Endriss - Fair Division Flashcards
What is the fair division problem?
The problem of dividing one ore seerveral goods amongst two or more agents in a way that satisfies a suitable fairness criterion
What are divisible / non-divisible goods?
Goods that can either be divided or not be divided (cakes vs books for example)
What does a fair division try to establish?
A desirable allocation of goods to agents: for each agent, we need to specify which (part of) which item they should obtain
What are the formal criteria for measuring and comparing the fairness, as well as efficiency, of alternative allocations of goods?
1) Pareto efficiency
2) Social welfare orderings
3) Proportionality
4) Envy-freeness
What is pareto efficiency
1) The most fundamental criterion
2) An allocation is Pareto efficient if no other feasible allocation would make any agent better off without making at least one agent worse off
When is agreement A pareto dominated by agreement A’
Agreement A is pareto dominated by agreement A’ if the utility in agreement A is equal to or lower then the utility in agreement A’ for all i in N, and this inequality is strict in at least one case
Is pareto efficiency a strong criterion?
No, there can be many alternative solutions that are all pareto efficient. It also does not have to be fair. We want stronger criteria to identify solutions
What is the welfarist approach?
Looking at the utility vectors instead of the allocations (social welfare orderings)
What is a social welfare ordering?
These are binary relations used to rank different allocations based on the utilities they provide to the community of agents. They are complete, reflexive, and transitive.
What is complete, reflexive and transitive?
A binary relation of utility vectors that is reflexive, transitive, and complete ensures that every utility vector is comparable to itself and others in a consistent and all-encompassing manner, enabling rational and coherent decision-making in evaluating different allocations.
What are collective utility functions (CUF)?
These functions map utility vectors to a real number, representing the overall welfare of an allocation.
What is the utilitarian CUF?
This social welfare ordering sums the utilities of all agents. It represents the total happiness or satisfaction derived from an allocation across all individuals.
Allocation A: (3, 2, 1)
Allocation B: (2, 2, 2)
The utilitarian welfare for A is 6 (3+2+1) and for B is 6 (2+2+2). Both allocations have equal utilitarian social welfare.
What is the egalitarian CUF?
This focuses solely on the welfare of the worst-off individual in society. The aim is to maximize the minimum utility anyone receives
Between the utilities (1, 5, 10) and (3, 3, 3):
Egalitarian welfare prefers the second scenario where the lowest utility is 3, higher than the lowest utility of 1 in the first scenario.
What is the Nash CUF?
This ordering is based on the product of the utilities of all agents. It seeks a balance between efficiency and equality, preferring allocations that improve the situation of the worst-off without sacrificing too much total utility
For the utilities (3, 2, 1) and (2, 2, 2):
The Nash welfare for the first is 3×2×1=6 and for the second is 2×2×2=8. Allocation B is preferred under Nash because it increases overall utility productively.
What is the Leximin CUF?
This ordering looks at the least well-off individual’s utility first; if there’s a tie, it moves to the second least well-off, and so on. It maximizes the minimum utility to address fairness most directly.
Comparing utilities (1, 2, 3) vs. (2, 2, 2):
Leximin would prefer the second allocation because the worst-off person is better off (2 vs. 1).
