Pizza Roll Theorem

Let P be a potluck containing a nonempty set of pizza rolls R where |R| > n where n is the total number of potluckers. Suppose that at time t0, no potlucker has yet taken a pizza roll. Then, if there exists an individual A such that A takes a pizza roll at time t0, it follows that for every other individual X at the potluck, there exists a time tx > t0 at which X will also take a pizza roll.

Proof.
Assume the setup as given, and suppose A is the first person to take a pizza roll. Prior to A’s action, for each other individual Y, the following belief holds:

• Y desires a pizza roll, but
• Y refrains from taking one because Y believes that taking the first pizza roll would make them violate the implicit social norm against appearing greedy a.k.a. being big.

Once A takes the first pizza roll, two things happen:

1. The norm against being the first to take pizza rolls is now satisfied, because someone else has already broken the ice.
2. Y observes A’s action and updates their belief: “If A was not punished socially for taking a pizza roll, then taking a pizza roll is now socially permissible.”

Therefore, for all Y not equal to A, Y now sees they need not wait (since pizza rolls are a finite, desirable resource) and that taking a roll carries no further social cost. Hence, for each Y, there exists a time ty (typically soon after t0) at which T will take at least one pizza roll.

Corollary. The only stable state with no pizza rolls taken is the state where nobody has taken one yet. As soon as one person defects from this equilibrium, the cascade to universal pizza roll consumption is instantaneous in the limit of social rationality.

Future work: What decision making processes lead to A starting the pizza roll process?