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Approachability in population games

  • * Corresponding author: Dario Bauso

    * Corresponding author: Dario Bauso 
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  • This paper reframes approachability theory within the context of population games. Thus, whilst a player still aims at driving her average payoff to a predefined set, her opponent is no longer malevolent but instead is extracted randomly at each instant of time from a population of individuals choosing actions in a similar manner. First, we define the notion of 1st-moment approachability, a weakening of Blackwell's approachability. Second, since the endogenous evolution of the population's play is then important, we develop a model of two coupled partial differential equations (PDEs) in the spirit of mean-field game theory: one describing the best-response of every player given the population distribution, the other capturing the macroscopic evolution of average payoffs if every player plays her best response. Third, we provide a detailed analysis of existence, nonuniqueness, and stability of equilibria (fixed points of the two PDEs). Fourth, we apply the model to regret-based dynamics, and use it to establish convergence to Bayesian equilibrium under incomplete information.

    Mathematics Subject Classification: Primary: 91A16, 91A22; Secondary: 35Q89.

    Citation:

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  • Figure 1.  Payoff space of Prisoner's dilemma: State space $ X = conv\{(3, 3), (1, 1), (0, 4), (4, 0)\} $ (boundary a solid line), supporting hyperplane $ H $ (dot-dashed line) passing through the barycenter, vector field $ dx(t) $ pointing towards $ (\frac{3}{2}, \frac{7}{2}) $ for those who cooperate (region below $ H $) and towards $ (\frac{5}{2}, \frac{1}{2}) $ for those who defect (region above $ H $), $ conv\{(\frac{3}{2}, \frac{7}{2}), (\frac{5}{2}, \frac{1}{2})\} $ is set of approachable points with population strategy $ q = ((\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})) $, barycenter is self-confirmed with uniform distribution over $ X $

    Figure 2.  Regret space of the Prisoner's dilemma: State space $ X = conv\{(-1, 0), (0, 1)\} $ (solid line), initial distribution $ \rho(x, 0) $ (grey area), and vector field $ dx(t) $ converging to $ y = (-0.5, 0.5) $

    Figure 3.  Regret space of the coordination game: State space $ X = conv\{(-1, 0), (0, 1), (0, -2), (2, 0)\} $ (boundary a solid line), and vector field $ dx(t) $ converging to $ (1, 0) $ (grey area) and $ (0, -1) $ (white area), approachable point is $ y = (0, -1) $, set of approachable points is $ conv\{(1, 0), (0, -1)\} $ (dashed line) with mixed population strategy $ q = (\frac{2}{3}, \frac{1}{3}) $

    Figure 4.  Regret space of parametric game with $ a< 0 < b $: State space $ X = conv\{(0, a), (-a, 0), (-b, 0), (0, b)\} $ (boundary a solid line), vector field $ dx(t) $ converging to $ (0, a) $ which is also an approachable vertex with population strategy $ q = (1, 0) $, supporting hyperplane $ H $ (dot-dashed line) intersects $ X $ only at one point (the vertex)

    Figure 5.  Regret space of parametric game with $ 0<b < a $: State space $ X = conv\{(0, a), (-a, 0), (-b, 0), (0, b)\} $ (boundary a solid line), supporting hyperplane $ H $ (dot-dashed line) passing through the vertex $ (-b, 0) $, vector field $ dx(t) $ converging to $ (0, b) $ left of $ H $ and to $ (-b, 0) $ right of $ H $, $ conv\{(0, b), (-b, 0)\} $ is set of approachable points with population strategy $ q = (0, 1) $, vertex $ (-b, 0) $ is not self-confirmed, while vertex $ (0, a) $ is self-confirmed with population strategy $ q = (1, 0) $

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