  We present a series of results regarding conceptually simple algorithms for
bipartite matching in various online and related models. We first consider a
deterministic adversarial model. The best approximation ratio possible for a
one-pass deterministic online algorithm is $1/2$, which is achieved by any
greedy algorithm. D\"urr et al. recently presented a $2$-pass algorithm called
Category-Advice that achieves approximation ratio $3/5$. We extend their
algorithm to multiple passes. We prove the exact approximation ratio for the
$k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the
approximation ratio converges to the inverse of the golden ratio
$2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is
extremely fast --- the $5$-pass Category-Advice algorithm is already within
$0.01\%$ of the inverse of the golden ratio.
  We then consider a natural greedy algorithm in the online stochastic IID
model---MinDegree. This algorithm is an online version of a well-known and
extensively studied offline algorithm MinGreedy. We show that MinDegree cannot
achieve an approximation ratio better than $1-1/e$, which is guaranteed by any
consistent greedy algorithm in the known IID model.
  Finally, following the work in Besser and Poloczek, we depart from an
adversarial or stochastic ordering and investigate a natural randomized
algorithm (MinRanking) in the priority model. Although the priority model
allows the algorithm to choose the input ordering in a general but well defined
way, this natural algorithm cannot obtain the approximation of the Ranking
algorithm in the ROM model.
