Envy Free and Efficient Allocations

for Graphical Valuations

Neeldhara Misra and Aditi Sethia

The 8th International Conference on *Algorithmic Decision Theory*

Wed, 16th October 2024

Envy Free and Efficient Allocations

for Graphical Valuations

*Outline*

I. Definitions & Setup

II. Motivation and Known Results

III. Our Contributions

IV. Future Directions

Envy Free and Efficient Allocations

for Graphical Valuations

*Outline*

*I. Definitions & Setup*

II. Motivation and Known Results

III. Our Contributions

IV. Future Directions

Envy Free and Efficient Allocations

for Graphical Valuations

*agents*

*objects*

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

*agents*

*objects*

*I. Definitions & Setup*

utility

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

🤩

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

👀

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

😍

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

Valuations are *additive* if

\(u_k(S) = \sum_{e \in S} u_k(e)\)

Valuations are *binary* if

\(u_k(e) \in \{0,1\}~~\forall e\)

Valuations are *identical* if

\(u_k(e) = u_e~~\forall k\)

Valuations are *monotone* if

\(u_k(T) \leqslant u_k(S),\) for any \(T \subseteq S\)

Envy Free and Efficient Allocations

for Graphical Valuations

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*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

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*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

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*I. Definitions & Setup*

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

Given an allocation \(X\),

if...

\(u(~~~~~) < u(~~~~~)\)

*envies*

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

Given an allocation \(X\),

if...

\(u(~~~~~) < u(~~~~~)\)

*strongly
envies*

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

Given an allocation or partial allocation \(X_1, \ldots, X_n\),

agent \(\textcolor{IndianRed}{i}\) is said to *envy* agent \(\textcolor{SeaGreen}{j}\) if

\[ \textcolor{IndianRed}{u_i}\left(\textcolor{IndianRed}{X_i}\right) < \textcolor{IndianRed}{u_i}\left(\textcolor{SeaGreen}{X_j}\right), \]

and agent \(\textcolor{IndianRed}{i}\) is said to *strongly envy* agent \(\textcolor{SeaGreen}{j}\) if

for some \(\textcolor{DodgerBlue}{x} \in \textcolor{SeaGreen}{X_j}\),

\[ \textcolor{IndianRed}{u_i}\left(\textcolor{IndianRed}{X_i}\right) < \textcolor{IndianRed}{u_i}\left(\textcolor{SeaGreen}{X_j} \setminus \{\textcolor{DodgerBlue}{x}\}\right). \]

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

An EFX allocation is an allocation \(X_1, \ldots, X_n\)

so that no agent strongly envies another.

For all \(i,j\), \(u_i\left(X_i\right) \geq u_i\left(X_j \backslash\{x\}\right), \forall x \in X_j\)

An EF allocation is an allocation \(X_1, \ldots, X_n\)

so that no agent envies another:

For all \(i,j\), \(u_i\left(X_i\right) \geq u_i\left(X_j\right)\)

**Envy-Freeness (EF).**

**Envy-Freeness up to any item (EFX).**

Envy Free and Efficient Allocations

for Graphical Valuations

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*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

if you see a matrix where every column has two ones...

...every row is a vertex and every column is an edge!

*I. Definitions & Setup*

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*recall the valuation matrix in our example...*

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

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Envy Free and Efficient Allocations

for Graphical Valuations

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

Envy Free and Efficient Allocations

for Graphical Valuations

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

\( u_{\textcolor{SeaGreen}{k}}(X \cup \{\textcolor{IndianRed}{e}\}) = u_{\textcolor{SeaGreen}{k}}(X), \forall X \subseteq E \)

An object \({\color{IndianRed}e}\) is *irrelevant *to an agent \({\color{SeaGreen}k}\) if:

Envy Free and Efficient Allocations

for Graphical Valuations

If an item is not irrelevant for agent \(k\) is said to be relevant for agent \(k\).

*Graphical Valuations.*

Every item is relevant to at most 2 agents and

for every pair of agents there exists at most one item relevant to both of them.

Envy Free and Efficient Allocations

for Graphical Valuations

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

\( u_{\textcolor{SeaGreen}{k}}(X \cup \{\textcolor{IndianRed}{e}\}) = u_{\textcolor{SeaGreen}{k}}(X), \forall X \subseteq E \)

An object \({\color{IndianRed}e}\) is *irrelevant *to an agent \({\color{SeaGreen}k}\) if:

Envy Free and Efficient Allocations

for Graphical Valuations

This class of valuations can be represented via a graph \(G=(V, E)\)

where the vertices represent the agents, i.e., \(V=\{1, \ldots, n\}\),

and the set of edges \(E\) is the set of items, where the item represented by the edge \((i, j)\) is only relevant to both agents \(i\) and \(j\).

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

Envy Free and Efficient Allocations

for Graphical Valuations

*I. Definitions & Setup*

*are orientations*

*some*

Envy Free and Efficient allocations

for Graphical Valuations

Envy Free and Efficient Allocations

for Graphical Valuations

*Outline*

I. Definitions & Setup

*II. Motivation and Known Results*

III. Our Contributions

IV. Future Directions

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

Example showing non-existence of EFX orientations

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

It is an NP-complete problem to decide whether a graph has an EFX orientation.

This remains true even for symmetric graphs.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

It is an NP-complete problem to decide whether a graph has an EFX orientation.

This remains true even for symmetric graphs.

*...is there a characterization of graphs that exhibit EFX orientation regardless of the assigned valuations?*

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

A graph is **strongly EFX-orientable** if,

for *any* assigned *monotone* valuation,

there exists an EFX orientation.

A graph \(G\) is **0-1 strongly EFX-orientable** if,

for *any* assigned *additive* 0-1 valuation,

there exists an EFX-orientation.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

Any graph of chromatic number \(\chi(G) \leqslant 2\)

is strongly EFX-orientable.

A graph is **strongly EFX-orientable** if,

for *any* assigned *monotone* valuation,

there exists an EFX orientation.

All strongly EFX-orientable graphs have chromatic number \(\chi(G) \leqslant 3\).

A graph is **strongly EFX-orientable** if,

for *any* assigned *monotone* valuation,

there exists an EFX orientation.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

*There exist graphs of \(\chi(G)=3\) which are not strongly EFX-orientable,

as well as graphs with \(\chi(G)=3\) which are strongly EFX-orientable,

so this bound is sharp.

**strongly EFX-orientable** if,

for *any* assigned *monotone* valuation,

there exists an EFX orientation.

**strongly EFX-orientable** if,

for *any* assigned *monotone* valuation,

there exists an EFX orientation.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

A graph \(G\) is \(0-1\) strongly EFX-orientable if and only if, for every subgraph \(H \subseteq G\) such that \(H\) is a forest consisting of trees \(T_1, T_2 \ldots T_k\),

for every \(1 \leqslant i \leqslant k\) there exists \(x_i \in T_i\)

such that \(\bigcup_{i=1}^k N_H\left(x_i\right)\) forms an independent set on \(G\),

where \(N_H(x)\) denotes the neighbors of \(x\) in \(H\).

A graph \(G\) is **0-1 strongly EFX-orientable** if,

for *any* assigned *additive* 0-1 valuation,

there exists an EFX-orientation.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

Any bipartite graph is strongly EFX-orientable.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

Any bipartite graph is strongly EFX-orientable.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

Preprint · 2024

*On the Structure of EFX Orientations on Graphs*

Any bipartite graph is strongly EFX-orientable.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

There is always an EFX allocation for graphs.

In particular this implies that there are cases

where any EFX allocation must assign items to agents for which they are irrelevant.

Envy Free and Efficient Allocations for Graphical Valuations

*II. Motivation and Known Results*

EC · 2023

*EFX Allocations on Graphs*

There is always an EFX allocation for graphs.

In particular this implies that there are cases

where any EFX allocation must assign items to agents for which they are irrelevant.

*...what about EF?*

*...what about EFX allocations that minimize waste?*

Envy Free and Efficient Allocations

for Graphical Valuations

*Outline*

I. Definitions & Setup

II. Motivation and Known Results

*III. Our Contributions*

IV. Future Directions

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

Envy-free allocations

on graphical valuations.

EFX allocations that maximize welfare

on graphical valuations.

*Two themes.*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

Given a graphical allocation instance,

there is an EF allocation *if and only if *there is an EF orientation.

For \(\{0,1\}\)-graphical instances,

an EF allocation can be found efficiently, if it exists.

Deciding whether an EF allocation exists is NP-hard,

even for symmetric \(\{0,1, d\}\)-graphical valuations.

Given a graphical allocation instance

with a bounded number of distinct utilities,

the problem of finding an EF allocation is FPT

in the vertex cover number of the associated graph \(G\).

*envy-freeness*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

Given a graphical allocation instance,

there is an EF allocation *if and only if *there is an EF orientation.

*envy-freeness*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

Given a graphical allocation instance,

there is an EF allocation *if and only if *there is an EF orientation.

*envy-freeness*

If

envies

in the new allocation,

then

envied

in the old allocation.

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*envy-freeness*

For \(\{0,1\}\)-graphical instances,

an EF allocation can be found efficiently, if it exists.

When edges are asymmetric

(valued at 0 by one agent and 1 by the other),

orient them towards the agent who values the edge more.

*By the previous result, it is enough to find an orientation.*

When edges are symmetric 0-0,

(neither agent cares for the item in question),

orient them arbitrarily.

Left with components with 1-1 edges only: cases based on trees/non-trees.

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*envy-freeness*

Deciding whether an EF allocation exists is NP-hard,

even for symmetric \(\{0,1, d\}\)-graphical valuations.

*By a reduction from the multi-colored independent set problem.*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*envy-freeness*

Deciding whether an EF allocation exists is NP-hard,

even for symmetric \(\{0,1, d\}\)-graphical valuations.

*By a reduction from the multi-colored independent set problem.*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*envy-freeness*

even for symmetric \(\{0,1, d\}\)-graphical valuations.

*By a reduction from the multi-colored independent set problem.*

selector gadgets

*one of these*

heavy *edges is oriented towards the global vertex in any EF orientation.*

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*envy-freeness*

*by a reduction to ILP with a small number of variables...*

Given a graphical allocation instance

with a bounded number of distinct utilities,

the problem of finding an EF allocation is FPT

in the vertex cover number of the associated graph \(G\).

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

For \(\{0,1\}\)-graphical instances,

a non-wasteful EFX allocation always exists and can be found in polynomial time.

Therefore, the Price of EFX with respect to Utilitarian welfare is 1.

The price of EFX with respect to Utilitarian welfare is \(\infty\)

even for \(\{0,1, d\}\)-graphical valuations.

*EFX*

Given an instance of graphical valuations,

deciding the existence of a utilitarian welfare-maximizing *and *EFX allocation (UM+EFX) is NP-Hard.

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

For \(\{0,1\}\)-graphical instances,

a non-wasteful EFX allocation always exists and can be found in polynomial time.

Therefore, the Price of EFX with respect to Utilitarian welfare is 1.

*EFX*

A graph \(G\) is \(0-1\) strongly EFX-orientable if and only if, for every subgraph \(H \subseteq G\) such that \(H\) is a forest consisting of trees \(T_1, T_2 \ldots T_k\),

for every \(1 \leqslant i \leqslant k\) there exists \(x_i \in T_i\)

such that \(\bigcup_{i=1}^k N_H\left(x_i\right)\) forms an independent set on \(G\),

where \(N_H(x)\) denotes the neighbors of \(x\) in \(H\).

*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

The price of EFX with respect to Utilitarian welfare is \(\infty\)

even for \(\{0,1, d\}\)-graphical valuations.

*EFX*

welfare-optimal: root takes all for \(d^2\).

EFX: root cannot get more than one item; \(\leqslant d\).

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*III. Our Contributions*

Envy Free and Efficient Allocations for Graphical Valuations

*EFX*

Given an instance of graphical valuations,

deciding the existence of a utilitarian welfare-maximizing *and *EFX allocation (UM+EFX) is NP-Hard.