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# Approachability in population games

• * Corresponding author: Dario Bauso
• 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: • • 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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