Proportional allocation
An allocation of objects is called proportional (PROP) if every agent ''i'' values his bundle at least 1/''n'' of the total. Formally, for all ''i'' (where ''M'' is the set of all goods): * . A proportional division may not exist. For example, if the number of people is larger than the number of items, then some people will get no item at all and their value will be zero. Nevertheless, such a division exists with high probability for indivisible items under certain assumptions on the valuations of the agents.Deciding whether a PROP allocation exists: cardinal utilities
Suppose the agents haveDeciding whether a PROP allocation exists: ordinal rankings
Suppose the agents have ordinal rankings on items. An allocation is called necessary-proportional (or sd-proportional) if it is proportional according to ''all'' valuations consistent with the rankings. It is called possibly-proportional if it is proportional according to ''at least one'' set of consistent valuations. * Pruhs and Woeginger present a polytime algorithm for deciding whether a necessary-proportional allocation exists, when agents have strict rankings. The algorithm is simpler when there are two agents. * Aziz, Gaspers, Mackenzie and Walsh extend this algorithm to agents with weak preferences, and with possibly differentRelation to other fairness criteria
With additive valuations: * EveryPROP1 allocations
An allocation is called proportional up to the best ''c'' items (PROPc) if for every agent ''i'', there exists a subset of at most ''c'' items that, if given to ''i'', brings the total value of ''i'' to at least 1/''n'' of the total. Formally, for all ''i'' (where ''M'' is the set of all goods): * . An equivalent definition is: the value of each agent ''i'' is at least (1/''n'' of the total) minus (the most valuable ''c'' items not assigned to ''i''): * PROP0 is equivalent to proportionality, which might not exist. In contrast, a PROP1 allocation always exists and can be found e.g. byFinding efficient PROP1 allocations
Conitzer, Freeman and Shah proved that, in the context of fair public decision making, a PROP1 allocation that is also PE. Barman and Krishnamurthy presented a strongy-polynomial-time algorithm finding a PE+PROP1 allocation for ''goods'' (objects with positive utility). Branzei and Sandomirskiy extended the condition of PROP1 to ''chores'' (objects with negative utility). Formally, for all ''i'': * . They presented an algorithm finding a PE+PROP1 allocation of chores. The algorithm is strongly-polynomial-time if either the number of objects or the number of agents (or both) are fixed. Aziz, Caragiannis, Igarashi and Walsh extended the condition of PROP1 to ''mixed valuations'' (objects can have both positive and negative utilities). In this setting, an allocation is called PROP1 if, for each agent ''i'', if we remove one negative item from i's bundle, or add one positive item to i's bundle, then i's utility is at least 1/''n'' of the total. Their Generalized Adjusted Winner algorithm finds a PE+EF1 allocation for two agents; such an allocation is also PROP1. Aziz, Moulin and Sandomirskiy presented a strongly-polynomial-time algorithm for finding an allocation that is fractionally-PE (stronger than PE) and PROP1, with general mixed valuations, even if the number of agents or objects is not fixed, and even if the agents have different entitlements.Relation to other fairness criteria
With additive valuations: * Every EF1 allocation is also PROP1, but the opposite is not necessarily true even with two agents. * The same is true for any ''c ≥'' 1: Every EF''c'' allocation is also PROP''c'', but the opposite is not necessarily true even with two agents.PROP*(''n''-1) allocations
An allocation is called proportional from all except ''c'' items (PROP*''c'') for an agent ''i'' if there exists a set of at most ''c'' items that, if removed from the set of ''all'' items, then ''i'' values his bundle at least 1/''n'' of the remainder. Formally, for all ''i'': * . PROP*(''n''-1) is slightly stronger than PROP1: when ''n''=2, PROP*(''n''-1) is equivalent to EF1, but PROP1 is weaker. A PROP*(''n''-1) allocation always exists and can be found e.g. byRelation to other fairness criteria
With additive valuations: * EF1 implies PROP*(''n''-1). The opposite implication is true when ''n''=2, but not when ''n''>2. Thus, the relation between EF1 and PROP*(''n''-1) is analogous to the relation between envy-freeness and proportionality, which shows that PROP*(''n''-1) is a more natural relaxation of proportionality than PROP1. * Moreover, for any integer ''c'' ≥ 0, EF''c'' implies PROP*((''n''-1)''c''). The opposite implication is true when ''n''=2, but not when ''n''>2. The followingPROPx allocations
An allocation is called proportional up to the worst item (PROPx) if for every agent ''i'', for any subset with one item not allocated to ''i'', if the subset is given to ''i'', then it brings his value to at least 1/''n'' of the total. Formally, for all ''i'': * An equivalent definition is: the value of each agent ''i'' is at least (1/''n'' of the total) minus (the ''least'' valuable item not assigned to ''i''): * Obviously, PROPx is stronger than PROP1. Moreover, while PROP1 allocations always exist, PROPx allocations may not exist.PROPm allocations
An allocation is called proportional up to the maximin item (PROPm) if the value of each agent ''i'' is at least (1/''n'' of the total) minus (the ''maximin'' item not assigned to ''i''), where the maximin item the maximum over all the other ''n''-1 agents ''j'', of the least-valuable item allocated to ''j''. Formally: * Obviously, PROPx is stronger than PROPm, which is stronger than PROP1. A PROPm allocation exists when the number of agents is at most 5.References
{{reflist Fair item allocation