Probabilistic-serial procedure

The probabilistic serial procedure (PS), also called serial eating algorithm, is a procedure for fair random assignment. It yields a randomized allocation of indivisible items among several agents that is both envy-free and Pareto efficient. It was suggested by Hervé Moulin and Anna Bogomolnaia.[1] The PS-lottery algorithm is a method to instantiate the PS procedure's ex-ante assignment outcome by a probability distribution over deterministic assignments.[2]

Description

For each item, a loaf of bread (or other food) is created. Initially, each agent goes to their favourite item and starts eating it. It is possible that several agents eat the same loaf at the same time.

Whenever a loaf is fully eaten, each of the agents who ate it goes to the loaf of their favourite remaining item and starts eating it in the same way, until all loaves are consumed.

For each item, the fraction of that item eaten by each agent is recorded. These fractions define a probability distribution; one of the agents is chosen at random according to this distribution, and given the item.

An important parameter to PS is the eating speed of each agent. In the simplest case, when all agents have the same entitlements, it makes sense to let all agents eat in the same speed all the time. However, when agents have different entitlements, it is possible to give the more privileged agents a higher eating speed. Moreover, it is possible to let the eating speed change with time.

Example

There are four agents and four items (denoted w,x,y,z). The preferences of the agents are:

  • Alice and Bob prefer w to x to y to z.
  • Carl and Dana prefer x to w to z to y.

The agents have equal rights so we apply PS with equal and uniform eating speed of 1 unit per minute.

Initially, Alice and Bob go to w and Carl and Dana go to x. Each pair eats their item simultaneously. After 1/2 minute, Alice and Bob each have 1/2 of w, while Carl and Dana each have 1/2 of x.

Then, Alice and Bob go to y (their favourite remaining item) and Carl and Dana go to z (their favourite remaining item). After 1/2 minute, Alice and Bob each have 1/2 of y and Carl and Dana each have 1/2 of z.

Based on the eaten fractions, item w is given to either Alice or Bob with equal probability and the same is done with item y; item x is given to either Carl or Dana with equal probability and the same is done with item z.

See also
  • Random priority - an alternative mechanism with different properties.

References
  1. Bogomolnaia, Anna; Moulin, Hervé (2001). "A New Solution to the Random Assignment Problem". Journal of Economic Theory. 100 (2): 295. doi:10.1006/jeth.2000.2710.
  2. Aziz, Haris (2020). "Simultaneously Achieving Ex-ante and Ex-post Fairness". arxiv. arXiv:2004.02554.
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