Abstract
We study the properties of scoring allocation correspondences and rules, due to Baumeister et al. [7], that are based on a scoring vector (e.g., Borda or lexicographic scoring) and an aggregation function (e.g., utilitarian or egalitarian social welfare) and can be used to allocate indivisible goods to agents. Extending their previous results considerably and solving some of their open questions, we show that while necessary duplication monotonicity (a notion inspired by the twin paradox [21] and false-name manipulation [1]) fails for most choices of scoring vector when using leximin social welfare, possible duplication monotonicity holds for a very wide range of scoring allocation rules. We also show that a very large family of scoring allocation rules is monotonic. Finally, we show that a large class of scoring allocation correspondences satisfies possible Pareto-optimality, which extends a result of Brams et al. [12].
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Notes
- 1.
Possible and necessary duplication monotonicity are inspired by the notions of possible and necessary winner in voting [9, 10, 18, 29] that have been used not only in fair division [3, 7] but were also applied, e.g., to strategy-proofness in judgment aggregation [8] and to stability concepts in hedonic games [19, 26].
- 2.
This is the price we pay for the simplicity of eliciting only ordinal preferences on single items (as opposed to ordinal, or even cardinal, preferences on all shares).
- 3.
Responsive set extensions are the most suitable in our context as they precisely capture the uncertainty in choosing the “right” scoring vector.
- 4.
Note that while \(\succeq ^{\mathrm {nec}}\) is a partial order, \(\succeq ^{\mathrm {pos}}\) is neither transitive nor antisymmetric in general.
- 5.
While enabling such cheating might not seem like a desirable property, this really is a common tradeoff: Procedures that behave predictably and naturally with respect to changing inputs will always be easier to manipulate than ones that behave chaotically. We believe that one should rather err on the side of regularity. Compare the situation to voting: Under most reasonable voting rules, cheaters who manage to improperly submit multiple ballots actually increase the chances of their favored candidates being selected. But that should not be held against the voting rule: It almost necessarily comes with trying to give equal weight to all votes, which is a principle that should not be carelessly abandoned in the name of deterring cheaters.
- 6.
more precisely, a family of tie-breaking relations \({>}^T_n\) on \(\Pi (G,n)\) for all \(n\ge 1\) (note that this property is really about how the orders \({>}^T_{n}\) and \({>}^T_{n+1}\) interact)
- 7.
Note that this condition, which is taken from Baumeister et al. [7], is more a technical device than a substantive suggestion for how to choose tie-breaking mechanisms. In the definition of duplication monotonicity the last agent gets duplicated. This choice is arbitrary, but this is justified by the fact that scoring allocation correspondences are anonymous, so the choice does not matter for our results, as long as no tied winning allocations occur. However, after applying a tie-breaking mechanism, the resulting allocation rule will no longer be anonymous, by necessity—non-trivial anonymous allocation rules do not exist. The definition here is carefully chosen to match the particular choice of duplicating the last agent, so we can give succinct statements of our main theorems that hold even in cases of tied winners. In those (and only those) cases, the results do depend on the arbitrary choices here, so are not entirely natural.
- 8.
The proof for lexicographic scoring also supposes that a winning allocation for \(F_{\mathrm{lex},\mathrm{leximin}}\) can give more than one item only to agents whose individual utility is minimal among all agents. That assumption is not correct, as Example 2 illustrates.
- 9.
In their terminology, maxsum divisions, maxmin divisions, and equimax divisions.
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This work was supported in part by DFG grant RO 1202/14-2.
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Kuckuck, B., Rothe, J. (2019). Monotonicity, Duplication Monotonicity, and Pareto Optimality in the Scoring-Based Allocation of Indivisible Goods. In: Lujak, M. (eds) Agreement Technologies. AT 2018. Lecture Notes in Computer Science(), vol 11327. Springer, Cham. https://doi.org/10.1007/978-3-030-17294-7_13
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