Social Choice Lab

Updated one typo on 9 November, 2006.

Purpose

The reasons I'm asking you to do this lab are:

I want you to play with a bunch of social welfare functions so that you get familiar with the concept.
I want you to see that the various voting mechanisms choose very different candidates (so that you do not mistakenly believe that all of them will select the same "most preferred" candidate).
I want you to see the problems that arise when we try to combine utilities from different agents (the interpersonal comparison of utility problem).
I want you to try and develop a couple of social choice mechanisms yourself.

The Problem

Consider a society of eight individuals who are trying to reach a consensus on which of five alternatives they want. Each agent has a different utility function defined for each of the five alternatives. The utilities for each of the agents are:

Agent	U(a₁)	U(a₂)	U(a₃)	U(a₄)	U(a₅)
A	8	6	4	2	0
B	-5	10	5	0	-10
C	19	3	2	1	20
D	-12	-6	-9	0	-3
E	2	5	0	6	1
F	-12	-11	-10	-13	-14
G	10	-4	8	0	9
H	58	64	61	70	67

Your job is to help these agents choose one of the alternatives from the set {a₁,a₂,a₃,a₄,a₅}. You should notice that these utility functions impose a preference pattern for each of the agents. Namely,

Agent	1st Choice	2nd Choice	3rd Choice	4th Choice	5th Choice
A	a₁	a₂	a₃	a₄	a₅
B	a₂	a₃	a₄	a₁	a₅
C	a₅	a₁	a₂	a₃	a₄
D	a₄	a₅	a₂	a₃	a₁
E	a₄	a₂	a₁	a₅	a₃
F	a₃	a₂	a₁	a₄	a₅
G	a₁	a₅	a₃	a₄	a₂
H	a₄	a₅	a₂	a₃	a₁

The Lab

I want you to determine what choice is made when the following social choice functions are applied:

Voting Schemes

Majority Rule (Plurality Protocol)
Pairwise Elimination (Binary Protocol) for the following orders (in case of a tie, pick the one randomly).

((((a₁,a₂),a₃),a₄),a₅); compare 1 and 2, then compare the winner to 3, then the winner of this to 4, and the winner of this to 5.
((((a₅,a₄),a₃),a₂),a₁); compare 5 and 4, then compare the winner to 3, then the winner of this to 2, and the winner of this to 1.

Borda Protocol

Single step (tally all the scores, and pick the winner)
Multiple step (tally all the scores, eliminate the choice with the lowest score, and repeat until a winner is chosen).

Try a voting scheme you make up (or one that you find in the literature).

Utility Schemes

Sum: Construct the utility for society by adding up the utiliities from each individual agent.
Weighted Sum: Add up the utilities from each individual agent, but weight the agents by their "social status"

Try the weightings w_A=8, w_B=7, ... w_H=1.
Try the weightings w_A=1, w_B=2, ... w_H=8.
Try the weightings w_A= w_B = , ... , = w_H=4.

Since utilities are unique only up to a positive affine transformation, alter the utilities for each agent using the transformation below. Then repeat the Sum and Weighted Sum exercises with these new utilties. The transformation sets the utility of the least desired alternative to zero and the utility of the most desired alternative to one; all other utilities are appropriately scaled between these values. The transformation is defined below:

Let U'(a_i) = U(a_i) - min_aj U(a_j).
Change U'(a_i) <-- U'(a_i) / max _aj U'(a_j)

Try a utility schems you make up (or one that you find in the literature).

What you'll turn in:

I'm not very interested in whether you complete this lab (this is assumed), but rather in how you analyze your results. Use the scientific method (observe a phenomena, generate a hypothesis, test your hypothesis, and present supporting data).You should submit a report that summarizes the social choice functions you made up, the results of each of the schemes above, and an analysis of the results (what was good about each scheme, what was bad, why were the schemes good/bad).