A Comment on Continuously Secure Games∗ Guilherme Carmona





Konrad Podczeck

University of Surrey

Universität Wien

October 30, 2013

Abstract We introduce the class of continuous diagonal games and establish the existence of equilibrium for such games using Barelli and Meneghel's (2013) notion of continuous security. Furthermore, we use a specic continuous diagonal game, a Bertrand duopoly featuring asymmetric rms with convex costs, to illustrate that other known approaches to establish existence of equilibrium are not applicable. We also amend the denition of continuous security which is needed for the main result in Barelli and Meneghel (2013) to hold.

1 Introduction The idea that players might locally secure payos by choosing a suitable action is the heart of the existence results for Nash equilibria in discontinuous games, as established in the seminal papers of Reny (1999) and, more recently, of McLennan, Monteiro, and Tourky (2011). This idea has been generalized by Barelli and Meneghel (2013) under the notion of continuous security, which, in particular, allows each player to secure

Thanks to Paulo Barelli and Idione Meneghel for helpful comments. Financial support from Fundação para a Ciência e Tecnologia is gratefully acknowledged. Any remaining errors are, of course, ours. † Address: University of Surrey, School of Economics, Guildford, GU2 7XH, UK. Email: [email protected]. ‡ Address: Institut für Wirtschaftswisenschaften, Universität Wien, Hohenstaufengrasse 9, A-1010 Wien, Austria. Email: [email protected]

1

payos along a closed correspondence, instead of allowing just for a single action as in Reny (1999), or, more generally, for some nite set of actions as in the notion of multiply security in McLennan, Monteiro, and Tourky (2011).

This generalization

is important for at least two reasons. First, it naturally relates with the KakutaniFan-Glicksberg xed point theorem on which Nash's original existence result is based (and thus with the starting point of the theory on existence of Nash equilibrium; see Carmona (2011) and Carmona (2013b) for more on this). Second, it greatly facilitates

1

the application of the existence results for discontinuous games.

In this note, we show that with the notion of continuous security it is possible to handle applications which are not covered by previous general existence results for discontinuous games. This was left open by Barelli and Meneghel (2013). In fact, they showed just that the games in their economic applications satisfy the assumptions of their Proposition 4.1, but, as we show here, there are games satisfying these assumptions but fail to be continuously secure.

Furthermore, while Barelli and Meneghel

(2013) have shown that their Proposition 4.1 can be proven using their main existence result, we show here that it can also be established by applying a xed point theorem to a subcorrespondence of the best-reply correspondence of a game satisfying its assumptions. As this subcorrespondence of the best-reply correspondence is already constructed in Barelli and Meneghel's (2013) proof, it seems more natural to apply a xed point theorem directly to it instead of doing this indirectly via continuous security. In summary, the notion of continuous security is not needed to establish Proposition 4.1. The goal of this note is precisely to show that the notion of continuous security is needed to establish existence of equilibrium in a class of economic games. Specically, we will consider a class of

continuous diagonal

two-player games that include Bertrand

duopolies with convex cost functions, where convexity is allowed to be strict. This class of games is, in particular, such that players have a common strategy space and payo functions are the composition of one of two (tie and no tie) continuous

1 For

example, Reny (1999), using his notion of better-reply security, simplied Arrow and Debreu's (1954) proof of existence of competitive equilibrium for pure exchange economies. Reny's (1999) proof can be further simplied by using the notion of continuous security; see e.g. Carmona (2013a, Section 3.2).

2

outcome functions and a continuous utility function dened on outcomes.

Despite

these continuity assumptions, discontinuities naturally arise because the tie outcome function is used when players choose the same action and, in contrast, the no tie outcome function is used when the players choose dierent actions.

The diculty

with such class of games, illustrated here with the Bertrand duopoly application, arises because, due to the strict convexity of the cost functions, a rm may prefer to share the market with the other rm rather than to be the sole seller; in other words, discontinuity points of the payo function may give the highest possible payo, which violates payo security as dened in Reny (1999) or multiply security as in dened in McLennan, Monteiro, and Tourky (2011). However, as we will show, this diculty is easily handled using Barelli and Meneghel's (2013) idea of continuous security. Indeed, under the assumption that the tie outcome function is continuous, each rm can secure its payo at a given price vector by just choosing the same price as its rival in case the latter deviates slightly. Using this approach, we show that any quasiconcave continuous diagonal game has a pure-strategy Nash equilibrium provided that the tie outcome function is ecient. We then use a specic Bertrand duopoly, featuring asymmetric rms with convex costs, to illustrate both our existence result and that other known approaches to establish existence of equilibrium are inapplicable.

2 Continuous security A (normal-form) game

G = (Xi , ui )i∈N

players, and a pure strategy space

i ∈ N,

where

X=



i∈N

Xi .

Xi

For each

is given by a nite set

and a payo function

i ∈ N , Xi

subset of a Hausdor locally convex space and



j̸=i

Xj ,

and given

x ∈ X , x−i

A strategy prole

(Xi , ui )i∈N

if

x∗ ∈ X

ui (x∗ ) ≥ ui (xi , x∗−i )

ui

is bounded.

x

onto

for all

i∈N

all

i ∈ N , α ∈ N,

G = (Xi , ui )i∈N , and

we let

for each

and

x i ∈ Xi . x∗

We write

X−i

of the game

G=

As we do not consider

a Nash equilibrium.

Bi (x, α) = {yi ∈ Xi : ui (yi , x−i ) ≥ α}

x ∈ X. 3

for

X−i .

pure strategy Nash equilibrium

mixed strategy Nash equilibria, we simply call such an Given a game

ui : X → R

of

is a nonempty, convex and compact

for the projection of is a

N = {1, . . . , n}

for

Denition 1. A game G = (Xi , ui )i∈N is continuously secure if whenever x ∈ is not a Nash equilibrium of

G,

there is an

closed non-empty-valued correspondence (a) for all

x′ ∈ V

subspace of

and

i ∈ N , φi (x′ )

α ∈ Rn ,

φi : V ⇒ Xi

a neighborhood for each

i∈N

V

of

∏ i∈N

x,

Xi

and a

such that

is convex or included in a nite-dimensional

Xi ,

(b) for each

x′ ∈ V

(c) for each

x′ ∈ V ,

and

i ∈ N , φi (x′ ) ⊆ Bi (x′ , αi ),

there exists

i∈N

such that

and

x′i ̸∈ coBi (x′ , αi ).

The denition of continuous security as stated here diers slightly from that in

2

Barelli and Meneghel (2013). The latter does not include condition (a).

In the introduction we claimed that there are games that satisfy the conditions in Proposition 4.1 in Barelli and Meneghel (2013) but fail to be continuously secure. The example below illustrates this. Actually, in this example continuous security fails already in the form as stated in Barelli and Meneghel (2013), i.e., without requiring condition (a). For convenience of reading, here is a statement of Barelli and Meneghel (2013, Proposition 4.1).

Proposition 1. Let G = (Xi , ui )i∈N be a game such that Xi is a nonempty, compact, convex subset of a metrizable locally convex space and ui : X → {0, 1} is upper semicontinuous and ui (·, x−i ) is quasiconcave for each i ∈ N and x−i ∈ X−i . Then G has a Nash equilibrium.

Example 1.

Let

u1 = χ{(1,1/2)}

G = (Xi , ui )i∈N

and

u2 = χD ,

the indicator function for

be given by

writing

A ⊆ X.

N = {1, 2}, Xi = [0, 1]

D = {(x1 , x2 ) ∈ X : x1 = x2 }

for

i = 1, 2,

and

χA

for

Thus player 1 receives a payo of 1 if he plays

2 As

we have argued in Carmona and Podczeck (2013), the importance of this dierence follows from the fact that the main existence result in Barelli and Meneghel (2013), Theorem 2.2, does not go through given the denition of continuous security in that paper. The reason is that the correspondence Φ dened in its proof, and constructed using securing correspondences such as φ above, is not necessarily closed-valued. This is so because the convex hull of a compact set need not be closed in an innite-dimensional space. To solve this problem, one can require, as we did in Denition 1, φ(y) to be either convex (as in Barelli and Soza (2009)) or contained in a nitedimensional subspace of Xi (as in McLennan, Monteiro, and Tourky (2011)). This point was made independently in Reny (2013). 4

1 and player 2 plays 1/2, and receives a zero payo otherwise; player 2 receives a payo of 1 when she and player 1 choose the same action, and receives a zero payo otherwise. Clearly, all assumptions in Proposition 4.1 in Barelli and Meneghel (2013) are satised. However,

G is not continuously secure.

is not a Nash equilibrium of

α, V ,

and

x′2 ̸= 1/2,

φi , i = 1, 2,

G,

as required in Denition 1. Since there exists

Therefore, condition (c) for the case i.e.,

x = (1/2, 1/2), which

and suppose by way of contradiction that there are

condition (b) implies that

u2 (x) < α2 ,

In fact, let

α2 > 1.

α1 ≤ 0. x′ = x

Hence

coB1 (x′ , α1 ) = X1

and the quasiconcavity of

x′ ∈ V

for all

with

x′ ∈ V .

u2 (x1 , ·)

imply

But this means that condition (b) fails, contradicting our

assumption. This contradiction establishes that

G

is not continuously secure.

We next show that Proposition 4.1 in Barelli and Meneghel (2013) can be established by applying a xed point theorem to a subcorrespondence of the best-reply correspondence of a game satisfying its assumptions.

This can be shown with the

following modication of the proof in Barelli and Meneghel (2013). As in their proof, let, for all

i ∈ N , φi : X−i ⇒ Xi

be any upper hemicontinuous correspondence with

nonempty and closed values that satises all

x−i ∈ Fi ,

where

φi (x−i ) = {xi ∈ Xi : ui (xi , x−i ) = 1}

Fi = {x−i ∈ X−i : maxxi ∈Xi ui (xi , x−i ) = 1}.

for

By Theorem 5.2

in Borges (1967) or Theorem 2.4 in Tan and Wu (2002), such correspondence exists. Note that player i's best-reply correspondence at it equals

φi (x−i )

if

x−i ∈ Fi .

Thus

φi

x−i

equals

Xi

if

x−i ∈ X−i \ Fi ,

and

is a subcorrespondence of the best-reply cor-

respondence. Since, in the context of Barelli and Meneghel (2013, Proposition 4.1), the best-reply correspondence has closed and convex values,

co φi

is also a subcor-

respondence of the best-reply correspondence. Now by Theorem 17.35 in Aliprantis and Border (2006), values. Hence

Ψ=

co φi ∏ i∈N

is upper hemicontinuous with nonempty, convex and closed

co φi

has a xed point, which is a Nash equilibrium of

G.

3 Applications We rst establish an abstract existence results for a class of two-player games that covers, in particular, Bertrand duopolies. We then use a specic instance of a Bertrand duopoly, featuring asymmetric rms with convex costs, to show that other methods to establish existence of equilibrium are not applicable.

5

The class of games we consider are continuous diagonal games dened as follows. A

continuous diagonal game C = (P, Y, θ, γ, π1 , π2 ) is dened by a nonempty, convex

and compact subset

P

of a Hausdor locally convex space, a compact Hausdor

topological outcome space tie outcome function diagonal of

P 2,

Y,

γ : P2 \ D → Y ,

and, for each

θ : P → Y,

a non-

D = {x ∈ P 2 : x1 = x2 }

is the

a continuous tie outcome function

i = 1, 2,

where

a continuous utility functions

πi : P 2 × Y → R.

The interpretation of these elements arises more naturally in the normal-form game

GC = (Xi , ui )i∈N GC

is

N = {1, 2}

C

that

induces and which we now describe. The set of players in

and players have a common action space

player's payo of a strategy prole by

x

x∈X

and the outcome corresponding to

of a tie (i.e.

x1 = x2 )

payo function

and by

ui : X → R

γ

x.

Xi = P , i = 1, 2.

Each

is that player's utility of the pair formed This outcome is determined by

in case of no tie. Thus, for each

is dened by setting, for each

  π (x, θ(x )) i 1 ui (x) =  πi (x, γ(x))

i ∈ N,

θ

in case

player

i's

x ∈ X,

if

x1 = x2 ,

if

x1 ̸= x2 .

The class of continuous diagonal games covers several economic games including, in particular, Bertrand duopolies, Hotelling location games and timing games. General formalizations of diagonal games have also been presented in Reny (1999), Bagh (2010b) and Bich and Laraki (2012) and used to address these economic applications. Our formalization diers from theirs in that we explicitly introduce outcome functions, which has the advantage of allowing us to dene ecient continuous diagonal games  this notion is key for our existence result below. Of the above economic applications, we focus on Bertrand duopolies dened as follows. A

R++ ,

Bertrand duopoly

B = (¯ p, d, c1 , c2 , s)

is

a continuous demand function

function

ci : R+ → R+

satisfying

for each

i = 1, 2,

s1 (p) + s2 (p) = d(p)

B = (¯ p, d, c1 , c2 , s)

d : [0, p¯] → R+

for all

dened by a choke-o price with

a continuous cost

and a continuous sharing rule

p ∈ [0, p¯].

  (d(x ), 0) 1 γ(x) =  (0, d(x2 )) 6

s : [0, p¯] → R2+

To see that a Bertrand duopoly

is a continuous diagonal game, set

y1 + y2 ∈ d(P )}, θ = s,

d(¯ p) = 0,

p¯ ∈

P = [0, p¯], Y = {y ∈ R2+ :

if

x1 < x2 ,

if

x1 > x2 ,

for all

x ∈ P 2 \ D,

πi (x, y) = xi yi − ci (yi )

and

for all

i ∈ N, x ∈ P 2

and

y ∈Y.

Slightly abusing terminology, we say that a continuous diagonal game

concave if ui (·, xj ) is quasiconcave for all i ∈ N we say that

C

is

ecient

and

xj ∈ P , where j ̸= i.

if there exists a closed correspondence

C

is

quasi-

Furthermore,

Λ : P2 ⇒ Y

with

nonempty values such that

(i)

(ii)

θ(c) ∈ Λ(c, c) γ(x) ∈ Λ(x)

(iii) For all

for all

c ∈ P,

πi ((c, c), θ(c))

c ∈ P,

for all

x ∈ P 2 \ D,

and

there does not exist

for all

i∈N

and

In the case of a Bertrand duopoly

it is natural to consider

for all

x ∈ [0, p¯]2 ,

ways of allocating total demand to the two rms. it then follows that a exist

θ′ ∈ R2+

B

such that

equilibrium of

Λ

πi ((c, c), θ′ ) ≥

dened by

Λ(x) = Λ,

With such a specication of

p ∈ [0, p¯],

there does not

θ1′ + θ2′ = d(p), pθi′ − ci (θi′ ) ≥ ps(p) − ci (s(p)) for some

j ∈ N.

for some

which represents the possible

is ecient provided that, for all

pθj′ − cj (θj′ ) > ps(p) − cj (s(p))

and

such that

πj ((c, c), θ′ ) > πj ((c, c), θ(c))

B,

{y ∈ Y : y1 + y2 = d(min{x1 , x2 })}

θ′ ∈ Λ(c, c)

j ∈ N.

An

for all

equilibrium of C

i∈N

is a Nash

GC .

Theorem 1. If C = (P, Y, θ, γ, π1 , π2 ) is a quasiconcave, ecient continuous diagonal game, then GC is continuously secure and, consequently, C has an equilibrium. Proof. that

x

where

We start by showing that

GC

is not a Nash equilibrium of

j ̸= i.

is continuously secure. Let

GC .

We rst show that for each

Set

ε > 0, i ∈ N

αi = ui (x∗i , xj ) − ε/2;

and (ii)

hence,

and

and suppose

vi (x) = supx′i ∈Xi ui (x′i , xj ) for all i ∈ N , ε > 0 there exists (α, Vˆ , (φi )i∈N ) satisfying

Let

conditions (a) and (b) in Denition 1 with To see the above, let

x∈X

αi > vi (x) − ε

for all

i ∈ N.

x∗i ∈ Xi be such that ui (x∗i , xj ) > vi (x)−ε/2.

αi > vi (x) − ε.

x∗i ̸= xj

We consider two cases: (i)

x∗i = xj .

In case (i), since continuous, let such that

xj ∈ Xj : xˆj ̸= x∗i } → R x′j 7→ πi ((x∗i , x′j ), γ(x∗i , x′j )) : {ˆ

Oi ⊆ Xj ∩ {ˆ xj ∈ Xj : xˆj ̸= x∗i }

πi ((x∗i , x′j ), γ(x∗i , x′j )) > πi ((x∗i , xj ), γ(x∗i , xj )) − ε/2.

open neighborhood of

x,

and let

φˆi (x′ ) = {x∗i } 7

in

Xj

V i = Xi × O i ,

an

be an open neighborhood of

for all

x′ ∈ V i .

Let

xj

is

Hence, for all

x′ ∈ V i

and

yi ∈ φˆi (x′ ), ui (yi , x′j ) = πi ((x∗i , x′j ), γ(x∗i , x′j )) > πi ((x∗i , xj ), γ(x∗i , xj )) − ε/2 =

ui (x∗i , xj ) − ε/2 = αi . In case (ii), since a neighborhood of

x′j ∈ Oi . all

such that

V i = Xi × O i ,

Let

x′ ∈ V i .

xj

x′j 7→ πi ((x′j , x′j ), θ(x′j )) : Xj → R

Hence, for all

πi ((x′j , x′j ), θ(x′j )) > πi ((xj , xj ), θ(xj )) − ε/2

an open neighborhood of

x′ ∈ V i

there exists

there exists a net

{xr }r

ui (xr ) ≥ vi (x) − εr For any

i∈N

such that

converging to

for all

x

and

ui (x′ ) < vi (x) − ε.

and a net

π(xr , βr ) = u(xr )

π(x, β) = limr u(xr ).

r

for all

Thus, for all

{εr }r

πi (x, β) ≥ πi (x, θ(x1 )) with

β ∈ Λ(x)

x ∈ D

for all

However,

j ∈ N,

equilibrium of there exists exists

GC ,

ε>0

i∈N

x such that,

Suppose not; then

{βr }r

Since

Y

is compact,

β = limr βr .

converges; let

π = (π1 , π2 ),

and so

C

j∈N

ui (x) = πi (x, θ(x1 ))

and

such that

for all

πj (x, β) > πj (x, θ(x1 )).

is ecient.

βr ∈ Λ(xr )

for all

πj (x, β) >

i ∈ N.

Hence,

But this, together

r

Λ

and

is closed),

This contradiction establishes that

j ∈ N.

uj (x) = πj (x, β)

implies that

i ∈ N.

i ∈ N , πi (x, β) ≥ vi (x) ≥ ui (x).

i∈N

contradicts the assumption that for all

of

and, by the continuity of

(which, in turn, follows because

uj (x) = πj (x, β)

for all

converging to zero such that

Suppose, in order to reach a contradiction, that there is This implies that

for

i ∈ N.

taking a subnet if needed, we may assume that

uj (x).

φi = φˆi |Vˆ

r, let βr = γ(xr ) if xr ̸∈ D and βr = θ(xr,1 ) if xr ∈ D.

We have that

be

for all

φˆi (x′ ) = {x′j }

and let

ε > 0 and an open neighborhood U

We next show that there exists

x′ ∈ U ,

x,

Oi

yi ∈ φˆi (x′ ), ui (yi , x′j ) = πi ((x′j , x′j ), θ(x′j )) >

and

πi ((xj , xj ), θ(xj )) − ε/2 = ui (x∗i , xj ) − ε/2 = αi . ˆ =V1∩V2 Hence, to conclude the argument, let V for each

is continuous, let

for all

uj (x) = vj (x)

j ∈ N,

for all

together with

j ∈ N.

πj (x, β) ≥ vj (x)

But this implies that

x

for all

is a Nash

contradicting our assumption. This contradiction establishes that

and an open neighborhood

such that

ui (x′ ) < vi (x) − ε.

U

of

x

such that, for each

x′ ∈ U ,

Hence, corresponding to such

there

ε > 0,

let

(α, Vˆ , (φi )i∈N ) satisfy conditions (a) and (b) in Denition 1 with αi > vi (x) − ε for all i ∈ N , and let V = Vˆ ∩U . Since GC is quasiconcave, this means that (α, V, (φi |V )i∈N ) satises conditions (a)-(c) in Denition 1. This shows that

GC

is continuously secure.

As action sets are non-empty and compact, it now follows from Theorem 2.2 of Barelli and Meneghel (2013)with their notion of continuous security replaced by

8

the one in Denition 1 abovethat

GC

(and, therefore,

C)

has an equilibrium.

We use the following concrete example of a Bertrand duopoly with convex costs both to illustrate Theorem 1 and how other approaches to establish existence of equilibrium are inapplicable.

Example 2.

B = (¯ p, d, c1 , c2 , θ)

Let

be such that

p¯ = 2, d(p) = 2 − p, θ2 (p) =

min{p/2, d(p)} and θ1 (p) = d(p)−θ2 (p) for all p ∈ P = [0, p¯], c1 (q) = 0 and c2 (q) = q 2 for all q ∈ R+ . Note that, for each p ∈ P , the unique solution to max ˆ θˆ2 (p− θˆ2 ) θ2 ∈[0,d(p)]

is

θ2 (p).

Hence,

B

is ecient. The following notation is useful in what follows and, in

particular, to show that prots),

B is quasiconcave.

Dene

π ¯im = maxpˆ∈P πim (ˆ p) (highest monopoly prots) and πis (p) = psi (p)−ci (si (p))

(sharing prots) for all

i∈N

π2m (p) = −4 + 6p − 2p2 ,

p ∈ P.

and

  p2 /4 s π2 (p) =  π m (p) 2

Then, the following implies that

i∈N

π2s (p) ≥ 0

since

dπim (p)

for all

dp

<0

p∈P

and (4)

follows by Theorem 1 that We next show that strategy Since

(1, p2 ) ∈ P 2

for all

B

GB

B

p ∈ P , π1m (p) = 2p−p2 ,

We have that, for all

  2p − 3p2 /2 s π1 (p) =  0

and

all

πim (p) = pd(p)−ci (d(p)) (monopoly

if

p ≤ 4/3,

if

p > 4/3,

if

p ≤ 4/3,

if

p > 4/3.

is quasiconcave: (1)

p ∈ P,

(2)

πim

is strictly concave for

0 ≤ π1s (p) ≤ π1m (p)

π2s (p) > π2m (p)

implies that

for all

π2m (p) < π ¯2m .

is not multiply secure.

such that

p2 > 1. then

Then

(1, p2 )

3

(i) For all

z∈V

i ∈ N,

and

Hence, it

Suppose not and consider the

u1 (1, p2 ) = π1m (1) = 1 and u2 (1, p2 ) = 0.

is not a Nash equilibrium.

supposed to be multiply secure, then there exists an open neighborhood and, for each

(3)

has an equilibrium.

u2 (1, 1) = π2s (1) = 1/4,

α ∈ R2

p ∈ P,

a nite set

i∈N

{yi1 , . . . , yiJi } ⊆ P

there exists

j ∈ {1, . . . , Ji }

Since

V

of

GB

is

(1, p2 ),

such that such that

ui (yij , z−i ) ≥ αi ,

and

3 Since

multiply restrictional security is a necessary condition for the existence of a Nash equilibrium (see footnote 2 in McLennan, Monteiro, and Tourky (2011)), the game in such example will necessarily be multiply restrictionally secure. 9

(ii) For all

z∈V

there exists

α1 ≤ 1.

Note rst that

u1 (ˆ p) = pˆ1 q

for some

z1 < 1

if

z1 < y2j

z1 < 1 But

and

α1 ≤ 1.

Since

ui (z) < αi .4

pˆ ∈ P 2 ,

either

and, hence, and

y2j < 1

α2 ≤ 0

pˆ1 ≤ pˆ2

or

and

pˆ1 q ≤ pˆ1 d(ˆ p1 ) = π1m (ˆ p1 ) ≤ π ¯1m = π1m (1) = 1,

Second, we have that

z1 ̸∈ {y21 , . . . , y2J2 }.

u1 (ˆ p) = 0

Then, for all

α2 ≤ 0.

Indeed, take

if

z1 > y2j

(recall that

in this latter case). Hence, it follows by (i) that

contradicts (ii) in the case of

z∈V

j ∈ {1, . . . , J2 }, u2 (z1 , y2j ) = 0

u2 (z1 , y2j ) = π2m (y2j ) = 2(2 − y2j )(y2j − 1) < 0

and

α1 ≤ 1

such that

Indeed, for all

q ∈ [0, d(ˆ p1 )].

it then follows by (i) that such that

i∈N

α2 ≤ 0.

z = (1, p2 ).

Another method of establishing Theorem 1 would consists in showing that every game satisfying its assumptions is a potential game (Rosenthal (1973) and Monderer

GB

and Shapley (1996)). In fact, a necessary condition for that

u1 (x, y) − u1 (z, y) = u2 (y, x) − u2 (y, z)

and

z = 2,

this equality becomes

it follows that

GB

for all

π1s (1) = π2s (1);

to be a potential game is

x, y, z ∈ P .

since

In the case

π1s (1) = 1/2

x=y=1

π2s (1) = 1/4,

and

is not a potential game.

Existence of equilibrium can also be established, in general, using the notion of strategic complementarities or increasing dierences (Bulow, Geanakoplos, and Klemperer (1985) but see also Theorem 12.6 in Fudenberg and Tirole (1991)). To show that

GB

has increasing dierences requires establishing that

ui (xi , x˜j ) − ui (˜ xi , x˜j ) ≽i

where

P

are

for all

is an order on

i∈N

Xi = P

and

and

≽j

≥ (the standard order on R) or ≤.

can be seen by letting it follows that

xi , xj , x˜i , x˜j ∈ P is an order on When

xi = xj = 2, x˜j = 0

and

ui (xi , xj ) − ui (˜ xi , xj ) ≥

with

xi ≽i x˜i

Xj = P .

GB

xj ≽j x˜j ,

Two natural order on

≽i =≽j =≥, this inequality fails, which x˜i = x∗i ,

where

πim (x∗i ) = π ¯im .

ui (xi , xj ) = 0, ui (˜ xi , xj ) = π ¯im > 0, ui (xi , x˜j ) = 0

and the inequality fails. Thus,

and

and

In fact,

ui (˜ xi , x˜j ) = 0, 5

does not have increasing dierences.

The method consisting of applying a xed point theorem to the best-reply correspondence of

GC

cannot be applied to Example 2.

correpondence is empty-valued for all

In fact, player 1's best-reply

0 < p2 ≤ 1.

4 Note

that multiply security imposes stronger requirements than those in (i) and (ii). can also let one or both ≽i and ≽j be equal to ≤ and still show that GB does not have increasing dierences. In fact, if ≽i =≽j =≤, let xi = x∗i , xj = 0, x˜i = x˜j = 2; if ≽i =≥ and ≽j =≤, let xi = x∗i , xj = 0, x˜i = 0 and x˜j = 2; if ≽i =≤ and ≽j =≥, let xi = 0, xj = 2, x˜i = x∗i and x˜j = 0; in all these cases, the inequality above becomes π¯im ≤ 0 and, thus, it fails. 5 We

10

Finally, we note that known existence results specic to Bertrand oligopolies can not be applied to Example 2. Theorem 3 in Hoernig (2007) requires, in particular, that the sharing rule be norm tie-decreasing which, in the case of a Bertrand duopoly, means that, for all

p ∈ P,

either

πim (p) ≤ πis (p) ≤ 0

or

0 ≤ πis (p) ≤ πim (p).

this condition fails for the example above can be seen by noting that, at

π2m (1) = 0

and

π2s (1) = 1/4.

That

p = 1,

Moreover, Theorem 3 in Bagh (2010a) and Proposition

2 in Dastidar (1995) cannot be applied because both require all rms to have strictly convex cost functions.

References Aliprantis, C.,

and

K. Border (2006):

Innite Dimensional Analysis.

Springer,

Berlin, 3rd edn.

and G. Debreu (1954): Existence of an Equilibrium for a Competitive Economy, Econometrica, 22, 265290.

Arrow, K.,

Bagh, A. (2010a): Pure Strategy Equilibria in Bertrand Games with Discontinuous

Demand and Asymmetric Tie-Breaking Rules,

Economics Letters, 108, 277279.

(2010b): Variational Convergence: Approximation and Existence of Equilibrium in Discontinuous Games,

Barelli, P.,

and

Journal of Economic Theory, 145, 12441268.

I. Meneghel (2013):

Problem in Discontinuous Games,

Barelli, P.,

and

A Note on the Equilibrium Existence

Econometrica, 81, 813824.

I. Soza (2009): On the Existence of Nash Equilibria in Discon-

tinuous and Qualitative Games, University of Rochester.

Bich, P.,

and R. Laraki (2012):

A Unied Approach to Equilibrium Existence in

Discontinuous Strategic Games, Paris School of Economics.

Borges, C. (1967): A Study of Multivalued Functions,

matics, 23, 451461.

11

Pacic Journal of Mathe-

Bulow,

J.,

J.

Geanakoplos,

and

P.

Klemperer

Oligopoly: Strategic Substitutes and Complements,

(1985):

Multimarket

Journal of Political Economy,

93, 488511. Carmona, G. (2011): Understanding Some Recent Existence Results for Discon-

tinuous Games,

Economic Theory, 48, 3145.

Existence and Stability of Nash Equilibrium. World Scientic Pub-

(2013a):

lishing, Singapore. (2013b): Reducible Equilibrium Properties: Comments on Recent Existence Results, University of Surrey.

Carmona, G.,

and K. Podczeck (2013):

Existence of Nash Equilibrium in Games

with a Measure Space of Players and Discontinuous Payo Functions, University of Surrey and Universität Wien. Dastidar, K. (1995): On the Existence of Pure Strategy Bertrand Equilibrium,

Economic Theory, 5, 1932. Fudenberg, D.,

and

J. Tirole (1991):

Game Theory. MIT Press, Cambridge.

Hoernig, S. (2007): Bertrand Games and Sharing Rules,

Economic Theory,

31,

573585.

McLennan, A., P. Monteiro,

and

R. Tourky (2011): Games with Discontin-

uous Payos: a Strengthening of Reny's Existence Theorem,

Econometrica,

79,

16431664.

Monderer, D.,

and L. Shapley (1996):

Potential Games,

Games and Economic

Behavior, 14, 124143. Reny, P. (1999): On the Existence of Pure and Mixed Strategy Equilibria in Dis-

continuous Games,

Econometrica, 67, 10291056.

(2013): Nash Equilibrium in Discontinuous Games, University of Chicago. Rosenthal, R. (1973): A Class of Games Possessing Pure-Strategy Nash Equilib-

ria,

International Journal of Game Theory, 2, 6567. 12

Tan, K.-K.,

and

Z. Wu (2002): An Extension Theorem and Duals of Gale-Mas-

Colell's and Shafer-Sonnenschein's Theorems, in

Set Valued Mappings With Ap-

plications in Nonlinear Analysis. Taylor & Francis, London.

13

A Comment on Continuously Secure Games

Oct 30, 2013 - Wien, Austria. Email: [email protected]. 1 ... share the market with the other rm rather than to be the sole seller; in other words,.

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