Endriss - Fair Division (Axioms)

Question 1

What are ‘axioms’ in the context of social welfare orderings (SWOs)?

Answer 1

Formal definitions of the properties that we would like our SWO of choice to have. It helps us with choosing the best SWO according to our fairness criteria

Question 2

What are the axioms?

Answer 2

1) Anonymity
2) Unanimity
3) Separability
4) Pigou-Dalton principle
5) Zero Independence (ZI)
6) Scale Independence (SI)

Question 3

What is the anonymity axiom?

Answer 3

Anonymity implies that the outcome of the division does not depend on who the agents are. Their identities should not affect the allocation of resources.

Question 4

What is the unanimity axiom?

Answer 4

If all agents prefer one allocation over another, then the society (considered collectively) should also prefer that allocation.

Suppose three agents have to choose between:
Allocation X: (2, 2, 4)
Allocation Y: (1, 1, 1)
If all agents prefer every component of X over Y, then X should be unanimously chosen over Y.

Question 5

What is the separability axiom?

Answer 5

Adding the same utility vector to different allocations should not change their relative preference order

Suppose society prefers Allocation A: (2, 2, 4) over Allocation B: (1, 3, 1). If we add (1, 1, 1) to both allocations, making them A: (3, 3, 5) and B: (2, 4, 2), A should still be preferred over B

Question 6

What is the pigou-dalton principle axiom?

Answer 6

A utility transfer from a richer to a poorer agent that reduces inequality should lead to a preferred allocation.

Original Allocation A: (3, 5)
After Transfer Allocation B: (4, 4) (1 unit transferred from the richer to the poorer)
Allocation B should be considered an improvement over Allocation A.

Question 7

What is the zero independence axiom?

Answer 7

Preferences should not change if the same amount is added to each agent’s utility.

If Allocation A: (3, 3, 3) is preferred over Allocation B: (1, 2, 5), adding 2 units to each (making A: (5, 5, 5) and B: (3, 4, 7)) should not change the preference order.

Question 8

What is the scale independence axiom?

Answer 8

Multiplying all utilities by the same positive factor should not change social preferences.

Suppose Allocation A: (2, 2, 4) is preferred over Allocation B: (1, 4, 2). Multiplying each allocation by 2 (making A: (4, 4, 8) and B: (2, 8, 4)) should maintain the preference for Allocation A over B.