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On a Special Two-Person Dynamic Game

Author

Listed:
  • Akio Matsumoto

    (Department of Economics, Chuo University, 742-1 Higashi-Nakano, Tokyo 192-0393, Japan)

  • Ferenc Szidarovszky

    (Department of Mathematics, Corvinus University, Föván tur 8, 1093 Budapest, Hungary)

  • Maryam Hamidi

    (Department of Industrial and Systems Engineering, Lamar University, 2210 Cherry Engineering Building, Beaumort, TX 77710, USA)

Abstract

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper.

Suggested Citation

  • Akio Matsumoto & Ferenc Szidarovszky & Maryam Hamidi, 2023. "On a Special Two-Person Dynamic Game," Games, MDPI, vol. 14(6), pages 1-22, October.
    • Handle: RePEc:gam:jgames:v:14:y:2023:i:6:p:67-:d:1266257
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    References listed on IDEAS

    as
    1. Farebrother, R W, 1973. "Simplified Samuelson Conditions for Cubic and Quartic Equations," The Manchester School of Economic & Social Studies, University of Manchester, vol. 41(4), pages 396-400, December.
    2. Akio Matsumoto & Ferenc Szidarovszky, 2018. "Dynamic Oligopolies with Time Delays," Springer Books, Springer, number 978-981-13-1786-6, March.
    3. Akio Matsumoto & Ferenc Szidarovszky, 2016. "Game Theory and Its Applications," Springer Books, Springer, edition 1, number 978-4-431-54786-0, March.
    4. Luca Gori & Luca Guerrini & Mauro Sodini, 2014. "Hopf Bifurcation in a Cobweb Model with Discrete Time Delays," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-8, June.
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