The chaotic duopolists revisited

Puu, Tönu

doi:10.1016/s0167-2681(97)00064-4

Cited by 135 publications

(64 citation statements)

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“…A similar example might serve to illustrate a situation where products are strategic complements: the (marginal) profit of one firm increases with the competitors' actions. A Cournot game with three oligopolists and unimodal reaction functions has been recently studied by Puu [21]. Also in this case, the model extends a duopoly model proposed by the same author in [19].…”

Section: Introductionmentioning

confidence: 92%

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Multistability in a dynamic Cournot game with three oligopolists

Agiza

Bischi

Kopel

1999

Mathematics and Computers in Simulation

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The time evolution of a dynamic oligopoly game with three competing firms is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional non-invertible map. For the symmetric case of identical players a complete analytical study of the stability conditions for the fixed points, which are Nash equilibria of the game, is given. For the situation of several coexisting stable Nash equilibria a numerical study of their basins of attraction is provided. This gives, evidence of the occurrence of global bifurcations at which the basins are transformed from simply connected sets into nonconnected sets, a basin structure which is peculiar of non-invertible maps. The presence of several coexisting attractors (or multistability) is observed even when complex attractors exist. Two different routes to complexity are presented: one related to the creation of more and more complex attractors; the other related to the creation of more and more complex structures of the basins. Starting from the benchmark case of identical players, the effects of heterogeneous behavior of the players, causing the loss of the symmetry properties of the dynamical system, are investigated through numerical explorations. # 1999 IMACS/ Elsevier Science B.V. All rights reserved.

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Section: Introductionmentioning

confidence: 92%

“…In this particular case, the decision of each firm only depends on the competitors' realized quantities of the previous period. This is the case which has been considered by Puu [20,21].…”

Section: The Modelmentioning

confidence: 99%

Multistability in a dynamic Cournot game with three oligopolists

Agiza

Bischi

Kopel

1999

Mathematics and Computers in Simulation

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“…Such dynamics have been studied extensively under the assumption of naive expectations (e.g. Theocharis (1960), Puu (1998), Tramontana et al (2009)) and adaptive expectations (e.g. Okuguchi (1970), Bischi and Kopel (2001)), for different assumptions about demand and production structure.…”

Section: Related Literaturementioning

confidence: 99%

Best response dynamics with level-n expectations in two-stage games

Dawid¹,

Heitmann²

2014

Journal of Economic Dynamics and Control

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This paper analyzes behavior in repeatedly played two-stage games, where players choose actions in both stages according to best replies using 'level-n expectations' about the opponent's actions in both stages. Level-n expectations are recursively defined in a way that a player holding level n expectations correctly predicts the action of an opponent holding level n − 1 expectations.A general conceptual framework to study such dynamics for two-stage games is developed and it is shown that, contrary to results for single-stage games, the fixed points of the dynamics depend on the level of the expectations. In particular, for level-zero expectation, fixed points correspond to a Nash equilibrium of a simultaneous move version of the game, whereas (under certain conditions) fixed points converge towards the subgame perfect equilibrium of the two-stage game if the level of expectations goes to infinity. The approach is illustrated using a two-stage duopoly game, where firms in the first stage invest in activities reducing their marginal costs and in the second stage engage in Cournot competition. An increase in the level of expectations leads in the long run to higher cost reducing activities and higher output of the firms, however to lower profits. Level-two expectations are sufficient to move the fixed-point of the dynamics to a close neighbourhood of the subgame-perfect equilibrium.

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“…Puu [6] conducted some of the first investigations in such economic games and thus he has concluded that various complex dynamics could result from a Cournot duopoly such as the presence of attractors with a fractal dimension. Various efforts were exerted then to study the dynamics of oligopoly models with considering more firms and various amendments [7,8].…”

Section: Introductionmentioning

confidence: 99%

Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products

2018

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This paper proposes a Cournot game organized by three competing firms adopting bounded rationality. According to the marginal profit in the past time step, each firm tries to update its production using local knowledge. In this game, a firm's preference is represented by a utility function that is derived from a constant elasticity of substitution (CES) production function. The game is modeled by a 3-dimensional discrete dynamical system. The equilibria of the system are numerically studied to detect their complex characteristics due to difficulty to get an explicit form for those equilibria. For the proposed utility function, some cases with different value parameters are considered. Numerical simulations are used to provide an experimental evidence for the complex behavior of the evolution of the system. The obtained results show that the system loses its stability due to different types of bifurcations.

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The chaotic duopolists revisited

Cited by 135 publications

References 1 publication

Multistability in a dynamic Cournot game with three oligopolists

Multistability in a dynamic Cournot game with three oligopolists

Best response dynamics with level-n expectations in two-stage games

Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products

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