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A Survey of the Latest Advances in Oligopoly Games

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Abstract

One of the most important problems of game theory—the game of firms in an oligopoly market—is considered. The survey covers classical and modern formulations for the game-theoretic problem of choosing optimal player’s strategies and the recent methodological achievements in oligopoly games with applications, including publications over the past five years.

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Notes

  1. In the English literature, Stackelberg game is synonymous with hierarchical game. However, in a game with multiple leaders, the hierarchy becomes ambiguous.

  2. Some studies consider a conjectural price variation as a supposed change in price in response to the unit quantity increase of player i. When analyzing this approach, the variation ρ will be referred to as the quantity variation.

  3. Most studies of quantity oligopoly consider the inverse demand function. Therefore, it will be briefly called the demand function. When referring to the function Q(P), the term “direct demand function” will be used.

  4. The parameter for correcting the step of the dynamic process is fractional in the range [0, 1] or fractal. This shows an analogy of the finite-difference approach with fractal differential equations; see the discussion below.

  5. In [25], it has the form ui = –2 – \(\frac{{C_{{{{i}_{{{{Q}_{i}}{{Q}_{i}}}}}}}^{{''}}}}{b}\) because it was obtained for a linear demand function: in this case, \(P_{Q}^{'}\) = \(P_{{{{Q}_{i}}}}^{'}\) = –b and \(P_{{Q{{Q}_{i}}}}^{{''}}\) = 0. In addition, formula (12) is presented for the conjectural variations that do not depend on players’ actions (i.e., for \(\rho _{{ij{{Q}_{i}}}}^{'}\) = 0), whereas [25] described a more general case where \(\rho _{{ij{{Q}_{i}}}}^{'}\) ≠ 0.

  6. This approach is conceptually close to the inclusive best response in aggregative games [5], which represents the optimal response of a player to the total action of all players (including himself), i.e., \({{\tilde {r}}_{i}}(Q)\), in contrast to the best response to the environment ri(Q–i).

  7. In the literature concerning vertical interaction, the retailer and supplier are called the downstream firm and upstream firm, respectively.

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    M. I. Geraskin

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This paper was recommended for publication by D.A. Novikov, a member of the Editorial Board

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Geraskin, M.I. A Survey of the Latest Advances in Oligopoly Games. Autom Remote Control 84, 565–578 (2023). https://doi.org/10.1134/S000511792306005X

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