Do you recognize this? You want to plan a meeting or a rehearsal with your band, and you have to send out not one but a couple of Doodles or other planners to do so. New research from Case Western Reserve University reveals just how challenging it is to find a suitable meeting time as the number of participants grows by showing the maths behind your calendar.

From the press release:

In a world where organizing a simple meeting can feel like herding cats, new research from Case Western Reserve University reveals just how challenging finding a suitable meeting time becomes as the number of participants grows.

The study, published in the European Physical Journal B, dives into the mathematical complexities of this common task, offering new insights into why scheduling often feels so impossible.

“If you like to think the worst about people, then this study might be for you,” quipped researcher Harsh Mathur, professor of physics at the College of Arts and Sciences at CWRU. “But this is about more than Doodle polls. We started off by wanting to answer this question about polls, but it turns out there is more to the story.”

Researchers used mathematical modeling to calculate the likelihood of successfully scheduling a meeting based on several factors: the number of participants (m), the number of possible meeting times (τ) and the number of times each participant is unavailable (r).

What they found: As the number of participants grows, the probability of scheduling a successful meeting decreases sharply.

Specifically, the probability drops significantly when more than five people are involved — especially if participant availability remains consistent.

“We wanted to know the odds,” Mathur said. “The science of probability actually started with people studying gambling, but it applies just as well to something like scheduling meetings. Our research shows that as the number of participants grows, the number of potential meeting times that need to be polled increases exponentially.

“The project had started half in jest but this exponential behavior got our attention. It showed that scheduling meetings is a difficult problem, on par with some of the great problems in computer science.”

‘More to the story’

Interestingly, researchers found a parallel between scheduling difficulties and physical phenomena. They observed that as the probability of a participant rejecting a proposed meeting time increases, there’s a critical point where the likelihood of successfully scheduling the meeting drops sharply. It’s a phenomenon similar to what is known as “phase transitions” in physics, Mathur said, such as ice melting into water.

“Understanding phase transitions mathematically is a triumph of physics,” he said. “It’s fascinating how something as mundane as scheduling can mirror the complexity of phase transitions.”

Mathur also noted the study’s broader implications, from casual scenarios like sharing appetizers at a restaurant to more complex settings like drafting climate policy reports, where agreement among many is needed.

“Consensus-building is hard,” Mathur said. “Like phase transitions, it’s complex. But that’s also where the beauty of mathematics lies — it gives us tools to understand and quantify these challenges.”

Mathur said the study contributes insights into the complexities of group coordination and decision-making, with potential applications across various fields.

Joining Mathur in the study were physicists Katherine Brown, of Hamilton College, and Onuttom Narayan, of the University of California, Santa Cruz.

Abstract of the study:

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.