#### Title

### On Passing the Buck

#### Type of Submission

Poster

#### Keywords

Random walk, cover tour

#### Abstract

Imagine there are n>1 people seated around a table, and person S starts with a fair coin they will flip to decide whom to hand the coin next -- if "heads" they pass right, and if "tails" they pass left. This process continues until all people at the table have "touched" the coin. Curiously, it turns out that all people seated at the table other than S have the same probability 1/(n-1) of being last to touch the coin. In fact, Lovasz and Winkler ("A note on the last new vertex visited by a random walk," J. Graph Theory, Vol. 17 Iss. 5 (1993), 593-596) showed that this situation and the one where a person is permitted to pass the coin to anyone else with uniform probability 1/(n-1) are the only scenarios where everyone at the table other than S have the same probability 1/(n-1) of touching the coin last. This begs the question -- what is the probability that a person will touch the coin last in scenarios that lie outside these two? We consider a version where the table has two sides, and the "passing rule" involves handing the coin to someone on the opposite side of the table with uniform probability. What is the resulting probability that a particular person touches the coin last in this two-sided situation?

#### Campus Venue

Stevens Student Center Lobby

#### Location

Cedarville, OH

#### Start Date

4-11-2018 11:00 AM

#### End Date

4-11-2018 2:00 PM

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

On Passing the Buck

Cedarville, OH

Imagine there are n>1 people seated around a table, and person S starts with a fair coin they will flip to decide whom to hand the coin next -- if "heads" they pass right, and if "tails" they pass left. This process continues until all people at the table have "touched" the coin. Curiously, it turns out that all people seated at the table other than S have the same probability 1/(n-1) of being last to touch the coin. In fact, Lovasz and Winkler ("A note on the last new vertex visited by a random walk," J. Graph Theory, Vol. 17 Iss. 5 (1993), 593-596) showed that this situation and the one where a person is permitted to pass the coin to anyone else with uniform probability 1/(n-1) are the only scenarios where everyone at the table other than S have the same probability 1/(n-1) of touching the coin last. This begs the question -- what is the probability that a person will touch the coin last in scenarios that lie outside these two? We consider a version where the table has two sides, and the "passing rule" involves handing the coin to someone on the opposite side of the table with uniform probability. What is the resulting probability that a particular person touches the coin last in this two-sided situation?