Simultaneous eating algorithm

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. In the context of random assignments,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 SE 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. The important thing is that the integral of the eating speed of each agent equals the total number of items that the agent should receive (in the assignment setting, each agent should get exactly 1 item, so the integral of all eating-speed functions should be 1).

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 SE 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 fractions 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.

Other examples can be generated at the MatchU.ai website.

The description below assumes that all agents have risk-neutral preferences, that is, their utility from a lottery equals the expected value of their utility from the outcomes.

The SE algorithm has been extended in many ways.

• Katta and Sethuraman[7] present Extended PS (EPS), which allows weak ordinal preferences (rankings with indifferences). The algorithm is based on repeatedly solving instances of parametric network flow.
• Bogomolnaia[3] presented a simpler definition of the PS rule for weak preferences, based on the leximin order.
• Yilmaz[8] allows both indifferences and endowments.
• Athanassoglout and Sethuraman[9] present the controlled consuming (CC) rule, which allows indifferences and fractional endowments of any quantity.
• Budish, Che, Kojima and Milgrom[10] present Generalized PS, which allows multiple units per item, more items than agents, each agent can get several units, upper quotas, and bi-hierarchical constraints on the feasible allocations.
• Ashlagi, Saberi and Shameli[11] present another Generalized PS, which allows lower and upper quotas, and distributional constraints (constraints on the probability distribution and not only the final allocation).
• Aziz and Stursberg[12] present Egalitarian Simultaneous Reservation (ESR), which allows not only fair item allocation but also general social choice problems, with possible indifferences.
• Aziz and Brandl[13] present Vigilant Eating (VE), which allows even more general constraints.

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[14] 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. The algorithm uses separation oracles.
• A different algorithm, based on an ex-ante 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. It cannot be decomposed into EF1 allocations.
• 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[15] shows:

• The PS-lottery algorithm, in which the allocation is ex-ante sd-EF, and the lottery is done only among deterministic allocations that are sd-EF1, i.e., the EF and EF1 guarantees hold for any cardinal utilities consistent with the ordinal ranking. Moreover, the outcome is sd-PO 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.
• With binary utilities, the PS-lottery algorithm is group-strategyproof, ex-ante PO, ex-ante EF and ex-post EF1.
• These combinations of properties are best possible: it is impossible to guarantee simultaneously ex-ante sd-EF, ex-post EF1 and ex-post PO; or ex-ante PO and ex-ante sd-EF.
• Checking whether a given random allocation can be implemented by a lottery over EF1 and PO allocations is NP-hard.

Babaioff, Ezra and Feige[16] show:

• A polynomial-time algorithm for computing allocations that are ex-ante proportional, and ex-post both PROP1 and 1/2-fraction maximin-share (and also 1/2-fraction truncated-proportional share).
• These properties are nearly optimal - it is impossible to guarantee more than PROP ex-ante, and more than n/(2n-1) truncated-proportional share ex-post.

Hoefer, Schmalhofer and Varricchio[17] extend the notion of "Best-of-Both-Worlds" lottery to agents with different entitlements.

The page on fair random assignment compares PS to other procedures for solving the same problem, such as the random priority rule.