Duopolistic competition with multiple scenarios and different attitudes toward uncertainty

• DOI: http://doi.org/10.1111/itor.12474
• Published date: 01 May 2018
• Date: 01 May 2018
• Author: M.A. Caraballo,L. Monroy,A.M. Mármol

Intl. Trans. in Op. Res. 25 (2018) 941–961

DOI: 10.1111/itor.12474

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Duopolistic competition with multiple scenarios and different

attitudes toward uncertainty

L. Monroya, M.A. Caraballoband A.M. M´

armola

aDepartamento de Econom´

ıa Aplicada III and IMUS, Universidad de Sevilla, Avda. Ram´

on y Cajal 1, 41018 Sevilla, Spain

bDepartamento de Econom´

ıa e Historia Econ´

omica, Universidad de Sevilla, Spain

E-mail: lmonroy@us.es [Monroy]; mcaraba@us.es [Caraballo]; amarmol@us.es[M ´

armol]

Received 21 September 2016; receivedin revised form 8 September 2017; accepted 14 September 2017

Abstract

In this paper, we address duopolistic competition when the firms have to assess the results of the interaction

at different scenarios. Specifically,we consider the case in which the scenarios are identified with several states

of nature and, therefore, the firms face uncertainty on their results. The probability of occurrence of the

scenarios is unknown by the firms and they make their output decision beforeuncertainty is resolved. Within

this framework, we analyze competition betweenfir ms whenthese fir ms exhibit extremeand neutral attitudes

toward uncertainty with respect to the final profits. Forthe variety of cases that can occur, we characterize the

sets of equilibria, and provide procedures to determine them. The analysis proposed can also be applied to

study situations in a deterministic setting with simultaneous multiplescenarios, and to the analysis of multiple

criteria duopolistic competition.

Keywords:game theory; multiscenario games; Cournot duopoly; equilibria

1. Introduction

In this paper, we investigate a model of duopolistic competition where firms face different market

demands at several possible scenarios or states of nature. Only one will be realized as the true state,

and no information is available about the probability distributions of the occurrence of these states.

Specifically, we analyze the extension of a Cournot (1838) duopoly in which two firms producing

homogeneous products have to deal with uncertain demand and mayshow different attitudes toward

uncertainty.

Approaches based on subjective expected utility (Savage, 1954) could primarily be used to ad-

dress the identification of equilibria in a multiple scenario context. When firms seek to maximize

their expected profits with subjective probabilities distributions, equilibrium outcomes are strongly

determined by the vectors of probabilities. However, in the strategic situations studied in this

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942 L. Monroy et al. / Intl. Trans. in Op. Res. 25 (2018) 941–961

paper, uncertainty on the probability distributions and attitudes of the firms with respect to risk are

relevant issues that cannot be easily accommodated in the theory of expected utility.

Other analysis of duopoly games where firms act under uncertainty can be foundin the literature.

One line of research investigates incentives for duopolists to share their private information about

market uncertainty with its competitors. For instance, Novshek and Sonnenschein (1982), Vives

(1984), Li (1985), Gal-Or (1986), among others, analyzed how market uncertainty with either

unknown marketdemand or unknown constant marginal cost affects firm’s behavior. Morerecently,

Wu et al. (2008) addressa Cournot model with capacity constraints in which the uncertainty is about

uncertain demand conditions or production costs.

Related work on oligopolistic competition under uncertainty have focused on the conditions

for the existence of equilibria and their properties in an effort to provide a general and tractable

framework for the analysis of quantity competition under demand uncertainty. In this line of

research, Eichberger and Kelsey (2002) analyzed the effect of ambiguity in symmetric n-player

games with aggregate externalities. The application of their results to a Cournot oligopoly showed

that the total output in these models is lower when there is uncertainty. Lagerl¨

of (2007) obtains

the conditions on distribution functions of the stochastic demand intercept that guarantee the

existence of a unique equilibrium in a linear framework, while Einy et al. (2010) show some

examples in which a (Bayesian) Cournot equilibrium in pure strategies may not exist when firms

have incomplete information about demand and costs. Moreover, they provide sufficient conditions

for existence and uniqueness of Cournot equilibria in a certain class of industries. De Frutos

and Fabra (2011) and Lepore (2012) consider two-stage games to analyze competition between

firms in which firms make capacity investments under demand uncertainty prior to competing in

prices.

The majority of these papers have assumed that firms are risk neutral. Nevertheless, evidence

shows that in the presence of uncertainty, firms often exhibit different risk attitudes. The incorpo-

ration of these rational, but uncertain, beliefs will certainly influence the outcomes of the decision

processes. This issue has been analyzed by Asplund (2002), who studied competition in prices and

quantities between risk-aversefirms. Fontini (2005) analyzed a Cournot oligopolyunder uncertainty

using the Choquet expected utility model (Schmeidler, 1989) with optimistic and pessimistic firms,

and showed that when uncertainty is low, optimistic firms make higher profits than pessimistic

firms, and when uncertainty is high, only optimistic firms participate in the market producing too

much and facing losses.Eichberger et al. (2009) also addressed ambiguity in strategic games. In their

model, uncertainty is defined over the other players’ actions. Formally, individualspartially distrust

their own beliefs about other players’ behavior and place themselves in the best and worst cases

depending on their relative optimism and pessimism. In a recent paper, Chronopoulos et al. (2014)

analyze the impact of risk aversion and uncertainty on the optimal investment timing decision in

a duopolistic competition. They consider the case where the two competing firms exhibit the same

level of risk aversion.

In the above-mentioned papers, uncertainty is analyzed by means of a random variable in the

corresponding parameter of the model or by considering a probability distribution on the state of

nature, and expected utility theory is applied then to make decisions. By contrast, we develop our

study in a context where the probabilities of occurrence of these states cannot be ascertained by

firms. Our approachis essentially based on the idea that uncertainty cannot be modeled globally for

every action of one of the firms. Due to the strategic interdependence of the environment in which

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies