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Two-stage distributionally robust noncooperative games: Existence of Nash equilibrium and its application to Cournot–Nash competition

  • *Corresponding author: Atsushi Hori

    *Corresponding author: Atsushi Hori 

This work was supported by JST SPRING, Grant Number JPMJSP2110

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  • Two-stage distributionally robust stochastic noncooperative games with continuous decision variables are studied. In such games, each player solves a two-stage distributionally robust optimization problem depending on the decisions of the other players. Existing studies in this area have been limited with strict assumptions, such as linear decision rules, and supposed that each player solves a two-stage linear distributionally robust optimization with a specifically structured ambiguity set. This limitation motivated us to generalize and analyze the game in a nonlinear case. The contributions of this study are (ⅰ) demonstrating the conditions for the existence of two-stage Nash equilibria under convexity and compactness assumptions, and (ⅱ) consideration of a two-stage distributionally robust Cournot–Nash competition as an application, as well as an investigation into the conditions for the existence of market equilibria in an economic sense. We also report some results of numerical experiments to illustrate how distributional robustness affects the decision of each player in the Cournot–Nash competition.

    Mathematics Subject Classification: 91A15, 90C33.

    Citation:

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  • Figure 1.  Results of numerical experiments in the Cournot–Nash competition ($ N = 2 $)

    Table 1.  Recent works on distributionally robust games

    Finite strategies (mixed strategies)1 Continuous pure strategies2
    One-stage Qu and Goh [35]
    Loizou [29,30]
    Peng et al. [34]
    Sun and Xu [50]
    Liu et al. [28]
    Two-stage ———— Li et al. [25] (linear case)
    Chen et al. [7] (Cournot competition)
    this paper (nonlinear case)
     | Show Table
    DownLoad: CSV

    Table 2.  Boxed pigs

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (p_d(\xi)-6,\xi-p_d(\xi)-2) $ $ (\xi-p_s(\xi)-6,p_s(\xi)) $
    (big pig) Wait $ (p_d(\xi),\xi-p_d(\xi)-2) $ $ (0,0) $
     | Show Table
    DownLoad: CSV

    Table 3.  Stochastic Nash equilibrium under $ P_1 $ (bold fonts)

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (3.0938,1.1562) $ $ \mathit{\boldsymbol{(0.4516,5.7984)}} $
    (big pig) Wait $ \mathit{\boldsymbol{(9.0938,1.1562)}} $ $ (0,0) $
     | Show Table
    DownLoad: CSV

    Table 4.  Distributionally robust Nash equilibrium (bold fonts)

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (-0.3021,-0.9479) $ $ (-3.8495,4.5995) $
    (big pig) Wait $ (5.6979,-0.9479) $ $ \mathit{\boldsymbol{(0,0)}} $
     | Show Table
    DownLoad: CSV
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