Equal Shares and Then Some

Algorithmic Frontiers of Fairness

@FSTTCS 2023

Neeldhara Misra

*Indian Institute of Technology, Gandhinagar *

This talk is based on

...with additional inputs borrowed from the

wonderful website on the method of equal shares.

Neeldhara Misra

*Indian Institute of Technology, Gandhinagar *

*Proportional Participatory Budgeting with Additive Utilities *Dominik Peters , Grzegorz Pierczyński , and Piotr Skowron

The conference organizers want to build a registration kit.

₹ 1000 → Total Budget

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A set of *candidates* \([m]\),

a set of *voters* \([n]\)*.*

PB v. Multi-Winner Voting

Voters *express preferences* over candidates.

A subset of \(k\) candidates are chosen,

in a way that makes all voters happy.

Hopefully.

A set of *candidates* (**projects**)\([m]\),

a set of *voters *(**stakeholders**) \([n]\)*.*

PB v. Multi-Winner Voting

Voters *express preferences* over candidates.

A subset of \(k\) **projects** are chosen,

in a way that makes all **stakeholders** happy.

This is the so-called *unit-cost *setting.

A set of *candidates* (**projects**)\([m]\) with associated **costs**,

a set of *voters *(**stakeholders**) \([n]\)*.*

PB v. Multi-Winner Voting

Voters *express preferences* over candidates.

A subset of **projects** with a total budget of at most \(k\) are chosen,

in a way that makes all **stakeholders** happy.

If we are going to pick \(k\) winners,

a set of \(\frac{n}{k}\) voters who *all*

approve at least one candidate

should not be completely unrepresented, i.e,

at least one representative among the winners.

Justified representation (JR)

If we are going to pick \(k\) winners,

a set of \({\color{IndianRed}L} \cdot \frac{n}{k}\) voters who *all*

approve at least \({\color{IndianRed}L}\) candidates;

should not be unrepresented (in a stronger sense), i.e,

have at least one \({\color{IndianRed}L}\) representatives among the winners.

Extended Justified representation (EJR)

Given a ballot profile \(\mathbf{A}=\left(A_1, \ldots, A_n\right)\) over a set of candidates \(C\)

and a target *committee size* \(k\),

we say that a set of candidates \(W\) of size \(|W|=k\) provides justified representation for \((\mathbf{A}, k)\) if the following holds:

Justified representation (JR)

for every subset of voters \(N^* \subseteq N\) with \(\left|N^*\right| \geqslant \frac{n}{k}\),

either \(\bigcap_{i \in N^*} A_i = \emptyset\)

or \(A_i \cap W \neq \emptyset\) for some \(i \in N^*\).

Given a ballot profile \(\mathbf{A}=\left(A_1, \ldots, A_n\right)\) over a set of candidates \(C\)

and a target *committee size* \(k\),

we say that a set of candidates \(W\) of size \(|W|=k\) provides justified representation for \((\mathbf{A}, k)\) if the following holds:

Extended Justified representation (JR)

for every subset of voters \(N^* \subseteq N\) with \(\left|N^*\right| \geqslant \ell \cdot \frac{n}{k}\),

either \(|\bigcap_{i \in N^*} A_i| < \ell\)

or \(A_i \cap W \geqslant \ell\) for some \(i \in N^*\).

If the chosen candidates are \(W\), and a voter \(i \in [n]\) voted for \(A \subseteq C\), then her PAV score on W is:

\(s_i(W) := 1 + \frac{1}{2} + \cdots + \frac{1}{|A \cap W|}\).

Proportional Approval Voting (PAV)

The total PAV score of a committee is:

\(\sum_{i \in [n]} s_i(W)\)

PAV satisfies* EJR 🎉

If the chosen candidates are \(W\), and a voter \(i \in [n]\) voted for \(A \subseteq C\), then her PAV score on W is:

\(s_i(W) := 1 + \frac{1}{2} + \cdots + \frac{1}{|A \cap W|}\).

Proportional Approval Voting (PAV)

The total PAV score of a committee is:

\(\sum_{i \in [n]} s_i(W)\)

Determining a winning set for PAV is NP-complete,

even when every voter approves only two candidates.

If the chosen candidates are \(W\), and a voter \(i \in [n]\) voted for \(A \subseteq C\), then her PAV score on W is:

\(s_i(W) := 1 + \frac{1}{2} + \cdots + \frac{1}{|A \cap W|}\).

Proportional Approval Voting (PAV)

The total PAV score of a committee is:

\(\sum_{i \in [n]} s_i(W)\)

PAV's EJR guarantees

break (dramatically) beyond the unit-cost scenarios.

Proportional Approval Voting (PAV)

PAV's EJR guarantees

break (dramatically) beyond the unit-cost scenarios.

The Method of Equal Shares

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Rule 1: We divide the budget equally among the voters.

The Method of Equal Shares

Rule 1: We divide the budget equally among the voters.

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The Method of Equal Shares

Rule 2: We select the project with the highest number of votes.

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The Method of Equal Shares

Rule 2: We select the project with the highest number of votes.

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The Method of Equal Shares

Rule 3: The cost of the project is divided equally among its supporters.

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The Method of Equal Shares

Rule 3: The cost of the project is divided equally among its supporters.

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The Method of Equal Shares

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The Method of Equal Shares

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Rule 4: We do not count voters with no money.

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The Method of Equal Shares

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The Method of Equal Shares

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The Method of Equal Shares

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*Underutilized budget: 1000 - 700 - 270 = 30.*

Equal Shares: The Method

**Approval voting**, where each voter votes for some of the projects, but votes for each of them with the *same strength*.

**Utilities**, where each voter assigns a utility number to each project

(0 or higher); each voter has the same number of *total points*.

The Input

*or*

Equal Shares: The Method

The overall budget is divided equally among the voters.

The Mechanism

If a project is not affordable* among voters who support it,

then delete the project from the system.

Equal Shares: The Method

The overall budget is divided equally among the voters.

We remove all projects from consideration that cost more than the combined share of all voters that voted for the project.

The Mechanism

Equal Shares: The Method

The overall budget is divided equally among the voters.

We remove all projects from consideration that cost more than the combined share of all voters that voted for the project.

If no projects remain, the computation of the method is finished.

The Mechanism

Equal Shares: The Method

The overall budget is divided equally among the voters.

We remove all projects from consideration that cost more than the combined share of all voters that voted for the project.

If projects remain, we calculate the **effective vote count** of every project.

The Mechanism

Equal Shares: The Method

The overall budget is divided equally among the voters.

If projects remain, we calculate the **effective vote count** of every project.

Select the project with the *highest effective vote count*.

**Split the cost** of this project as equally as possible among supporters.

The Mechanism

Equal Shares: The Method

We do not count voters if they have already spent their entire budget share.

The Effective Vote Count

Voters who still have money left,

but not enough money to pay for the project when its cost is equally divided,

*will count as a fraction*.

Splitting Costs Equally

The cost of a project is split in such a way that

*the maximum payment of any voter is as small as possible*

among voters who approved it.

Equal Shares: The Method

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Equal Shares: The Benefits

Higher and more equal voter satisfaction.

Reduced bias towards the most popular categories.

Mathematical fairness guarantees.

ES v. other voting systems for participatory budgeting

Equal Shares: The Benefits

Support \(\implies\) Success

A project is guaranteed to be funded if

its exclusive support is proportional to its relative cost. 👍

Consider a project proposal with a cost that would consume 5% of the overall available budget.

Then, if at least 5% of voters vote for that project *and for no other projects*, then this proposal will win under the Method of Equal Shares.

Equal Shares: The Benefits

Theorem. Let \(P\) be a project and let \(\operatorname{cost}(P)\) be the cost of the proposal.

Suppose \(B\) is the overall budget, and \(n\) is the total number of voters.

If at least \(n \cdot {\color{IndianRed}\operatorname{cost}(P) / B}\) voters vote for \(P\) and no other projects,

then \(P\) will be among the projects selected by the Method of Equal Shares.

\(n \cdot \operatorname{cost}(P) / B\) voters

have a budget of at least \(B/n\) at the start.

\(\geqslant \underbrace{n \cdot \frac{\operatorname{cost}(P)}{B}}_{\text {number of voters }} \cdot \underbrace{\frac{B}{n}}_{\text {budget share }} = \operatorname{cost}(P) .\)

Their total leverage

Support \(\implies\) Success

Equal Shares: The Benefits

Theorem. Let \(P\) be a project and let \(\operatorname{cost}(P)\) be the cost of the proposal.

Suppose \(B\) is the overall budget, and \(n\) is the total number of voters.

If at least \(n \cdot {\color{IndianRed}\operatorname{cost}(P) / B}\) voters vote for \(P\) and **no other projects**,

then \(P\) will be among the projects selected by the Method of Equal Shares.

\(n \cdot \operatorname{cost}(P) / B\) voters

have a budget of at least \(B/n\) at the start.

\(\geqslant \underbrace{n \cdot \frac{\operatorname{cost}(P)}{B}}_{\text {number of voters }} \cdot \underbrace{\frac{B}{n}}_{\text {budget share }} = \operatorname{cost}(P) .\)

Their total leverage

Support \(\implies\) Success

Equal Shares: The Benefits

Groups with similar votes

Any group of voters who voted for similar projects can expect to be represented in the outcome

to an extent that is proportional to the group size.

A group of 20% of the voters can

expect to influence 20% of the budget spending.

Equal Shares: The Benefits

Groups with identical votes

Suppose that \(t\) out of the \(n\) voters submitted an identical ballot, that is, they all voted for the exact same set of projects.

Suppose: \(\operatorname{cost}\left(P_1\right)+\cdots+\operatorname{cost}\left(P_k\right) \leqslant \frac{t}{n} \cdot B,\).

In this case, the Method of Equal Shares will select all the projects.

Equal Shares: The Benefits

Groups with identical votes

Suppose that \(t\) out of the \(n\) voters submitted an identical ballot, that is, they all voted for the exact same set of projects.

Suppose: \(\operatorname{cost}\left(P_1\right)+\cdots+\operatorname{cost}\left(P_k\right) > \frac{t}{n} \cdot B,\).

In this case, the Method of Equal Shares will still be reasonable.

Equal Shares: The Benefits

Groups with identical votes

Equal Shares: The Benefits

Groups with identical votes

Theorem. Let \(T=\left\{P_1, P_2, \ldots, P_k\right\}\) be a set of projects, and

suppose \(t\) of the \(n\) voters voted for all of these projects

(and only these projects).

Then the Method of Equal Shares will select a subset \(T^{\prime} \subseteq T\) of projects such that for every project \(P^* \in T\) that was not selected by the Method, we have:

\(\sum_{P \in T^{\prime}} \operatorname{cost}(P)+\operatorname{cost}\left(P^*\right)>t / n \cdot B\).

Equal Shares: The Benefits

Groups of voters with overlapping votes

In practice, groups of voters with strong agreement

will not always vote for *exactly* the same set of projects.

In general, if \(T=\) \(\left\{P_1, P_2, \ldots, P_k\right\}\) is a set of projects, we write

\(\operatorname{cost}(T)=\operatorname{cost}\left(P_1\right)+\operatorname{cost}\left(P_2\right)+\) \(\ldots+\operatorname{cost}\left(P_k\right)\)

for the total cost of the projects in \(T\).

Equal Shares: The Benefits

Groups of voters with overlapping votes

In practice, groups of voters with strong agreement

will not always vote for *exactly* the same set of projects.

Suppose that a group of \(t\) voters all approve the projects in the set \(T\),

but some of the voters also approve some other projects not in \(T\).

Assume that \(\operatorname{cost}(T) \leqslant t / n \cdot B\).

Then, the Method of Equal Shares guarantees that at least one of the \(t\) voters will be sufficiently represented in the outcome \(W\).

Equal Shares: The Benefits

Groups of voters with overlapping votes

In practice, groups of voters with strong agreement

will not always vote for *exactly* the same set of projects.

Suppose that a group of \(t\) voters all approve the projects in the set \(T\),

but some of the voters also approve some other projects not in \(T\).

Assume that \(\operatorname{cost}(T) \leqslant t / n \cdot B\).

In other words, there exists some member \(i\) of the group with:

\(\operatorname{cost}\left(A_i \cap W\right) \geqslant \operatorname{cost}(T){\color{white}-\operatorname{cost}\left(P_j\right)} \text{ for some } P_j \in T\)

Equal Shares: The Benefits

Groups of voters with overlapping votes

will not always vote for *exactly* the same set of projects.

Suppose that a group of \(t\) voters all approve the projects in the set \(T\),

but some of the voters also approve some other projects not in \(T\).

Assume that \(\operatorname{cost}(T) \leqslant t / n \cdot B\).

In other words, there exists some member \(i\) of the group with:

\(\operatorname{cost}\left(A_i \cap W\right) \geqslant \operatorname{cost}(T){\color{IndianRed}-\operatorname{cost}\left(P_j\right)} \text{ for some } P_j \in T\)

Good-to-know Axioms

Core · FJR · EJR

Good-to-know Axioms

Core · FJR · EJR

Suppose that \(S\) can come up with a set \(T\) of projects

such that \(T\) can be funded with a \(|S|/|N|\) fraction of the budget.

Suppose further that each voter in \(S\) approves all the projects in \(T\); this means the group is cohesive.

EJR demands that the voting rule must select a set \(W\) such that at least one voter in \(S\) approves at least \(|T|\) of the funded projects in \(W\).

Good-to-know Axioms

Core · FJR · EJR

We say that a set \(S \subseteq N\) of voters **blocks** an outcome \(W\)

if there is a set \(T\) of projects affordable with a \(\frac{|S|}{|N|}\) fraction of the budget such that each member of \(S\) *strictly prefers* \(T\) to \(W\).

An outcome \(W\) is in the core if it is not blocked by any coalition \(S\).

EJR is a special case of the core where

the set \(T\) needs to be unanimously liked by the members of \(S\),

so that \(S\) is a “cohesive” group.

Good-to-know Axioms

Core · FJR · EJR

We say that a set \(S \subseteq N\) of voters **blocks** an outcome \(W\)

if there is a set \(T\) of projects affordable with a \(\frac{|S|}{|N|}\) fraction of the budget such that each member of \(S\) *strictly prefers* \(T\) to \(W\).

An outcome \(W\) is in the core if it is not blocked by any coalition \(S\).

For the approval-based case, it is unknown whether

there always exists an outcome in the core

(even under the unit cost assumption).

*For general additive utilities, we already know that the core might be empty.*

Good-to-know Axioms

Core · FJR · EJR

FJR strengthens EJR by weakening the cohesiveness requirement.

FJR requires that if a group \(S \subseteq N\) of voters can propose a set \(T\)

of projects that is affordable with \(S\)'s share of the budget,

and each voter has utility at least \(\ell\) for the set \(T\),

then at least one voter in \(S\) has utility at least \(\ell\) in the chosen outcome \(W\).

In the approval case, we see that FJR

does not insist that \(T\) is unanimously approved by the group \(S\) (like in EJR),

but it just requires that \(T\) is very popular among \(S\).

Good-to-know Axioms

Core · FJR · EJR

Both PAV and Equal Shares fail FJR.

That there does indeed exist a rule satisfying FJR,

which works for arbitrary costs.

*(Not polynomial time, though.)*

Equal Shares fails EJR, as does any efficient mechanism.

Equal Shares does satisfy EJR *up to one project* 😀

Equal Shares: The Benefits

Other Nice Things

Polynomial time computability

Independence of clones

*(if a losing project is cloned; no clone wins,
if a winning project is cloned; at least one of the clones win.)*

Voter Monotonicity

*(more approvals on a winning project don't hurt the project)*

Discount Monotonicity

*(outcomes unchanged if a winning project becomes cheaper
or a losing project becomes more expensive)*

Equal Shares: The Benefits

Other Not-So-Nice Things

Pareto-optimality

Budget limit monotonicity

*(a winning project should stay winning if the available budget is increased)*

Strategyproofness

*(a voter should not be able to improve the outcome by changing their ballot)*

Equal Shares: The Story

Equal Shares: The Story

Equal Shares: The Open Problem

The 74th constitutional amendment of India in 1992 brought with itself greater responsibility for the local government.

It defined 18 new tasks in the functional domain of urban local government such as slum improvement, urban planning and emphasized

greater citizen participation in local decision-making.

Equal Shares: Other Directions

A nice algorithm [sic] that satisfies FJR?

Axiomatic extensions to account for

scenarios involving groups and interactions?

Are structural domain restrictions useful,

as they turn out to be for multiwinner voting?

Does the core exist?

Fix the not-so-nice list for Equal Shares, especially PO?

#### Equal Shares

By Neeldhara Misra

# Equal Shares

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