  Pilgrim's monopoly is a probabilistic process giving rise to a non-negative
sequence $T_1, T_2,\ldots$ that is infinitely exchangeable, a natural model for
time-to-event data. The one-dimensional marginal distributions are exponential.
The rules are simple, the process is easy to generate sequentially, and a
simple expression is available for both the joint density and the multivariate
survivor function. There is a close connection with the Kaplan-Meier estimator
of the survival distribution. Embedded within the process is an infinitely
exchangeable ordered partition processes connected to Markov branching
processes in neutral evolutionary theory. Some aspects of the process, such as
the distribution of the number of blocks, can be investigated analytically and
confirmed by simulation. By ignoring the order, the embedded process can be
considered as an infinitely exchangeable partition process, shown to be closely
related to the Chinese restaurant process. Further connection to the Indian
buffet process is also provided. Thus we establish a previously unknown link
between the well-known Kaplan-Meier estimator and the important Ewens sampling
formula.
