# Envy and Fairness#

Consider cake cutting: Alice and Bob want to split a cake, and the cake has different toppings (strawberries, chocolate, etc.) over different regions. One canonical mechanism is I-cut-you-choose:

One agent cuts the cake;

The other picks which slice they want;

First agent gets the remaining cake.

I-cut-you-choose is not dominant-strategy incentive compatible for the cutting agent but is for the choosing agent.

Proof. The cutter’s optimal strategy depends on the chooser’s preferences. The chooser directly chooses their outcome.

Why, has this mechanism been used? There is some notion of fairness in this:

(Envy-Free)

An allocation is envy-free if no agent envies another agent’s allocation.

The I-cut-you-choose mechanism can always lead to an envy-free allocation for both players.

Proof. The cutter can always cut the cake into two pieces that are equal in their perspective, so no matter which slice they get the allocation is envy-free for them. Then, the choose chooses their favorite slice, so it clearly is envy-free for them.

## Formalizing Fair Allocations#

Goal: from a set of items \(A\), choose some allocation for each of \(n\) agents. Each agent \(i\) has a valuation function \(v_i: \mathcal{P}(A) \to \mathbb{R}\). Then, there are several measures of fairness (defined in this setting):

Envy-Free: for all \(i,j\) we have \(v_i(A_i) \geq v_i(A_j)\);

Proportional: for all \(i, v_i(A_i) \geq v_i(A)/n\);

Equitable: for all \(i,j\) we have \(v_i(A_i) = v_j(A_j)\);

Perfect: for all \(i, v_i(A_i) = v_i(A)/n\).

Another possibility: competitive equilibrium from equal incomes, which is similar to competitive equilibrium except give everyone a “fake dollar” to spend.

In cases with indivisible goods, often none of them can be satisfied. In particular, if there is a single good (and both individuals gain strictly positive utility from it), none of these can be satisfied. A compromise:

Another compromise can be to approximately satisfy these conditions: let \(\overline{a}\) be the best item and \(\underline{a}\) be the worst item. Then,

Envy free up to one item: \(v_i(A_i) \geq v_i(A_j \setminus \overline{a})\);

Can always be satisfied.

Envy free up to any item: \(v_i(A_i) \geq v_i(A_j \setminus \underline{a})\)

Open problem for whether or not this can always be satisfied.

\(\alpha\)-Maximin Share: \(v_i(A_i) \geq \alpha \max_{A^i} \min_j v_i(A_j^i)\)

Can always be satisfied for \(\alpha \leq 3/4\).

Approximate competitive equilibrium from equal incomes: only require approximate equilibrium.

Can always be satisfied but computationally difficult to compute

Finally, we can directly seek to maximize some “fair” measure of overall utility. Some examples are:

Utilitarian/Social Welfare: \(\max_{A_1,...,A_n} \sum_i v_i(A_i)\);

Nash social welfare: \(\max_{A_1,...,A_n} \prod_i v_i(A_i)\);

Middle ground between Utilitarian and Egalitarian.

Competitive equilibrium from equal incomes maximizes Nash social welfare;

Egalitarian/maximin: \(\max_{A_1,...,A_n} \min_i v_i(A_i)\).

## Dominant Resource Fairness#

Suppose there are \(m\) resources with fixed capacities and \(n\) users that want to run as many identical tasks as possible. Each task demands some vector of resources to run. For now, suppose resources are infinitesimally small. How can resources be allocated fairly and efficiently without transfers?

For any allocation, each user has some fraction of total capacity of each resource (given by point-wise division of the user’s usage vector by the capacity vector). Then, a user’s dominant resource is the resource they have the highest fraction of usage of. Dominant resource fairness requires equal shares of dominant resources.

Algorithm to get DRF Allocation

Every agent submits their ratio of demands;

Algorithm computes the maximal allocation subject to DRF.

Every user prefers the DRF allocation to just getting \(1/n\) of every resource.

Proof. As the allocation is maximal, some resource \(j^*\) is exhausted so some agent \(i^*\) gets at least \(1/n\) of \(j^*\). Then, each agent has at least \(1/n\) of their dominant resource.

The DRF mechanism is envy-free.

Proof. If \(i\) envies \(j\), then \(j\) must have more of every resource so in particular, \(j\) must have more of \(i\)’s dominant resource than \(i\) does, a contradiction.

The DRF mechanism is Pareto-optimal and strategyproof.

Proof. Pareto-Optimal: As the allocation is maximal, some resource ie exhausted and no agent can be made happier without making some other agent sadder.

Strategyproof: for any misreport, Pareto-optimality of the allocation reached by truthful reporting means that less of the dominant resource is received.

## DRF in Practice#

Some of the assumptions made during the setup might break:

Tasks are not identical, arrive over time, and depart;

While resources are not exhausted, select user \(i\) with minimal dominant share and add their next feasible task. Not strategyproof (can clump tasks together) but works well in practice.

Tasks might have flexibility in resource use;

Tasks might have more specific criteria;

Tasks not infinitesimally small;

Users might have different priority levels;

Weighted DRF.

Some tasks have zero demand for some resources.

Find the maximal DRF allocation and then recurse on remaining feasible tasks.