Convergence and Hardness of Strategic Schelling Segregation WINE - - PowerPoint PPT Presentation

• H. Echzell, T. Friedrich, P

. Lenzner, L. Molitor, M. Pappik, F . Schöne, F . Sommer, D. Stangl

Convergence and Hardness

• f Strategic Schelling Segregation

WINE Conference 2019

Algorithm Engineering Research Group

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

agents

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

neighborhood agents

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

neighborhood agents "I am happy if at least a fraction τ of my neighborhood is of my type."

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

neighborhood agents "I am happy if at least a fraction τ of my neighborhood is of my type." e.g. τ = 1

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

neighborhood agents "I am happy if at least a fraction τ of my neighborhood is of my type." e.g. τ = 1

4

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

neighborhood agents "I am happy if at least a fraction τ of my neighborhood is of my type." e.g. τ = 1

4

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Convergence and Hardness of Strategic Schelling Segregation

Schelling Segregation

Thomas Schelling (1921-2016) "economics Nobel prize" winner Micromotives and Macrobehavior (1978)

https://www.bostonglobe.com/

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Convergence and Hardness of Strategic Schelling Segregation

Theoretical Approaches

Young et al. (2001) Brandt et al. (STOC 2012) Bhakta et al. (SODA 2014) Barmpalias et al. (FOCS 2014) Immorlica et al. (SODA 2017) Omidvar et al. (PODC 2017) Stochastic Models many more...

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Convergence and Hardness of Strategic Schelling Segregation

Theoretical Approaches

Young et al. (2001) Brandt et al. (STOC 2012) Bhakta et al. (SODA 2014) Chauhan et al. (SAGT 2018) Elkind et al. (IJCAI 2019) Brederek et al. (AAMAS 2019) Agarwal et al. (AAAI 2020) Barmpalias et al. (FOCS 2014) Immorlica et al. (SODA 2017) Omidvar et al. (PODC 2017) Stochastic Models many more... Game Theoretic Models

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Convergence and Hardness of Strategic Schelling Segregation

Theoretical Approaches

Young et al. (2001) Brandt et al. (STOC 2012) Bhakta et al. (SODA 2014) Barmpalias et al. (FOCS 2014) Immorlica et al. (SODA 2017) Omidvar et al. (PODC 2017) Stochastic Models many more... Chauhan et al. (SAGT 2018) Elkind et al. (IJCAI 2019) Brederek et al. (AAMAS 2019) Agarwal et al. (AAAI 2020) Game Theoretic Models

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

undirected (simple) graph G = (V, E)

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

undirected (simple) graph G = (V, E) set of agents A with partitioning P(A)

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

undirected (simple) graph G = (V, E) set of agents A with partitioning P(A) placement pG : A → V (injective) neighborhood NpG(a) := adjacent agents

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

undirected (simple) graph G = (V, E) set of agents A with partitioning P(A) placement pG : A → V (injective) neighborhood NpG(a) := adjacent agents intolerance threshold τ ∈ [0, 1]

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

always: N+

pG(a) := neighbors with same type as a

costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else a N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else N+

pG(a) =

N−

pG(a) =

a N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

1 vs. 1 Schelling Game (1-1-SG) costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else N+

pG(a) =

N−

pG(a) =

a a N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

1 vs. 1 Schelling Game (1-1-SG) costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else N−

pG(a) =

N+

pG(a) =

N+

pG(a) =

N−

pG(a) =

a a N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

1 vs. 1 Schelling Game (1-1-SG) costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else costpG(a) = 1

12

costpG(a) = 0 N−

pG(a) =

N+

pG(a) =

N+

pG(a) =

N−

pG(a) =

a a τ = 1

3

N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

4

Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

1 vs. all Schelling Game (1-k-SG) always: N+

pG(a) := neighbors with same type as a

1 vs. 1 Schelling Game (1-1-SG) costpG(a)    max(0, τ −

|N+

pG(a)|

|N+

pG(a)|+|N− pG(a)|) if NpG(a) = ∅

τ else costpG(a) = 1

12

costpG(a) = 0 N−

pG(a) =

N+

pG(a) =

N+

pG(a) =

N−

pG(a) =

a a τ = 1

3

N+

pG(a), N− pG(a) ⊆ NpG(a)

cost pnr

τ τ

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

What do discontent agents do?

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

What do discontent agents do? Jump Schelling Game (JSG): "jump to empty node to decrease costs" if costpG(a) > costp′

G(a)

pG → p′

G

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

What do discontent agents do? Jump Schelling Game (JSG): "jump to empty node to decrease costs" Swap Schelling Game (SSG): "swap position to decrease costs" if costpG(a) > costp′

G(a)

if costpG(a) > costp′

G(a) and costpG(b) > costp′ G(b)

pG → p′

G

pG → p′

G

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Convergence and Hardness of Strategic Schelling Segregation

Strategic Schelling Segregation

What do discontent agents do? Jump Schelling Game (JSG): "jump to empty node to decrease costs" Swap Schelling Game (SSG): "swap position to decrease costs" 1-k-JSG 1-1-JSG 1-k-SSG 1-1-SSG if costpG(a) > costp′

G(a)

if costpG(a) > costp′

G(a) and costpG(b) > costp′ G(b)

pG → p′

G

pG → p′

G

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

What do we mean by "convergence"?

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

What do we mean by "convergence"? swap-/jump-stable: pG such that no other placement p′

G can be reached via swap/jump

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

What do we mean by "convergence"? swap-/jump-stable: pG such that no other placement p′

G can be reached via swap/jump

improving response cycle (IRC): sequence of placements p1

G, ..., pk G

such that pi

G can be reached via swap/jump from pi−1 G

pk

G = p1 G (upto type similarity)

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

What do we mean by "convergence"? swap-/jump-stable: pG such that no other placement p′

G can be reached via swap/jump

improving response cycle (IRC): sequence of placements p1

G, ..., pk G

such that pi

G can be reached via swap/jump from pi−1 G

pk

G = p1 G (upto type similarity)

not weakly acyclic: there is an unstable placement pG from which no stable placement p′

G

can be reached

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

1 − k−SSG 1 − 1−SSG 1 − k−JSG 1 − 1−JSG ∆−regular arbitrary |P(A)| = 2 |P(A)| = 2, ∆ = 2 |P(A)| = 2, τ ≤ 1

2

Previous results by Chauhan et al (SAGT 2018):

guaranteed convergence 7

Convergence and Hardness of Strategic Schelling Segregation

Convergence

1 − k−SSG 1 − 1−SSG 1 − k−JSG 1 − 1−JSG |P(A)| = 2 |P(A)| = 2, ∆ = 2

• τ ≤ 1

∆

∆−regular

guaranteed convergence

• improving response cycle

× not weakly acyclic

• τ > 6

∆

τ ≤ 2

∆

τ ≤ 1

∆

• τ > 2

∆

• τ > 2

∆

|P(A)| = 2, τ ≤ 1

2

× else × else × ×

arbitrary Our results:

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Convergence and Hardness of Strategic Schelling Segregation

Convergence

1 − k−SSG 1 − 1−SSG 1 − k−JSG 1 − 1−JSG |P(A)| = 2 |P(A)| = 2, ∆ = 2

• τ ≤ 1

∆

∆−regular

guaranteed convergence

• improving response cycle

× not weakly acyclic

• τ > 6

∆

τ ≤ 1

∆

|P(A)| = 2, τ ≤ 1

2

× else × else × ×

arbitrary Our results: τ ≤ 2

∆

• τ > 2

∆

• τ > 2

∆

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

costpG(a) = 5

6 − 2 3 = 1 6

costp′

G(a) = 0

c b a IRC for ∆ = 3, τ > 2

3 (e.g. τ = 5 6):

Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

costpG(b) = 5

6

costp′

G(b) = 5

6 − 1 2 = 1 3

IRC for ∆ = 3, τ > 2

3 (e.g. τ = 5 6):

a c b Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

costpG(c) = 5

6 − 2 3 = 1 6

costp′

G(c) = 0

IRC for ∆ = 3, τ > 2

3 (e.g. τ = 5 6):

a b c Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

costpG(a) = 5

6

costp′

G(a) = 5

6 − 1 2 = 1 3

a b c IRC for ∆ = 3, τ > 2

3 (e.g. τ = 5 6):

Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump IRC

IRC for τ > 2

∆:

almost for free: not weakly acyclic on arbitrary graphs ∆ − 2 . ∆ − 2 . . ∆ − 2 . ∆ − 2 Theorem Neither 1 − k−JSG nor 1 − 1−JSG are guaranteed to converge for any τ > 2

∆ on ∆-regular graphs.

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Convergence and Hardness of Strategic Schelling Segregation

Jump Potential

Theorem The 1 − k−JSG is guaranteed to converge in O(|E|) for any τ ≤ 2

∆ on

every ∆-regular graph.

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Convergence and Hardness of Strategic Schelling Segregation

Jump Potential

Proof (sketch): again search a potential function Φ Φ(pG) =

(u,v)∈E wpG(u, v) 1 2 − 1 2∆ < c < 1 2

wpG(u, v) =      1 if c if 0 ow. Theorem The 1 − k−JSG is guaranteed to converge in O(|E|) for any τ ≤ 2

∆ on

every ∆-regular graph.

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Convergence and Hardness of Strategic Schelling Segregation

Jump Potential

Proof (sketch): again search a potential function Φ Φ(pG) =

(u,v)∈E wpG(u, v) 1 2 − 1 2∆ < c < 1 2

let pG → p′

G by jump of a ∈ A

jump: costpG(a) > costp′

G(a)

wpG(u, v) =      1 if c if 0 ow. Theorem The 1 − k−JSG is guaranteed to converge in O(|E|) for any τ ≤ 2

∆ on

every ∆-regular graph.

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Convergence and Hardness of Strategic Schelling Segregation

Jump Potential

Proof (sketch): again search a potential function Φ Φ(pG) =

(u,v)∈E wpG(u, v) 1 2 − 1 2∆ < c < 1 2

let pG → p′

G by jump of a ∈ A

jump: costpG(a) > costp′

G(a)

• bservation: |N+

pG(a)| ≥ 2 or |N+ p′

G(a)| = 0 never happen

wpG(u, v) =      1 if c if 0 ow. Theorem The 1 − k−JSG is guaranteed to converge in O(|E|) for any τ ≤ 2

∆ on

every ∆-regular graph.

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Convergence and Hardness of Strategic Schelling Segregation

Jump Potential

Proof (sketch): again search a potential function Φ Φ(pG) =

(u,v)∈E wpG(u, v) 1 2 − 1 2∆ < c < 1 2

let pG → p′

G by jump of a ∈ A

jump: costpG(a) > costp′

G(a)

• bservation: |N+

pG(a)| ≥ 2 or |N+ p′

G(a)| = 0 never happen

2 cases: |N+

pG(a)| < |N+ p′

G(a)| and |N+

pG(a)| = |N+ p′

G(a)| = 1 using regularity

wpG(u, v) =      1 if c if 0 ow. Theorem The 1 − k−JSG is guaranteed to converge in O(|E|) for any τ ≤ 2

∆ on

every ∆-regular graph.

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Convergence and Hardness of Strategic Schelling Segregation

Hardness

Optimal placement Is there a pacement with at least k content agents?

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Convergence and Hardness of Strategic Schelling Segregation

Hardness

Optimal placement Is there a pacement with at least k content agents? Surprise: NP-complete in general (reductions for τ = 1

2 and τ ≈ 1)

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Convergence and Hardness of Strategic Schelling Segregation

Hardness

Optimal placement Is there a pacement with at least k content agents? Surprise: NP-complete in general (reductions for τ = 1

2 and τ ≈ 1)

Proof (sketch): transform it to unary encoded SUBSET SUM Theorem There is an O(|V|2) time algorithm for optimal placements in 1−k−SSG and 1 − 1−SSG on 2-regular graphs for |P(A)| = 2 and τ > 1

2.

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Convergence and Hardness of Strategic Schelling Segregation

Hardness

Optimal placement Is there a pacement with at least k content agents? Surprise: NP-complete in general (reductions for τ = 1

2 and τ ≈ 1)

Proof (sketch): transform it to unary encoded SUBSET SUM Theorem There is an O(|V|2) time algorithm for optimal placements in 1−k−SSG and 1 − 1−SSG on 2-regular graphs for |P(A)| = 2 and τ > 1

2.

Theorem It is NP-complete to decide the optimal placement problem for 1 − k−SSG and 1 − 1−SSG on 2-regular graphs for τ > 1

2 and an arbitrary

number of types. Proof (sketch): reduction from 3-PARTITION

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Convergence and Hardness of Strategic Schelling Segregation

Summary & Future Work

Convergence highly depends on cost model, number of types, swap

• r jump, ...

Hardness of optimal placements, even on simple graphs for an ar- bitrary number of types

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Convergence and Hardness of Strategic Schelling Segregation

Summary & Future Work

Convergence highly depends on cost model, number of types, swap

• r jump, ...

Hardness of optimal placements, even on simple graphs for an ar- bitrary number of types Future work more precisely characterize convergence existence of stable placements (Elkind et al. IJCAI 2019) if it converges, how segregated is the stable placement?

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Convergence and Hardness of Strategic Schelling Segregation

Summary & Future Work

Convergence highly depends on cost model, number of types, swap

• r jump, ...

Hardness of optimal placements, even on simple graphs for an ar- bitrary number of types Future work more precisely characterize convergence existence of stable placements (Elkind et al. IJCAI 2019) if it converges, how segregated is the stable placement? Thank you very much and let’s be happy polygons. https://ncase.me/polygons/

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