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]

Description

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

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

For each item, the fraction of that item eaten by each agent is recorded. These fractions are considered as probabilities. Based on these probabilities, a lottery is done. The type of lottery depends on the problem:

• If each agent is allowed to receive any number of items, then a separate lottery can be done for each item. Each item is given to one of the agents who ate a part of it, chosen at random according to the probability distribution for that item.
• If each agent should receive exactly one item, then there must be a single lottery that picks an assignment by some probability distribution on the set of deterministic assignments. To do this, the n-by-n matrix of probabilities should be decomposed into a convex combination of permutation matrices. This can be done by the Birkhoff algorithm. It is guaranteed to find a combination in which the number of permutation matrices is at most n²-2n+2.

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.
• Chana 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 Chana 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 Chana and Dana each have 1/2 of x.

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

The matrix of probabilities is now:

Alice: 1/2 0 1/2 0

Bob : 1/2 0 1/2 0

Chana: 0 1/2 0 1/2

Dana: 0 1/2 0 1/2

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 Chana or Dana with equal probability and the same is done with item z. If it is required to give exactly 1 item per agent, then the matrix of probabilities is decomposed into the following two assignment matrices:

1 0 0 0 ||| 0 0 1 0

0 0 1 0 ||| 1 0 0 0

0 1 0 0 ||| 0 0 0 1

0 0 0 1 ||| 0 1 0 0

One of these assignments is selected at random with a probability of 1/2.

Properties

Fairness

PS satisfies a fairness property called stochastic-dominace envy-freeness (sd-envy-free). Informally it means that each agent, considering the resulting probability matrix, weakly prefers his/her own row of probabilities to the row of any other agent. Formally, for every two agents i and j:

• Agent i has a weakly-higher probability to get his best item in row i than in row j;
• Agent i has a weakly-higher probability to get one of his two best items in row i than in row j;
• ...
• For any k ≥ 1, agent i has a weakly-higher probability to get one of his k best items in row i than in row j.

Note that sd-envy-freeness is an ex-ante fairness notion: it is fair only before the lottery takes place. The algorithm is of course not ex-post fair: after the lottery takes place, the unlucky agents may envy the lucky ones. But this is inevitable in allocation of indivisible objects.

Efficiency

PS satisfies an efficiency property called stochastic-dominace Pareto efficiency (sd-efficiency, also called: ordinal efficiency). Informally it means that, considering the resulting probability matrix, there is no other matrix that all agents weakly-sd-prefer and at least one agent strictly-sd-prefers.

Here, the ex-ante notion of sd-efficiency is stronger than the ex-post notion: sd-efficiency implies that every allocation selected by the lottery is sd-Pareto-efficient.

Strategy

PS is not a truthful mechanism: an agent who knows that his most preferred item is not wanted by any other agent, can manipulate the algorithm by eating his second-most preferred item, knowing that his best item will remain intact.

Improvements

As explained above, the allocation determined by PS is fair only ex-ante but not ex-post. Moreover, when each agent may get any number of items, the ex-post unfairness might be arbitrarily bad: it is possible that one agent will get all the items while other agents get none.

The PS-lottery algorithm is an improvement of PS in which the lottery is done only among deterministic allocations that are envy-free-except-one-item (EF1). This guaratees that the ex-post allocation is "not too unfair". Moreover, the EF1 guarantee holds for any cardinal utilities consistent with the ordinal ranking, i.e., it is stochastic-dominance EF1 (sd-EF1).[2]

The algorithm uses as subroutines both the PS algorithm and the Birkhoff algorithm.

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.