Scheduling meetings: are the odds in your favor?

Abstract

Abstract: 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 ℓ possible meeting times succeeding, where each participant rejects r of the ℓ options. For large ℓ and fixed r/ℓ, we can carry out a saddle-point expansion and obtain analytical results for the probability of success. Despite the thermodynamic limit of large ℓ, 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/ℓ of rejecting any meeting time, only agree with each other if m→∞. For m→∞,ℓ 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∼1/ℓ1/m. If the participants’ availability is approximated as being unchanging from one week to another, i.e., ℓ 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. Graphical abstract: (Figure presented.)