A Nash Equilibrium in Electoral Competition Models∗
Shino Takayama†
Yuki Tamura‡
Abstract
Since the introduction of better-reply security by Reny (1999), the literature studying
the existence of a pure strategy Nash equilibrium (PSNE) in discontinuous games has
substantially grown. From our interest in electoral competition games, which may be
neither quasiconcave, we introduce a weak notion of better-reply security, which is also
applicable to non-quasiconcave games. Our condition for feeble better-reply security is
simple, easy to verify and particularly useful in the class of electoral competition games.
We also provide necessary and sufficient conditions for the existence of a PSNE in the
class of electoral competition games. Finally, this paper demonstrates why a PSNE fails
to exist when a particular type of discontinuity exists in a model.
Key Words: Noncooperative games, discontinuous payoffs, pure strategy Nash equilibrium,
existence of equilibrium, better-reply security, electoral competitions.
1
Introduction
Many important games such as Hotelling’s model, various auction games and Bertrand competition have discontinuous payoffs. Since the introduction of better-reply security by Reny
(1999), the literature studying conditions for the existence of a pure strategy Nash equilibrium
(PSNE) has grown significantly.1
∗
We would like to thank Andrew McLennan, and Rabee Tourky. All errors remaining are our own.
Takayama (corresponding author); School of Economics, University of Queensland, St Lucia, QLD 4072,
Australia; email: s.takayama@economics.uq.edu.au; tel: +61-7-3346-7379; fax: +61- 7-3365-7299.
‡
Tamura; School of Economics, University of Queensland, St Lucia, QLD 4072, Australia; e-mail:
yuki.tamura@uqconnect.edu.au
1
Among many, recently, Bich and Laraki (2013) use the ideas of approximation games and a sharing equilibrium in Simon and Zame (1990) to show the existence of an approximate equilibrium in the class of quasiconcave
games with discontinuous payoffs. Barelli, Govindan and Wilson (2014) study a version of Colonel Blotto games
to show the existence result. Prokopovych (2013) proposes a new form of the better-reply security condition, the
strong single deviation property.
†
1
In the literature, Reny (1999) also provides sufficient conditions, namely payoff security and
reciprocal upper semicontinuity, for quasiconcave and compact games to be better-reply secure
so that the game has a PSNE. Bagh and Jofre (2006) generalize the condition of reciprocal
upper semicontinuity by allowing for the payoffs of all players to jump down at some point as
long as the payoff of some player jumps up, relative to his old payoff, somewhere in his strategy
set. On the one hand, Carmona (2009) establishes the conditions, weak upper semicontinuity
and weak payoff security, which together are sufficient for the existence of a PSNE. Later,
Carmona (2011) proposes a condition weak better-reply security and states that his existence
theorem implies the existence results of both Bagh and Jofre (2006) and Carmona (2009). The
result in Carmona (2011) is the most general among them. On the other hand, McLennan,
Monteiro and Tourky (2011) weaken the hypotheses for Reny (1999)’s existence theorem and
propose the MR-security condition.
However, with regard to the applicability, as pointed out by Reny (2013)2 , some of the
hypotheses in precedent works are difficult to verify due to their global nature for a practical
purpose. In this paper, we propose a concept of feeble better-reply security. This concept is
developed from weak better-reply security in Carmona (2011) and MR-security in McLennan,
Monteiro and Tourky (2011) (hereafter MMT).3 Our feeble better-reply security is equivalent
to Carmona (2011)’s weak better-reply security if the game is quasiconcave.
Further, we study the existence of a PSNE in two types of electoral competition models. In
the class of electoral competition models the players’ expected payoff functions do not always
satisfy continuity or quasi-concavity. The first model, developed by Roemer (1997), considers a one-dimensional two-party model with uncertainty about the median voter’s bliss point.
Also, in Roemer’s model, the parties are sorely interested in the policy after the election. In the
second model, we extend Roemer’s model to a model of mixed motivation, where the parties
are also interested in winning the election. Feeble better-reply security allows us to simplify the
proof of existence in these models, which have been increasingly important in the literature of
political science (Ball, 1999, Aragones and Palfrey, 2005, Saporiti, 2008, Bernhardt, Duggan
and Squintani, 2009, Drouvelis, Saporiti and Vriend, 2014, Takayama, 2014). We also provide
necessary and sufficient conditions for the existence of a PSNE. Finally, we demonstrate why
a PSNE fails to exist when a particular type of discontinuity exists in a model.
Our proofs using feeble better-reply security are brief. Proving the PSNE existence by
using the MR-security condition of MMT requires us to construct an operator that restricts a
2
See footnote 2 in Reny (2013).
Citing from page 1661 in McLennan, Monteiro and Tourky (2011), they do not know of analogs of the two
conditions by Carmona (2009) in their setting. We believe that our feeble better-reply security is an analog of
weak better-reply security by Carmona (2011) in their setting, which the two conditions in Carmona (2009) are
equivalent to.
3
2
set of strategies for which payoff security is satisfied.4 Our feeble better-reply security focuses
on a local neighborhood of each strategy profile, which secures a certain payoff and does not
requires such a restriction operator.
The organization of this paper is as follows. The second section defines the concept, feeble better-reply security, and then provides the main theorem. The third section provides the
existence results in the electoral competition models. The last section concludes with the discussion on the relationship of feeble better-reply security with other existence conditions in
the literature.
2
The Theorems
Consider a noncooperative normal form game
G = (X1 , . . . , XN , u1 , . . . , uN ),
where, for each i ∈ {1, . . . , N }, the player i’s strategy set Xi is a nonempty compact convex
subset of a metrizable locally convex topological vector space, and player i’s payoff function
Q
ui is a bounded function from the set of strategy profiles X = N
i=1 Xi to R.
In order to prove that feeble better-reply security implies the existence of a PSNE, we first
define two bounds on payoffs. Then, for each strategy profile which is not a PSNE, we examine
if such a strategy profile satisfies the following two conditions. The first condition relates to
a payoff security as is done in better-reply security of Reny (1999). In the second condition,
since we do not impose a condition of quasi-concavity in our existence theorem, we set a
restriction on a convex hull of an upper contour set of a given player. Finally, we apply a
variant of the fixed point principle stated in McLennan, Monteiro and Tourky (2011).
2.1
The Main Theorem
For each i ∈ N , define a function, ui : X → R, by ui (di , x−i ) = lim
inf ui (di , x0−i ), which
0
x−i →x−i
is the payoff that a strategy di can almost guarantee to player i if his opponents play any
strategies close enough to x−i . We also define another function, fi : X−i → R, by fi (x−i ) =
sup ui (di , x−i ), which denotes the highest payoff that player i can obtain against x−i .
di ∈Xi
Define
B̄i (x) = {yi ∈ Xi : ui (yi , x−i ) ≥ fi (x−i )}; and
C̄i (x) = con B̄i (x),
where con Z is the convex hull of the set Z.
4
The proofs in these models by using the MR-security are available upon request.
3
Definition 1 (Feeble Better-Reply Security). For a set Z ⊂ X, a game G is feebly better-reply
secure on Z if the following conditions hold:
1. for each i ∈ N and any z ∈ Z, yi ∈ B̄i (z) is nonempty; and
2. for any z ∈ Z, there is some player i ∈ N such that zi ∈
/ C̄i (z).
The game G is feebly better-reply secure at x ∈ X if there is some neighborhood Ux of x
such that it is feebly better-reply secure on Ux . Then, the game G is feebly better-reply secure
if it is feebly better-reply secure at each x ∈ X which is not a PSNE.
Let Γ = {(x, u) ∈ X × RN : u(x) = u} be the graph of the game’s vector payoff function,
and let Γ̄ be its closure. We now review the definition for better-reply security introduced in
Reny (1999).
Definition (Better-reply security). A game G is better-reply secure if whenever (x, α) ∈ Γ̄
and x is not a PSNE, there is some player i such that he has a strategy di ∈ Xi satisfying
ui (di , x−i ) > αi .
In comparison with the condition of better-reply security in Reny (1999), by specifying the
payoff values to be secured, in terms of the applicability, it becomes easier to verify. On the
other hand, condition 2 of feeble better-reply security is the additional condition to be satisfied,
which makes it harder to satisfy in general. However, similar to the result of C-security in
McLennan, Monteiro and Tourky (2011), when G is quasiconcave, C̄i (z) is replaced by B̄i (z),
so that it becomes a trivial condition.
Theorem 1. If a game G is feebly better-reply secure, then it has a pure strategy Nash equilibrium.
To prove Theorem 1, we use the following version of fixed point theorem which is proved
in MMT.
Lemma (Lemma 7.1 in McLennan, Monteiro and Tourky (2011)). Let X be a nonempty convex subset of a topological vector space and let P : X ⇒ X be a set-valued mapping. If there
T
is a finite closed cover H1 , . . . , HJ of X such that z∈Hj P (z) 6= ∅ for each j = 1, . . . , J, then
there exists x∗ ∈ X such that x∗ ∈ con P (x∗ ).
Proof of Theorem 1. For each i, let Pi (x) = B̄i (x) and let P (x) = P1 (x) × . . . × PN (x), so
that P : X ⇒ X. Aiming at a contradiction, suppose that G is feebly better-reply secure and
has no PSNE. Then, consider a particular x ∈ X. Let Ux be an open neighborhood of x such
that G is feebly better-reply secure on Ux . Since, X is a compact subset of a metric space, there
is a closed subset ZUx ⊂ Ux such that G is feebly better-reply secure on ZUx . Also, because X
4
is a compact subset of a metric space, X =
on ZUx ,
\
S
Pi (z) =
z∈ZUx
ZUx . Since G is feebly better-reply secure
x∈X
\
B̄i (z) 6= ∅,
z∈ZUx
for each i. Hence, for each x ∈ X,
\
P (z) =
z∈ZUx
\
P1 (z) × . . . × PN (z)
z∈ZUx
=
\
P1 (z) × . . . ×
z∈ZUx
\
PN (z) 6= ∅.
z∈ZUx
Therefore, by Lemma 7.1 in McLennan, Monteiro and Tourky (2011), there exists x∗ ∈ X
such that, for each i ∈ N , x∗i ∈ con Pi (x∗ ) = C̄i (x∗ ), contradicting condition 2 of feeble
better-reply security.
Recently, Reny (2013) provides some generalizations of the equilibrium existence results
in Reny (1999), Barelli and Meneghel (2013) and McLennan, Monteiro and Tourky (2011).
Reny (2013) proposes the concept of point security with respect to a subset of players by using
ordinal preference relations. Reny (2013) expresses his conditions by using its local nature
in order to overcome practical difficulties in proving conditions proposed in precedent works.
Our feeble better reply security also uses a local nature of strategy profiles, although our aim
is to use it in electoral competition games to obtain simpler proofs. In the next section, we
provide the simple proofs by using the concept.
In the first game of the next section, players have a continuous but not quasiconcave payoff
function. The following example shows that when a payoff is continuous but not quasiconcave,
the violation of feeble better-reply security may lead to non-existence of a PSNE.
Example 1. There are two players with strategy sets Xi = [−1, 1], i = 1, 2. The payoff of
player i, ui : X1 × X2 → R, is given by
u1 (x1 , x2 ) = |x1 − x2 |,
and
u2 (x1 , x2 ) = |x1 + x2 |.
Then, the best response correspondence for each player i, BRi (x−i ) : X−i ⇒ Xi , is






if x2 < 0,
if x1 < 0,


1
−1
BR1 (x2 ) =
BR2 (x1 ) =
{−1, 1} if x2 = 0,



−1
if x2 > 0,
5
{−1, 1}



1
if x1 = 0,
if x1 > 0.
Hence, non-existence of a PSNE is evident in the lack of an intersection in the two best
response correspondences. Moreover, this example violates the condition 2 of feeble betterreply security. To see that consider a strategy profile x = (x1 , x2 ) = (0, 0). It is obvious that x
is not a PSNE. In this profile, given an opponent’s strategy x−i = 0, each player i maximizes
his payoff at xi = −1 and xi = 1, implying yi ∈ C̄i (x) for any yi ∈ Xi . Hence, this indicates
that, for each neighborhood Ux of x, there is a strategy z ∈ Ux such that each player i holds
zi ∈ C̄i (z), violating condition 2 of feeble better-reply security.
3
3.1
The Existence Proofs in Electoral Competition Models
Roemer’s Model
Consider an economy with two goods: a private good and a public good. In this economy,
there is a continuum of voters, and all of them have the same quasilinear preferences over the
two goods, which are represented by a direct utility function, u : R × R → R, given by
u(b, g) = b + ϕ(g),
(1)
where ϕ(g) is increasing and strictly concave, b is the individual voter’s consumption level of
the private good, and g is per capita value of the public good. In this model, income, H ⊂ R,
is the only characteristic which determines preferences over the policies. For instance, a voter
whose income is h ∈ H can purchase h units of the private good. We assume h is distributed
according to a probability measure F on H with the mean of h̄. Since the public good is
entirely financed by tax revenue, the political issue is a flat tax rate. Then, we define a voter’s
indirect utility function, v : H × [0, 1] → R, by
v(h, t) = (1 − t)h + ϕ(th̄).
(2)
Note that v(h, t) is continuous and strictly concave in t. Hence, v(h, t) is single-peaked in t.
By using (2), we define a monotonic bliss point of a voter whose income is h, θh ∈ [0, 1], given
by
θh = argmax v(h, t).
(3)
t∈[0,1]
We assume that there are two political parties, Party L and Party R. Following Roemer
(1997), each party has objectives to maximize the utility of some voter whose income is hL
and hR , respectively. We assume that hL < hm < hR , where hm is the randomly distributed
median voter’s income whose cumulative distribution function is defined by a function G,
which is assumed to have no mass-point. As a consequence of (2) and the assumption of the
6
distribution of the median voter’s income, for each party i = L, R, there is a single ideal policy
θhi , which are located as θhR < θhm < θhL , where θhm is a median voter’s bliss point. For
simplicity, we write θhR = θR , θhm = θm and θhL = θL , respectively.
In the game, each party i independently and simultaneously announces a policy xi ∈ Xi =
[0, 1]. Then, each voter casts their vote for the party whose announced policy gives a higher
utility according to the indirect utility function defined by (2). When the distribution of income
does not have a mass-point, the set of voters whose preferences are indifferent between the two
policies has measure zero. Finally, parties implement their announced policies if they win the
election.
Given a pair of announced policies (xL , xR ), let π : X → [0, 1] be the probability that L
wins the election. From (2), π(xL , xR ) is indeed
π(xL , xR ) = Pr[v(hm , xL ) ≥ v(hm , xR )].
(4)
Then, by using the cumulative distribution function of the median voter’s income, we can write
π(xL , xR ) by




1 − G(σ(xL , xR )) if xL < xR
π(xL , xR ) = p
(5)
if xL = xR



G(σ(xL , xR ))
if xL > xR ,
where p ∈ [0, 1] is a constant value and σ : X → R is the identity function that relates to the
cut-off wealth level at which the probability of L winning the election becomes either zero or
one. Now, we compute σ(xL , xR ). For a given m, he strictly prefers L to R if and only if
(xR − xL )hm > ϕ(xR h̄) − ϕ(xL h̄).
Hence, σ is given by
σ(xL , xR ) =




0
if



1
if
ϕ(xR h̄)−ϕ(xL h̄)
xR −xL
if
ϕ(xR h̄)−ϕ(xL h̄)
xR −xL
ϕ(xR h̄)−ϕ(xL h̄)
xR −xL
ϕ(xR h̄)−ϕ(xL h̄)
xR −xL
≤ hL
∈ (hL , hR )
(6)
≥ hR .
Unlike Roemer (1997) who applies an equal-sharing rule when the two parties announce
an identical policy, we assume the winning probability for L becomes some constant at such a
situation.
By the probability of winning function, π(xL , xR ), and the indirect utility function, v(h, xi ),
we define an objective function, EΠi : X → R, i = L, R, of the party by
EΠi (xL , xR ) = π(xL , xR )v(hi , xL ) + (1 − π(xL , xR ))v(hi , xR ).
7
(7)
As proved in Roemer (1997), EΠi is continuous but may not be quasiconcave. Throughout,
we denote Roemer’s model as G.
Assumption 1. When there exists some xi ∈ Xi such that EΠi (xL , xR ) > v(hi , x−i ) holds,
the sets of maximizers of EΠi (xL , x−i ) in response to x−i are convex.
Assumption 1 is necessary to assure the existence of a PSNE. There are some sufficient
conditions to make this assumption hold, and Roemer (1997) assumes the decreasing hazard
rate, which indeed assumes the following:
Assumption 2. Let G be twice differentiable. The hazard rate
G0 (x)
G(x)
is decreasing in x.
Assumption 2 is a relatively common assumption in the literature of electoral competition
models (e.g. Ortuño-Ortı́n (1997), Roemer (1997), Ortuño-Ortı́n (2002), Bernhardt, Duggan
and Squintani (2009) and Takayama (2014)) and in the literature of Auction theory, it assumes
a more general version of monotone hazard rates.
Theorem 2. If an electoral competition game G satisfies Assumption 1, it has a PSNE.
Proof. Let x = (xL , xR ) ∈ X be a strategy profile which is not a PSNE. Since the objective
functions in G are continuous, ui (di , x−i ) = ui (di , x−i ). Then, in response to an opponent
player’s strategy, each player i has a best response to it, and hence condition 1 is satisfied.
Let d : X → R+ be the Euclidean distance on X, and let Ux (δ) = {y ∈ X : d(y, x) < δ}
be an open neighborhood about x with radius δ > 0. Because (xL , xR ) is not a PSNE, there
is some player i who has a strategy which is different from xi but is a best response to x−i .
Hence, we have fi (x−i ) > EΠi (xL , xR ). Then, by Assumption 1, for each player i, we define
a set of strategies Ai whose payoff equals to fi (x−i ). We claim that (xL , xR ) 6∈ A = AL × AR .
This is, in fact, obvious. If (xL , xR ) ∈ A, this indicates that (xL , xR ) is a PSNE, contradicting
the initial assumption that (xL , xR ) is not a PSNE.
Then, by continuity, we can choose a sufficiently small δ > 0 such that, for any (zL , zR ) ∈
Ux (δ), fi (z−i ) > EΠi (zL , zR ) holds. This indicates zi 6∈ C̄i (z). Therefore, the game G is
feebly better-reply secure, hence by Theorem 1, G admits a PSNE.
3.2
A Model of Mixed Motivations: Sufficient Conditions
In the new setting for this section, parties are interested in winning the election as well as in
ideology, i.e., the policy implemented after the election. Let k i ≥ 0 be the intrinsic value
that party i = L, R places on holding office. We assume that the values of k i ’s are common
knowledge.
8
The objective functions, EΠM
i : X × R → R, for L and R, respectively, are
L
L
EΠM
L (xL , xR , k ) = π(xL , xR )(v(hL , xL ) + k ) + (1 − π(xL , xR ))v(hL , xR ) and
R
R
EΠM
R (xL , xR , k ) = π(xL , xR )v(hR , xL ) + (1 − π(xL , xR ))(v(hR , xR ) + k ).
(8)
We denote this model as G M . Let (x∗L , x∗R ) ∈ X be a PSNE of G M . Notice that in response
to x−i , xi which yields the winning probability zero to party i is strictly dominated by other
policies that yield a strictly positive winning probability. Symmetrically, a policy xi that yields
the winning probability one also cannot be supported as an equilibrium. Thus, we remark the
following:
Condition (Condition on Winning Probability). In response to x−i for each i = L, R, xi is
optimally chosen to yield a winning probability in (0, 1).
Lemma 1. A PSNE (x∗L , x∗R ) satisfies:
(I) x∗L < θL and θR < x∗R .
(II) x∗R ≤ x∗L .
Proof. To prove (I), suppose θL < x∗L . Then, L would profitably deviate to θL . To see that,
notice that
∗
L
M ∗
∗
L
EΠM
L (θL , xR , k ) − EΠL (xL , xR , k )
= π(θL , x∗R )(v(hL , θL ) − v(hL , x∗R )) − π(x∗L , x∗R )(v(hL , x∗L ) − v(hL , x∗R ))
+ (π(θL , x∗R ) − π(x∗L , x∗R ))k L > 0,
where the last inequality follows from the two facts: (1) since θm < θL < x∗L , π(θL , x∗R ) >
π(x∗L , x∗R ) > 0, by Condition on Winning Probability, and (2) v(hL , x) is strictly concave and
single-peaked at θL . Therefore, θL < x∗L does not hold.
Alternatively, suppose x∗L = θL . If x∗R ≥ θL , R would profitably deviate to some strategy
θL − ε, where ε > 0 is sufficiently
small. Otherwise, if x∗R < θL , since (θL , x∗R ) is a PSNE, we
M
∗
L
∂EΠL (xL ,xR ,k ) must have
= 0. Because v(hL , θL ) − v(hL , x∗R ) + k L > 0, this requires
∂xL
xL =θL
that σL0 (θL , x∗R )G0L (σ(θL , x∗R )) = 0. However, by θm < θL , for some ε > 0, π(θL − ε, x∗R ) >
π(θL , x∗R ), which is a contradiction. Symmetrically, we can prove the second statement in (I)
for R.
To prove (II), we first show that x∗R ≤ θL . Aiming at a contradiction, suppose θL < x∗R . As
π(θL , x∗R ) = 1 and v(hL , xL ) is maximized at θL , L would profitably deviate to θL . However, as
is shown in (I), θL cannot be an equilibrium strategy. Similarly, we can conclude that θR ≤ x∗L .
Then, suppose, by contradiction, x∗L < x∗R . there is some player i who would profitably deviate
to x∗−i . To show it, we consider the two cases.
9
Case 1: θR < x∗R ≤ θm . Since θR ≤ x∗L < x∗R ≤ θm , π(x∗L , x∗R ) < 12 . Then,
∗
∗
L
M
∗
∗
L
EΠM
L (xR , xR , k ) − EΠL (xL , xR , k )
= π(x∗L , x∗R )(v(hL , x∗R ) − v(hL , x∗L )) + (p − π(x∗L , x∗R ))k L > 0,
if p > π(x∗L , x∗R ). Otherwise,
∗
∗
R
M
∗
∗
R
EΠR
L (xL , xL , k ) − EΠR (xL , xR , k )
= (1 − π(x∗L , x∗R ))(v(hR , x∗L ) − v(hR , x∗R )) + (π(x∗L , x∗R ) − p)k R > 0,
Case 2: θm < x∗R ≤ θL . By Condition on Winning Probability, we only need to consider the
∗
L
∂EΠM
L (xL ,xR ,k ) case of x∗L ∈ (IR (x∗R ), x∗R ). Then, the first order condition implies
= 0,
∂xL
∗
xL =xL
which implies
vL0 (hL , x∗L ) =
σL0 (x∗L , x∗R )G0L (σ(x∗L , x∗R ))(v(hL , x∗L ) − v(hL , x∗R ) + k L )
.
1 − G(σ(x∗L , x∗R ))
Then, since x∗L < θL , implying v 0 (hL , x∗L ) > 0, and σ 0 (x∗L , x∗R ) < 0, it must be the case that
0 < k L < v(hL , x∗R ) − v(hL , x∗L ).
(9)
However, x∗R is a profitable deviation for L, because by (9),
L
∗
∗
L
M
∗
∗
EΠM
L (xR , xR , k ) − EΠL (xL , xR , k )
= pk L + (1 − G(σ(x∗L , x∗R )))(v(hL , x∗R ) − v(hL , x∗L ) − k L ) > pk L .
As Ball (1999) points out, generally a party wants to choose a policy between their opponent’s policy and their own ideal policy. However, as the degree of an office-motive increases,
the party obtains an incentive for “undercutting” behavior such that they would be better off by
choosing a policy infinitesimally close to that of the opponent. Then, if one party is relatively
more office-motivated while the other is more policy-motivated, the former party is willing to
choose the same position of the policy-motivated party’s, forcing the latter to randomize their
announced policy in order not to be predictable. Consequently, a PSNE may not exist.5 One
extreme example is when k L → ∞ and k R = 0.6 The existence of a PSNE depends on how
different are k L and k R relative to the electoral uncertainty.
Let
L
R
x
bL (xR ) = argmax EΠM
bR (xL ) = argmax EΠM
L (xL , xR , k ) and x
R (xL , xR , k ).
xL ∈[xR ,θL )
5
6
xR ∈(θR ,xL ]
However, it is often shown that there exists a mixed strategy equilibrium.
See Saporiti (2008) for more details.
10
Condition (Conditions on Discontinuities). Each intrinsic value k i , for i = L, R, in G M respectively satisfies the following conditions: if xL = xR 6= θm , then
L
EΠM
xL (xR ), xR , k L ) ≥ lim supx̃L →(xR )− EΠM
L (b
L (x̃L , xR , k ); and
R
EΠM
bR (xL ), k R ) ≥ lim supx̃R →(xL )+ EΠM
L (xL , x
R (xL , x̃R , k ).
Theorem 3. If Assumption 1 and Conditions on Discontinuities hold, an electoral competition
model G M has a PSNE.
Proof. Let x = (xL , xR ) ∈ X be a strategy profile which is not a PSNE. Except in the case of
xR = xL , the objective functions are continuous and so the proof is exactly same as the one
of Theorem 2. Thus, we focus on the case of xR = xL . Without loss of generality, let i = L.
Take a sufficiently small δ. We consider a δ-neighborhood of (xL , xR ) with xL = xR . Then,
by Conditions on Discontinuities, we have
EΠM
xL (xR ), xR , k L ) ≥ lim sup EΠM
xL , xR , k L ).
L (b
L (e
x
eL →(xR )−
Therefore, condition 1 of feeble better-reply security holds. For condition 2 of feeble betterreply security, by Conditions on Discontinuities, since, for any x0L such that x0L < xR , the
payoff of L is strictly less than fL (xR ). Therefore, by choosing a sufficiently small δ, condition
2 of feeble better-reply security holds.
3.3
A Model of Mixed Motivations: Necessary Conditions
Here, we show that Conditions on Discontinuities are necessary for the existence of a PSNE. If
Conditions on Discontinuities do not hold, then there is a party which can increase the payoff
by choosing a strategy that is infinitesimally smaller than that of the opponent. However, such
an undercutting behavior decreases the payoff of the opponent. Therefore, the opponent is
willing to randomize their announced policy in order not to be predictable. As a result, a PSNE
does not exist. Drouvelis, Saporiti and Vriend (2014, p.92) provide analogous conditions for
the existence of a PSNE. In fact, their conditions are a special case on the following result
since they specify the distribution of the median voter’s bliss point in their model.
Theorem 4. If an electoral competition model G M admits a PSNE, Conditions on Discontinuities hold.
Proof. Suppose there is a PSNE denoted by (x∗L , x∗R ). Then, by definition of a PSNE, we have
∗
∗
L
0
∗
L
EΠM
EΠM
L (xL , xR , k ) = max
L (xL , xR , k ) and
0
xL ∈XL
∗
∗
R
∗
0
R
EΠM
EΠM
R (xL , xR , k ) = max
R (xL , xR , k ).
0
xR ∈XR
11
(10)
Since the arguments for L and R are analogous, without loss of generality, we clarify the
condition for L. Set x∗L = x
bL (x∗R ). Then, by definition of x
bL , we have
∗
∗
L
0
∗
L
EΠM
EΠM
L (xL , xR , k ) = 0 max
L (xL , xR , k ).
∗
xL ∈[xR ,1]
Then, from (10), as (x∗L , x∗R ) is a PSNE, it must satisfy that
∗
∗
L
M
EΠM
xL , x∗R , k L ).
L (xL , xR , k ) ≥ lim sup EΠL (e
x
eL →(x∗R )−
Otherwise, L profitably deviates to a strategy which is infinitesimally smaller than x∗R , contradicting the hypothesis that (x∗L , x∗R ) is a PSNE.
4
Discussion with Related Results
Among the recent equilibrium existence results in the literature, Barelli and Meneghel (2013)
and McLennan, Monteiro and Tourky (2011) study a game that may not be quasiconcave.
It is not difficult to show that when players’ strategy spaces are metric and locally convex,
feeble better-reply security is a sufficient condition for MR security in McLennan, Monteiro
and Tourky (2011).7 In a relation with the generalized payoff secure in Barelli and Soza
(2009), Carmona (2011) proposes the concept, weak better-reply security and states that the
conditions of weak better-reply security and generalized better-reply security are equivalent
when players’ strategy spaces are metric and locally convex, and the game is quasiconcave.
Barelli and Meneghel (2013) propose continuous security which is the most general condition
before Reny (2013), and generalized better-reply security is sufficient for continuous security.
The first condition of feeble better-reply security is comparable to the one of weak betterreply security, if a game G is quasiconcave and thus our existence theorem also allows us to
obtain the results of Bagh and Jofre (2006) and Carmona (2009). Lemma 1 of Carmona (2011)
shows that if the game is quasiconcave and compact, and satisfies a generalized payoff security,
then there exist an open neighborhood Vx of x and an upper hemicontinuous correspondence
ϕi with nonempty, closed and convex values such that
ϕi (z) ⊆ B̄i0 (z) = {yi ∈ Xi : ui (yi , z−i ) > fi (z−i )}
for each i and for each z ∈ Vx . Our condition does not require obtaining a correspondence ϕi .
7
Example 2.1 in Tian (2013) can be an example for MR-security but not feeble better-reply security.
12
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