Scheduling meetings: are the odds in your favor?

Polling all the participants to find a time when everyone is available is the ubiquitous method of scheduling meetings nowadays. We examine the probability of a poll with m participants and \(\ell \) possible meeting times succeeding, where each participant rejects r of the \(\ell \) options. For large \(\ell \) and fixed \(r/\ell ,\) we can carry out a saddle-point expansion and obtain analytical results for the probability of success. Despite the thermodynamic limit of large \(\ell ,\) the ‘microcanonical’ version of the problem where each participant rejects exactly r possible meeting times, and the ‘canonical’ version where each participant has a probability \(p = r/\ell \) of rejecting any meeting time, only agree with each other if \(m\rightarrow \infty .\) For \(m\rightarrow \infty ,\) \(\ell \) has to be \(O(p^{-m})\) for the poll to succeed, i.e., the number of meeting times that have to be polled increases exponentially with m. Equivalently, as a function of p, there is a discontinuous transition in the probability of success at \(p \sim 1/\ell ^{1/m}\). If the participants’ availability is approximated as being unchanging from one week to another, i.e., \(\ell \) is limited, a realistic example discussed in the text of the paper shows that the probability of success drops sharply if the number of participants is greater than approximately 4.