A picking sequence is a protocol for fair item assignment. Suppose "m" items have to be divided among "n" agents. One way to allocate the items is to let one agent select a single item, then let another agent select a single item, and so on. A picking-sequence is a sequence of "m" agent-names, where each name determines what agent is the next to pick an item.
As an example, suppose 4 items have to be divided between Alice and Bob. Some possible picking sequences are:
Advantages.
A picking-sequence has several merits as a fair division protocol:
Welfare maximization.
How should the picking sequence be selected? Bouveret and Lang study this question under the following assumptions:
They show picking-sequences that maximize the expected utilitarian welfare (sum of utilities) or the expected egalitarian welfare (minimum utility) in various settings.
Kalinowski et al show that, when there are two agents with a Borda scoring function, and each ranking is equally probable, the "round robin" sequence (ABABAB...) attains the maximal expected sum-of-utilities.
Fairness with different entitlements.
Brams and Kaplan study the problem of allocating cabinet ministries among parties. There is a coalition of parties; each party has a different number of seats in the parliament; larger parties should be allocated more ministries or more prestigious ministries. This is a special case of fair item assignment with different entitlements. A possible solution to this problem is to determine a picking sequence, based on the different entitlements, and let each party pick a ministry in turn. Such a solution is used in Northern Ireland, Denmark and the European parliament.
Brams assumes that each agent has a strict ordering on the items, and has responsive preferences on bundles of items. This means that, at each point in the picking sequence, there is a single remaining item which is the "best item" for the agent. An agent is called "sincere" ("truthful") if, at each point, he picks his best item. If agents have complete information on each other's preferences (as is common among parties), it may not be rational for them to choose truthfully; it may be better for them to make "sophisticated" ("strategic") choices. Thus, the picking sequence induces a sequential game and it is interesting to analyze its subgame-perfect equilibrium. Several results are proved:
Determining the picking-sequence.
Given the agents' different rights, what would be a fair picking sequence?
Brams suggests to use divisor methods, similar to the ones used for apportionment of congress seats among states. The two most commonly used methods are the ones proposed by Daniel Webster and Thomas Jefferson. Both methods start in the same way:
Competitive equilibrium.
Picking sequences can be used to find allocations that satisfy a strong fairness and efficiency condition called competitive equilibrium.
See also.
The round-robin item allocation protocol is a special case of a picking sequence in which the sequence is cyclic: 1, 2, ..., "n", 1, 2, ..., "n", ...
Schoolyard games often need to select "teams". When using the "ABBA" selection, the "A" team will declare "first pick" and the B team will declare "Second-Two": List of traditional children's games
References.
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