Can Picking First Make Up for Getting Less?

When equal claims produce unequal bundles

Dividing heirlooms or shared season tickets creates an arithmetic problem: objects cannot always be split evenly. Without payments, earlier choices offer a possible form of compensation for receiving fewer objects.

In their preprint, Who Picks First? Compensatory Fairness When Equal Division Is Impossible, Aleksei Chernulich and M. Bumin Yenmez study people with equal claims who report rankings of distinct goods. Any grouping of objects is feasible. Money, repeated allocations and scheduling constraints are outside the main model.

The central results are mathematical, not findings from an experiment with people. Additional simulations compare picking orders under specified assumptions about values.

What counts as compensation?

The authors compare bundles object by object. One bundle dominates another if every object in the second can be matched to a distinct object in the first that the person likes at least as much. This deliberately avoids assuming how much better one object is than another. Some bundles therefore remain incomparable.

Their framework combines three fairness requirements:

• Envy-freeness up to one object, or EF1: for every comparison, removing a suitable object from the other person's bundle leaves a bundle the observer's own bundle dominates.
• No envy toward larger bundles, up to any object, or NEL: someone receiving fewer objects must dominate the larger bundle after whichever single object is removed—not merely after a convenient removal.
• Alternating advantage under unanimity, or AAU: when everyone ranks objects identically, people receiving equally many objects must alternate who has the better first object, second object, and so on.

These are ordinal guarantees, based on rankings. With strict rankings, the dominance comparisons hold for every positive additive valuation consistent with those rankings. Allocating every object while satisfying EF1 also ensures that bundle sizes differ by at most one; that balance is a consequence, not an initial quota assumption.

The block–snake solution

The proposed procedure begins with a lottery ordering the participants. Those earlier in the draw receive the smaller bundles; the remaining participants receive one extra object each.

In every complete round, the smaller-bundle group picks before the larger-bundle group. Within each group, the order reverses between rounds. Finally, the larger-bundle group takes its extra objects.

For three friends sharing seven tickets, suppose Alice and Bob receive two each and Carol receives three. The sequence is Alice–Bob–Carol, then Bob–Alice–Carol, then Carol. If everyone ranks the tickets from 1, best, to 7, worst, Alice receives {1, 5}, Bob {2, 4}, and Carol {3, 6, 7}. Alice and Bob each dominate even Carol's two best tickets.

The authors prove that these three fairness requirements select exactly the block–snake sequences among fixed picking orders. A broader characterization also derives fixed picking orders from three structural conditions: allocating everything, treating object names as irrelevant, and leaving allocations unchanged when received objects only rise in their owners' rankings relative to other objects.

The lottery supplies equal chances of occupying each position, not equal expected welfare.

Fairness is not equality of value

Picking earlier can overcompensate someone receiving fewer objects. In the seven-ticket example, assigning common values 7 through 1 gives Alice and Bob 10 points each, versus Carol's 8.

The appendix compares orders using common equally spaced values, common steeply declining values, and two stochastic models with private or partly shared values. Each stochastic comparison uses 100,000 simulated draws and sincere picking.

Block–snake has the lowest estimated average absolute gap between the two groups' mean normalized utilities among the orders compared in every reported stochastic problem. But this is not universal superiority: some common-value benchmarks favor other orders. In two reported problems with relatively few objects per person, the larger-bundle group still comes out ahead in roughly 60–78% of simulated draws under block–snake. These are model-dependent group averages, not observed behavior or guarantees for every pair.

Ties and strategic reporting

Indifference creates another complication. If an early picker likes two objects equally, taking one arbitrarily may deprive a later picker who strictly prefers it. The authors' forward-looking extension preserves earlier commitments while resolving ties in favor of the earliest later picker who cares. It requires rankings in advance and preserves EF1, NEL and efficiency under object-by-object dominance.

That efficiency standard is limited: it does not exclude mutually beneficial exchanges whose appeal depends on preference intensity.

Incentives remain a separate obstacle. An agent may profit by choosing a contested object first and collecting a favorite later. The authors prove that no EF1 draft is even weakly strategy-proof once at least two people receive more than one object. For two people each receiving at least two, the impossibility extends to every non-wasteful, object-neutral rule.

Compensation alone can coexist with truthful reporting if smaller-bundle recipients choose their entire bundles first, but this generally sacrifices EF1. Whether block–snake's compensation survives strategic play remains open: its guarantees concern allocations under reported rankings, not a demonstrated outcome of strategic interaction.