Download Games as Social Conventions

Games as Social Conventions
Micha÷Króly
November 30, 2011
Abstract
I propose a set-valued solution concept for non-cooperative game theory, based
on a recent generalization of the von Neumann-Morgenstern Stable Set. The concept
extends that of the Nash Equilibrium, and its rationale of a ‘stable social convention’,
allowing for the fact that conventions need not determine individuals’ exact strategy
choices, but may permit ‡exibility of behaviour within the limits of convention.
In particular, a restriction of the players’strategy sets is said to constitute a ‘stable
convention’when any player who believes others will adhere to it (but is ignorant of
their exact strategy choices) …nds a strategy irrational if and only if it violates the
convention.
I apply the concept to a simultaneous-move quantity-price competition game, and
…nd that Cournot is a stable convention when production costs are relatively high and
di¢ cult to recover for unsold output. In contrast, Bertrand competition is never stable.
1
Introduction
The purpose of this paper is to apply a re-interpretation of one of game theory’s earliest
solution concepts, the von Neumann - Morgenstern (vN-M) Stable Set, to a classic, ongoing problem in oligopoly theory, namely the discrepancy between the Cournot model of
quantity choice and the alternative price-setting Bertrand speci…cation.
Recalling the debate, Cournot seems, at …rst, to exhibit more desirable properties. Not
only the existence (Novshek [1985]), but also the uniqueness (Friedman [1977]) of Nash
Equilibrium holds under fairly general conditions, leading to plausible comparative statics
results (Dixit [1986], Novshek [1980]). Indeed, Cournot has attracted considerable attention
within empirical Industrial Organization (see e.g. Aiginger [1996], Brander and Zhang
[1990], Domowitz et al. [1987], Genesove and Mullin [1998], Haskel and Martin [1994]), as
well as experimental Game Theory (Feinberg and Husted [1993], Raab and Schipper [2009],
Morrison and Kamarei [1990]). While the said literature does not unanimously support the
y
This work was supported by the Economic and Social Research Council [grant number ES/G016321/1].
Department of Economics, The University of Manchester, [email protected]
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Cournot framework, it does provide considerable evidence for its key testable implications,
such as relatively high price-cost margins that, in addition, are decreasing in the number of
…rms and demand elasticity. It will also be of special importance for the present study that
Cournot appears to be a particularly good description of utility industries (Puller [2007],
Willems et al. [2009]).
In contrast, given constant marginal costs and unbounded capacity, Bertrand competition yields the paradoxical zero-pro…t result regardless of the number of …rms (see
Harrington [1989] for a modern treatment and Baye and Morgan [2002] for an extension to
discontinuous costs), which has relatively little empirical support (see Bresnahan [1989]).
For di¤erent cost / capacity speci…cations, the model is typically indeterminate, either
because of non-existence of equilibria (e.g. Edgeworth [1925], Shapiro [1989]), or due to
their multiplicity (Dastidar [1995], Hoernig [2007]).
Nevertheless, the Cournot model was often described as being ‘right for the wrong
reasons’(a term which, according to Vives [1989], was initially coined by Fellner). Although
it yields an outcome consistent with the stylized facts, it does so without allowing for price
choices, which, in reality, is an essential part of entrepreneurial decision-making (Aiginger
[1999]). Indeed, a common view is that …rms set both quantities and prices (see Judd
[1996]).
Unsurprisingly then, considerable research e¤ort has been devoted to demonstrating
that although …rms do not directly compete in quantities alone, their seemingly more
complex strategic interaction implicitly complies with the simpli…ed Cournot framework
or, as Tirole [1988] put it, has the Cournot reduced form. This was often achieved by
imposing a dynamic structure on the original static problem, for instance, by introducing
an initial stage of the game at which the …rms commit to a certain mode of competition,
either directly, as in Singh and Vives [1984], or e.g. via capacity choice, as in Kreps and
Scheinkman [1983]. However, the scope of this approach is limited by the need to justify
the availability of such commitments in reality.
An interesting alternative was provided by Klemperer and Meyer [1986, 1989], who
considered static models with rich strategy sets, encompassing both Cournot and Bertrand
strategies1 . While this leads to multiplicity of equilibria, their number is dramatically
reduced in the presence of demand uncertainty, where the surviving equilibria exhibit
Cournot play when the marginal costs are relatively steep. Even so, this goes only halfway
to show that Cournot is right ‘for the right reasons’, because the fact that players choose
a particular pro…le of Cournot strategies in equilibrium does not mean that the entire
Cournot game is valid.
The present paper presents a potential remedy to this and similar problems, by formulating a solution concept o¤ering insights into the emergence and stability of games within
more general strategic environments, in a way analogous to Nash predictions regarding the
1
Klemperer and Meyer [1986] considers a simple union of Cournot (quantity) and Bertrand (price)
strategies, whereas Klemperer and Meyer [1989] allows for more general ‘supply-functions’, committing
sellers to possibly di¤erent quantities depending on the selling price.
2
games’speci…c outcomes. The postulated notion can also mirror the standard equilibrium
comparative statics, in that it makes it possible to study the e¤ect of changes in parameter
values / assumptions on the shape of ‘social conventions’.
The last notion is traditionally used to justify the Nash Equilibrium concept, by arguing that it describes an established way of playing the game, possibly arrived at via an
underlying dynamic process of repeated play (see Schotter [2008] for an overview of the related literature, and Young [1993] for a seminal study of the stochastic processes involved).
Nevertheless, in this paper I take the view that conventions need not determine individuals’
exact strategy choices, but may permit ‡exibility of behaviour within the limits of convention. Thus, I use an extended notion of ‘convention’, denoting, more generally, a collection
of the players’restricted strategy sets (or their Cartesian Product), rather than a point in
the strategy space. In this sense, the paper will establish the requirements for competition
à la Cournot to become a conventional way of playing a more general price-quantity game,
where the convention does not restrict the players to any speci…c quantity choices.
In fact, the present solution concept is based on one of the earliest and most profound
formalizations of the notion of ‘standards of behavior’- the vN-M Stable Set, particularly
its recent generalization by Luo [2001], further axiomatized in Luo [2009], in which the
dominance relation is conditional on the set of available alternatives. More precisely, I will
call a restriction of the players’ strategy sets a ‘stable convention’ when any player who
believes others will adhere to it (but is ignorant of their exact strategy choices) …nds a
strategy irrational if and only if it violates the convention. Indeed, the ‘necessary’ and
‘su¢ cient’parts of the de…nition re‡ect the vN-M internal and external stability requirements. Thus, the current paper can be seen as an application of a recent interpretation
of one of game theory’s earliest solution concepts to an even older, but still live problem
within oligopoly theory.
Having laid down the details of the postulated concept in Section 2, in Section 3 I
apply it to a simultaneous-move variant of the quantity-price model of Moreno and Ubeda
[2006]. The latter is a re…nement of Kreps and Scheinkman [1983] in which the …rms set
reservation (rather than exact) prices. This feature is particularly suitable for the present
study, as it means that, depending on how the players’default strategy sets are restricted,
one can obtain either Cournot- or Bertrand-equivalent games.
It turns out that Cournot constitutes a stable convention if and only if production
costs are relatively high and di¢ cult to recover for unsold output. In contrast, Bertrand
competition is never stable. Thus, the entire Cournot game, rather than just its equilibrium
outcome, may be justi…ed as a valid simpli…cation of oligopolistic competition when players
set reservation, rather than exact prices, an assumption applicable to many modern-day
trading platforms (see Section 3.2). Expecting others to behave in some way compliant
with Cournot competition is su¢ cient to deter any violation of the convention. In this
sense, when the derived conditions are satis…ed, the Cournot model is indeed ‘right for the
right reasons’.
3
2
The concept
2.1
De…nition and basic properties
Consider a normal-form game G = N; fSi gi2N ; fUi gi2N , where N = f1; 2; :::; ng is the
set of players, Si is the set of (possibly mixed) strategies of player i and Ui (s) 2 R is the
associated payo¤ when a strategy pro…le s 2 j2N Sj is played. Also let S i
j2N nfig Sj
be the set of strategy pro…les of players other than i.
De…nition 1 A product set i2N Si0 is a stable convention when for each i 2 N we have
Si0 Si and when it is true that si 2 Si nSi0 if and only if there exists a s0i 2 Si0 such that
Ui s0i ; s0 i
Ui si ; s0 i for all s0 i 2 S 0 i , where the inequality is strict for some s0 i .
In other words, no conventional strategy of any player i (one in Si0 ) is weakly dominated by another conventional strategy against the set S 0 i of other players’conventional
strategy pro…les, while every unconventional strategy (i.e. not in Si0 ) is weakly dominated
by some conventional strategy.
De…nition 1 lays down what is, formally, a set-valued solution concept - one that assigns
to every game G a collection of stable conventions, each of which determines a set of strategy
pro…les i2N Si0 . An outcome / solution is therefore a game G0 contained in G (in terms
of strategy sets), that could be argued to describe a conventional, self-sustainable way of
playing G, just as it is the case for a Nash Equilibrium.
Remark 2 Every strict Nash Equilibrium of G constitutes a stable convention, where the
strategy sets Si0 are singletons.
The main source of motivation for the present concept is that any conjectures on behalf
of the players about the strategies others are going to choose (or the respective probabilities)
must be based on their knowledge of the structure of the game, i.e. the formation of these
conjectures can only happen once the structure of the game has already been determined.
Conversely, the structure of the strategic interaction should be determined a priori, or in
the absence of the said conjectures2 . This re‡ects the view that, in reality, the reason
why people obey a social convention is not that they expect others to take some speci…c
conventional actions, and hence they decide, as a best-response to these actions, to do
something which also happens to be within the limits of the convention. Instead, people
adhere to social conventions simply because they expect others to do so, whatever their
speci…c strategy choices might be. Only once they accept a convention (and believe it to
be universally accepted) they decide on the exact action they wish to take within its limits.
2
This is somewhat analogous to the problem raised by Aumann and Dreze [2009], namely that the
formation of subjective probabilities about the counterparts’ strategy choices in games is in con‡ict with
specifying the set of strategies that are available to the players as a basis for the construction of these
subjective estimates. This is because the latter speci…cation would, in turn, a¤ect the counterparts’ play
and the corresponding estimates.
4
That is not to say that acceptance of a convention is seen as itself constituting a
strategic decision, i.e. that individuals could anticipate the equilibrium conjectures and
exact actions following from a given convention, and strategically ‘choose’ to adhere to
the convention that they think will result in the most favourable equilibrium outcome.
This would presuppose the existence of a two-stage ‘meta-game’, in which the strategy of
player j entails selecting a set Sj0
Sj , and a function prescribing a particular strategy
s0j 2 Sj0 depending on fSi0 gi2N nfjg , i.e. the convention that players jointly choose to follow
at stage one. However, such a commitment to not playing certain strategies is questionable,
as it assumes the existence of institutions / instruments that make it possible in reality.
Furthermore, an equilibrium outcome of the meta-game would entail a choice of speci…c
strategies at stage two, making the fact that these belong to some hypothetical, restricted
strategy sets super‡uous for the description of the outcome. For instance, the question of
whether …rms actually play the Cournot game would be irrelevant if their decision process
was e¤ectively an analysis of the general price-quantity game, followed by a choice of
speci…c price-quantity strategies that may, or may not, implement an outcome consistent
with Cournot play.
For those reasons, the approach taken here is inherently static, and the question it
strives to address is simply this: Under what conditions would a given social convention
be stable / self sustainable, based solely on the fact that it is believed to be universally
accepted? In particular, the absence of any additional knowledge or conjectures leads
to the use of weak (rather than strict) dominance in De…nition 1. To see why, consider
an individual who gets to choose between two options, the …rst one of which is either
equally good or strictly better than the second, but is completely ignorant of which of those
possibilities is the case (including their respective likelihoods). Naturally, it is irrational to
choose the second option. Similarly, strategies weakly dominated by conventional strategies
against the counterparts’conventional play will never be used, and must be the ones that
lie outside the convention if the latter is to be self-sustainable based on rational behaviour.
The fact that every unconventional strategy is dominated by some conventional one
ensures that any unilateral violation of the convention is irrational. In comparison, requiring that one conventional strategy may not dominate another prevents any strategies that
are ‘redundant’ in this sense from entering a stable convention. This captures the natural tendency of economic agents to simplify the strategic problems they have to face and
underlines the potential role of social conventions in reducing the excessive complexity of
strategic interactions. Problems associated with the computational complexity of equilibria
are well-known and it is often questioned if, in reality, individuals would be able to arrive
at equilibrium outcomes that are identi…ed by researchers only with the greatest di¢ culty
(see Papadimitriou [2007] for an overview of this issue). In fact, computational constraints
become increasingly important in empirical Industrial Organization (see e.g. the discussion
related to the ‘curse of dimensionality’in the seminal studies of Pakes and McGuire [1994,
2001], and a recent paper by Doraszelski and Satterthwaite [2010]). The posited approach
could o¤er the advantage of reducing the computational burden that algorithms / agents
5
have to face through a decrease of the number of strategies that need to be considered.
Both requirements embedded in De…nition 1 may be better understood by means of
the following example.
Example 3 Consider the following two-player normal-form game:
A
B
C
D
(0; 6)
(4; 2)
(1; 1)
E
(2; 5)
(0; 4)
(1; 5)
F
(1; 6)
(3; 0)
(4; 3)
There exists no Nash Equilibrium in pure strategies, and no strategies can be eliminated as
(weakly) dominated or non-rationalizable. However, if we eliminate B, then D is weakly
dominated by F , and if we eliminate D, then B is strictly dominated by C. Neither of
the remaining strategies is then weakly dominated by another, i.e. removing both B and D
results in a stable convention.
A crucial feature that distinguishes the stable convention from other set-valued solution
concepts is that the criterion of rationality utilized here is based on ignorance about other
players’ choices within their restricted strategy sets, i.e. a form of strategic uncertainty.
For instance, in Example 3 the column player knows only that the counterpart will not
play B, but is completely ignorant as to whether she will choose A or C. Based on that,
it is irrational to choose D, since it would yield either 6 or 1 (with unknown respective
likelihoods), whereas a conventional strategy F would give 6 or 3.
In general, any unconventional strategy must be matched and at times bettered against
all pro…les of conventional (possibly mixed) rivals’strategies by a single conventional strategy. This contrasts the two other, closely related set-valued concepts of sets Closed Under
Rational Behavior (CURB) proposed by Basu and Weibull [1991], and Preparation (prep)
sets introduced by Voorneveld [2004, 2005]. These require only that any strategy outside the restricted strategy set prescribed by the solution is matched (prep), or bettered
(CURB) against all combinations of rivals’ restricted strategies (and their mixtures) by
possibly di¤ erent strategies within the player’s restricted set3 . To illustrate, suppose the
payo¤ of the row-player associated with strategy-pro…le fC; Eg in Example 3 is equal to
1 instead of 1. Then strategy B can no longer lie outside the convention, as it is no longer
(weakly) dominated by C against fE; F g (so that the convention is not stable anymore).
However, B could still lie outside a CURB or a prep set and satisfy the corresponding
requirements with respect to the same strategies fE; F g, because B is bettered by A when
the rival plays E, and it is bettered by C when the rival plays F .
3
A more common, equivalent way of stating those de…nitions is that the players’restricted strategy sets
must contain all (CURB) or at least one (prep) best-response to any pro…le of the counterparts’restricted
strategies (and their mixtures). See Voorneveld [2004] for more details.
6
The notion of ‘simplicity’ embedded in the concepts of minimal CURB / prep sets
(ones that are not proper supersets of other CURB / prep sets) is, in turn, somewhat
analogous to the requirement that no conventional strategy can weakly dominate another
for the convention to be stable. Without this condition, which eliminates the ‘redundant’
strategies, any game as a whole would constitute a stable convention. However, as opposed to minimal CURB / prep sets, one stable convention may well contain another. For
instance, the entire game in Example 3 is stable (as no strategy is weakly dominated by
another), and so includes the only other stable convention fA,Cg fE,Fg.
Finally, it should be noted that any strict Nash Equilibrium is clearly both minimal
CURB and minimal prep, since any non-equilibrium strategy is bettered by the equilibrium
one against equilibrium play of others. This means (recall Remark 2) that all the described
concepts coincide at single-valued solutions.
In addition to the above analogies, a much more direct parallel can be drawn between
the present concept and one of the earliest and most profound formalizations of ‘standards
of behaviour’. This is explained below.
2.2
Relationship to von Neumann - Morgenstern stable sets
De…nition 1 can be linked to the vN-M stable set concept, by expressing it using the general
systems framework proposed by Luo [2001]. To see this, consider the following instance of
a general system:
A
S;
; S
(1)
j2N Sj
A S
and let:
A
i
fs
i
2S
i
j 9si 2 Si : (si ; s i ) 2 Ag
where A is a conditional dominance relation on S such that for any s1 ; s2 2 S and A S
we have s1 A s2 if and only if for some i 2 N it is true that for all a i 2 A i we have
Ui s1i ; a i
Ui s2i ; a i , and the inequality is strict for some a i .
The general stable set S 0
S of system (1) is then de…ned as the vN-M stable set of an
0
S
; i.e. one in which the unconditional dominance relation coincides
abstract game S;
with
S0 .
That is to say, S 0 must satisfy:
1. [internal stability] @s; s0 2 S 0 : s
S0
s0
2. [external stability] 8s 2 SnS 0 9s0 2 S 0 : s0
S0
s
0
Firstly, observe that S 0 must be in Cartesian product form, i.e.: S 0
j2N Sj . Other0
wise, any strategy pro…le not in S but consisting of strategies that are each part of some
strategy pro…le in S 0 could only be dominated by an element of S 0 if the internal stability
requirement was violated.
7
Clearly, the internal stability requirement is then equivalent to the ‘if’part of De…nition
1, which states that one conventional strategy cannot weakly dominate another. Similarly,
any S 0 satisfying the ‘only if’part of the De…nition must be externally stable, because any
strategy pro…le s 2
= S 0 includes a strategy si weakly dominated by some s0i 2 Si0 , so that
0
s0 S s for any strategy pro…le s0 2 S 0 which includes s0i . Conversely, for any strategy
si 2
= Si0 , where S 0 is generally stable, consider a strategy pro…le
s
fsj gj2N ; where sj = si for i = j and otherwise sj 2 Si0 :
0
There must exist a s0 2 S 0 such that s0 S s and, as ensured by the internal stability of S 0 ,
such that s0j 2 Sj0 for i 6= j. Consequently si must be weakly dominated by some s0i 2 Si0 ,
satisfying the ‘only if’ part of De…nition 1. Thus, the convenient terms of internal and
external stability will, henceforth, be used in reference to the respective parts of De…nition
1. In summary, the above discussion establishes the following result.
0
Proposition 4 Any stable convention S 0 is a vN-M stable set of an abstract game (S; S ),
A
and any general vN-M stable set S 0 of a general system (S;
) is a stable convention.
A S
Naturally, one of the most well-known properties of vN-M stable sets is their nonuniqueness, which will hence also be inherent in the present solution concept. Indeed, it
may often be impossible to characterize the complete set of stable conventions in the game
under consideration, as this is generally more di¢ cult than identifying the set of Nash
Equilibria.
For this reason, rather than to say something speci…c about the outcome of a strategic
interaction, a more practical application of the current notion might be to use it as a
model evaluation or selection criterion for non-cooperative game theory. For instance, one
might be faced with a modelling choice between two ways in which a complex strategic
problem may be simpli…ed for the purpose of tractability, by reducing the players’strategy
sets and focusing only on selected aspects of the problem at hand. Suppose further that
the two alternative speci…cations can be given equally plausible justi…cations. One may
then invoke the present concept to determine which one (if any) of the postulated model
set-ups is a stable convention, and hence constitutes a more justi…able way of framing the
problem by rational agents. This brings us back to the earlier mentioned classic debate
within oligopoly theory.
3
3.1
Application: Cournot vs. Bertrand
Background
The classic Kreps and Scheinkman [1983] speci…cation has two …rms simultaneously choosing quantities of output fqi gi=1;2 to be produced at a cost given by a twice-continuously
8
di¤erentiable, convex function C, satisfying C (0) = 0 < C 0 (0). This is interpreted as
‘building capacity’, and the associated choice of the rival may be observed, before the …rms
simultaneously select the exact prices pi at which they wish to o¤er their outputs. The
authors also assume that the inverse market demand function P ( ) is twice continuously
di¤erentiable, strictly decreasing and concave so long as it remains positive, and equal
to zero thereafter. It turns out that the Cournot equilibrium outcome is also the unique
subgame-perfect equilibrium (SPNE) outcome of the two stage game, in the sense that
…rms choose the Cournot equilibrium quantities fqic gi=1;2 as their capacities at stage one,
and then name prices equal to P (q1c + q2c ). However:
1. As demonstrated by Davidson and Deneckere [1986], the result relies on a particular
(‘e¢ cient’) rationing rule, applied when the lower-priced …rm is unable to satisfy the
entire demand at its price due to the previously self-imposed capacity constraint.
Speci…cally, the rule states that the residual demand of the more expensive …rm is
minimized and is equal to the market demand at its price reduced by the other …rm’s
capacity. Davidson and Deneckere consider an alternative, ‘proportional’ rationing
rule, whereby the higher-priced …rm only loses a fraction of the corresponding market
demand equal to the proportion of the market demand associated with the lower
price satis…ed by the cheaper seller. They show that in this case, as well as for most
intermediate rationing rules between the ‘proportional’and ‘e¢ cient’extremes, the
Cournot outcome is no longer sustained as a SPNE.
2. For a continuum of stage one quantity (capacity) choices other than fqic gi=1;2 , only
mixed strategy equilibria in prices exist, which is subject to the usual criticism (see
e.g. Rubinstein [1991]).
3. For a continuum of stage one quantity (capacity) choices other than fqic gi=1;2 , the
subsequent price equilibria exhibit aggregate sales smaller than the aggregate stage
one quantity (capacity). Thus, the two-stage game does not have a Cournot reduced
form, in the sense of Tirole [1988].
A recent modi…cation of the Kreps and Scheinkman model, proposed by Moreno and
Ubeda [2006], generalizes it to n 2 players and resolves di¢ culties 1: and 2: above, by
assuming …rms set reservation, rather than exact prices (the di¤erence is explained below
in the context of the current framework). While the Cournot equilibrium outcome is still
sustained as a SPNE (without resorting to mixed strategies o¤ the equilibrium path), the
last of the above three di¢ culties remains, and the results do not justify the Cournot game
as a representation of the way …rms compete4 .
4
Other variants of the Kreps and Scheinkman [1983] framework are capable of addressing problems
1 3, but only by means of making somewhat ‘special’ assumptions about market demand. When the
demand is perfectly inelastic, as in Acemoglu et al. [2009] or Fabra et al. [2006], the market demand is the
same regardless of the price and all rationing rules from ‘e¢ cient’to ‘proportional’coincide. Alternatively,
9
3.2
The model
The current framework is the same as that of Moreno and Ubeda, except …rms are assumed
to make both quantity and pricing decisions simultaneously, so that the quantity choice
need not be interpreted as a capacity precommitment. Thus, a …rm’s strategy is a nonnegative quantity-price pair, i.e.:
Si
f(qi ; pi ) j qi ; pi
0g
In addition, production technology is no longer required to be the same for all …rms, where
the individual cost functions Ci satisfy the same assumptions as C in the models described
above. The outcome of a particular strategy pro…le is the same as the one resulting from
the …rms choosing the corresponding quantities (capacities) and prices in the two-stage
game of Moreno and Ubeda. Speci…cally, prices pi are still the minimum at which the
…rms declare themselves willing to sell their corresponding outputs qi . This gives rise to
an aggregate supply correspondence S (p), de…ned for a pro…le of quantity-price strategies
and a market price p as follows:
S (p) =
X
j2fi2N j pi <pg
qj ;
X
j2fi2N j pi pg
qj
Although S (p) may not be single-valued, the equilibrium price p is uniquely determined
by the market-clearing condition:
P
1
(p ) 2 S (p )
As opposed to the Kreps and Scheinkman speci…cation, output is sold exclusively at the
market equilibrium price. In particular, when the quantity demanded P 1 (p ) is insuf…cient to match the entire produce of …rms with reservation prices not greater than p ,
only …rms which set prices strictly below p are guaranteed to sell their entire output. Any
residual demand is distributed among the …rms characterized by pi = p according to a
tie-breaking rule, which is left unspeci…ed, as it turns out that it does not in‡uence the
results in any way. More precisely, let xi denote the quantity of output sold by …rm i. We
then have xi = 0 for pi > p , xi = qi for pi < p , and …nally, for pi = p , the exact value of
xi depends on the tie-breaking speci…cation, but lies within the following interval:
max 0; P
1
(p )
X
j2fl2N nfigj pl p g
qj
; min qi ; P
1
(p )
X
j2fl2N j pl <p g
qj
Although this entails an implicit rationing rule (…rms with pi < p sell …rst), it is the
only appropriate rule in the present circumstances and by no means arbitrary. Indeed, one
uniformly elastic demand, as in Madden [1998], ensures that the two-stage game has the Cournot reduced
form.
10
cannot imagine a situation in which output o¤ered for sale below the prevailing market
price remains unsold.
As argued by Moreno and Ubeda, this type of pricing speci…cation describes more realistically situations in which most of the trading occurs at the market clearing price, where
the authors point to utility industries as an example. In my view, con…ning the game to a
single stage makes the model even more widely applicable to modern-day trading platforms.
For instance, in online auction sites, sellers can o¤er their products at a minimum/starting
price, speci…ed simultaneously with other auction parameters5 . Due to the ease of search
and comparison o¤ered by these sites, one then strongly expects the …nal selling prices of
close substitutes to be equal or marginally di¤erentiated. In fact, a similar mechanism is
intrinsic to many other markets, from the opening/closing stock market auctions to online
betting exchanges, i.e. wherever IT-based intermediaries can play a role similar to the
mythical ‘Cournot auctioneer’.
The last issue that should be considered at this point is whether production costs
incurred by the …rms are ‘sunk’, i.e. incurred at the time of making an o¤er, based on
the values of qi , or depend only on the actual sales xi
qi . The two respective payo¤
speci…cations are:
[sunk costs] :
i
= p xi
Ci (qi )
[sales-dependent costs] :
i
= p xi
Ci (xi )
Although the models described in Section 3.1 adhere to the …rst alternative (as is natural
when interpreting quantity as capacity precommitment), it is an empirical matter which
speci…cation is appropriate for simultaneous price-quantity decisions. For instance, some
(but not all) types of goods may need to be produced ‘up-front’, e.g. before setting up
an auction, so as to enable the seller to deliver them within a reasonable time frame.
However, in the latter case it may still be possible to recover a certain fraction of the costs
of producing the output that has not been sold. For this reason, the present paper will,
more generally, consider both of the above extreme possibilities, as well as (for constant
unit costs) the ‘intermediate’cases, i.e. the generic payo¤ function that will be used is:
i
= p xi
[(1
) Ci (xi ) + Ci (qi )]
where
2 [0; 1] is the proportion of the production cost that is sunk and cannot be
recovered for unsold output.
5
The ‘Dutch Auction’format allows sellers to list several items at a single reservation price, where the
result is determined in a way consistent with the present model speci…cation. This is supported by major
auction sites, with the exception of eBay, which restricted its use in 2009. However, eBay sellers may still
emulate the format by posting several copies of the same single-item listing.
11
3.3
Results
A convenient feature of the current model speci…cation is that by restricting the original
strategy sets Si accordingly, one can reduce the game to a Cournot or a Bertrand model.
0
0
In particular, a convention S 0
Si is compatible
i2N Si of restricted strategy sets Si
0
0
with the Cournot speci…cation
only
if
for
any
strategy
pro…le
f(q
;
p
)g
2
S 0 , the total of
i i i2N
P
players’outputs Q = i2N qi0 is sold at the corresponding demand price p = P (Q). This
in turn happens only when the market-clearing price p is in excess of any of the players’
respective reservation prices p0i .
Furthermore, it should be the case that each player’s strategy set Si0 includes a continuum
of quantities up to the optimal monopoly output (larger quantities are strictly dominated
in a Cournot game under the present assumptions). More precisely:
Notation 5 Let qim denote the optimal monopoly output of player i, i.e.:
qim = arg max fqP (q)
q 0
In addition, let:
Qm =
P
De…nition 6 A convention S 0
compatible when we have:
j2N
qjm ; and Qmi =
0
i2N Si
8
i2N
P
m
j2N nfig qj
of restricted strategy sets Si0
8i 2 N 8qi0 2 [0; qim ] 9p0i
qi0 ; p0i
Ci (q)g
2S
0
qi0 ; p0i 2 Si0
P
: P ( i2N qi0 )
:
Si is Cournot(C1)
max p0i
i2N
(C2)
Despite the fact that players’ strategies are speci…ed in terms of both quantities and
prices, satisfying the above de…nition means that there is a one-to-one mapping between
such quantity-price pairs and their corresponding quantities in the classic Cournot game, in
the sense that they each yield the same payo¤s in their respective games. In other words,
a Cournot-compatible restriction of the current game may be thought of as taking the
original Cournot model and ‘labelling’every quantity strategy with an additional number
(price) without any consequence for the payo¤s. Whether or not these labels are present,
we are then dealing with the same game.
We now have the apparatus required to investigate the question of whether the Cournot
model is a justi…able restriction of a more general quantity-price competition framework.
More speci…cally, I will now establish the necessary and su¢ cient conditions for the existence of a Cournot-compatible restriction of the model de…ned in Section 3.2, which is also
stable in the sense of De…nition 1.
12
Proposition 7 For = 1 (sunk costs), there exists a Cournot-compatible stable convention if and only if the following two conditions hold for every player i 2 N :
P Qmi
m
P (Q ) +
For
Ci0 (0)
0
(IS1)
qim P 0 (Qm )
0
(ES1)
= 0 (sales-dependent costs), a Cournot-compatible stable convention does not exist.
It is easy to see that condition IS1 is both necessary and su¢ cient for the existence of
an internally stable (satisfying the ‘if’part of De…nition 1) Cournot-compatible convention.
On the one hand, if it did not hold for some player i, then producing nothing would be
dominated by a su¢ ciently small (and hence also in Si0 ) quantity of output. On the other
hand, if it does hold for every player, then a convention comprising exclusively quantities
below the monopoly optimal ones is internally stable, because no such level of output
dominates another in a Cournot game.
Condition ES1 is similarly linked to the external stability (‘only if’) requirement, albeit
in a somewhat more complicated way. Intuitively, the potential appeal of Cournot competition to the participating …rms is that it ensures that one’s entire output is always sold.
This is particularly important when production costs are sunk and the …rms tend to sell
relatively small quantities of output at relatively high prices. To see why, observe that it
is then better to sell everything at a lower (but still relatively high) price, than to leave
some of the potentially expensive produce unsold, but succeed in charging more for the
remainder.
Low quantities and large prices can, in turn, be associated with an ine¢ cient production
technology and a small number of …rms in the industry. In such case, even the optimal
monopoly outputs are low (individually and in total), as re‡ected by condition ES1 (recall
that the inverse demand P ( ) is concave, making the inequality true for su¢ ciently small
quantities qim ). In particular, the condition states that the marginal revenue of each …rm
is non-negative even when everyone produces their largest rationalizable output. This
happens when the marginal costs of production rise quickly, as the …rms only want to
produce while their monopolistic marginal revenue is at least as large. When the number
of players is not too big, then even the substantially smaller ‘oligopolistic’marginal revenue
is still non-negative.
In contrast, when the monopoly outputs are high and there are many …rms, Cournot
competition is potentially risky, as it entails a possibility of a damaging ‘coordination failure’scenario, in which a very large aggregate output means the …rms sell their substantial
produce at a price signi…cantly below their average cost level. Hence, in such case Cournot
is no longer a stable convention.
For more insight into the origin of condition ES1, note that it states that when every
…rm produces qim ; the decrease in revenue due to a marginal reduction of sales (for a
…xed price) is greater than the decrease in revenue brought about by an increase in the
13
competitors’ production by the same marginal amount, subject to a price adjustment
allowing the increased aggregate output to sell entirely. In other words, a surge in the
counterparts’ production is less damaging to players who compromise on their products’
prices, than to ones who would rather maintain the original unit price at the cost of being
left with the residual customer demand. Indeed, the fact that this holds when everyone
behaves as a monopolist implies the same is true for lower production levels (see Proof).
Proposition 7 establishes that Cournot competition can be a stable convention when
costs are sunk, but not when they are sales-dependent. This suggests that, in general,
Cournot may be stable when the proportion of costs that are impossible to recover for
unsold output is su¢ ciently large. Intuitively, when increases, it becomes less appealing
to sell less at the same price (while losing a proportion of the unsold output’s cost),
compared to selling the original amount cheaper. Indeed, the next proposition con…rms
this intuition for constant unit costs of production.
Proposition 8 For 2 (0; 1) and Ci (q) = qci , ci > 0 (constant unit costs), Cournot
competition constitutes a stable convention if and only if the following two conditions hold
for every player i 2 N :
P Qmi
m
P (Q ) +
qim P 0 (Qm )
(1
ci
0
(IS2)
) ci
0
(ES2)
A question that naturally arises is: When Cournot competition is stable, is it the unique
stable convention? Alternatively, can anything be said about the set of all existing stable
conventions? The next result will be of use here.
Proposition 9 Suppose the requirements of Proposition 7 orPthose of Proposition 8 are
satis…ed. Then for any strategy pro…le f(qi0 ; p0i )gi2N ; such that i2N qi0 Qm ; of any stable
convention S 0 , we have:
8i 2 N s:t: qi0 qim : xi = qi0
In other words, whenever players o¤er in total less than the sum of their monopolyoptimal quantities, anyone o¤ering no more than their individual monopoly-optimal quantity will be able to sell the entire output. This also means that if every …rm i 2 N chooses a
qi0 qim , then the total of the …rms’outputs will be sold at the corresponding demand-price,
precisely as in Cournot. However, this does not preclude o¤ering quantities in excess of
qim at prices that do not guarantee selling all output, even if other …rms produce relatively
little. Still, the fact is interesting in the light of the common view (Maggi [1996], Shapiro
[1989]) that Cournot is a more apt description of oligopolistic competition than Bertrand
when the …rms are faced with capacity constraints. This is formally demonstrated here, as
we see that if the …rms’production capacities were not greater than qim , then (under the
speci…ed conditions) any stable convention would always entail the entire output being sold
in the Cournot manner. Either way, Proposition 9 provides the framework with potential
14
testable implications, which will be brie‡y discussed in Section 3.4. At this point, it is
worth recalling the fact (noted in the introductory literature review) that the predictions
of the Cournot model are consistent with the data from energy industries. These are typically put forth as a prime example of reservation, rather than exact pricing (as discussed
in Section 3.2), and are also associated with capacity constraints. The present model suggests such …rms may adapt Cournot as a stable convention, thereby providing a potential
explanation for the observed empirical pattern.
To illustrate the above considerations, consider the following ‘textbook’ example of
linear demand and constant unit costs.
Example 10 Suppose that P (Q) = max f
Q; 0g and for each i 2 N : Ci (qi ) = qi c;
where > c > 0; > 0; so that i = p xi c [(1
) xi + qi ] :
It follows that 8i 2 N : qim = (
c) =2 , so that condition(s) IS2 reduce to n
condition(s) ES2 are equivalent to:
c
n
n
3, while
1
1+2
In other words, Cournot competition is a stable convention when the unit cost of production
is su¢ ciently large, where the required cost threshold is increasing in the number of players
and decreasing in the proportion of sunk costs.
It is interesting to compare the results in Propositions 7 and 8 to those of Moreno and
Ubeda [2006]. They show that, despite the multiplicity of equilibria at the price competition
stage, a pure strategy equilibrium always exists, while any pure-strategy equilibrium of
the two-stage game yields the Cournot outcome. This improvement on the Kreps and
Scheinkman [1983] result (where pure strategy equilibria in prices may fail to exist) is
achieved by introducing reservation, rather than exact pricing.
It turns out that reservation prices are equally instrumental in providing an alternative
justi…cation for Cournot competition. In fact, the current framework demonstrates that
the entire Cournot game, rather than its outcome alone, may constitute a self-sustaining
convention within a more general form of price-quantity competition. This is achieved
without restructuring the original problem with respect to the timing of moves, and relying
on equilibrium prices to follow any given vector of revealed outputs (capacities). Instead,
the belief that others are willing to sell their (unknown) outputs relatively cheap is su¢ cient
to make it rational for each …rm to set its own reservation price similarly low, provided the
conditions of Proposition 7 or 8 hold. This results in a strategic interaction equivalent to
Cournot competition.
The results are also analogous to those of Klemperer and Meyer [1986, 1989], who
found that, in the presence of demand uncertainty, equilibria of the one-shot competition
game with general strategy sets exhibit Cournot-like strategies when the number of …rms
is small and the (increasing) marginal cost curves are steep relative to demand. This in
15
turn can generally be related to the total of monopoly-optimal outputs being relatively
small, which is what ensures that Cournot competition is externally stable in the present
context. Interestingly, Klemperer and Meyer rely on demand uncertainty, which means
that equilibrium strategies must perform well under di¤erent levels of consumer demand.
In comparison, the current results are obtained under uncertainty about the actions of the
competitors. Hence, each strategy in a Cournot-compatible stable convention must perform
well against di¤erent levels of the competitors’aggregate output, which a¤ects selling prices
and players’payo¤s in a way similar to demand variations.
In the same vein, I will now apply De…nition 1 to evaluate the alternative, Bertrand
speci…cation of oligopolistic competition. To this end, let p0i denote the monopoly ‘breakeven’price, i.e. the smallest price such that the associated monopoly pro…t is non-negative.
To fully mimic the Bertrand behaviour, it is necessary that each player’s strategy set Si0
now includes at least a continuum of prices from p0i up to the optimal monopoly price
P (qim ). Furthermore, the output level for each quantity-price strategy in Si0 must be such
that the …rm is committed to satisfying the entire customer demand at the associated price.
Finally, costs must be sales-dependent, so that a player whose price is undercut does not
have to pay for producing the unsold output. More precisely:
0
De…nition 11 A convention S 0
i2N Si of restricted strategy sets is Bertrand-compatible
when = 0 and for each i 2 N we have:
8p0i 2 p0i ; P (qim ) 9qi0
8
qi0 ; p0i
2
Si0
:
:
qi0 ; p0i 2 Si0
qi0
P
1
p0i
(B1)
(B2)
Note that condition B2 means that trade occurs at a price p equal to the minimum of
the players’reservation prices, as the market-clearing condition P 1 (p ) 2 S (p ) is then
satis…ed (recall the model speci…cation in Section 3.2). The result below easily extends
to 2 (0; 1], and is also valid for any tie-breaking rule applied when two or more players
charge the same lowest price.
Proposition 12 Bertrand competition never constitutes a stable convention.
Proof. See the Appendix.
The above Proposition states that no Bertrand-compatible restriction of the original
game can satisfy the stability requirements of De…nition 1. The reasoning behind this
result is fairly simple. If a player produces less then the market demand at the associated
reservation price, than this strategy cannot be weakly dominated by a Bertrand price above
the unconventional one, since the former is better when rivals set their conventional prices
16
in between the two. Furthermore, the same is true for any conventional price below the
unconventional one, since the latter yields higher pro…ts when the competitors’Bertrand
prices are above the demand price of the unconventional output, at least as long as the last
price lies below the monopoly optimal one. Thus, the external stability criterion is shown
to fail through an argument similar to the one which establishes the internal stability of a
Bertrand game.
It follows that Bertrand competition cannot be justi…ed as a stable convention within
the general quantity-price game under a minimum (rather than exact) pricing regime. In
other words, strategies that do not entail a commitment to meet the entire demand at the
chosen price cannot be ruled out as irrational based solely on the fact that others will select
some Bertrand-style strategies. This suggests that the earlier mentioned drawbacks of the
Bertrand framework, such as its indeterminate or paradoxical zero-pro…t predictions with
just two players, would not apply to markets in which reservation pricing is possible, as
the …rms would not frame their strategic interaction in the postulated way.
Naturally, this result depends on the way the general price-quantity model is speci…ed.
For instance, if the …rms were to choose the exact prices at which to o¤er their output,
as in the Kreps and Scheinkman [1983] model, then some Bertrand-compatible convention
may well be stable. However, in such a case no restriction of the players’strategy sets could
be compatible with the Cournot framework, making such a pricing speci…cation unsuitable
for the comparative analysis of Cournot and Bertrand as alternative simpli…cations of
oligopolistic competition.
Finally, it should be stressed that, while neither Cournot nor Bertrand are stable when
the conditions of Proposition 7 or Proposition 8 fail to hold, this does not mean that other,
possibly more complex stable conventions cannot then exist. Indeed, given that the vN-M
stable sets, on which the present solution concept is based, are typically non-unique, this
would be hardly surprising. Unfortunately, a complete characterization of the set of all
stable conventions of the present game is a task beyond the reach of the present study.
3.4
Possible Extensions
Having established the above results, a natural direction for future work would be to seek
empirical veri…cation for its main testable implications, namely the fact that, when the
requirements of Proposition 7 or Proposition 8 are satis…ed and players are observed to
o¤er quantities below the monopoly optimum, the associated reservation prices should
allow the entire aggregate output to sell at the corresponding demand price. If that was
not the case, then it would serve as an indication that players do not adhere to any
stable convention, thereby contradicting the postulated solution concept and its particular
application discussed here.
Such a veri…cation would be fairly straightforward in an experimental context, where
the demand / cost speci…cations, as well as any rules of the game imposed on the subjects
could be easily manipulated. Using empirical data from markets mentioned in Section 3.2
17
(e.g. online auction sites) is more complicated, because one would have to obtain reliable
estimates of the said demand / cost speci…cations (possibly based on proxy measures). Even
then, the estimated functions might not often comply with the assumptions of the model,
unless it is …rst generalized on a theoretical level. Such extensions might include allowing
for increasing (rather than constant) marginal costs for 2 (0; 1), non-zero …xed costs,
and relaxing the assumption of a concave demand in favour of, more generally, concave
revenue. In addition, it would be useful to extend the players’strategy sets, so that, more
realistically, they can submit more than one quantity-price o¤er6 .
The postulated solution concept could also be applied to other problems in Game
Theory / Industrial Organization. For instance, in the light of the well-known controversies
associated with mixed-strategy pricing (see Maggi [1996]), and mixed-strategies in general
(Rubinstein [1991]), one could investigate the conditions for either the set of purely-mixed,
or degenerate (pure) price strategies to constitute a stable convention within the general
mixed-strategy pricing game. Another example might be the issue of the timing of moves in
leader-follower games, typically addressed by adding an initial stage to the original game,
in which the …rms decide when they wish to move and commit to that choice (see Amir
[1995], Hamilton and Slutsky [1990], Normann [2002]). While the consequences of using
the present concept in a dynamic setting should be carefully thought through, it is, in
principle, possible to apply it to general strategy sets (i.e. including the timing decision),
to establish the stability conditions for a convention in which a particular …rm (e.g. the
more e¢ cient one) leads and the other one follows, regardless of what levels of output they
speci…cally choose to produce within that convention.
4
Concluding Remarks
In this paper, I proposed a set-valued solution concept for non-cooperative game theory,
based on a recent generalization of the von Neumann-Morgenstern Stable Set, and applied
it to a long-standing problem in oligopoly theory - the discrepancy between the Cournot
model of quantity choice and the alternative price-setting Bertrand speci…cation.
The postulated concept builds on the premise that, since players make conjectures about
the counterparts’strategy choices based on their knowledge of the game, the structure of
the latter should be determined independently of such speci…c conjectures. In particular,
a restriction of the players’ strategy sets is said to constitute a stable convention when
expecting others to choose some conventional strategies ensures that each player’s set
of irrational choices coincides with unconventional behaviour. More precisely, a strategy
lies outside the convention if and only if it is weakly dominated by some strategy within a
6
Indeed, this would provide a new application / interpretation for the old concept of ‘supply functions’,
as seen in Klemperer and Meyer [1989]. Twenty years ago, it was di¢ cult to o¤er a clear-cut economic
intuition for this speci…cation, but the ability to submit an arbitrary number of quantity - reservation price
o¤ers in online auctions / …nancial markets is nothing but a submission of a supply function.
18
stable convention, so long as others act in a conventional manner. In fact, it was shown that
any stable convention constitutes a von Neumann-Morgenstern stable set for a dominance
relation over strategy pro…les speci…ed accordingly.
The concept was then used to revisit the Cournot vs. Bertrand controversy in the
modelling of oligopolistic competition. In particular, a game was considered in which …rms
simultaneously set quantities of output and minimum prices at which they would be willing
to sell it. It turned out that a restriction of the default strategy sets equivalent to a Cournot
game is a stable convention when production costs are high (and rise steeply) relative to
the number of …rms, while being di¢ cult to recover when output remains unsold. This
is in line with the fact that Cournot is supported by data from utility industries, which
generally match these requirements and constitute a prime example of reservation (rather
than exact) pricing.
In this way, the present paper illustrated how the belief that others will o¤er to sell
their produce relatively cheap makes it irrational to do otherwise, leading to Cournot competition. Thus, the entire Cournot game, rather than just its equilibrium outcome, can be
justi…ed as a description of a more general simultaneous-move quantity-price competition.
In this sense, under certain conditions the Cournot model may indeed be ‘right for the
right reasons’.
Appendix
Proof of Proposition 7.
In order to be stable, a Cournot-compatible convention
0
S0
i2N Si must not include quantities in excess of the monopoly output, which, due to
the concavity of pro…ts, are dominated by other conventional strategies, violating internal
stability. Thus, Cournot-compatibility in this case means that every (qi0 ; p0i ) 2 Si0 satis…es:
P qi0 + Qmi
p0i
(recall part C2 of De…nition 6).
Condition IS1 is then necessary and su¢ cient for internal stability of Si0 , ensuring that no
qi0 2 [0; qim ] is dominated by any other output in a Cournot-compatible game.
I will now show that ES1 is necessary for the external stability of S 0 . To this end, consider
an unconventional strategy s^i = (^
qi ; p^i ) 2 Si nSi0 comprising an output q^i = qim and a
reservation price p^i marginally above P (Qm ). The only strategy s0i 2 Si0 that could match
it entails qi0 = q^i = qim , since any other quantity is outperformed if the remaining …rms
choose to produce nothing. Let:
^
Q
i
=P
1
(^
pi )
q^i
So that the unconventional strategy is equaled by s0i as long as the aggregate output Q i
^ i . For Q i > Q
^ i ; the
from other players’ conventional strategies is not greater than Q
19
di¤erence between pro…ts resulting from s0i and those associated with s^i is:
(Q i ) = [qim P (qim + Q i )
Thus, for
Ci (qim )]
(Q) to be non-negative for all Q
0
^
Q
^
= qim P 0 qim + Q
i
i
i
p^i P
1
(^
pi )
Q
i
Ci (qim )
^ i , it is necessary that:
>Q
+ p^i = qim P 0 P
1
(^
pi ) + p^i
0
which holds for p^i arbitrarily close to P (Qm ) only if:
qim P 0 P
1
(P (Qm )) + P (Qm )
qim P 0 (Qm )
0,
P (Qm )
which amounts to condition ES1.
In addition, it is necessary that production costs are ‘sunk’. Otherwise, the di¤erence
^ i is:
between s0i and s^i for Q i > Q
(Q i ) = [qim P (qim + Q i )
Ci (qim )]
p^i P
1
(^
pi )
Q
Ci P
i
1
(^
pi )
Q
i
leading to:
0
^
Q
i
= qim P 0 P
1
Ci0 (qim )
(^
pi ) + p^i
0
or, for p^i arbitrarily close to P (Qm ) :
qim P 0 (Qm ) + P (Qm )
Ci0 (qim )
0
Substituting qim P 0 (qim ) + P (qim ) for Ci0 (qim ), based on the monopoly pro…t maximizing
condition, this becomes:
qim P 0 (Qm )
P 0 (qim ) + [P (Qm )
P (qim )]
0
which is never the case for a concave P ( ) :
It remains to show that ES1 is not only necessary, but also su¢ cient to ensure the external
stability of a Cournot-compatible convention S 0 , characterized as follows:
Si0
qi0 ; p0i : qi0 2 [0; qim ] ^ p0i
P qi0 + Qmi
Suppose that:
q^i = qi0 2 (0; qim ] and p^i > P q^i + Qmi
Hence, for Q
we have:
i
^
Q
i
p~ = P (Qm )
both strategies once again yield equal payo¤s, while for Q
(Q i ) = qi0 P qi0 + Q
p^i max P
i
20
1
(^
pi )
Q i; 0
i
^
>Q
i
so that for Q
i
^ i; P
2 Q
1 (^
pi )
0
:
(Q i ) = qi0 P 0 qi0 + Q
+ p^i
i
and, using the concavity of P ( ) :
@2
= P 0 qi0 + Q
@Q i @qi0
i
+ qi0 P 00 qi0 + Q
i
< 0;
@2
= qi0 P 00 qi0 + Q
@Q2 i
i
<0
which implies that:
0
(Q i ) > qim P 0 (Qm ) + p~
0
where the last inequality follows from condition ES1. Thus, for Q
i
^ i; P
Q
2
1 (^
pi )
any increase in Q i will improve the payo¤ from using s0i relative to that of s^i , so that the
former will outperform the latter. Trivially, this will continue to hold for Q i P 1 (^
pi ),
since the revenue generated by the unconventional strategy is then equal to zero, while
both strategies entail the same costs.
^ i all …rms’entire outputs
Consider now the case of q^i > qim = qi0 and p^i > p~: For Q i Q
^ i ; P 1 (^
are sold, so q m is better than q^i : For Q i 2 Q
pi ) ; similarly to the previous case:
i
0
(Q i ) = qim P 0 (qim + Q i ) + p^i > qim P 0
P
m
j qj
+ p~
0
and for Q i P 1 (^
pi ) the revenue generated by s^i is equal to zero, while the associated
costs are higher than Ci (qim ).
Finally, suppose q^i > qim = qi0 and p^i p~. For Q i P 1 (~
p) q^i the entire outputs are
0
1
sold and si outperforms s^i . For Q i > P (~
p) q^i the conventional strategy gives a pro…t
of qim P (qim + Q i ) Ci (qim ), while the unconventional one yields at most p~q^i Ci (^
qi ) ;
which is the case when other players’ conventional prices are all at their maximum of p~
and either p^i < p~ or p^i = p~ and the tie-breaking rule is such that it allocates the maximum
possible demand to player i. In other words, the existing output qim is sold at a price of
at most p~, i.e. not greater than P (qim + Q i ) ; which, for the additional output q^i qim ; is
less than the average cost of its production. This is because condition IS1 implies:
p~ = P (Qm ) < P Qmi
Ci0 (0)
and Ci ( ) is convex. Consequently, s^i is, again, outperformed by s0i , which completes the
proof.
21
Proof of Proposition 8. Based on the above proof of Proposition 7, it is clear that
condition IS2 and the exclusion of quantities in excess of the monopoly output, are both
necessary and su¢ cient for internal stability. Analogously, I will now show that ES2 is
necessary for the external stability of S 0 , by considering the case of q^i = qi0 = qim and p^i
^ i = P 1 (^
marginally above p~ = P (Qm ). For Q i > Q
pi ) q^i ; the di¤erence between
0
pro…ts resulting from si and those associated with s^i is:
(Q i ) = [qim P (qim + Q i )
ci qim ]
p^i P
Thus, for
1
(^
pi )
Q
(Q) to be non-negative for all Q
0
^
Q
(1
i
i
1
(^
pi )
Q
i
+ ci qim
^ i , it is necessary that:
>Q
^
= qim P 0 qim + Q
i
) ci P
i
+ p^i
(1
) ci
0
which is the same as:
qim P 0 P
1
(^
pi ) + p^i
(1
) ci
This holds for p^i arbitrarily close to P (Qm ) only if:
qim P 0 P
1
(P (Qm )) + P (Qm )
(1
) ci
, qim P 0 (Qm ) + P (Qm )
(1
) ci
which amounts to condition ES2.
I will now show that ES2 is not only necessary, but also su¢ cient to ensure the external
stability of a Cournot-compatible convention as follows:
qi0 ; p0i : qi0 2 [0; qim ] ^ p0i
Si0
P qi0 + Qmi
Suppose that:
It follows that for Q
^
Q
i
(Q i ) = qi0 P qi0 + Q
q^i = qi0 2 (0; qim ] , p^i > P q^i + Qmi
both strategies yield equal payo¤s, while for Q
i
ci qi0
i
1
p^i max P
+ (1
so that for Q
i
^ i; P
2 Q
0
1 (^
pi )
(^
pi )
i
^
>Q
) ci max P
i
+ p^i
1
(^
pi )
Q i ; 0 + ci qi0
(1
) ci
and, using the concavity of P ( ) :
@2
= P 0 qi0 + Q
@Q i @qi0
i
+ qi0 P 00 qi0 + Q
22
i
< 0;
:
Q i; 0 +
:
(Q i ) = qi0 P 0 qi0 + Q
i
@2
= qi0 P 00 qi0 + Q
@Q2 i
i
<0
which leads to:
0
Thus, for Q
s0i
i
(Q i ) > qim P 0 (Qm ) + P (Qm )
^ i; P
2 Q
1 (^
pi )
any increase in Q
(1
i
) ci
0
will improve the payo¤ from using
relative to that of s^i , so that the former will outperform the latter. For Q
using s^i leads to selling no output, so that we have:
(Q i ) = qi0 P qi0 + Q
P
1 (^
pi )
ci qi0 + ci qi0 =
i
= qi0 P qi0 + Q
i
i
(1
qi0 qim P 0 (Qm ) + P (Qm )
) ci
(1
) ci
0
where the last inequality again follows from condition ES2.
^ i all …rms’entire outputs
Consider now the case of q^i > qim = qi0 and p^i > p~: For Q i Q
^ i ; P 1 (^
are sold, so qim is better than q^i : For Q i 2 Q
pi ) we have:
0
(Q i ) = qim P 0 (qim + Q i ) + p^i
and for Q
i
P
1 (^
pi )
(1
) ci > qim P 0 (Qm ) + P (Qm )
(1
) ci
0
) ci
0
the revenue generated by s^i is equal to zero, so that:
(Q i ) = qim [P (qim + Q i )
(1
) ci ]
qim qim P 0 (Qm ) + P (Qm )
(1
Finally, suppose q^i > qim = qi0 and p^i
p~. For Q i
P 1 (~
p) q^i the entire outputs
0
1
are sold and si outperforms s^i . For Q i > P (~
p) q^i the conventional strategy gives
a pro…t of qim [P (qim + Q i ) ci ], while the unconventional one yields at most q^i (~
p ci ) ;
independent of the tie-breaking rule. In other words, the existing output qim is sold at a
price of at most p~, i.e. not greater than P (qim + Q i ) ; which, for the additional output
q^i qim ; is less than ci . This is because condition IS2 implies:
p~ = P (Qm ) < P Qmi
ci
Hence, s^i is, again, outperformed by s0i , which completes the proof.
Proof of Proposition 12. Consider any player i such that qim = maxj2N qjm and an
unconventional strategy s^i characterized by p^i = 0 and q^i = qim + "0 ; "0 > 0. Suppose
the minimum of the remaining players’ conventional (Bertrand-compatible) prices is p~ =
P (qim + "0 ) : Then the pro…t of …rm i resulting from s^i is ^ 0 = p~q^i Ci (^
qi ). By the
concavity of monopoly payo¤s, any price below p~ yields a pro…t below ^ 0 , where the latter
is positive for "0 su¢ ciently small.
Thus, the only conventional strategy that could match s^i in this situation is s0i = (qi0 ; p0i )
such that qi0 = q^i and p0i = p~. Furthermore, this occurs only if the price-tie is broken by
23
allocating the entire demand to i (for instance, as the most cost-e¢ cient …rm, based on a
particular ‘winner takes all’sharing rule). Suppose this is indeed the case and consider an
alternative situation, in which the minimum of the remaining players’conventional prices
is p = p~ "1 : In this case, s0i gives zero pro…t at best (assuming costs are not sunk), while s^i
yields ^ 1 = p^
qi Ci (^
qi ) : Since ^ 1 can be arbitrarily close to the optimal monopoly pro…t,
s^i will outperform s0i for "0 and "1 su¢ ciently small.
Finally, note that if it was not possible for some player j 6= i to price-undercut P (qim ),
then internal stability would require that Si0 = (qim ; P (qim )), i.e. no price competition à la
Bertrand would be possible due to the monopolistic position of …rm i.
Proof of Proposition 9. Consider the case of = 1, and suppose the players’conventional strategy choices are such that player i is only able to sell q~i out of her entire output
qi > q~i , due to the fact that the total Q i of other players’output is o¤ered at reservation
prices below pi = P (~
qi + Q i ). In particular, the payo¤ of player i equals:
qi ; Q i )
i (~
= q~i P (~
qi + Q i )
Ci (qi )
di¤erentiating with respect to q~i yields:
@
qi ; Q i ) =@ q~i
i (~
= q~i P 0 (~
qi + Q i ) + P (~
qi + Q i )
Thus, any decrease in the reservation price pi , and the resulting increase in q~i , would make
the player better o¤, even if the quantity of output o¤ered below the new price was still
Q i , so long as the associated marginal revenue is not smaller than qim P 0 (Qm ) + P (Qm ),
which is non-negative by virtue of Condition ES1. We have:
@2
qi ; Q i ) =@ q~i2
i (~
< @2
qi ; Q i ) =@ q~i @Q i
i (~
<0
so that:
(~
qi < qim ^ q~i + Q
i
< Qm ) ) @
qi ; Q i ) =@ q~i
i (~
>0
This means that any strategy (qi ; pi ) such that qi < qim that fails to sell the entire output
when others o¤er less than (Qm qi ) below pi , is weakly dominated by some strategy
(qi0 ; p0i ) such that qi0 = qi and p0i < pi against the set of conventional strategy pro…les.
Consequently, strategy (qi ; pi ) could not be part of a stable convention.
The proof of the 2 (0; 1) is analogous, where the payo¤ of player i from (qi ; pi ) is:
qi ; Q i )
i (~
= q~i P (~
qi + Q i )
[(1
) ci q~i + ci qi ]
leading to:
@
qi ; Q i ) =@ q~i
i (~
= q~i P 0 (~
qi + Q i ) + P (~
qi + Q i )
>
whenever q~i < qim and q~i + Q
i
(1
) ci
qim P 0 (Qm )
+ P (Qm )
< Qm ; and by virtue of Condition ES2:
24
(1
) ci
0
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