Evolutionarily stable conjectures and other regarding preferences in duopoly games
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Abstract
We study the evolutionary selection of conjectures in duopoly games when players have other regarding preferences, i.e. preferences over payoff distributions. In both the Cournot and Bertrand duopoly games, the consistent conjectures are independent of other regarding preferences. Both duopoly games have evolutionarily stable conjectures that depend on other regarding preferences but that do not coincide with the consistent conjectures. For increasingly spiteful preferences, the evolutionarily stable conjectures implicate low quantities in the Cournot game and high prices in the Bertrand game, whereas the inverse relationships hold for the consistent conjectures. We discuss our findings in the context of ultimate and proximate causation.
Keywords
Duopoly Conjectural variations Other regarding preferences Evolutionary stabilityJEL Classification
D43 B52 C731 Introduction
In many games players maximize something other than profit. In the strategic delegation literature a firm’s owners incentivize managers to include market share or the relative profits with respect to rivals into their objectives (Vickers 1985; Fershtman et al. 1991). In mixed oligopoly models, a welfaremaximizing public firm competes with profitmaximizing private firms (De Fraja and Delbono 1989). The motive for maximizing other things than profit can also be nonstrategic. A large body of literature in experimental economics has demonstrated that players have other regarding preferences (ORPs) that can be included into the maximization problem by complementing the utility functions with such factors as concerns for efficiency or reciprocity (Cooper and Kagel 2016).Could it be the case that conjectures are consistent but that subjects are maximizing something other than profit? – Charles (Holt 1985)
Various models of strategic behavior also take beliefs into account. A traditional way to model beliefs in the industrial organization literature is to assume that firms conjecture that competitors react to variations in own strategies by nonzero variations in their strategies. As in the case of ORP models, the conjectural variations models can represent many different outcomes in a single framework (Varian 1992). However, conjectural variations have been criticized as imposing a dynamic process of belief formation into a static framework (e.g. Makowski 1987). Presently, conjectural variations are modeled in contexts that have some sort of an implicitly or explicitly dynamic nature (Figuières et al. 2004). In these models, the conjectural variations equilibria are interpreted as shortcuts to modeling outcomes of dynamic games (Dockner 1991; Cabral 1995). Alternatively the conjectures can be assumed to result from behavior that is boundedly rational, either in belief formation (Friedman and Mezzetti 2002; JeanMarie and Tidball 2006) or in shortterm fitness maximization in the presence of evolutionary selection pressure on the conjectures (Dixon and Somma 2003; Müller and Normann 2005; Possajennikov 2009). Common to these bounded rationality approaches is that they justify the consistent conjecture, i.e. a conjecture that represents correctly anticipating the competitor’s reaction (Bresnahan 1981).
The appeal of the evolutionary approach is that it can also justify the origin of the conjectures. In the models studying the evolutionary selection of conjectures in infinite populations, the evolutionarily stable conjecture coincides with the consistent conjecture (Müller and Normann 2005). Possajennikov (2009) argues that this occurs because the firstorder conditions of payoff maximization and fitness maximization coincide and the players solve the correct maximization problem.
Our motivation in this paper is to demonstrate cases when the consistent conjecture is not justified by evolutionary arguments. We use the indirect evolutionary approach to derive conjectures in a model where otherregarding (altruistic or spiteful) behavior is possible. We show that the evolutionarily stable conjecture does not coincide with the consistent conjecture if the players maximize a simple utility function that takes ORPs into account, thus contrasting with the results obtained by Müller and Normann (2005). We consider the two classical models of duopoly competition, quantity competition (Cournot) and price competition (Bertrand). Both games have evolutionarily stable conjectures with ORPs that do not coincide with the consistent conjectures.
The model of ORPs that we consider is a linear rule that imposes a concern for the total payoff distribution. Varying the ORP parameter allows the representation of altruism, spite, or selfregard. A concern for relative payoffs can be modeled with a change of the parameter. As this model has only a single parameter, it is selected over competing and more complex models of otherregarding behavior (such as the inequity aversion model of Fehr and Schmidt (1999) or the egocentric altruism model of Cox et al. 2007) for its analytical tractability. Although the ORP parameter is symmetric among the players, the model captures the essence of distributional preferences.
Some evolutionary models in the literature subject the ORP parameter to selection pressure and derive conditions for stability of altruism or spite (Bester and Güth 1998; Possajennikov 2000). In our model only the conjecture is shaped by selection pressure but the ORP parameter is free of that pressure. Thus, in an evolutionary sense the phenotypic variability is maintained in the model via the ORP parameter. We find that this variability is maintained also in the conjectures themselves, i.e. the conjectures (beliefs) depend directly on ORPs.
We analyze the Cournot duopoly game in Section 2 and the Bertrand duopoly game in Section 3. In each of these sections we first derive the consistent conjecture and then analyze the evolutionary stability. Section 4 discusses the results and Section 5 contains all the proofs.
2 The Cournot duopoly game
2.1 The model
Lemma 1
The conjectures are required to be constant in the strategic variables but they may depend on the other parameters a, b, c and t. To allow generality, we are only interested in symmetric conjectures r _{ i } = r _{ j } = r that are continuous in the full ranges of the parameters. The symmetry assumption of the conjectures is not restrictive because the payoff and utility functions are symmetric as well. By the chain rule, because the payoff function (1) is smooth, the utility function (2) is also smooth in the strategic variables.
Remark 1
2.2 Consistent conjectures
Proposition 1
Remark 2
Proposition 1 shows that the consistent conjecture does not depend on t in the Cournot duopoly game. Assuming ORPs, therefore, does not change the predictions of the consistent conjectures theory. This was already shown by Holt (1985), but we have generalized his result for nonconstant marginal cost and differentiated products. The consistent conjecture (7) is strictly negative for all feasible values of b and c, as is typical for Cournot competition (Bresnahan 1981).
2.3 Evolutionarily stable conjectures
Evolutionarily stable (ES) conjectures are formed by selection pressure. Players with different conjectures are matched in duopoly pairs and those players who have the highest fitness pass their conjectures to future generations of players. In practice this can happen, e.g., through cultural evolution or when managers switch between firms. Alternatively, the conjectures may evolve through individual evolutionary learning (Arifovic and Maschek 2006) where beliefs (i.e. the conjectures) are updated as if they undergo natural selection. In this section, we assume that the population from which the conjectures are selected is infinite.^{1} The conjectures thus derived conform to the concept of evolutionarily stable (ES) strategies, introduced by Maynard Smith (1982). A strategy is evolutionarily stable if it cannot be invaded by any alternative mutant strategy.
Proposition 2
Remark 3
The conjecture r ^{ E }(t) assumes generally negative values but becomes positive for a range of negative t and approaches unity as t →−1. This implies that, for sufficiently spiteful ORPs, the reaction functions slope up and the quantities decrease below those in the CNE (see Remark 1).
Corollary 1
The ES conjecture r ^{ E }(t)is strictly decreasing in t in the Cournot game.
Remark 4
Corollary 1 implies that r ^{ E }(t) crosses r ^{∗} only once and this is exactly at t = 0. This can be seen by observing that r ^{ E }(0) = r ^{∗}. This also implies that the point t = 0 where ORPs change between spiteful and altruistic also determines whether the equilibrium quantities with the ES conjectures are lower or higher than those in the CCE. With selfregarding preferences t = 0 the quantities (4) are equal as well.
3 The Bertrand duopoly game
3.1 The model
As in the Cournot case, the Bertrand payoff functions are quadratic and the reaction functions are linear. For the special case of zero conjectures, r _{ i },r _{ j } = 0, the corresponding equilibrium is again called the CournotNash equilibrium.
Lemma 2
We are again only interested in conjectures that are continuous in the parameters a, b, c and t. Also, as the game is symmetric, we can assume that the conjectures are symmetric as well, r _{ i } = r _{ j } = r.
Remark 5
3.2 The consistent conjectures
Proposition 3
Remark 6
Proposition 3 shows that (as in Remark 2) assuming ORPs does not change the consistent conjectures in the Bertrand duopoly game.
3.3 Evolutionarily stable conjectures
Proposition 4
Remark 7
As in the Cournot case, the evolutionarily stable conjecture in the Bertrand game coincides with the consistent conjecture exactly at t = 0. The evolutionarily stable conjecture is generally positive but obtains negative values as t approaches unity.
Corollary 2
The ES conjecture r ^{ E }(t)is strictly decreasing in t in the Bertrand game.
4 Discussion
Holt (1985) argues that the concept of CCE can be refuted on the basis of laboratory evidence. In the Cournot duopoly game the CCE predicts higher quantities than the CNE even in the presence of ORPs (see Fig. 1), but laboratory experiments show that quantities are usually equal to or lower than those in the CNE. For example, Huck et al. (2001) find that average quantities in oneshot Cournot duopoly markets roughly correspond to the CNE quantities, and Suetens and Potters (2007) survey Bertrand experiments and find that tacit collusion is more frequent with price than with quantity. Our results suggest that a variety of behavioral outcomes can be supported by assuming conjectures and/or otherregarding preferences in both the Cournot and the Bertrand duopoly games. While cooperative behavior can generally be explained by altruistic preferences, our novel result is that spiteful preferences also explain cooperative outcomes if we assume that the players have evolutionarily stable conjectures. Figure 1 implies that spiteful Cournot players with ES conjectures can actually choose quantities that are close to the CNE quantities or even close to the collusive quantities. On the other hand, altruistic players with ES conjectures would end up playing quantities that are higher than even the CCE quantities. Similar arguments hold for the Bertrand duopoly game as Fig. 2 demonstrates. These results shed new insights also to explaining the laboratory evidence of Huck et al. (2001) and others.
Burnham (2014) presents the concepts of proximate and ultimate mechanisms from evolutionary biology to economic theory. Ultimate mechanisms determine what kind of behavior survives in evolution in the long run. Proximate mechanisms explain what motivates behavior in the short run. In this view, the consistent conjectures are the products of proximate mechanisms and the ES conjectures arise from ultimate mechanisms. Whereas prior research has shown that the ultimate and proximate mechanisms produce the same conjectures if the duopolists are selfregarding (Müller and Normann 2005; Possajennikov 2009), we show that the ES conjectures are different from the consistent conjectures if we allow a simple type of other regarding behavior.
It should be noted that our models of Cournot and Bertrand competition are specific in terms of linear demand functions and quadratic costs. Also, the utility function model is restrictive because only a very simple linear utility function is used and the ORP parameter is assumed to be symmetric among the firms. These restrictions limit the generalizations that can be drawn from the results.
Unlike Bester and Güth (1998) and others who have modeled the evolution of altruism and spite in the linearquadratic context, we leave the ORP parameter untouched by evolutionary forces. This is in part a modeling decision: If both the ORP parameter and the conjecture were subject to selection pressure, we would lose analytical tractability. However, another rationale is that it allows us to explain situations where selection pressure focuses on nonzero conjectures but where there are degrees of freedom in terms of distortions in payoff maximization. Therefore, the conjecture is the expression of the tendency that evolves (Trivers 1971) whereas the ORP parameter allows, for example, the characterization of phenotypic variability. Our results then suggest that in two of the most studied duopoly games with linearquadratic payoffs, the Cournot and the Bertrand games, this variability in the ES conjectures is nonzero.
The assumption of the common ORP parameter t for both players can be justified on several grounds. Here, too, the assumption maintains analytic tractability. However, one can also quickly imagine situations where exogenous factors demand that assumption, such as when the individual utilities are common knowledge, or when firms explicitly agree on a common t parameter (as in colluding). Even in a population of dissimilar phenotypes, evolutionary forces may favor interactions between phenotypically similar individuals (Antal et al. 2009).
Müller and Normann (2005) argue that because the consistent conjecture is equal to the ES conjecture, evolutionary stability can rationalize consistency. Possajennikov (2009) points out that the ES conjecture is consistent because the players with consistent conjectures correctly anticipate the other player’s reaction function and therefore solve the correct firstorder conditions. Our results challenge both of these arguments. This is because in the presence of ORPs (i) evolutionary stability cannot rationalize consistency, and (ii) players with consistent conjectures do not solve the correct firstorder conditions.
Future research should extend our analysis by allowing the model of ORPs be more general or by allowing distortions in the symmetric ORP parameters. Also, the results could be extended to cover other games such as public goods games. Yet another extension would be an evolutionary model that allows the evolution of both ORPs and conjectures. The players could, for example, first develop evolutionarily stable conjectures that depend on the ORP parameters and then subject the ORPs to evolutionary pressure.
5 Proofs
5.1 Proof of lemma 1
5.2 Proof of proposition 1
5.3 Proof of proposition 2
At Step 1 we determine the ES conjecture candidate from the firstorder condition for fitness maximization (9) and show that the candidate satisfies the limit values. At Step 2 we show that the candidate is the best response against itself and that the population of players with r ^{ E }’s survives a mutant invasion, i.e. the candidate is an ES strategy (Maynard Smith 1982).
5.3.1 Step 1

First we note that the denominators of the candidates r ^{ E±} are always nonzero. Furthermore, the term A is a product of two negative factors and therefore positive. Thus the candidates r ^{ E±} are continuous for all feasible b, c and t.
 The positive root r ^{ E+} ∈ [−1,1]. This is shown by first examining limit behavior and then behavior between the limits. The positive root approaches 1 when t →−1 and − 1 when t → 1. We thus have to show that r ^{ E+} = −1 only at t = −1 and r ^{ E+} = 1 only at t = 1. Rearranging the equation r ^{ E+} − 1 = 0 such that the squareroot term is at the righthand side and all the other terms at the lefthand side, squaring each side and then moving all terms back to the lefthand side results in an equationthat holds only when t = −1 or t = 1. However, knowing that r ^{ E+} = −1 at t = 1, we can conclude that t = −1 is the unique solution of the equation r ^{ E+} − 1 = 0. Using a similar procedure for the equation r ^{ E+} + 1 = 0 yields the conclusion that this equation holds exactly at t = 1.$$2 (1 + b + c) (1 + t^{2})^{2} \left( b + b t + 2 (1 + c) t^{2}\right) = 0 $$

For the negative root, there are values of b, c and t that yield the candidate out of its limits. For example, when b = 1/2, c = 0 and t = 1/3 we have r ^{ E−} = −3. This is enough to show that r ^{ E−}∉[−1,1] for some values of b, c and t.
5.3.2 Step 2
5.4 Proof of corollary 1
5.5 Proof of lemma 2
and see that all the coefficients of c are strictly negative and thus the denominator cannot assume nonnegative values. The limit values of the denominator are − (−1 + b ^{2})(2 + c)^{2}(−1 + r _{ i } r _{ j }) as t →−1 and (−1 + b ^{2})(4b ^{2} − (2 + c)^{2})(−1 + r _{ i } r _{ j }) as t → 1 and these limits are strictly negative.
5.6 Proof of proposition 3
5.7 Proof of proposition 4
This proof is similar to the Proof of Proposition 2.
5.7.1 Step 1
The firstorder condition (17) is of the form P(r)/Q(r) and the solution r ^{ E,1} is obtained from P(r) = 0. P(r) is a seconddegree polynomial in r. The denominator of the solution to P(r) = A(t)r ^{2} + B(t)r + C(t) = 0 is zero for some t, namely, when A(t) = 0. These zeros are given by t ^{−} and t ^{+}. For these zeros in the denominator we consider the reduced form P(r) = B(t)r + C(t) = 0, which has r ^{ E,2} as the unique solution.
5.7.2 Step 2
When t ∈ (−1,1) ∖{t ^{−}}∪{t ^{+}}, the function g _{ i }(r, r ^{ E,1}) is a rational function of the form P(r)/Q(r) where P(r) and Q(r) are seconddegree polynomials in r. Taking its derivative in r leads again to a form (P ^{′}(r)Q(r) − P(r)Q ^{′}(r))/Q(r)^{2}, which simplifies to a rational function that has a firstdegree polynomial in r at the numerator and thirddegree polynomial at the denominator. Therefore, there is a unique maximizer to g _{ i }(r, r ^{ E,1}) and by substitution it can be shown to be r ^{ E,1}. The complicated expressions of P(r) and Q(r) are too long to be presented here, but are available from the Author upon request.
When t ∈{t ^{−}}∪{t ^{+}}, the function g _{ i }(r, r ^{ E,2}) is again of the form P(r)/Q(r) with seconddegree polynomials in r. The same arguments hold as above, i.e. we get the result that r = r ^{ E,2} is the unique maximizer of g _{ i }(r, r ^{ E,2}) in [−1,1].
5.8 Proof of corollary 2
Footnotes
Notes
Acknowledgements
I would like to thank two anonymous reviewers for comments that improved the manuscript significantly.
Compliance with Ethical Standards
Conflict of interests
The author declares that he has no conflicts of interest.
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