Simultaneous eating algorithm

The Probabilistic Serial algorithm (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 ex-ante envy-free and Pareto efficient. It was developed by Hervé Moulin and Anna Bogomolnaia.

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 ex-ante 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 guaranteed ex-ante: 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. 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.

Guaranteeing ex-post approximate fairness

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: theoretically it is possible that one agent will get all the items while other agents get none. Recently, several algorithms have been suggested, that guarantee both ex-ante fairness and ex-post approximate-fairness.

Freeman, Shah and Vaish show:

• The Recursive Probabilistic Serial (RecPS) algorithm, which returns a probability distribution over allocations that are all envy-free-except-one-item (EF1). The distribution is ex-ante EF, and the allocation is ex-post EF1. A naive version of this algorithm yields a distribution over a possibly exponential number of deterministic allocations, a support size polynomial in the number of agents and goods is sufficient, and thus the algorithm runs in polynomial time.
• A different algorithm, based on rounding the max-product allocation, which attains ex-ante group envy-freeness (GEF; it implies both EF and PO), and ex-post PROP1+EF₁¹_. This is the only allocation rule that achieves all these properties.
• These combinations of properties are best possible: it is impossible to guarantee simultaneously ex-ante EF (even PROP) and ex-ante PO together with ex-post EF1; or ex-ante EF (even PROP) together with ex-post EF1 and fractional-PO.
• The RecPS can be modified to attain similar guarantees (ex-ante EF and ex-post EF1) for bads.

Aziz shows:

• The PS-lottery algorithm, in which the lottery is done only among deterministic allocations that are sd-EF1, i.e., the EF1 guarantee holds for any cardinal utilities consistent with the ordinal ranking. The outcome is sd-efficient both ex-ante and ex-post. The algorithm uses as subroutines both the PS algorithm and the Birkhoff algorithm. The ex-ante allocation is equivalent to the one returned by PS; this shows that the outcome of PS can be decomposed into EF1 allocations.
• Checking whether a given random allocation can be implemented by a lottery over EF1 and PO allocations is NP-hard.

Babaioff, Ezra and Feige present a polynomial-time algorithm for computing allocations that are ex-ante proportional, and ex-post both PROP1 and 1/2-fraction maximin-share.

See also

• Random priority - an alternative mechanism with different properties.