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 $$\ell $$ ℓ possible meeting times succeeding, where each participant rejects r of the $$\ell $$ ℓ options. For large $$\ell $$ ℓ and fixed $$r/\ell ,$$ 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 $$\ell ,$$ 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 $$p = r/\ell $$ p = r / ℓ of rejecting any meeting time, only agree with each other if $$m\rightarrow \infty .$$ m → ∞ . For $$m\rightarrow \infty ,$$ m → ∞ , $$\ell $$ ℓ has to be $$O(p^{-m})$$ 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 $$p \sim 1/\ell ^{1/m}$$ p ∼ 1 / ℓ 1 / m . If the participants’ availability is approximated as being unchanging from one week to another, i.e., $$\ell $$ 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