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Mental control model in a duopoly game

The article examines a game-theoretic model describing the interaction between two firms selling identical products, a classic duopoly model. The model expresses competition between the firms, as the players’ objective functions express their utility (profit) and are mutually inconsistent, i. e., a noncooperative, zero-sum game is considered. The players’ objective functions are formed based on the linear inverse demand function for the product and the linear cost functions of the participating firms. The solution to the game is a Nash equilibrium, determined using a system of first-order necessary optimality conditions. Unlike the classical approach, the model incorporates two new aspects. First, players are considered reflexive, meaning each player hypothesizes the counterparty’s likely response, which is formally expressed as a conjectural variation – the expected change in one player’s output in response to a single increase in the output of the other. Second, an aggregative game in a duopoly model, in which the players’ objective functions are interdependent, is analyzed from the perspective of controlling the behavior of one player by manipulating their mental representations of the other player’s intentions. The analysis of the game is based on a reflexive model of player behavior, in which players sequentially predict each other’s representations, resulting in the identification of a mental model of one player that is optimal from the perspective of the other player. An iterative procedure is proposed for inducing this mental model in the controlled player by creating phantom instances of the controlling player. Numerical experiments illustrate various aspects of the mental control process, whereby the technical characteristics of the player firms, such as their production volumes, interact with the mental behavioral parameters of decision makers, forming a cyber-physical system. Read more...

Conjectural variations in the oligopoly game with different types of reflexive behavior and successive number of players

A game-theoretic model of oligopoly with reflexive player behavior and quantity conjectural variations is proposed. The model generalizes two types of player reflexion –symmetric and asymmetric. It allows to describe the forecasts regarding competitors’ actions more flexibly and to study a wider range of market equilibrium options. For symmetric reflexions, the model is examined in terms of the influence of reflexivity rank and the number of players, revealing the dependence of equilibria on the depth of players’ strategic thinking. For asymmetric reflexions, the model is analyzed depending on the type of representation and the number of players. The goal of the study is to develop a method for calculating conjectural variations for both symmetric and asymmetric player representations. Unlike existing methods for calculating conjectural variations, which ignore the interactions between the number of players and the depth and type of reflexivity, this study yields new results for linear demand and cost functions. These results have significant practical implications and allow for the calculation of game equilibria with arbitrary reflexivity ranks and an arbitrary number of players, while also taking into account the type of reflexion of the analyzed player. This expands the toolkit for modeling real-world market situations. Thus, this study makes a significant contribution to the development of reflexive game theory by offering an adequate mathematical framework for analyzing strategic interactions in oligopoly settings, taking into account the heterogeneous expectations of players, which is necessary for forecasting market dynamics. To visualize the analytical results, graphical interpretations of the patterns of change in hypothetical variations are presented, which are used to formulate conclusions. Read more...