75 research outputs found
Monotonicity and Competitive Equilibrium in Cake-cutting
We study the monotonicity properties of solutions in the classic problem of
fair cake-cutting --- dividing a heterogeneous resource among agents with
different preferences. Resource- and population-monotonicity relate to
scenarios where the cake, or the number of participants who divide the cake,
changes. It is required that the utility of all participants change in the same
direction: either all of them are better-off (if there is more to share or
fewer to share among) or all are worse-off (if there is less to share or more
to share among).
We formally introduce these concepts to the cake-cutting problem and examine
whether they are satisfied by various common division rules. We prove that the
Nash-optimal rule, which maximizes the product of utilities, is
resource-monotonic and population-monotonic, in addition to being
Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium
condition. Moreover, we prove that it is the only rule among a natural family
of welfare-maximizing rules that is both proportional and resource-monotonic.Comment: Revised versio
Universal characterization sets for the nucleolus in balanced games
We provide a new mo dus op erandi for the computation of the nucleolus in co op-
erative games with transferable utility. Using the concept of dual game we extend
the theory of characterization sets. Dually essential and dually saturated coalitions
determine b oth the core and the nucleolus in monotonic games whenever the core
is non-empty. We show how these two sets are related with the existing charac-
terization sets. In particular we prove that if the grand coalition is vital then the
intersection of essential and dually essential coalitions forms a characterization set
itself. We conclude with a sample computation of the nucleolus of bankruptcy games
- the shortest of its kind
Apportionment and districting by Sum of Ranking Differences
Sum of Ranking Differences is an innovative statistical method that ranks competing solutions based on a reference point. The latter might arise naturally, or can be aggregated from
the data. We provide two case studies to feature both possibilities. Apportionment and districting are two critical issues that emerge in relation to democratic elections. Theoreticians
invented clever heuristics to measure malapportionment and the compactness of the shape
of the constituencies, yet, there is no unique best method in either cases. Using data from
Norway and the US we rank the standard methods both for the apportionment and for the
districting problem. In case of apportionment, we find that all the classical methods perform
reasonably well, with subtle but significant differences. By a small margin the Leximin
method emerges as a winner, but—somewhat unexpectedly—the non-regular Imperiali
method ties for first place. In districting, the Lee-Sallee index and a novel parametric method
the so-called Moment Invariant performs the best, although the latter is sensitive to the function’s chosen parameter
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