## Introduction

In the last semester of my bachelor's journey, I decided to take a course about algorithmic game theory. In the end, we did this fun, small project with Henry Fleischmann, where we managed to design a mechanism that addresses the *gerrymandering* problem, proving it has the desired properties. The following is a quick summary of our project.

## The Model

We describe our model of the problem. Our goal is to design a *fair* rule to get a redistricting map based on a selected group of commissioners while being aware that

- There are three types of commissioners, i.e., independent commissioners, Democrats, and Republicans.
- Some groups of commissioners might collude together, trying to manipulate the outcome.

### Social-Choice Function

The usual model of a voting scheme is known as the *social-choice function* under the context of algorithmic game theory. The basic idea is to collect everyone's preferences profile and return a final result, and in our project, we want to design a social-choice function that collects commissioners' preferences for maps and returns one at the end.

### Fairness

As one might expect, defining *fairness* is not a trivial thing to do. In our example, we want to achieve at least the following.

#### Group Strategy Proof

The *group strategyproofness* is a generalized notion of strategyproofness, where even if a group of voters colludes to misreport their preferences, there's no way they can improve their utility (i.e., achieving their goal such as selecting a map that is biased to a particular party). In other words, even if a group of people can collude, being truthful is the best strategy.

#### Unbiased

We want to always select an unbiased map such that no particular party benefits. We refer to such an unbiased map as a *neutral map*. Notice that this is a desired property directly controlling the outcome of the voting rule to produce a *fair* outcome.

We see that by combining group strategyproofness and unbiased map property, there's no way to manipulate the mechanism.

## Positional-Scoring Rule

Now we describe our technical contribution. Assuming that we have three maps, biased toward Democrats, Republicans, and neutral respectively. Then, by considering a simple positional-scoring rule with score $\langle 1, 0, -1\rangle$ on commissioners' preference on these three types of maps, we prove the following.

### Main Theorem

**Theorem** *Supposes that the commission is composed of an equal number of Democrats and Republicans and at least one Independent commissioner, $w_D$, or $w_R$. ^{2} Then, the positional-scoring voting rule with scores $\langle 1, 0, -1\rangle$, respectively, is group strategy-proof and chooses a neutral map.*

The equal number of Democrats and Republicans is referred to as the *balanced* case.^{3}

- We assume that even an independent commissioner has his/her preference as a
*weak*Democrat/Republican. This is a fair assumption since the preference of an independent commissioner is expected to be either "*neutral map*$\succ$*Democrats map*$\succ$*Republicans map*" (i.e., weak*Democrats*, $w_D$) or "*neutral map*$\succ$*Republicans map*$\succ$*Democrats map*" (i.e., weak*Republicans*, $w_R$).↩ - It's possible to relax the result to the unbalanced case with some adjustments to the scoring. See the paper for details.↩