  We study the polynomial-time approximability of the optimal online stochastic
bipartite matching algorithm, initiated by Papadimitriou et al. (EC'21). Here,
nodes on one side of the graph are given upfront, while at each time $t$, an
online node and its edge weights are drawn from a time-dependent distribution.
The optimal algorithm is $\textsf{PSPACE}$-hard to approximate within some
universal constant. We refer to this optimal algorithm, which requires time to
think (compute), as a philosopher, and refer to polynomial-time online
approximations of the above as philosopher inequalities. The best known
philosopher inequality for online matching yields a $0.652$-approximation. In
contrast, the best possible prophet inequality, or approximation of the optimum
offline solution, is $0.5$.
  Our main results are a $0.678$-approximate algorithm and a
$0.685$-approximation for a vertex-weighted special case. Notably, both bounds
exceed the $0.666$-approximation of the offline optimum obtained by Tang, Wu,
and Wu (STOC'22) for the vertex-weighted problem. Building on our algorithms
and the recent black-box reduction of Banihashem et al. (SODA'24), we provide
polytime (pricing-based) truthful mechanisms which $0.678$-approximate the
social welfare of the optimal online allocation for bipartite matching markets.
  Our online allocation algorithm relies on the classic pivotal sampling
algorithm (Srinivasan FOCS'01, Gandhi et al. J.ACM'06), along with careful
discarding to obtain negative correlations between offline nodes. Consequently,
the analysis boils down to examining the distribution of a weighted sum $X$ of
negatively correlated Bernoulli variables, specifically lower bounding its mass
below a threshold, $\mathbb{E}[\min(1,X)]$, of possible independent interest.
Interestingly, our bound relies on an imaginary invocation of pivotal sampling.
