Exploring the Use of Fuzzy Constraint Satisfaction Problems to Evaluate the Happiness of Society. Peter Ballen Computer Systems Lab, 2009-2010

Hard Constraint Satisfaction All constraints are imperative and inflexible. A solution is only valid if all tuples are satisfied. Ex: 4-Colored Map Problem, N-Queens, Sudoku Map Source:math.utah.edu Sudoku: topnews.in

Hard Constraint Satisfaction The problem: Real life doesn't give easy constraints with perfect solutions. Hard Constraint Satisfaction cannot “compromise”. It can find only a perfect solution, not an optimal one. True/False Source: fredmckinnon.com

Soft Constraint Satisfaction Tuples receive values between 0 (fully violated) and 1 (fully satisfied). Solutions are rated on how many tuples are satisfied. Optimal solution is solution that most satisfied tuples, not necessarily all of them. VS

Society Model Black “Voters” are placed randomly on a 1x1 square. A white “Proposal” dot is placed on the square. Voters' satisfaction is a function of the distance between that Voter and the white dot. Less distance = Greater Satisfaction = Green Rated between 0 and 1. Independent of other voters End satisfaction is average of Voters' satisfaction.

Democracy of One The Voter V is black, the two proposals are in white. Green = Happier Proposal A is rated at 85.8% satisfaction. Proposal B is rated at 27.9% satisfaction.

Actual Democracy Four Voters (V1-V4) Proposal A makes V1 very happy, but V2, V3, & V4 unsatisfied. Proposal B makes V2 and V3 happy, but V1 and V4 unsatisfied. Proposal O is optimal solution. (72%) Everyone sacrifices.

Sympathy Tuples are sympathetic if increasing one's satisfaction increases the other's. A & B are sympathetic. Tuples are antagonistic if increasing one's satisfaction decreases the other. A & C are antagonistic.

Sympathetic Society In Society example, sympathetic Voters are Voters close to one another, so that moving the white dot towards one by necessity moves it towards another. More sympathetic Voters will translate into a happier society. By adjusting the range Voters are allowed to be placed in, we can force sympathy into society.

Sympathetic Societies Range [.45,.55] Satisfaction = 96.12% Range [.3,.7] Satisfaction = 84.44%

Antagonistic Societies Range [.1,.9] Satisfaction = 69.36% Range [0,1] Satisfaction = 57.57%

Finding the Solution Previous pictures were generated with brute force. Look at every possible solution, the pick out the best one. Why are we wasting time looking at the lower-left corner?

Finding the Solution Instead, we use pruning. First, find the optimal x-value, ignoring the y- values. Then, prune away all the other x- values. Finally, look at what's left to find the best solution. Notice that we never actually look most of the values.

Process at Work Find optimal x- values, ignoring y- values (draw the black and blue lines) Prune away the waste (everything in gray) Look at what's remaining to find best solution (the blue dot)

Comparison of Methods Blue line (Brute) is growing linearly. Red line (Pruning) grows, but at substantially slower rate. Difference becomes more pronounced as population size gets larger.

Prioritization Not all constants are given same level of importance. In the model, this means not all Voters get the same number of Votes. Each Voter gets a different proportion of the total vote. Scales Source: http://commons.wikimedia.org/wiki/File:Unbalanced_scales.svg

Prioritized Society The left Voter (the larger dot) is four times as important as the right Voter. Optimal solution (white dot) is slanted towards the left as a result.

Changing Priorities As the priority assigned to each dot changes, the optimal solution reacts. Green: Both equal priority. Pink: Right 60%; Left 40% Blue: Right 80%, Left 20% (as seen on last slide) Yellow: Right 95%, Left 5% (notice that left Voter is almost completely ignored)

Priority at the Fringe Voter in lower right corner is given large priority. Observe how much this shifts optimal result towards the edge.

Case Study – Problem In response to the failing economy, we must take drastic action. We propose that we substantially increase unemployment benefits (money given to unemploymed men and women from the government to help them buy food and search for work). To offset the cost of these benefits, we recommend raising the national income on individuals making more than $50,000 visually. How would this proposal be received?

Case Study - Options Multiple potential options: Lower taxes, lower unemployment benefits Raise taxes, raise unemployment benefits. Lower taxes, raise unemployment benefits, fund the benefits through the defect Raise taxes, lower unemployment benefits, use the increased revenue in other areas. Maintain current levels

Case Study – Collect Data Consider each state as one homogeneous Voter. States priority is based on the number of House representatives. States with higher tax rates favor higher taxes and vice versa. States with a higher unemployment favor more benefits.

Case Study – Initial Observations Data is difficult to evaluate. Where does the optimal solution lie?

This dot is South Carolina. High tax rate, low unemployment. Favors increasing taxes and spending the money on other issues.

This dot is Pennsylvania Low tax rate, high unemployment. Favors increasing benefits with defect spending. Notice that both states don't like proposal, but for different reasons.

The pink dot is Virginia. Favors raising benefits and raising taxes. Virginia likes proposal.

The blue dot is the weighted optimal solution. Each state's priority is its number of Representativ es in House.

The red dot is the unweighted optimal solution. Each state's has equal priority.

Case Study - Conclusions Both the House and Senate favor increasing income taxes to increase unemployment benefits. The Senate prefers a bill with slightly more benefits and slightly less taxes than the House. However, both fall in the middle of the spectrum. The proposal would be well received.