Scheduling meetings: are the odds in your favor?

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 $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> possible meeting times succeeding, where each participant rejects r of the $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> options. For large $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> and fixed $$r/\ell ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we can carry out a saddle-point expansion and obtain analytical results for the probability of success. Despite the thermodynamic limit of large $$\ell ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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 $$p = r/\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math> of rejecting any meeting time, only agree with each other if $$m\rightarrow \infty .$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> For $$m\rightarrow \infty ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> has to be $$O(p^{-m})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$p \sim 1/\ell ^{1/m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . If the participants’ availability is approximated as being unchanging from one week to another, i.e., $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> 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