Müller, W.

Normann, H.T.

English

Research Papers in Economics

J Evol Econ (2006) 17: 53–61
DOI 10.1007/s00191-006-0042-4
REGULAR ARTICLE
Conjectural variations and evolutionary stability
in ﬁnite populations
Wieland Müller·Hans-Theo Normann
Published online: 2 December 2006
© Springer-Verlag 2006
Abstract Recentlyithasbeenshownthatconsistentconjecturesareevolution-
arily stable. In this note we show that this ﬁnding depends on the use of the
inﬁnite population ESS (Maynard-Smith, Evolution and the theory of games,
Cambridge University Press, Cambridge, UK, 1982). When applying the ﬁnite-
population ESS (Schaffer, J Theor Biol 132:469–478, 1988) we show that the
conjectures surviving in the long run are not consistent.
Keywords Consistent conjectures·Duopoly·Evolutionary stability·
Finite population·Indirect evolutionary approach
JEL Classiﬁcation D43
1 Introduction
Dixon and Somma (2003)a sw e l la sM ü l l e ra n dN o r m a n n( 2005) have recently
shown that consistent conjectures (Bresnahan 1981) are evolutionarily stable.
Possajennikov (2004) extends these results to general two-player games. In
this note, we show that this ﬁnding depends on assuming an inﬁnite population
and,accordingly,the use of the inﬁnite-populationESS (Maynard-Smith1982).
Here, we assume a ﬁnite population of size N < ∞ and apply the ﬁnite-
population ESS as introduced by Schaffer (1988). By virtue of the properties
W. Müller (B )
Department of Economics, Tilburg University, Postbus 90153,
5000 LE Tilburg, The Netherlands
e-mail: w.mueller@uvt.nl
H.-T. Normann
Department of Economics, Royal Holloway, Egham, Surrey TW20 0EX, UK
e-mail: hans.normann@rhul.ac.uk54 W. Müller, H.-T. Normann
of this static ﬁnite-population ESS, we show that the conjectures surviving in
the long run are not consistent. Only when the population size N approaches
inﬁnity do the results provided here approach those reported in Müller and
Normann (2005). In this sense, the results described in this paper generalize
the ones reported in the earlier paper. They suggest that the evolutionary
support of consistent conjectures is weaker in the more realistic setting of a
ﬁnite population.
More generally, we show that the market outcome in a ﬁnite population will
be more competitive than in an inﬁnite population. In particular, when N → 2,
the market result approaches the perfectly competitive market outcome. This
relates our results to several papers in the literature. Vega-Redondo (1997)
considers an imitation dynamic in a Cournot market with a ﬁnite number of
players who, as time proceeds, imitate the behavior of the player who had the
highest proﬁt in the previous period. This dynamic is shown to converge to
the competitive outcome. Furthermore, Güth and Huck (1997) consider an
inﬁnite population of sellers who are repeatedly matched in pairs to act in
a symmetric heterogenous-goods market. The authors assume that, although
true proﬁt functions reﬂecting some particular market structure exist, sellers
may not be aware of them. Instead, sellers may entertain some beliefs about
the market structure which leads them to maximize so-called conjectural proﬁt
functions. Güth and Huck (1997) show that conjectural proﬁt functions ne-
glecting strategic interdependence are (neutrally) evolutionarily stable. More
precisely, it turns out that each seller in the evolutionary equilibrium thinks
he would act in a monopoly market and, accordingly, produces monopoly
quantities with regard to his conjectural proﬁt function. Surprisingly, these
individual quantities give rise to the competitive outcome with regard to the
true proﬁt functions.
2 Assumptions
Consider two ﬁrms i = 1, 2 in a heterogeneous-goods market where ﬁrms
choose quantities. The inverse demand functions are given by
pi(qi,qj) = a − qi − θqj, i, j = 1,2; i  = j (1)
with 0 <θ≤ 1. The symmetric cost functions are
C(qi) = c(qi)
2 /2, i = 1,2,
with c ≥ 0. Firm i’s proﬁt reads
πi(qi,qj) =

a − qi − θqj

qi −
c
2
(qi)
2 .Conjectural variations and evolutionary stability in ﬁnite populations 55
Our assumptions on demand and cost in the Cournot case are equal to
Assumptions 1 and 2 in Bresnahan (1981), except that we assume that ﬁrms
are symmetric.1
3 Quantity competition
We brieﬂy reiterate Bresnahan’s (1981) deﬁnition of a consistent conjectures
equilibrium (CCE) with Cournot competition. Let ρi = ρi(qj), i  = j, denote
ﬁrm i’s best response function. From our assumptions, we know that a unique
and linear CCE exists (Bresnahan 1981, Theorem 1). We therefore restrict
attention to linear conjectures such that ri ∈[ − 1,1], i = 1,2, denotes ﬁrm i’s
conjectures about ﬁrm j’s reaction to qi. We impose the restriction ri ∈[ − 1,1]
to guarantee equilibrium quantities to be non-negative. The outcomes in a
symmetric Cournot duopoly range from perfect competition to joint monopoly
when the conjectural variation increases from −1t o1 .
Deﬁnition 1 A consistent conjectures equilibrium is a pair of quantities,
(q∗
1,q∗
2), and of conjectures, (r∗
1,r∗
2), such that
q∗
1 = ρ1(q∗
2), q∗
2 = ρ2(q∗
1), (2)
and
r∗
1 =
dρ2(q1)
dq1
, r∗
2 =
dρ1(q2)
dq2
. (3)
That is, ﬁrms’ quantities have to constitute a Nash equilibrium (conditions
(2)), and a ﬁrm’s conjecture about the other ﬁrm’s behavior has to be equal to
the slope of the other ﬁrm’s reaction function (conditions (3)).
In Müller and Normann (2005) we show that, for the above market, the
symmetric consistent-conjectures equilibrium is given by
q∗
i =
2a
2(1 + θ)+ c + A
,( i = 1,2), r∗ =
−2 − c + A
2θ
(4)
where A :=

(2 + c)2 − 4θ2 > 0.
Instead of imposing a consistency condition as in Deﬁnition 1, one can make
conjectures subject to evolutionary selection. For this purpose, we ﬁrst derive
ﬁrms’ outputs given their conjectures. Since conjectures determine proﬁts,
they also determine reproductive success and one can study the evolution-
ary selection of conjectures in a second step. The underlying assumption is
that, if ﬁrms differ in evolutionary success, the individual characteristics of
a more successful ﬁrm will spread within the population more quickly than
1With asymmetric demand and cost functions, the evolutionary analysis below is extremely messy
and cumbersome. Bresnahan (1981) shows that the model may also allow for ﬁxed costs which,
from his Assumption 3, should not be too large.56 W. Müller, H.-T. Normann
the characteristics of a less successful one. This leads to a dynamic process
that determines the long-run distribution of individual characteristics within
an economy.
Consider the two steps more formally. Refer to ﬁrm i’s (constant) conjec-
ture, ri, as to ﬁrm i’s type. Firm types may be completely arbitrary and types
are known whenever two ﬁrms compete with each other. First, we derive ﬁrms’
behavior given their types. We assume here that ﬁrm i will choose its quantity
qi in order to maximize proﬁts, that is, we assume that players act rationally
given their types. Within strategic games this implies that the chosen strategy
proﬁle is a Nash equilibrium, denoted by (q∗
i (ri,rj), q∗
j(ri,rj)). In the second
step of evolutionary analysis, the types (conjectures) are the strategies and the
evolutionary success function, that is, ﬁrm proﬁts
π∗
i (r1,r2) ≡ πi(q∗
1(ri,rj),q∗
2(ri,rj))
evaluated at equilibrium strategies, are the payoff functions. To ﬁnd the types
that survive in the long run, in Müller and Normann (2005) we apply the ESS
concept as introduced for an inﬁnite population by Maynard-Smith (1982)a n d
show that the conjecture r∗ given in Eq. 4 is evolutionarily stable. For this
reason, let us denote r∗
inﬁnite := (−2 − c + A)/2θ with A as deﬁned above.
In what follows, we will apply the ESS concept for a ﬁnite population as
deﬁned by Schaffer (1988) for the case of a two-player game.2 For this purpose,
consider a population of size N ≥ 2. Each time, two out of the N individuals
might be called to play a duopoly game. The question is whether a single
mutant with conjecture r can invade the population if all other individuals
adhere to the conjecture r∗. This idea is captured by Shaffer’s ESS deﬁnition
used in condition (6) in the following deﬁnition.
Deﬁnition 2 An equilibrium with evolutionarily stable conjectures in a ﬁnite
population is a pair of quantities, (q∗
1,q∗
2), and a conjecture, r∗, such that
q∗
1 = ρ1(q∗
2), q∗
2 = ρ2(q∗
1), (5)
and
π∗
1(r,r∗) ≤

N − 2
N − 1

π∗
2(r∗,r∗) +
1
N − 1
π∗
2(r∗,r) for all r  = r∗. (6)
That is, an equilibrium with evolutionarily stable conjectures requires a Nash
equilibrium in outputs given the types (5), and an evolutionarily stable prefer-
ence typer∗ such that nor-mutant invading a “society” ofr∗-players can spread
(Eq. 6).
2For another application of this ESS concept see e.g. Hehenkamp et al. (2004).Conjectural variations and evolutionary stability in ﬁnite populations 57
We now solve for an equilibrium of this kind. The system of ﬁrst-order con-
ditions ∂πi(qi,qj)/∂qi = a − θqj − qi(2 + θri + c) = 0 (i = 1,2) can be solved
for equilibrium strategies
q∗
i (ri,rj) =
a

2 + θ

rj − 1

+ c

4(1 + c) + θ (2 + c)

ri + rj

+ θ2 
rirj − 1

+ c2.
Substituting q∗
i (ri,rj) and q∗
j(ri,rj) in πi(.) yields the evolutionary success
π∗
i (ri,rj) of type ri, given that the opponent exhibits type rj:3
π∗
i (ri,rj) = πi(q∗
i (ri,rj),q∗
j(ri,rj))
=
(q∗
i )2
2
(c + 2(1 + θri)).
Following Schaffer (1988), an ESS solves the maximization problem
max
r

π∗
1(r,r∗) −

1 −
1
N − 1

π∗
2(r∗,r∗) −
1
N − 1
π∗
2(r∗,r)

.
Since π∗
2(r∗,r∗) is a constant, this problem reduces to
max
r

π∗
1(r,r∗) −
1
N − 1
π∗
2(r∗,r)

.
Solving the ﬁrst-order condition and using symmetry (r = r∗) yields the
following
Proposition 1 The unique evolutionarily stable conjecture of the quantity game
is
r∗
ﬁnite =
−(2N − c + Nc+ 2θ − 2) + B
2θ (N − 1)
where B =

(c + 2)2 + N (c + 2θ + 2)(c − 2θ + 2)(N − 2).
(Another candidate can be excluded as it violates the assumptions set out in
Section 2.) A few observations are in order.
• First, the conjecture in Proposition 1 does not coincide with the one given
in Eq. 4. The proposition says that the evolutionarily stable conjecture
in the ﬁnite-population case is not the consistent conjecture. Importantly,
whereas in general r∗
inﬁnite > r∗
ﬁnite, we ﬁnd that limN→∞r∗
ﬁnite = r∗
inﬁnite.
3Note that the game with types ( ri,  rj) does not have an equilibrium if 4(1 + c) +
θ (2 + c)

 ri + rj

+ θ2 
 ri rj − 1

+ c2 = 0. For such ( ri, rj) we proceed as in Possajennikov (2000)
by extending the ﬁtness function by continuity in the ﬁrst argument in the sense of π∗
i ( ri, rj) =
limri→ ri limrj→ rj π∗
i (ri,rj). This limit does always exist on the extended real line R ∪{ ± ∞ }and, as
a result, the function π∗
i (ri,rj) is differentiable with respect to the ﬁrst argument at rj = ri.58 W. Müller, H.-T. Normann
• Second, we have q∗
i (r∗
inﬁnite,r∗
inﬁnite)<q∗
i (r∗
ﬁnite,r∗
ﬁnite). This follows from
r∗
inﬁnite > r∗
ﬁnite and ∂q∗
i (r,r)/∂r =− θa/(c + θ + rθ + 2)2 < 0. This means
that the market outcome in the case of a ﬁnite population will be more
competitive than in the case of an inﬁnite population.
• Third, we ﬁnd limN→2r∗
ﬁnite =− 1. This implies that, for the border case
of N → 2, we approach the perfectly competitive market outcome. In
this case, the evolutionarily stable conjecture is independent of underlying
market parameters which is generally in contrast to the analysis with an
inﬁnitely large population in Müller and Normann (2005). However, for
the special case of homogenous goods (θ = 1) and constant marginal cost
(c = 0), the consistent conjecture is −1, and therefore ﬁnitely and inﬁnitely
large population analysis yield the same results again.
4 Price competition
In Müller and Normann (2005) we also consider price competition in addition
to quantity competition. For this purpose, we invert the demand system
(1) and impose θ<1 strictly. We then show that the consistent-conjectures
equilibrium for the price-setting market is given by
p∗
i =
a

2(1 − θ2) + c(c + 2 + A)

(2 + c + 2θ)(1 + c − θ) + (c + 1 + θ)A
,
r∗
inﬁnite =
2(1 − θ2) + c(1 + θ2) + A

θ2 − 1

2θ

1 − θ2 + c
 (7)
where A is deﬁned as above for Cournot competition. Moreover, we show that
the conjecturer∗
inﬁnite is evolutionarily stable when applying the ESS concept as
introduced for an inﬁnite population.
Applying the general deﬁnition of a ﬁnite-population ESS as deﬁned in
Eq. 6 for arbitrary N ≥ 2 turns out to be extremely cumbersome. We therefore
state here only the result for the extreme case of N = 2. Proceeding as above
one can proof the following
Proposition 2 The unique evolutionarily stable conjecture of the price game in
the case of a population of size N = 2 is r∗
ﬁnite =− 1.
(The proof is given in the Appendix). We observe the following results.
First, it turns out that the evolutionarily stable conjectures (in the ﬁnite
population) in the Bertrand game are the same as in the Cournot game when
N = 2. In general, this is not true in the case of the inﬁnite-population ESS.
Second, this proposition says that the evolutionarily stable conjecture of the
Bertrand game is inconsistent in the ﬁnite-population case (see Eq. 7). Third,
again in contrast to Müller and Normann (2005), the evolutionarily stable
conjecture is independent of underlying market parameters.Conjectural variations and evolutionary stability in ﬁnite populations 59
5 Discussion
Why do the outcomes differ when we apply the ﬁnite-population ESS in-
stead of the inﬁnite-population ESS? The answer seems simple. In general,
evolutionary forces direct individuals to maximize relative payoffs. In inﬁ-
nite populations, individuals cannot affect the average payoff of all other
individuals. This implies that maximization of relative payoffs coincides with
maximization of absolute payoffs for the inﬁnite-population ESS.
However, for the ﬁnite-population ESS, this is not the case. Here individ-
uals’ concern for relative payoffs is not tantamount to a concern for absolute
payoffs. In a population of only two players, we saw that players produce the
perfectly competitive output. The reason is that each action directly affects the
other player’s proﬁt, which forces players to maximize relative payoffs. With a
large number of payers, the likelihood of affecting the payoff of other players
becomes smaller and it becomes zero in the limit N →∞ . In our model, the
higher the quantity in the Cournot game, the higher a player’s relative payoff
in a ﬁnite population. In turn, the quantity in the Cournot game is the higher
the lower is a player’s conjecture.4 It follows that, the smaller the conjecture,
the larger the output and, consequently, the larger a player’s relative payoff.
Finally, we believe that the results derived here would hold in a more
general framework that, in particular, would not make use of a parametrized
demand system. However, we leave this to future research.
Acknowledgements We thank Werner Güth and an anonymous referee for very helpful com-
ments. The ﬁrst author acknowledges ﬁnancial support from the Netherlands Organisation for
Scientiﬁc Research (NWO) through a VIDI grant.
Appendix
Proof of Proposition 2
To analyze price competition, we need to invert the inverse demand functions
given in Eq. 1 yielding
qi(pi, pj) =
a(1 − θ)− pi + θpj
1 − θ2 , i, j = 1,2; i  = j.
and impose θ<1 strictly. Firm i’s proﬁt reads
πi(pi, pj) = pi

a(1 − θ)− pi + θpj
1 − θ2

−
c
2

a(1 − θ)− pi + θpj
1 − θ2
2
4This can be easily seen by inspecting a player’s reaction function. For the sake of simplicity
consider the Cournot game when N = 2. In this case, player i’s reaction function is given by
qi(qj,ri) =

a − θqj

/(2 − c + θri). Thus, the lower is a player’s conjecture ri, the higher the
output in the Cournot game.60 W. Müller, H.-T. Normann
To compute the equilibrium prices, we have to solve the system of ﬁrst order
conditions given conjectures. The solution of this system of equations yields
p∗
1(r1,r2) and p∗
2(r1,r2), that is, the optimal prices given the conjectures of the
ﬁrms. These equilibrium prices are quite complex so we refrain from writing
them down. The proﬁts implied by p∗
1(r1,r2) and p∗
2(r1,r2) are
π(p∗
1(r1,r2), p∗
2(r1,r2))
= p∗
1

a(1 − θ)− p∗
1 + θp∗
2
1 − θ2

−
c
2

a(1 − θ)− p∗
1 + θp∗
2
1 − θ2
2
A conjecture r∗ is an ESS for N = 2 if it solves the following maximization
problem
max
r

π∗(r,r∗) − π∗(r∗,r)

.
Assuming a symmetric equilibrium, i.e.r = r∗, the ﬁrst-order condition is given
by
(r∗+1)(r∗θ(c−θ+1)−2+θ(θ+1)−c)(θ+1)(θ−1)2θ2a2
(2θr∗(c+2)(c−θ2+1)+(r∗)2θ2(2c+c2+θ2+1)−(c−3θ+θ2+2)(c+3θ+θ2+2))
2 = 0.
Hence, the two candidate solutions are given by by r∗
1 =− 1 and r∗
2 =
c−θ(θ+1)+2
θ(c−θ+1) .
To check the second-order condition, note that since 0 <θ<1
∂2 (π(r,r∗) − π(r∗,r))
∂r2


 
r=r∗=−1
=−
θ2a2 (c + 2(1 − θ))(θ + 1)2 (θ − 1)2

6θ (θ − c) + 4θ(θ2 − 1) + θc

cθ + 2

θ2 − c

− (c + 2)22 < 0.
Furthermore,
sign
	
∂2 (π(r,r∗) − π(r∗,r))
∂r2


 
r=r∗=
c−θ(θ+1)+2
θ(c−θ+1)


=− sign(2(θ − 1) − c) > 0.
   
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Conjectural variations and evolutionary stability in finite populations.

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