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In this article we analyze a partial-information Nash Q-learning algorithm for a general 2-player stochastic game. Partial information refers to the setting where a player does not know the strategy or the actions taken by the opposing player. We prove convergence of this partially informed algorithm for general 2-player games with finitely many states and actions, and we confirm that the limiting strategy is in fact a full-information Nash equilibrium. In implementation, partial information offers simplicity because it avoids computation of Nash equilibria at every time step. In contrast, full-information Q-learning uses the Lemke-Howson algorithm to compute Nash equilibria at every time step, which can be an effective approach but requires several assumptions to prove convergence and may have runtime error if Lemke-Howson encounters degeneracy. In simulations, the partial information results we obtain are comparable to those for full-information Q-learning and fictitious play.

This paper presents Bayesian parameter estimation for first order Grey system models' parameters (or sometimes referred to as hyperparameters). There are different forms of first-order Grey System Models. These include $GM(1,1)$, $GM(1,1| \cos(\omega t)$, $GM(1,1| \sin(\omega t)$, and $GM(1,1| \cos(\omega t), \sin(\omega t)$. The whitenization equation of these models is a first-order linear differential equation of the form \[ \frac{dx}{dt} + a x = f(t) \] where $a$ is a parameter and $f(t) = b$ in $GM(1,1|)$ , $f(t) = b_1\cos(\omega t) + b_2$ in $GM(1,1| cos(\omega t)$, $f(t) = b_1\sin(\omega t)+b_2$ in $GM(1,1| \sin(\omega t)$, $f(t) = b_1\sin(\omega t) + b_2\cos(\omega t) + b_3$ in $GM(1,1| \cos(\omega t), \sin(\omega t)$, $f(t) = b x^2$ in Grey Verhulst model (GVM), and where $b, b_1, b_2$, and $b_3$ are parameters. The results from Bayesian estimations are compared to the least square estimated models with fixed $\omega$. We found that using rolling Bayesian estimations for GM parameters can allow us to estimate the parameters in all possible forms. Based on the data used for the comparison, the numerical results showed that models with Bayesian parameter estimations are up to 45\% more accurate in mean squared errors., Comment: 11 pages, 6 figures