Math models are useful in the social sciences

Social sciences motivate
new math

Takeaways...

Source

Cake Cutting

Hesiod's Theogony:
Prometheus cuts ox in two, offers Zeus choice

Old Testament Book of Genesis:
Abraham offers Lot choice: land to left or right

Mining in the Ocean:
The United Nations Convention on the Law of the Sea applies a procedure similar to divide-and-choose for allocating areas in the ocean among countries.

Cake Cutting

Jack Sprat could eat no fat
His wife could eat no lean
So 'twixt them both they cleared the cloth,
And licked the platter clean

Mother Goose Nursery Rhyme

Cake Cutting

Outline

Outline

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow \mathbb{Z} $$

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

0/1 Valuations

$$ \{0,1\} $$

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Additive Valuations

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

For all subsets \( S \subseteq R \), \( \nu_\ell(S) = \sum_{o \in S} \nu_\ell(\{o\}) \).

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

If 🙋‍♀️ values the items 🍰 and 💐 at 1 each, then she values {🍰, 💐} at 2.

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😀 → {🍰,🍵}

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

👀 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😬 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😠 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

(between friends)

Task:

$$ \pi: A \rightarrow 2^R $$

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 → {🍰,🍵}

😍 → {🍔,🍟,☕️}

 Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

(between friends)

and a social network on \( A \)

Outline

\(f_j := (u_1(r_j),u_2(r_j),...,u_n(r_j)).\)

Define the signature of a resource

to be a \(n\)-tuple that collects all the utilities that agents have for said resource.

#signatures ~ \(z^n\)

\(f  = (u_1(   ),u_2(   ), \ldots\)

\(f  = (u_1(   ),u_2(   ), \ldots\)

Objects with the same signature are indistinguishable.

Parameters for Resource Allocation

Recap: the parameterized paradigm

Parameterized complexity: acknowledge the presence of additional structure, which manifests as a secondary measurement a parameter.

Design algorithms that restrict the combinatorial explosion
to a function of the parameter.

Undirected Graphs

Goal: Determine the existence of a
complete and locally envy-free allocation.

P

even when valuations are binary and identical.*

*for connected graphs

Undirected Graphs

Goal: Determine the existence of a
complete and locally envy-free allocation.

NP-hard

even when valuations are binary and nearly identical.

Directed Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

P

if the social network is a DAG

Undirected Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

NP-hard

even if the social network is a path

Undirected Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

NP-hard

even if* the social network is a star

*general utilities

Undirected Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

W[1]-hard

parameterized by the vertex cover number for 0/1 utilities

Complete Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

FPT
parameterized by #goods or #agents

Undirected Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

FPT
parameterized by #goods or #agents

Undirected Graphs

Goal: Determine the existence of a
efficient and locally envy-free allocation.

W[1]-hard
parameterized by #goods or #agents

Takeaway

undirected graphs

v/s

directed graphs

some contrasts, some similarities...

Outline

Reductions: a framework for showing hardness

Historically

annoying

problem

Our

annoying

problem

\(\leq\)

Recipe for solving

well-known annoying problem (e.g, SAT)

via any hypothetical algorithm (A) for solving problem X:

Cutting \( \ell \) vertices

Given a graph \( G \), partition its vertex set into three parts: \( (X \cup S \cup Y) \) so that:

a) \( |S| \leq k \)

b) \( |X| = \ell \)

c) There are no edges between \( X \) and \( Y \).

\( \Downarrow \)

\( G \)

\( S \)

\( Y \)

\( X \)

n happy agents

n greedy agents

one trigger agent

n happy agents

n greedy agents

one trigger agent

n happy agents

n greedy agents

one trigger agent

...based on the structure of the graph \( G \)

n happy agents

n greedy agents

one trigger agent

...based on the structure of the graph \( G \)

n happy agents

n greedy agents

one trigger agent

n happy agents

n greedy agents

one trigger agent

n happy agents

n greedy agents

one trigger agent

😡

n happy agents

n greedy agents

one trigger agent

😡

n happy agents

n greedy agents

one trigger agent

😎

n happy agents

n greedy agents

one trigger agent

😎

😕

😕

😕

😕

😕

😕

n happy agents

n greedy agents

one trigger agent

😎

😕

😕

😕

😕

😕

😕

n happy agents

n greedy agents

one trigger agent

😎

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Are any of the happy agents jealous?

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

They don't envy the 😀-agents, since they don't value the     good.

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Some of them will envy the agents who got the coveted goods     .

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Some of them will envy the agents who got the coveted goods     .

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Agents in \( X \cup S \) will envy the agents who got the coveted goods     .

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Agents in \( X \cup S \) will envy the agents who got the coveted goods     .

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

Agents in \( X \cup S \) will envy the agents who got the coveted goods     .

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

n happy agents

n greedy agents

one trigger agent

😎

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

😀

Takeaway

undirected graphs

v/s

directed graphs

some contrasts, some similarities...

Outline

future work