A basic test of fairness when we divide a manna  of private items  between n agents is the lowest welfare the rule guarantees to each agent, irrespective of others preferences. Two familiar examples are: the Equal Split Guarantee when the manna is divisible and preferences are convex; and 1/n of the utility of a heterogenous non atomic cake, if utilities are additive. The minMax utilityof an agent is that of her best share in the worst possible n-partition of . It is weakly below her Maxmin utility , that of her worst share in the best possible n-partition. The Maxmin guarantee is not feasible, even with two agents, if non convex preferences are allowed. The minMax guarantee is feasible in the very general class of problems where the manna is non atomic and utilities are continuous, but not necessarily additive or monotonic. The proof uses advanced algebraic topology techniques. And the minMax guarantee is implemented by the n-person version of Divide and Choose due to Kuhn (1967). When utilities are co-monotone (a larger part of the manna is weakly better for everyone, or weakly worse for everyone) better guarantees than minMax are feasible. In our Bid & Choose rules , agents bid the smallest size (according to some benchmark measure of the manna) of a share they find acceptable , and the lowest bidder picks such a share.The resulting guarantee is between the minMax and Maxmin utilities. 1
To divide a "manna" \Omega of private items (commodities, workloads, land, time intervals) between n agents , the worst case measure of fairness is the welfare guaranteed to each agent, irrespective of others ' preferences. If the manna is non atomic and utilities are continuous (not necessarily monotone or convex), we can guarantee the minMax utility: that of our agent's best share in her worst partition of the manna; and implement it by Kuhn's generalisation of Divide and Choose. The larger Maxmin utility -- of her worst share in her best partition -- cannot be guaranteed, even for two agents. If for all agents more manna is better than less (or less is better than more), our Bid & Choose rules implement guarantees between minMax and Maxmin by letting agents bid for the smallest (or largest) size of a share they find acceptable .
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A basic test of fairness when we divide a manna
To divide a "manna"
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\Omega
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(commodities, workloads, land, time intervals)
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is the lowest welfare the rule guarantees
, the worst case measure of fairness is the welfare guaranteed
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preferences. Two familiar examples are: the Equal Split Guarantee when the manna is divisible and preferences are convex; and 1/n of the utility of a heterogenous non atomic cake, if utilities are additive. The minMax utilityof an agent is that of her
' preferences. If the manna is non atomic and utilities are continuous (not necessarily monotone or convex), we can guarantee the minMax utility: that of our agent's
R
the worst possible n-partition of . It is weakly below her Maxmin utility , that
her worst partition of the manna; and implement it by Kuhn's generalisation of Divide and Choose. The larger Maxmin utility --
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the best possible n-partition. The Maxmin guarantee is not feasible, even with two agents, if non convex preferences are allowed. The minMax guarantee is feasible in the very general class of problems where the manna is non atomic and utilities are continuous, but not necessarily additive or monotonic. The proof uses advanced algebraic topology techniques. And the minMax guarantee is implemented by the n-person version of Divide and Choose due to Kuhn (1967). When utilities are co-monotone (a larger part of the manna is weakly better for everyone, or weakly worse for everyone) better guarantees than minMax are feasible. In
her best partition -- cannot be guaranteed, even for two agents. If for all agents more manna is better than less (or less is better than more),
R
, agents bid the smallest size (according to some benchmark measure of the manna)
implement guarantees between minMax and Maxmin by letting agents bid for the smallest (or largest) size
R
, and the lowest bidder picks such a share.The resulting guarantee is between the minMax and Maxmin utilities. 1
.