CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time......
Transcript of CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time......
![Page 1: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/1.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
CONVERGENCE TO EQUILIBRIAIN PLURALITY VOTING
Reshef Meir Maria PolukarovJeffrey S. Rosenschein Nicholas R. Jennings
Padova, 23 January 2014
![Page 2: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/2.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Once upon a time...
“Life is worthless without love!”- told Snow White the Seven Dwarfs
![Page 3: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/3.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Once upon a time...
...and fell asleep
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CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Once upon a time...
...for a long-long while.
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CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Who will wake Snow White up?
I Prince (P)
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CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Who will wake Snow White up?
I Prince (P)
I Prince Charming (C)
![Page 7: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/7.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Who will wake Snow White up?
I Prince (P)
I Prince Charming (C)
I Little Prince (L)
![Page 8: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/8.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Who will wake Snow White up?
I Prince (P)
I Prince Charming (C)
I Little Prince (L)
![Page 9: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/9.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Who will wake Snow White up?
I Prince (P)
I Prince Charming (C)
I Little Prince (L)
I Batman (B)
![Page 10: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/10.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have to choose
So they vote:
P C L B
1 2 3 4 5 6 7
![Page 11: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/11.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have to choose
So they vote:
P C L B
1 2 3 4 5 6 7
P, C, L are tied (2 points)
![Page 12: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/12.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have to choose
So they vote:
P C L B
1 2 3 4 5 6 7
C is elected (ties broken alphabetically)
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CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have to choose
So they vote:
P C L B
1 2 3 4 5 6 7
C is elected (ties broken lexicographically)
![Page 14: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/14.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
But wait a minute...
The voters have the following preferences regarding the outcome:
1 : P � B � L � C2 : P � B � C � L3 : C � L � P � B4 : C � B � P � L5 : L � . . .6 : L � . . .7 : B � . . .
![Page 15: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/15.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have incentives to strategise
So they may change their mind:
P C L B
2 3 4 5 6 1 7
C, L, B are tied (2 points)
![Page 16: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/16.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have incentives to strategise
So they may change their mind:
P C L B
2 3 4 5 6 1 7
B is elected (lexicographic tie-breaking)
![Page 17: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/17.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
... and change the outcome!
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CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
It’s not yet the end...
The voters have the following preferences regarding the outcome:
1 : P � B � L � C2 : P � B � C � L3 : C � L � P � B4 : C � B � P � L5 : L � . . .6 : L � . . .7 : B � . . .
![Page 19: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/19.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have incentives to strategise
So they may change their mind:
P C L B
2 4 3 5 6 1 7
L is elected (unique winner)
![Page 20: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/20.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
It’s not yet the end...
The voters have the following preferences regarding the outcome:
1 : P � B � L � C2 : P � B � C � L3 : C � L � P � B4 : C � B � P � L5 : L � . . .6 : L � . . .7 : B � . . .
![Page 21: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/21.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
The dwarfs have incentives to strategise
So they may change their mind:
P C L B
4 3 5 6 1 2 7
B is elected (lexicographic tie-breaking)
![Page 22: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/22.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
No more objections!
1 : P � B � L � C2 : P � B � C � L3 : C � L � P � B4 : C � B � P � L5 : L � . . .6 : L � . . .7 : B � . . .
P C L B
4 3 5 6 1 2 7
![Page 23: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/23.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
Happy end!
![Page 24: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/24.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
What are we after?
I Agents have to agree on a joint plan of actionor allocation of resources.
I Their individual preferences over available alternatives may vary,so they vote.
I Agents may have incentives to vote strategically.
I We study the convergence of strategic behaviour to stable decisions,from which no one will wish to deviate—equilibria.
I Agents may have no knowledge about the preferences of the othersand/or no communication.
![Page 25: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/25.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
What are we after?
I Agents have to agree on a joint plan of actionor allocation of resources.
I Their individual preferences over available alternatives may vary,so they vote.
I Agents may have incentives to vote strategically.
I We study the convergence of strategic behaviour to stable decisions,from which no one will wish to deviate—equilibria.
I Agents may have no knowledge about the preferences of the othersand/or no communication.
![Page 26: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/26.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
What are we after?
I Agents have to agree on a joint plan of actionor allocation of resources.
I Their individual preferences over available alternatives may vary,so they vote.
I Agents may have incentives to vote strategically.
I We study the convergence of strategic behaviour to stable decisions,from which no one will wish to deviate—equilibria.
I Agents may have no knowledge about the preferences of the othersand/or no communication.
![Page 27: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/27.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
What are we after?
I Agents have to agree on a joint plan of actionor allocation of resources.
I Their individual preferences over available alternatives may vary,so they vote.
I Agents may have incentives to vote strategically.
I We study the convergence of strategic behaviour to stable decisions,from which no one will wish to deviate—equilibria.
I Agents may have no knowledge about the preferences of the othersand/or no communication.
![Page 28: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/28.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Introduction
What are we after?
I Agents have to agree on a joint plan of actionor allocation of resources.
I Their individual preferences over available alternatives may vary,so they vote.
I Agents may have incentives to vote strategically.
I We study the convergence of strategic behaviour to stable decisions,from which no one will wish to deviate—equilibria.
I Agents may have no knowledge about the preferences of the othersand/or no communication.
![Page 29: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/29.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting setting
I V = {1, . . . ,n} – set of voters (or agents)
I C = {c1, . . . , cm} – set of candidates (or alternatives)
I L(C ) – set of all strict linear orders (transitive, antisymmetricand total relations) on C
I �i∈ L(C ) – agent i ’s private preference order over the candidates,for each i ∈ V
![Page 30: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/30.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting profile
I Pi ∈ L(C ) – vote of voter i (may or may not coincide with �i)I P = (P1, . . . ,Pn) ∈ L(C )n – voting profile
I P = (Pi ,P−i) where P−i – set of partial votes of a subset V \ {i}of all the agents but i
I P = (�1, . . . ,�n) – truthful profile
![Page 31: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/31.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting rule
I F : L(C )n → 2C \ {∅} – voting ruleI determines the winners of the election
![Page 32: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/32.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting rule
I F : L(C )n → C – resolute voting ruleI returns a single winnerI paired with a tie-breaking rule
I deterministic (e.g., lexicographic)I randomised
![Page 33: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/33.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting rule
I F : L(C )n → C – resolute voting ruleI returns a single winnerI paired with a tie-breaking rule
I deterministic (e.g., lexicographic)I randomised
![Page 34: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/34.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Plurality
I Each voter reports his top candidate:I Pi ∈ C
I Voters may have different weights: wi ∈ N, ∀i ∈ V .
I The score of a candidate c is the total weight of agents voting for him:
s(c) =∑
i∈V :Pi=c
wi
I The winner is selected from the candidates with the highest score.
![Page 35: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/35.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Plurality
I Each voter reports his top candidate:I Pi ∈ C
I Voters may have different weights: wi ∈ N, ∀i ∈ V .
I The score of a candidate c is the total weight of agents voting for him:
s(c) =∑
i∈V :Pi=c
wi
I The winner is selected from the candidates with the highest score.
![Page 36: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/36.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Plurality
I Each voter reports his top candidate:I Pi ∈ C
I Voters may have different weights: wi ∈ N, ∀i ∈ V .
I The score of a candidate c is the total weight of agents voting for him:
s(c) =∑
i∈V :Pi=c
wi
I The winner is selected from the candidates with the highest score.
![Page 37: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/37.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Plurality
I Each voter reports his top candidate:I Pi ∈ C
I Voters may have different weights: wi ∈ N, ∀i ∈ V .
I The score of a candidate c is the total weight of agents voting for him:
s(c) =∑
i∈V :Pi=c
wi
I The winner is selected from the candidates with the highest score.
![Page 38: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/38.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Game-theoretical interpretation
The (Plurality) voting game is a normal form game 〈V ,C ,F ,�〉 where:
I V – set of agents = set of voters
I C – set of strategies = set of candidates
I F – voting rule (paired with a tie-breaking rule)
I � – profile of voters’ preferences over the candidates
![Page 39: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/39.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Game-theoretical interpretation
The (Plurality) voting game is a normal form game 〈K ,C ,F ,�〉 where:
I K ⊆ V – set of agents = set of strategic voters
I B = V \K – sincere (non-strategic) voters
I C – set of strategies = set of candidates
I F (P), where P = (PK ,PB ), is an outcome
I Agent i ∈ K prefers profile P ′ over profile P if F (P ′) �i F (P)
![Page 40: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/40.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Game-theoretical interpretation
The (Plurality) voting game is a normal form game 〈K ,C ,F ,�〉 where:
I K ⊆ V – set of agents = set of strategic voters
I B = V \K – sincere (non-strategic) voters
I C – set of strategies = set of candidates
I F (P), where P = (PK ,PB ), is an outcome
I Agent i ∈ K prefers profile P ′ over profile P if F (P ′) �i F (P)
![Page 41: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/41.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Game-theoretical interpretation
The (Plurality) voting game is a normal form game 〈K ,C ,F ,�〉 where:
I K ⊆ V – set of agents = set of strategic voters
I B = V \K – sincere (non-strategic) voters
I C – set of strategies = set of candidates
I F (P), where P = (PK ,PB ), is an outcome
I Agent i ∈ K prefers profile P ′ over profile P if F (P ′) �i F (P)
![Page 42: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/42.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Game-theoretical interpretation
The (Plurality) voting game is a normal form game 〈K ,C ,F ,�〉 where:
I K ⊆ V – set of agents = set of strategic voters
I B = V \K – sincere (non-strategic) voters
I C – set of strategies = set of candidates
I F (P), where P = (PK ,PB ), is an outcome
I Agent i ∈ K prefers profile P ′ over profile P if F (P ′) �i F (P)
![Page 43: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/43.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
3 candidates with initial scores:
I sB (a) = 7
I sB (b) = 9
I sB (c) = 3
XXXXXXXXXXXw1 = 3w2 = 4
a b c
a
b
c
![Page 44: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/44.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
3 candidates with initial scores:
I sB (a) = 7
I sB (b) = 9
I sB (c) = 3
XXXXXXXXXXXw1 = 3w2 = 4
a b c
a
b
c
![Page 45: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/45.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
3 candidates with initial scores:
I sB (a) = 7
I sB (b) = 9
I sB (c) = 3
XXXXXXXXXXXw1 = 3w2 = 4
a b c
a (14, 9. 3)
b (11, 12, 3)
c
![Page 46: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/46.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
3 candidates with initial scores:
I sB (a) = 7
I sB (b) = 9
I sB (c) = 3
XXXXXXXXXXXw1 = 3w2 = 4
a b c
a (14, 9. 3) (10, 13, 3) (10, 9. 7)
b (11, 12, 3) (7, 16, 3) (7, 12, 7)
c (11, 9. 6) (7, 13, 6) (7, 9, 10)
![Page 47: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/47.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
Agents’ preferences:
I 1 : a � b � c
I 2 : c � a � bXXXXXXXXXXXw1 = 3
w2 = 4a b c
a (14, 9. 3) (10, 13, 3) (10, 9. 7)
b (11, 12, 3) (7, 16, 3) (7, 12, 7)
c (11, 9. 6) (7, 13, 6) (7, 9, 10)
![Page 48: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/48.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting as a normal form game
Agents’ preferences:
I 1 : a � b � c
I 2 : c � a � bXXXXXXXXXXXw1 = 3
w2 = 4a b c
a (14, 9. 3) (10, 13, 3) (10, 9. 7)b (11, 12, 3) (7, 16, 3) (7, 12, 7)
c (11, 9. 6) (7, 13, 6) (7, 9, 10)
![Page 49: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/49.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting in turns (a.k.a. “iterative voting”)
I Agents start from some initial profile (e.g., truthful).
I They change their votes in turns.
I At each step, a single agent makes a move.
I The game ends when there are no more objections.
I Implemented in polls via Doodle or Facebook.
![Page 50: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/50.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Voting in turns (a.k.a. “iterative voting”)
I Agents start from some initial profile (e.g., truthful).
I They change their votes in turns.
I At each step, a single agent makes a move.
I The game ends when there are no more objections.
I Implemented in polls via Doodle or Facebook.
![Page 51: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/51.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
Agents make rational moves to improve their state, when
I they do not know the preferences of the others,
I and cannot coordinate their actions.
⇒ The agents apply myopic (or, local) moves.
![Page 52: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/52.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
Agents make rational moves to improve their state, when
I they do not know the preferences of the others,
I and cannot coordinate their actions.
⇒ The agents apply myopic (or, local) moves.
![Page 53: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/53.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
3 : C � L � P � B
P C L B
1 2 4 5 6 3 7
![Page 54: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/54.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
3 : C � L � P � B
P C L B
1 2 4 3 5 6 7
B3→ L is an improvement move
(or better reply) of agent 3
![Page 55: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/55.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
3 : C � L � P � B
P C L B
1 3 2 4 5 6 7
B3→ P is a best
reply of agent 3
![Page 56: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/56.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Improvement moves
3 : C � L � P � B
P C L B
1 2 3 4 5 6 7
B3→ C is a restricted best reply
(which is unique) for agent 3
![Page 57: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/57.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Voting setting:
I Voting ruleI Plurality
I Tie-breaking ruleI DeterministicI Randomised
I Number of voters, number of candidatesI Agent types
I WeightedI Unweighted
![Page 58: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/58.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Voting setting:
I Voting ruleI Plurality
I Tie-breaking ruleI DeterministicI Randomised
I Number of voters, number of candidatesI Agent types
I WeightedI Unweighted
![Page 59: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/59.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Voting setting:
I Voting ruleI Plurality
I Tie-breaking ruleI DeterministicI Randomised
I Number of voters, number of candidates
I Agent typesI WeightedI Unweighted
![Page 60: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/60.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Voting setting:
I Voting ruleI Plurality
I Tie-breaking ruleI DeterministicI Randomised
I Number of voters, number of candidatesI Agent types
I WeightedI Unweighted
![Page 61: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/61.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Voting setting:
I Voting ruleI Plurality
I Tie-breaking ruleI DeterministicI Randomised
I Number of voters, number of candidatesI Agent types
I WeightedI Unweighted
![Page 62: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/62.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Dynamics:
I Initial stateI TruthfulI Arbitrary
I Improvement movesI Better repliesI Best repliesI Restricted best replies
![Page 63: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/63.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Dynamics:
I Initial stateI TruthfulI Arbitrary
I Improvement movesI Better repliesI Best repliesI Restricted best replies
![Page 64: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/64.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Variations of the game
Dynamics:
I Initial stateI TruthfulI Arbitrary
I Improvement movesI Better repliesI Best repliesI Restricted best replies
![Page 65: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/65.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Our results
We show how the convergence dependson all of these game/dynamic attributes.
![Page 66: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/66.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Deterministic tie-breaking
Theorem
If all agents have weight 1 and use restricted best replies,the game converges to a Nash equilibrium from any state.
![Page 67: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/67.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof sketch
(by Reyhani & Wilson 2012)
I ot – outcome at step t
I Restricted best replies at any step t are of 2 types:I type 1: a → b where a 6= ot−1 and b = otI type 2: a → b where a = ot−1 and b = ot
I We will show that there areI ≤ m moves of type 1 in total, andI ≤ m − 1 moves of type 2 for each voter.
![Page 68: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/68.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof
PWt = {c∣∣∃i ∈ K : ot
i→ c ⇒ c = ot+1} – potential winners at step t
Lemma
For t < t ′ we have PWt ′ ⊆ PWt .
![Page 69: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/69.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof of the lemma
I Let a → b at step t . Then, b = ot .
I Let c ∈ PWt .
I Consider the scores of b, c, y ∀y ∈ C \ {a, b}:
st−1(c) + 1 = st(c) + 1 � st(b)− 1 = st−1(b)
st−1(c) + 1 = st(c) + 1 � st(y) = st−1(y)
where c � c′ if s(c) > s(c′) or s(c) = s(c′) and c has a lower index.
![Page 70: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/70.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof of the lemma (contd.)
I If a → b at step t is of type 2, then followed by b → c at step t + 1results in the same scores as a → c at step t . Hence, c ∈ PWt−1.
I Otherwise, let a ′ = ot and note a ′ 6= a, b.
I We have:
st−1(c) + 1 � st−1(a′)
st−1(a′) � st−1(y) ∀y ∈ C
⇒ st−1(c) + 1 � st−1(a′) � st−1(a)
Hence, c ∈ PWt−1.
2
![Page 71: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/71.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof of the theorem (contd.)
I If a → b at step t is of type 1 then a /∈ PWt :I If a ∈ PWt then b → a makes a a winner,
a contradiction to a → b being of type 1.
I By the lemma, a /∈ PWt ′ for all t ′ > t⇒ the number of type 1 moves is bounded by m.
I At every improvement step ai→ b of type 2, it must hold that b �i a
⇒ each voter can make at most m − 1 steps of type 2.
2
![Page 72: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/72.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof of the theorem (contd.)
I If a → b at step t is of type 1 then a /∈ PWt :I If a ∈ PWt then b → a makes a a winner,
a contradiction to a → b being of type 1.
I By the lemma, a /∈ PWt ′ for all t ′ > t⇒ the number of type 1 moves is bounded by m.
I At every improvement step ai→ b of type 2, it must hold that b �i a
⇒ each voter can make at most m − 1 steps of type 2.
2
![Page 73: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/73.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
(Not restricted) best replies
3 candidates with initial scores (1, 0, 0)
2 voters with preferences
I 1 : a � b � c
I 2 : c � b � a
XXXXXXXXXXXvoter 1voter 2
a b c
a
b (1,2,0) (1,1,1)
c (1,1,1) (1,1,1)
Cycle from an arbitrary state!
![Page 74: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/74.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Better replies
4 candidates with initial scores (2, 2, 2, 0)
3 voters with preferences
I 1, 3 : d � a � b � c
I 2 : c � b � a � d
dcd(2, 2, 3, 2)1→ bcd(2, 3, 3, 1)
3→ bca(3, 3, 3, 0)
↑12→ bba(3, 4, 2, 0)
1→ cba(3, 3, 3, 0)2→ cca(3, 2, 4, 0)
Cycle from the truthful state!
![Page 75: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/75.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Better replies
4 candidates with initial scores (2, 2, 2, 0)
3 voters with preferences
I 1, 3 : d � a � b � c
I 2 : c � b � a � d
dcd(2, 2, 3, 2)1→ bcd(2, 3, 3, 1)
3→ bca(3, 3, 3, 0)
↑12→ bba(3, 4, 2, 0)
1→ cba(3, 3, 3, 0)2→ cca(3, 2, 4, 0)
Cycle from the truthful state!
![Page 76: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/76.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Better replies
4 candidates with initial scores (2, 2, 2, 0)
3 voters with preferences
I 1, 3 : d � a � b � c
I 2 : c � b � a � d
dcd(2, 2, 3, 2)1→ bcd(2, 3, 3, 1)
3→ bca(3, 3, 3, 0)
↑12→ bba(3, 4, 2, 0)
1→ cba(3, 3, 3, 0)2→ cca(3, 2, 4, 0)
Cycle from the truthful state!
![Page 77: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/77.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Weighted voters
I No convergence for 3+ voters, even whenstart from the truthful state and use restricted best replies
I Convergence for 2 voters, if they bothuse restricted best replies or start from the truthful state
![Page 78: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/78.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Randomised tie-breaking
I �i does not induce a complete order over the oucomes,which are sets of candidates.
I We augment agents’ preferences with cardinal utilities:I ui(c) ∈ R – utility of candidate c to voter i ,
I for multiple winners, ui(W ) =∑
c∈W ui (c)
|W | .
I A utility function u is consistent with a preference relation �i if
u(c) > u(c′)⇔ c �i c′
![Page 79: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/79.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Randomised tie-breaking
I To prove convergence, we must show it is guaranteed for anyutility function which is consistent with the given preference order.
I To disprove, it is sufficient to show a cycle for a specific assignmentof utilities: weak counterexample.
I If the counterexample holds for any profile of utility scales, it is strong.
![Page 80: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/80.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Weighted voters
3 candidates with initial scores (0, 1, 3)
2 voters weighted voters with preferences
I 1 : a � b � c
I 2 : b � c � aXXXXXXXXXXXw1 = 5
w2 = 3a b c
a (8,1,3) (5,4,3) (5,1,6)
b (3,6,3) (0,9,3) (0,6,6)
c (3,1,8) (0,4,8) (0,1,11)
![Page 81: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/81.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Weighted voters
3 candidates with initial scores (0, 1, 3)
2 voters weighted voters with preferences
I 1 : a � b � {b, c} � c
I 2 : b � {b, c} � c � aXXXXXXXXXXXw1 = 5
w2 = 3a b c
a (8,1,3) (5,4,3) (5,1,6)
b (3,6,3) (0,9,3) (0,6,6)
c (3,1,8) (0,4,8) (0,1,11)
No Nash equilibrium!
![Page 82: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/82.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Weighted voters
3 candidates with initial scores (0, 1, 3)
2 voters weighted voters with preferences
I 1 : a � b � {b, c} � c
I 2 : b � {b, c} � c � aXXXXXXXXXXXw1 = 5
w2 = 3a b c
a (8,1,3) (5,4,3) (5,1,6)
b (3,6,3) (0,9,3) (0,6,6)
c (3,1,8) (0,4,8) (0,1,11)
No Nash equilibrium!
![Page 83: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/83.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Unweighted voters
Theorem
If all agents have weight 1 and use restricted best replies,the game converges to a Nash equilibrium from the truthful state.
![Page 84: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/84.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Proof (skipped)
We show that in each step, an agent votes for a less preferred candidate.
I Clearly holds for the first step. Proceed by induciton.
Hence, each voter can make only m − 1 steps.
![Page 85: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/85.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Less restricted dynamics
I Arbitrary state:I weak counterexample with 3 unweighted agents,
even if they use restricted best replies
I Better replies:I strong counterexample with 3 unweighted agentsI weak counterexample with 2 agents,
even if they start from the truthful state
![Page 86: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/86.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Summary
Deterministic Tie breakingDynamics R. best reply Best reply Any better reply
Initial state Truth Any Truth Any Truth Any
Weighted (k > 2) X X X X X XWeighted (k = 2) V V V X V XNon-weighted V V X X X
Randomized Tie breakingDynamics R. best reply Any better reply
Initial state Truth Any Truth Any
Weighted X X X XNon-weighted V X X X
![Page 87: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/87.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Summary
Deterministic Tie breakingDynamics R. best reply Best reply Any better reply
Initial state Truth Any Truth Any Truth Any
Weighted (k > 2) X X X X X XWeighted (k = 2) V V V X V XNon-weighted V V V (#) X X X
(#) Reijngoud & Endriss, 2012
Randomized Tie breakingDynamics R. best reply Any better reply
Initial state Truth Any Truth Any
Weighted X X X XNon-weighted V X X X
![Page 88: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/88.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Other remarks
I Truth biased agentsI Vangelis’s talk tomorrow
I Quality of outcomesI Simina’s talk tomorrow
![Page 89: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/89.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Other remarks
I Truth biased agentsI Vangelis’s talk tomorrow
I Quality of outcomes
I Simina’s talk tomorrow
![Page 90: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/90.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Other remarks
I Truth biased agentsI Vangelis’s talk tomorrow
I Quality of outcomesI Simina’s talk tomorrow
![Page 91: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/91.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Other remarks
I Truth biased agentsI Vangelis’s talk tomorrow
I Quality of outcomesI Simina’s talk tomorrow
![Page 92: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/92.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Future work
![Page 93: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/93.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Past future work
![Page 94: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/94.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Past future work
I Rules other than Plurality
I Restricted Iterative Processes
I Iterative processes as a single-round gameI Today’s talks
![Page 95: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/95.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Past future work
I Rules other than Plurality
I Restricted Iterative ProcessesI Iterative processes as a single-round game
I Today’s talks
![Page 96: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/96.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Past future work
I Rules other than Plurality
I Restricted Iterative ProcessesI Iterative processes as a single-round game
I Today’s talks
![Page 97: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/97.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Past future work
I Rules other than Plurality
I Restricted Iterative ProcessesI Iterative processes as a single-round game
I Today’s talks
![Page 98: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/98.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Model
Future work
I Rules other than Plurality
I Restricted Iterative Processes
I Iterative processes as a single-round game
I Weak acyclicity?
I Dynamics leading to desirable outcomes?
![Page 99: CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTINGumberto/iterative/polukarov.pdf · Once upon a time... \Life is worthless without love!" - told Snow White the Seven Dwarfs. CONVERGENCE](https://reader036.fdocuments.net/reader036/viewer/2022081617/603cee66542ca343b6735aa0/html5/thumbnails/99.jpg)
CONVERGENCE TO EQUILIBRIA IN PLURALITY VOTING
Happy end!
THE END