We have the following table representing data

First Steps in Defining a “Loserless” System

KNOWLES 1

Introduction

We have the following table representing data collected from department stores (I made these numbers up, but it illustrates the idea). Pants_ID Pants_Quality Pants_Cost 1

5

-$60

2

7

-$60

3

4

-$30

Just by looking at these, it is clear to tell that pair #2 has the highest quality and that pair #3 has both the lowest price and the best quality per cost. If we plot these using V (t , w) = wt 1 +(1−w) t 2 we can produce the following graph (weight on the x-axis, value on the y-axis):

Using this system, pair #3 (tan) is the optimal choice for most weights, pair #2 (purple) is the optimal choice for higher weights (weighted towards quality), and pair #1 (blue) is always suboptimal. Looking at these plots, pair #1 is always a “loser,” allowing us to discard it from our consideration altogether. But what if we have a case where we are not able to disregard items? Student_ID Student_Grade Student_Attendance 1

88%

96%

2

89%

87%

3

92%

100%

If we plot these in the same manner as before, we produce the following:

First Steps in Defining a “Loserless” System

KNOWLES 2

Student #3 is obviously always the optimal choice with this model. However, since we are not able to discard items in this case (as they represent human beings who 1 deserve a fairer system), an alternative value function is required, such as V (t , w)= p∣w−0.5−s∣ t1 t2 where p = and ∣s∣