2013/10/16/walmart-well-winno-matter-what-environment-were-in>.
1
recent local elections on March 30, 2014, the Turkish Prime Minister Recep Tayyip
Erdogan announced that ìIíll resign if my party does not come Örst.î2 In sports,
Muhammad Ali was well-known for his pre-match statements like ìIíll beat him so
bad heíll need a shoehorn to put his hat on.î3
One obvious reason behind such public statements is to discourage the rival(s)
by committing to being agressive. According to Dixit (1987), the intuition thus
suggests that at least the ìfavoriteî player should have a strict incentive to make a
pre-contest announcement to overcommit to his action. This intuition turns out to be
incomplete. In an interesting paper, Fu et al. (2013) shows that when players su§er
from a credibility loss in case of a failed public statement, both the favorite and the
underdog prefer remaining silent prior to the contest.4 In light of this, these authors
have o§ered a signaling-based explanation where players may claim to be strong. In
this paper, remaining in the realm of complete information, I o§er a complementary
explanation by noting that players may also gain credibility by making a successful
public statement. That is, the presence of an audience can present a ìstickî as in
Fu et al. (2013) and a ìcarrotî for pre-contest communications. In particular, it is
clear that if it only provided the carrot, then unlike in Fu et al. (2013), players would
always make public statements. Thus, whether or not public statements are made in
equilibrium should depend on the amount of credibility gained relative to credibility
lost as well as the degree of competition. This is what we investigate in this paper.
Our model follows the one suggested by Fu et al.
(2013): a standard two-
player Tullock (1980) contest where, to demonstrate their resolution and psychological
strength, players may engage in pre-contest communication by publicly stating their
desire to win the contest. We suppose that communication involves not only an audience cost, e.g., a reputation loss, in case of a false statement (as assumed in theirs)
but also an audience reward, e.g., a credibility gain, in case of a true statement.
2
More details can be found at <http://www.robert-schuman.eu/en/eem/1153-is-recep-tayyiperdogan-s-justice-and-development-party-mowing-towards-victory-in-turkey>.
3
For more information see <http://www.independent.co.uk/sport/general/others/trash-talkthe-best-insults-in-boxing-2281927.html?action=gallery&ino=10>.
4
In particular, they show that communication can only be beneÖcial if it deters rivalís entry into
the contest, which is feasible only if there is entry cost. We here ignore this case because we suppose
that either players are not allowed to quit the contest or quitting is more costly than entering.
2
Moreover, in contrast to theirs, we suppose that neither player has the option to quit,
or equivalently the contest is unavodiable.
For symmetric players, we Önd that, in equilibrium, pre-contest communication
which is not feasible for su¢ciently low audience reward, becomes feasible otherwise.
More precisely, for a given audience cost, while both players remain silent for sufÖciently low audience reward, only one player and both players make pre-contest
announcements for intermediate and high audience reward respectively. These make
sense because the decision as to whether to communicate or not hinges upon a simple
trade-o§ between an increase in the probability of winning and an increase in the
e§ort cost. Our Öndings for asymmetric players are qualitatively similar except that
when only one player communicates in equilibrium, the identity of the communicator
is noteworthy. We Önd that while only the favorite may communicate in equilibrium
if the audience reward is intermediate, only the underdog may do so if the audience
cost is high in addition to intermediate audience reward.
1.1
Related Literature
As we examine the playersí incentives to engage in costly actions (i.e., preannouncements) in pursuit of the corresponding future beneÖts, our paper relates to the extant
literature on commitment. Ever since Schelling (1960), this literature is mainly concerned with the roots of incentives of the players interacting dynamically to commit
themselves to certain costly actions. Despite being vast, commitment is typically
modelled such that one of the players is given the opportunity to move Örst.5 In contrast, in our paper, commitment occurs through public announcements, which reáects
playersí self-conÖdence in winning.
The purpose of publicly committing to being aggressive is also studied in the
audience cost literature, where the standard example involves democratic leaders who
make strong public threats in an international crisis bargaining in order to get better
deals. The conventional view is that such threats are costly in that democratic leaders
making public threats will be punished by voters (i.e., audience) in the upcoming
elections if they later back down. Thus, it has been argued that democratic leaders
5
Prominent works focusing on time commitment include: Baik and Shogren (1992); Dixit (1987);
Leininger (1993); Morgan (2003); Morgan and V·rdy (2007); and Yildirim (2005).
3
make such threats in order to credibly signal their resolve,6 which, in turn, might lead
to better deals for them.
The rest of the paper is organized as follows. The model is set up in Section 2. All
formal results appear in Sections 3 and 4, where the results regarding the symmetric
case and the asymmetric case, respectively. Section 5 discusses the results, and Section
6 concludes.
2
The Model
Consider a two-stage contest (t = 1; 2) in which two risk-neutral players (i = 1; 2)
compete for a prize which is normalized to one (V = 1). At t = 1 (communication
stage), the players simultaneously decide whether or not to send a public message
of conÖdence that displays how comfortable they feel about winning the subsequent
contest. Let Ii = 1 denotes player iís sending the message and Ii = 0 denotes his
remaining silent. The sender of the message of conÖdence enjoys an audience reward
of & 2 R+ if he wins the contest and bears an audience cost of ' 2 R+ if he loses.
Upon observing their communication actions, at t = 2 (contest stage), the players
compete for the Öxed prize by simultaneously exerting e§orts (xi ; xj ) as in a standard
Tullock contest.7 SpeciÖcally, player i wins the contest according to the following
ratio form:
p(xi ; xj ) =
8
>
<
>
:
xi
x1 + x2
if
1=2
if
(xi ; xj ) 6= (0; 0)
:
(xi ; xj ) = (0; 0)
Let Ci (xi ) = ci xi be the cost of his e§ort. Then, the expected payo§ of player i at
the contest stage is written
. i (xi ; xj ; si ) = p(xi ; xj )(1 + &Ii ) $ (1 $ p(xi ; xj ))('Ii ) $ ci xi ;
(1)
where with probability p he wins the prize and, if he previously sent a message
of conÖdence, receives the audience reward while bearing the audience cost with
probability 1 $ p. Rearranging terms player iís payo§ can be written as:
. i (xi ; xj ; si ) = p(xi ; xj )(1 + (& + ')Ii ) $ ('Ii + ci xi ):
6
(2)
See Fearon (1994, 1997); Schultz (1999); and Smith (1998).
See CorchÛn (2007), Nitzan (1994), Konrad (2009), and Dechenaux et al. (2012) for a detailed
survey of contest literature.
7
4
Equation (2) suggests a modiÖed contest with a winning prize of 1 + & + ' and an
e§ort cost ' +ci xi . Note that audience reward increases only the prize, while audience
cost increases both the prize and the cost of e§ort. Without loss of generality, we set
c1 = 1, 1 % c2 = c.
Since our model is a two-stage game with observed actions, we will use subgame
perfect Nash equilibrium as the solution concept. To develop a benchmark as well
as a Örst step toward understanding equilibrium communication incentives, we begin
our analysis by considering the symmetric players with equal marginal costs of e§ort.
3
Benchmark: Symmetric Case
Consider two symmetric players with equal marginal costs of e§ort: c1 = c2 = 1. We
solve the game by backward induction.
3.1
Contest Stage
At t = 2 (contest stage), player i sets his e§ort (xi ) to maximize
. i (xi ; xj ; si ) = p(xi ; xj )(1 + (& + ')Ii ) $ ('Ii + xi )
=
xi
(1 + (& + ')Ii ) $ ('Ii + xi ).
x1 + x2
The Örst order condition yields
xi = p(xi ; xj )(1 $ p(xi ; xj ))(1 + (& + ')Ii ).
Since p(xi ; xj ) + p(xj ; xi ) = 1, dividing the Örst order conditions for players i and j
side by side, it follows that
1!ij &
x!i
1 + (& + ')Ii
=
,
!
xj
1 + (& + ')Ij
(3)
where 1!ij denotes equilibrium e§ort ratio, or the relative e§ort of player i given the
e§ort of player j (i.e., his rival). Observe that 1!ij (Ii = 0) < 1!ij (Ii = 1) for a given Ij .
In words, Öxing player jís communication behavior, player i exerts relatively higher
e§ort and thereby becomes more aggressive if he previously sent a message than if he
previously remained silent. Being more aggressive in the state where he previously
5
sent a message, player i should be more likely to win the contest if he previously sent
a message. Formally, this is so because employing equilibrium e§ort ratio, player iís
equilibrium winning probability can be written as
p(x!i ; x!j )
or equivalently 1!ij =
1!ij
= !
,
1ij + 1
(4)
p(x!i ; x!j )
, which together with equation (3) implies
1 $ p(x!i ; x!j )
1 + (& + ')Ii =
p(x!i ; x!j )
' (1 + (& + ')Ij ).
1 $ p(x!i ; x!j )
Inserting this into the Örst order condition, player i0 s equilibrium e§ort cost is found
as
x!i = p2 (x!i ; x!j ) ' (1 + (& + ')Ij ),
(5)
suggesting that if player i sends a message, he incurs a higher cost e§ort. Using (3),
(4), and (5), player iís payo§ becomes
. i (x!i ; x!j ; si ) =
3.2
(1 + (& + ')Ii )3
$ 'Ii
(2 + (& + ')(Ii + Ij ))2
(6)
Communication Stage
The following proposition considers communication behavior in two limiting cases.
Proposition 1 Suppose the two players are symmetric, i.e., c1 = c2 = 1. If sending
a message only involves the audience cost (i.e., & = 0 < '), then remaining silent is
a dominant strategy for each player. If, on the other hand, sending a message only
involves the audience reward (i.e., & > ' = 0), sending one is a dominant strategy
for each player.
Proof. If sending a message only involves audience cost (& = 0 < '), utilizing
(6) yields the following payo§ matrix:
s1 ns2
S
D
S
D
1 $ 3' 1 $ 3'
,
4
4
1
1 $ '2 $ '
,
(2 + ')2
(2 + ')2
1 $ '2 $ '
1
,
2
(2 + ')2
(2 + ')
1 1
,
4 4
6
For player 1, remaining silent is a dominant strategy since
'(' + 1) (3' + 8)
;
4 (' + 2)2
0 < . 1 (D; S) $ . 1 (S; S) =
0 < . 1 (D; D) $ . 1 (S; D) =
(5' + 8)'
:
4 (' + 2)2
Moreover, due to the symmetry, it is a dominant strategy for player 2. So, if sending
a message only involves audience cost (& = 0 < '), remaining silent is a dominant
strategy for each player.
On the other hand, if sending a message only involves the audience reward (i.e., & >
' = 0), utilizing (6) gives the following payo§ matrix:
s1 ns2
S
D
S
D
1+& 1+&
,
4
4
1
(1 + &)3
,
(2 + &)2 (2 + &)2
(1 + &)3
1
2 ,
(2 + &)
(2 + &)2
1 1
,
4 4
For player 1, sending a message is a dominant strategy since
0 < . 1 (S; S) $ . 1 (D; S) =
0 < . 1 (S; D) $ . 1 (D; D) =
(&2 + 5& + 8)&
;
4 (& + 2)2
(4&2 + 11& + 8)&
:
4 (& + 2)2
Similarly, due to the symmetry, it is a dominant strategy for player 2. Thus, if sending
a message only involves the audience reward, sending one is a dominant strategy for
each player.
Proposition 1 highlights that, for communication to occur, sending a message
should involve the audience reward. SpeciÖcally, it shows that, if the presence of an
audience presents only a stick (audience cost), regardless of what a playerís rival does,
remaining silent gives him a larger payo§ than does sending a message of conÖdence.
This Önding corroborates the Önding of Fu et al. (2013), who, in a similar setting with
only audience cost, showed that communication can never be optimal under complete
7
information.8 The intuition is as follows. For a given communication behavior by
his rival, a player knows that if he sends a message of conÖdence, he should act
more aggressively in order to win the contest and not to incur the audience cost.
Acting more aggressively, in equilibrium, he ensures a higher winning probability at
the expense of an extra e§ort cost on him. Accordingly, he sends a message provided
that the beneÖt accrued from an increase in winning probability exceeds an extra
e§ort cost induced by it. Part (a) shows that, with a standard Tullock contest,
the accrued beneÖt always falls short of this extra cost. If, on the other hand, the
presence of an audience presents only a carrot (audience reward), part (b) shows that,
regardless of what a playerís rival does, sending a message of conÖdence gives him a
larger payo§ than does remaining silent. This follows because the audience reward
increases only the prize and, by revealed preference, a player is always better-o§ in a
contest with a higher prize.
While enlightening, Proposition 1 is restrictive in that it examines the communication behavior merely for the two limiting cases where sending a message involves
either the audience cost or the audience reward. In the next proposition, we examine
communication behavior when sending a message involves both.
Proposition 2 Suppose the two players are symmetric, i.e., c1 = c2 = 1. Then, for
a given audience cost, ' 2 (0; 1), there exist & = &(') and & = &(') such that, in
equilibrium,
(a) both players send a message if the audience reward is su¢ciently high (i.e., & 2
[&; 1)),
(b) both players remain silent if the audience reward is su¢ciently low (i.e., & 2
[0; &]),
(c) only one player sends a message if the audience reward is moderate (i.e., & 2
(&; &)),
%
&
%
&
where & satisÖes 4&3 + (8' + 11) &2 + 4' 2 + 6' + 8 & $ 5' 2 + 8' = 0 and &
%
&
satisÖes &3 $ (' $ 5) &2 $ (5' 2 + 6' $ 8)& $ 3' 3 + 11' 2 + 8' = 0. Moreover,
& < ' < &.
8
In particular, when communication involves only cost, they show that communication may
occur only if (i) it deters rivalís entry or (ii) there is incomplete information.
8
Proof. All omitted proofs are relegated to the ìAppendixî.
Parts (a) and (b) extend Proposition 1. In particular, Öxing an audience cost,
the audience reward must be high enough to overcome the incentive to remain silent;
otherwise no player sends a message.
Part (c) is interesting in that despite playersí being symmetric, only one of them
sends a message of conÖdence (i.e., only one remains silent) in equilibrium. This
follows because a playerís pre-contest communication incentives, besides hinging upon
the amount of the audience reward relative to the audience cost, also hinges upon
the communication behavior of his rival. As sending a message induces a player to
behave more aggressively, a player has better communication incentives if his rival
remains silent than if his rival sends a message. This implies that, for a given audience
cost, the amount of the audience reward that will provide a player with enough precontest communication incentives is lower when his rival remains silent than when he
communicates.
Proposition 2 highlights that introducing an audience reward can induce precontest communication without assuming any cost of entry as in Fu et al. (2013),
even when a fair amount of audience reward is introduced.
Armed with the insights from the symmetric case, we now extend our analysis to
asymmetric players where new strategic issues emerge and show that most results are
qualitatively the same.
4
Asymmetric Case
Suppose that the players are asymmetric. Without loss of generality, let player 2 have
a higher marginal cost, i.e., c1 = 1, 1 < c2 = c, making him an underdog and his rival
a favorite according to Dixit (1987).
4.1
Contest Stage
At t = 2 (contest stage), player i sets his e§ort (xi ) to maximize
. i (xi ; xj ; si ) = p(xi ; xj )(1 + (& + ')Ii ) $ ('Ii + ci xi )
=
xi
(1 + (& + ')Ii ) $ ('Ii + ci xi ),
x1 + x2
9
which yields the following Örst order condition
ci xi = p(xi ; xj )(1 $ p(xi ; xj ))(1 + (& + ')Ii ).
Following the same steps as in the symmetric case, we have
1!ij =
ci x!i
=p
2
(x!i ; x!j )
x!i
cj
1 + (& + ')Ii
= '
,
!
xj
ci
1 + (& + ')Ij
(7)
1!ij
ci
!
!
' ' (1 + (& + ')Ij ), where p(xi ; xj ) = !
cj
1ij + 1
Since c1 = 1, 1 < c2 = c, playersí equilibrium payo§s are given as
4.2
. 1 (x!1 ; x!2 ; s1 ) =
c2 (1 + (& + ')I1 )3
$ 'I1 ,
(c(1 + (& + ')I1 ) + 1 + (& + ')I2 )2
. 2 (x!2 ; x!1 ; s2 ) =
(1 + (& + ')I2 )3
$ 'I2 .
(c(1 + (& + ')I1 ) + 1 + (& + ')I2 )2
(8)
Communication Stage
Following the same steps as in the symmetric case, in order to build intuition, we Örst
consider the two limiting cases where either the audience cost or the audience reward
is present. The next proposition generalizes Proposition 1 to asymmetric players.
Proposition 3 Suppose the two players are asymmetric, i.e., c1 = 1, 1 < c2 = c. If
sending a message only involves the audience cost (i.e., & = 0 < '), then remaining
silent is a dominant strategy for each player. If, on the other hand, sending a message
only involves the audience reward (i.e., & > ' = 0), sending one is a dominant
strategy for each player.
Proof. If & = 0 < ', by (8), the corresponding payo§ matrix is
s1 ns2
S
D
S
D
$ (2c + 1) ' + c2 $(c2 + 2c)' + 1
,
(c + 1)2
(c + 1)2
c2
(' + 1)3
,
$'
(' + c + 1)2 (' + c + 1)2
c2 (' + 1)3
1
$
'
,
(c' + c + 1)2
(c' + c + 1)2
c2
1
,
2
(c + 1)
(c + 1)2
10
Since
0 < . 1 (D; S) $ . 1 (S; S) = ' (' + 1)
0 < . 1 (D; D) $ . 1 (S; D) = '
(2c + 1) ' + 3c2 + 4c + 1
(c + 1)2 (' + c + 1)2
(3c2 + 2c) ' + 3c2 + 4c + 1
,
(c' + c + 1)2 (c + 1)2
remaining silent is a dominant strategy for player 1. Similarly, since
(2c + 1) ' + 3c2 + 4c + 1
0 < . 1 (D; S) $ . 1 (S; S) = ' (' + 1)
(c + 1)2 (' + c + 1)2
0 < . 1 (D; D) $ . 1 (S; D) = '
(3c2 + 2c) ' + 3c2 + 4c + 1
,
(c' + c + 1)2 (c + 1)2
remaining silent is also dominant strategy for player 2. Hence, if sending a message
only involves audience cost (& = 0 < '), remaining silent is a dominant strategy for
each player. On the other hand, if & > ' = 0. by (8), the corresponding payo§
matrix is
s1 ns2
S
D
S
D
c2 (& + 1)
&+1
2 ,
(c + 1)
(c + 1)2
c2
(& + 1)3
,
(& + c + 1)2 (& + c + 1)2
c2 (& + 1)3
1
,
2
(c& + c + 1)
(c& + c + 1)2
c2
1
,
(c + 1)2 (c + 1)2
Since
&2 + (2c + 3) & + (c + 1) (c + 3)
0 < . 1 (S; S) $ . 1 (D; S) = c &
(c + 1)2 (c + & + 1)2
2
(c + 1)2 &2 + (2c2 + 6c + 3) & + (c + 1) (c + 3)
0 < . 1 (S; D) $ . 1 (D; D) = c &
,
(c + 1)2 (c + c& + 1)2
2
sending a message is a dominant strategy for player 1. Likewise, since
0 < . 2 (S; S) $ . 2 (S; D) = &
c2 &2 + (3c2 + 2c) & + (c + 1) (3c + 1)
(c + 1)2 (c + c& + 1)2
(c + 1)2 &2 + (3c2 + 6c + 2) & + (c + 1) (3c + 1)
0 < . 2 (D; S) $ . 2 (D; D) = &
,
(c + 1)2 (c + & + 1)2
11
sending a message is also a dominant strategy for player 2. Thus, if sending a message
only involves audience reward (& > ' = 0), sending one is a dominant strategy for
each player.
Having shown that introducing asymmetry has no impact on communication behavior in the two limiting cases, we now investigate communication behavior of the
favorite and the underdog when both the audience cost and the audience reward are
present.
Proposition 4 Suppose the two players are asymmetric, i.e., c1 = 1, 1 < c2 = c.
Then, for a given level of asymmetry, c, and the audience cost, ' 2 (0; 1), there exist
&1 < &2 ; &3 < &4 such that, in equilibrium,
(a) both players send a message if the audience reward is su¢ciently high (i.e., & 2
[&4 ; 1)),
(b) both players remain silent if the audience reward is su¢ciently low (i.e., & 2
[0; &1 ]),
(c) only the favorite sends a message if the audience reward is su¢ciently moderate
(i.e., & 2 [&1 ; &4 ]),
(d) only the underdog sends a message if the audience cost is su¢ciently high and the
(c#1)(c2 +1)
audience reward is su¢ciently moderate (i.e., ' 2 [
; 1), & 2 [&2 ; &3 ]).
c+1
Proposition 4 is revealing in several aspects. First, parts (a) and (b) show that
despite the playersí being asymmetric (i.e., the favorite and the underdog), if the audience reward is extreme; in equilibrium, they may exhibit the same communication
behavior as in the symmetric case. In particular, given the level of asymmetry and
the audience cost, the favorite and the underdog both send a message of conÖdence
if the audience reward is high enough to overcome the incentive to remain silent and
they both remain silent if it is low enough. When the level of asymmetry between
types is low enough, this is expected because, for the symmetric case, Proposition 2
has already revealed that the players follows the same communication pattern under
similar conditions. As the payo§s are continuous, this suggests that, for a given audience cost and an audience reward, introducing a small asymmetry does not disturb
equilibrium.
12
Second, parts (c) and (d) show that when the audience reward is intermediate,
only one player (i.e., either the favorite or the underdog) sends a message of conÖdence
in equilibrium. More precisely, part (c) shows that only the favorite engages in precontest communication for any given audience cost as long as the audience reward is
intermediate, whereas part (d) shows that only the underdog engages in pre-contest
communication for high enough audience cost. Put di§erently, when the audience
reward is intermediate, the favorite is willing to put his credibility at stake for any
audience cost, whereas the underdog is willing to do so only when the audience cost
is relatively high. The intuition is that, all else equal, the favorite, being a low-cost
player, exerts higher e§ort than the underdog, thereby more likely to win the contest.
So, by sending a message and thus committing to increasing his e§ort, the favorite
knows that he may discourage the underdog and ensure a reasonable increase in his
equilibrium likelihood of winning. Accordingly, in equilibrium, the favorite is willing
to send a message for any audience cost. On the other hand, all else equal, the
underdog, being a high-cost player, exerts lower e§ort than the favorite, thereby less
likely to win the contest. So, by sending a message and thus increasing his e§ort, the
underdog knows that he may discourage the favorite and ensure a reasonable increase
in his equilibrium winning likelihood only when the audience cost is relatively high.
5
Discussion
In many competitive and conáict settings, players often engage in pre-contest communication by publicly stating their desire to win an upcoming contest. As mentioned in
the Introduction, examples can be found in business, politics, and sports. Unlike the
extant literature that studies pre-contest communication incentives under incomplete
information, thereby coming up with signalling-based explanations, we have investigated these incentives under complete information. Our investigation has revealed
that players may still engage in pre-contest communication under complete information, thereby o§ering one complementary explanation that is not based on signalling.
More importantly, in our setup, playersí engagement in pre-contest communication
may occur even though there is no entry cost or it does not deter the rivalís entry.
This Önding contrasts with that of Fu et al. (2013) who argue that with complete
information, pre-contest communication is feasible only if there is entry cost. This
13
follows from our additional assumption. Their key assumption is that such communication can be costly as it results in public embarrassment or a credibility loss for the
sender if he loses. While we keep their assumption, we make an additional assumption
that such communication can also be beneÖcial for the sender if he wins in that it
may result in a public appreciation or a credibility gain.
Our investigation has also revealed the conditions which lead to distinct communication behavior in equilibrium. For both symmetric and asymmetric players, we
have shown that whereas both players remain silent if the audience reward is su¢ciently low, only one player (both players) communicates if the audience reward is
intermediate (high). Intuitively, Öxing an audience cost, the audience reward must
be high enough to overcome the incentive to remain silent; otherwise no player communicates. Interestingly, for aymmetric players, we Önd that when the audience cost
is low enough, while the underdog never communicates, the favorite may do so. One
implication is that if the asymmetry between types is too large, then the favorite
never communicates.
6
Concluding Remarks
This paper has investigated playersí incentives to communicate conÖdence before they
start to engage in their main competitive activities. As pre-contest communication involves audience cost in the case of loss, one obvious rationale behind such pre-contest
communication incentives is to discourage the rival(s) by committing to being aggressive. However, it has recently been argued that when the competition is modeled as a
standard Tullock contest, pre-contest communication is never feasible under complete
information (see, e.g., Fu et. al. (2013)). That is, with standard Tullock contest, under complete information, the commitment value of pre-contest communication falls
short of its cost.
In this paper we have shown that, for a given audience cost, introducing a fair
amount of an audience reward makes pre-contest communication feasible. SpeciÖcally,
only one player communicates if the audience reward is intermediate and both players
communicate if the audience reward is high. Interestingly, for intermediate audience
reward, while only the favorite may communicate for any given audience cost, only
the underdog may do so for high audience cost.
14
7
Appendix A
In this part, we give a list of functions and some properties that will later prove to
be useful.
f1 (c; &; ') =
%
&2
%
&
c2 + c &3 + 2 (c + 1)2 ' + 2c2 + 6c + 3 c2 &2
%
%
&
&
+ (c2 + c)2 ' 2 + 2 c3 + 3c2 $ 1 c' + c4 + 4c3 + 3c2 &
%%
&
&
$ 3c2 + 2c ' + 3c2 + 4c + 1 '
%
&
f2 (c; &; ') = (c + 1)2 &3 + 2 (c + 1)2 ' + 3c2 + 6c + 2 &2
+((c + 1)2 ' 2 $ 2(c3 $ 3c $ 1)' + 3c2 + 4c + 1)&
%
&
$c2 (2c + 3) ' + c2 + 4c + 3 '
&
&
2c2 $ 2c $ 1 ' + 2c3 + 3c2 &2
%
&
+ (c + ' + 1) ( c2 $ 4c $ 2 ' + c3 + 3c2 )&
%
&
$ (2c + 1) ' + 3c2 + 4c + 1 (' + 1) '
f3 (c; &; ') = c2 &3 +
%%
%
&
f4 (c; &; ') = c2 &3 + ( 2c2 $ 2c3 $ c4 ' + 3c2 + 2c)&2
%
%
&
&
+ 3c $ 2c2 + 4c $ 1 c' + 1 (c + c' + 1))&
%%
&
&
$c2 c2 + 2c ' + c2 + 4c + 3 (' + 1) '
Lemma 1 For each i 2 f1; 2; 3; 4g, let fi (c; &; ') be deÖned as above. If c 2 (1; 1)
and ' 2 (0; 1)
(a) there is a unique &i = &i (c; ') 2 [0; 1) that solves fi (c; &; ') = 0
(b) fi (c; &; ') , 0 if and only if & 2 [&i ; 1)
2
+1)
(c) &1 < &3 < &2 < &4 if ' 2 (0; (c#1)(c
) and &1 < &2 < &3 < &4 if ' 2
c+1
(c#1)(c2 +1)
( c+1 ; 1)
Proof. Let c 2 (1; 1) and ' 2 (0; 1) be given. For each i 2 f1; 2; 3; 4g,
fi (c; &; ') is a cubic function. Letting /i denote the discriminant corresponding to
fi (c; &; ') = 0, direct calculation gives
%
&%
&2
/1 = $c4 (c + 1)2 4c (c + 1)2 ' + (4c + 3) (c + 3)2 (c + 1)2 ' + c2
15
%
&%
&2
/2 = $c3 (c + 1)2 4 (c + 1)2 ' + (3c + 4) (3c + 1)2 (c + 1)2 ' + 1
/3 = $c2 (c + 1)2 (4 (c + 1)6 ' 3 +12c3 (c + 1)4 ' 2 +12c6 (c + 1)2 '+c6 (4c + 3) (c + 3)2 )
/4 = $c3 (c + 1)2 (4c3 (c + 1)6 ' 3 +12c2 (c + 1)4 ' 2 +12c (c + 1)2 '+(3c + 4) (3c + 1)2 )
Clearly, /i < 0 for each i 2 f1; 2; 3; 4g, which suggests that fi (c; &; ') = 0 has
only one real root. Observe that the constant terms are negative implying that
lim$!0 fi (c; &; ') < 0. Besides, it is easy to see that lim$!1 f1 (c; &; ') = 1 > 0.
Hence, the continuity of fi (c; &; ') ensures the uniqueness of &i = &i (c; ') 2 [0; 1)
that solves fi (c; &; ') = 0, showing part (a). Part (b) directly follows from part (a)
and the observation that lim$!1 fi (c; &; ') = 1 > 0. To see part (c), note that
(&' + (& + 1)2 ) (c2 + 1) + 2 (& + 1) (& + ' + 2) c
>0
1
(c2 $ 1) (& + ')
%
&
f2 (c; &; ') $ f4 (c; &; ') = (& + ') (& + ' + 2) 'c4 + 2'c3 + 2&c + & > 0
f1 (c; &; ') $ f2 (c; &; ') =
(&' + (' + 1)2 )(c2 + 1) + 2 (' + 1) (& + ' + 2) c
f3 (c; &; ') $ f4 (c; &; ') =
>0
1
(c2 $ 1) (& + ')
By part (a), these imply that &1 < &2 < &4 and &3 < &4 .
8
Appendix B
Proof of Proposition 2. By (6), the corresponding payo§ matrix is
s1 ns2
S
D
S
D
(1+#+$)3
(2+#+$)2
#$3$+1
#$3$+1
,
4
4
(1+#+$)3
1
,
2
(2+#+$)
(2+#+$)2
$'
$' ,
1
(2+#+$)2
1
1
4 , 4
First, to show part (a), note that both players send a message in equilibrium if and
only if 0 % . 1 (S; S) $ . 1 (D; S) = . 2 (S; S) $ . 2 (S; D). Since
%
&
%
&
&3 $ (' $ 5) &2 $ 5' 2 + 6' $ 8 & $ 3' 3 + 11' 2 + 8'
. 1 (S; S) $ . 1 (D; S) =
;
4 (2 + & + ')2
16
0 % . 1 (S; S) $ . 1 (D; S) is equivalent to
%
&
%
&
0 % &3 $ (' $ 5) &2 $ 5' 2 + 6' $ 8 & $ 3' 3 + 11' 2 + 8' .
|
{z
}
&f ($;')
Observe that the discriminant of the cubic equation on the right hand side is / =
%
&
$64 (' + 1) 16' 2 $ 4' + 7 < 0, implying that, for a given ', there exists a unique
%
&
& = &(') that solves f (&; ') = 0. Moreover, as lim f (&; ') = $ 3' 3 + 11' 2 + 8' <
$!0
0 and 0 < lim f (&; ') = 1 for 0 < ', the continuity of f (&; ') ensures that 0 < &
$!1
and . 1 (S; S)$. 1 (D; S) < 0 if & < & and . 1 (S; S)$. 1 (D; S) , 0 if & , &. So, (S,S) is
an equilibrium if & 2 [&; 1). Second, to show part (b), note that both players remain
silent in equilibrium if and only if 0 % . 1 (D; D) $ . 1 (S; D) = . 2 (D; D) $ . 2 (D; S) ,
where
. 1 (D; D) $ . 1 (S; D) = $
%
&
%
&
4&3 + (8' + 11) &2 + 4' 2 + 6' + 8 & $ 5' 2 + 8'
4 (2 + & + ')2
.
Clearly, 0 % . 1 (D; D) $ . 1 (S; D) if
%
&
%
&
0 , 4&3 + (8' + 11) &2 + 4' 2 + 6' + 8 & $ 5' 2 + 8' .
|
{z
}
&g($;')
%
&
Notice that g(&; ') is increasing in &. Since lim (&; ') = $ 5' 2 + 8' < 0 and 0 <
$!0
lim g(&; ') = 1 for 0 < ', the continuity of g(&; ') together with the monotonicity
$!1
guarantees that, for 0 < ', there exists a unique & = &('), which is positive, that
solves g(&; ') = 0 and that . 1 (D; D) $ . 1 (S; D) % 0 if & % & and . 1 (D; D) $
. 1 (S; D) > 0 if & > &. So, (D,D) is an equilibrium if & 2 [0; &]. Finally, to show
part (c), note that only one player sends a message in equilibrium if and only if (i)
0 % . 1 (S; D) $ . 1 (D; D) and (ii) 0 % . 2 (S; D) $ . 2 (S; S). (i) holds if & , & by part
(b) and (ii) holds if & % & by part (a). Thus, only one player sends a message if
& 2 [&; &].
Proof of Proposition 4. By (8), the corresponding payo§ matrix is
s1 ns2
S
S
c2 (#+$+1)
2
(c+1)
D
c2
(1+#+$+c)2
D
#+$+1
2 $ '
(c+1)
(1+#+$)3
, (1+#+$+c)2 $ '
$' ,
17
c2 (#+$+1)3
(1+c(#+$+1))2
$' ,
1
(1+c(1+#+$))2
c2
1
(c+1)2 , (c+1)2
We Örst show part (a). Both players send a message if and only if
0 % . 1 (S; S) $ . 1 (D; S) =
(c +
f3 (c; &; ')
2
1) (c + & + '
+ 1)2
; equivalently, 0 % f3 (c; &; '),
and
0 % . 2 (S; S) $ . 2 (S; D) =
f4 (c; &; ')
; equivalently, 0 % f4 (c; &; ').
(c + 1)2 (c& + c' + c + 1)2
By Lemma 1(a) and (b), 0 % f3 (c; &; ') holds if & 2 [&3 ; 1) and 0 % f4 (c; &; ') holds
if & 2 [&4 ; 1), where &3 < &4 by Lemma 1(c). Hence, both players send a message if
& 2 [&4 ; 1). To show part (b), note that both players remain silent if and only if
0 % . 1 (D; D)$. 1 (S; D) = $
(c +
f1 (c; &; ')
2
1) (c& + c' +
c + 1)2
; equivalently, 0 , f1 (c; &; '),
and
0 % . 2 (D; D) $ . 2 (D; S) = $
(c +
f2 (c; &; ')
2
1) (& + ' + c
+ 1)2
; equivalently, 0 , f2 (c; &; ').
By Lemma 1(a) and (b), 0 , f1 (c; &; ') holds if & 2 [0; &1 ] and 0 , f2 (c; &; ') holds
if & 2 [0; &2 ], where &1 < &2 by Lemma 1(c). So, both players remain silent if
& 2 [0; &1 ]. To show part (c), note that only the Örst player (i.e., the favorite) sends
a message if and only if
. 1 (S; D) $ . 1 (D; D) =
f1 (c; &; ')
; equivalently, 0 % f1 (c; &; '),
(c + 1)2 (c& + c' + c + 1)2
and
. 2 (S; D) $ . 2 (S; S) = $
(c +
f4 (c; &; ')
2
1) (c& + c' +
c + 1)2
; equivalently, 0 , f4 (c; &; '),
By Lemma 1(a) and (b), 0 % f1 (c; &; ') holds if & 2 [&1 ; 1) and 0 , f4 (c; &; ')
holds if & 2 [0; &4 ], where &1 < &4 by Lemma 1(c). Hence, only the favorite sends a
message if & 2 [&1 ; &4 ]. Finally, to show part (d), note that only the second player
(i.e., the underdog) sends a message if and only if
. 1 (D; S) $ . 1 (S; S) = $
(c +
f3 (c; &; ')
2
1) (& + ' + c
18
+ 1)2
; equivalently, 0 , f3 (c; &; '),
and
. 2 (D; S) $ . 2 (D; D) =
(c +
f2 (c; &; ')
2
1) (& + ' + c
+ 1)2
; equivalently, 0 % f2 (c; &; '),
By Lemma 1 (a) and (b), 0 , f3 (c; &; ') holds if & 2 [0; &3 ] and 0 % f2 (c; &; ') holds
2
+1)
if & 2 [&2 ; 1). Moreover, by Lemma 1(c), we have &2 > &3 if ' 2 [0; (c#1)(c
)
c+1
2
+1)
and &2 % &3 if ' 2 [ (c#1)(c
; 1). Hence, only the underdog sends a message if
c+1
2
+1)
' 2 [ (c#1)(c
; 1) and & 2 [&2 ; &3 ].
c+1
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