  We study multivariate approximation in the average case setting with the
error measured in the weighted $L_2$ norm. We consider algorithms that use
standard information $\Lambda^{\rm std}$ consisting of function values or
general linear information $\Lambda^{\rm all}$ consisting of arbitrary
continuous linear functionals. We investigate the equivalences of various
notions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and
$\Lambda^{\rm all}$ for the absolute error criterion, and show that the power
of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ for all
notions of algebraic and exponential tractability without any condition.
Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115
as posed by E.Novak and H.Wo\'zniakowski in the book: Tractability of
Multivariate Problems, Volume III: Standard Information for Operators, EMS
Tracts in Mathematics, Z\"urich, 2012.
