Resource monotonicity

The egalitarian rule is always RM:<ref name=moulin/>{{rp|47}} when there is more resource to share, the minimum utility that can be guaranteed to all agents increases, and all agents equally share the increase. In contrast, the utilitarian rule might be not RM.

The egalitarian rule is always RM:<ref name=moulin/>{{rp|47}} when there is more resource to share, the minimum utility that can be guaranteed to all agents increases, and all agents equally share the increase. In contrast, the utilitarian rule might be not RM.

For example, suppose there are two agents, Alice and Bob, with the following utilities:

For example, suppose there are two agents, [[Alice and Bob]], with the following utilities:

* <math>u_A(y_A) = y_A^2</math>

* <math>u_A(y_A) = y_A^2</math>

* When the pieces may be ''disconnected'', any allocation rule maximizing a concave welfare function of the ''absolute'' (not normalized) utilities is RM. In particular, the Nash-optimal rule, absolute-[[leximin]] rule and absolute-[[Utilitarian cake-cutting|utilitarian]] rule are all RM. However, if the maximization uses the ''relative'' utilities (utilities divided by total cake value) then most of these rules are not RM; the only one that remains RM is the Nash-optimal rule.<ref>{{Cite journal|last1=Segal-Halevi|first1=Erel|last2=Sziklai|first2=Balázs R.|date=2019-09-01|title=Monotonicity and competitive equilibrium in cake-cutting|url=https://doi.org/10.1007/s00199-018-1128-6|journal=Economic Theory|language=en|volume=68|issue=2|pages=363–401|arxiv=1510.05229|doi=10.1007/s00199-018-1128-6|s2cid=179618|issn=1432-0479}}</ref>

* When the pieces may be ''disconnected'', any allocation rule maximizing a concave welfare function of the ''absolute'' (not normalized) utilities is RM. In particular, the Nash-optimal rule, absolute-[[leximin]] rule and absolute-[[Utilitarian cake-cutting|utilitarian]] rule are all RM. However, if the maximization uses the ''relative'' utilities (utilities divided by total cake value) then most of these rules are not RM; the only one that remains RM is the Nash-optimal rule.<ref>{{Cite journal|last1=Segal-Halevi|first1=Erel|last2=Sziklai|first2=Balázs R.|date=2019-09-01|title=Monotonicity and competitive equilibrium in cake-cutting|url=https://doi.org/10.1007/s00199-018-1128-6|journal=Economic Theory|language=en|volume=68|issue=2|pages=363–401|arxiv=1510.05229|doi=10.1007/s00199-018-1128-6|s2cid=179618|issn=1432-0479}}</ref>

* When the pieces must be ''connected'', no Pareto-optimal proportional division rule is RM. The absolute-[[Equitable cake-cutting|equitable]] rule is weakly Pareto-optimal and RM, but not proportional. The relative-equitable rule is weakly Pareto-optimal and proportional, but not RM. The so-called ''rightmost mark'' rule, which is an variant of [[Divide and choose|divide-and-choose]], is proportional, weakly Pareto-optimal and RM - but it works only for two agents. It is an open question whether there exist division procedures that are both proportional and RM for three or more agents.<ref>{{Cite journal|last1=Segal-Halevi|first1=Erel|last2=Sziklai|first2=Balázs R.|date=2018-09-01|title=Resource-monotonicity and population-monotonicity in connected cake-cutting|url=http://www.sciencedirect.com/science/article/pii/S0165489618300520|journal=Mathematical Social Sciences|language=en|volume=95|pages=19–30|arxiv=1703.08928|doi=10.1016/j.mathsocsci.2018.07.001|s2cid=16282641|issn=0165-4896}}</ref>

* When the pieces must be ''connected'', no Pareto-optimal [[proportional division]] rule is RM. The absolute-[[Equitable cake-cutting|equitable]] rule is weakly Pareto-optimal and RM, but not proportional. The relative-equitable rule is weakly Pareto-optimal and proportional, but not RM. The so-called ''rightmost mark'' rule, which is an variant of [[Divide and choose|divide-and-choose]], is proportional, weakly Pareto-optimal and RM - but it works only for two agents. It is an open question whether there exist division procedures that are both proportional and RM for three or more agents.<ref>{{Cite journal|last1=Segal-Halevi|first1=Erel|last2=Sziklai|first2=Balázs R.|date=2018-09-01|title=Resource-monotonicity and population-monotonicity in connected cake-cutting|url=http://www.sciencedirect.com/science/article/pii/S0165489618300520|journal=Mathematical Social Sciences|language=en|volume=95|pages=19–30|arxiv=1703.08928|doi=10.1016/j.mathsocsci.2018.07.001|s2cid=16282641|issn=0165-4896}}</ref>

=== Single-peaked preferences ===

=== Single-peaked preferences ===