reshef meir - bilgi · 2015-11-10 · desiderata for voting models •theoretic criteria...
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Reshef Meir
Tehcnion-Israel Institute of Technology
Based on joint work(s) with Omer Lev, David Parkes,
Jeff Rosenschein, and James Zou
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Plurality voting - example
90 votes
20
85
45
??
Can theory explain/predict
voting behavior?
…
“truthful”
“strategic”
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Desiderata for voting models
• Theoretic criteria
(Rationality, equilibrium)
• Behavioral criteria
(voters’ beliefs and
capabilities)
• Scientific criteria:
(Robustness, Simplicity,
consistent with data,
Discriminative power)
(arguable)
Bounded
rationality
“Leader rule”
[Laslier’09]
Expected utility
[MW’93,MP’02,...]
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Our contribution
Epistemic model
(for limited information)
Behavioral model
(for limited capabilities)
Formal and
empirical
results
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Prospective scores 𝒔 • E.g. from a poll
• “world state”
Uncertainty level 𝑟𝑖 ≥ 0
[Simon’57]:
“...the state of information may as well be
regarded as a characteristic of the decision-
maker as a characteristic of his environment”
𝒔 = 90,20,85,45 𝜖 ℝ|𝒞|
Epistemic model
𝒞 = { }
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Prospective scores 𝒔 • E.g. from a poll
• “world state”
Uncertainty level 𝑟𝑖 ≥ 0
Voter 𝑖 considers as “possible” all states close
enough to 𝒔. 𝑆 𝒔, 𝑟𝑖 = {𝒔′: 𝒔′ − 𝒔 ≤ 𝑟𝑖} – Example I: “𝑎𝑑𝑑𝑖𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦"
𝒔 = (90,20,85,45) 90 + 𝑟𝑖
90 − 𝑟𝑖
Epistemic model
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Prospective scores 𝒔 • E.g. from a poll
• “world state”
Uncertainty level 𝑟𝑖 ≥ 0
Voter 𝑖 considers as “possible” all states close
enough to 𝒔. 𝑆 𝒔, 𝑟𝑖 = {𝒔′: 𝒔′ − 𝒔 ≤ 𝑟𝑖} – Example I: “𝑎𝑑𝑑𝑖𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦"
– Example II: “𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦"
𝒔 = (90,20,85,45) 90 (1 + 𝑟𝑖)
90/(1 + 𝑟𝑖)
Epistemic model
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Def. I (Local dominance): A candidate 𝑐′
S-dominates candidate 𝑐 if it is always weakly
better for 𝑖 to vote for 𝑐′.
𝒔 = (90,20,85,45)
𝒔 𝜖 ℝ|𝒞| : state (scores)
𝑆 = 𝑆 𝒔, 𝑟𝑖 : possible states
in every state 𝒔′ ∈ S
Behavioral model
Rational agents
avoid dominated
strategies!
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One-shot voting:
Vote for a candidate that
is not locally-dominated
Iterative setting: As long as your vote is locally
dominated, switch to a candidate that dominates
it. Otherwise – stay.
Rational agents
avoid dominated
strategies!
Behavioral model
Local dominance move
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Strategic voting (one shot)
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Strategic voting (one shot)
A B C D E F
𝑇 𝒔, 𝑟𝑖
A B C D E F
Lemma: All dominance relations in state 𝒔 are characterized by a
single threshold 𝑇 𝒔, 𝑟𝑖 : (depends on winner’s score)
𝑐 is dominated iff below the threshold or least preferred.*
2𝑟𝑖
𝑟𝑖
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Strategic voting (iterative)
A B C D E F
A B C D E F
𝑇 𝒔, 𝑟𝑖
𝑠[𝑖]
𝑐′
𝑠 𝑖 ∈ 𝒞: the vote of voter 𝑖 in state 𝒔
Local dominance move
A → B
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A B C D E F
𝑇 𝒔, 𝑟𝑖
A B C D E F
Strategic voting (iterative)
𝑐′
𝑠 𝑖 ∈ 𝒞: the vote of voter 𝑖 in state 𝒔
𝑠[𝑖]
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A B C D E F
𝑇 𝒔, 𝑟𝑖
A B C D E F
Strategic voting (iterative)
𝑠 𝑖 ∈ 𝒞: the vote of voter 𝑖 in state 𝒔
𝑠[𝑖]
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A B C D E F
A B C D E F
𝒔0 (initial state)
Strategic voting (iterative)
𝑠0[𝑖]
𝑠𝑡 𝑖 ∈ 𝒞 : the vote of voter 𝑖 in state 𝒔𝑡
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A B C D E F
A B C D E F
𝒔0 (initial state)
𝒔1
𝒔2
𝒔3
…
Strategic voting (iterative)
𝑠1[𝑖]
𝑠2[𝑖]
𝑠𝑡 𝑖 ∈ 𝒞 : the vote of voter 𝑖 in state 𝒔𝑡
Def. II (voting equilibrium):
A state 𝒔 where for every voter 𝑖, the
candidate 𝑠[𝑖] is not 𝑆(𝒔, 𝑟𝑖)-dominated.
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• Existence?
• Convergence?
• Properties?
Def. II (voting equilibrium):
A state 𝒔 where for every voter 𝑖, the
candidate 𝑠[𝑖] is not 𝑆(𝒔, 𝑟𝑖)-dominated.
Prop. [M., Polukarov, Rosenschein,
Jennings, AAAI’10]: “best-response in
voting converges to a Nash equilibrium.”
Independent of voting order
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Results
Main Theorem [M. AAAI’15]:
Any sequence 𝒔0 → 𝒔1 → 𝒔2 → ⋯ of Local-
dominance moves is acyclic (must converge).
In particular, a voting equilibrium always exists.
• From any initial state 𝒔0
• Uncertainty levels 𝑟𝑖 may be diverse
• Arbitrary order of players • For a nonatomic model: Also holds under simultaneous moves
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Main Theorem [M. AAAI’15]:
Any sequence 𝒔0 → 𝒔1 → 𝒔2 → ⋯ of Local-
dominance moves is acyclic (must converge).
In particular, a voting equilibrium always exists.
Results
Prop. [M., Polukarov, Rosenschein, Jennings, AAAI’10]:
“best-response in voting converges to a Nash equilibrium.”
𝑆(𝒔, 𝑟𝑖) = {𝒔} ⇒ Proof sketch:
Local-dominance ≡ Best response
Voting equilibrium ≡ Nash equilibrium ⇒
𝑟𝑖 = 0 for all 𝑖
⇒
Follows as a special case!
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• Existence?
• Convergence?
• Properties?
Results
Extensive computer simulations:
>100 distributions of preferences
>10K profiles in total
>1M simulations
Def. II (voting equilibrium):
A state 𝒔 where for every voter 𝑖, the
candidate 𝑠[𝑖] is not 𝑆(𝒔, 𝑟𝑖)-dominated.
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• Decisiveness
• Duverger Law
• Participation
• Welfare
Results (computer simulations)
[M., Lev, Rosenschein, EC’14]
More uncertainty
𝒔0
𝒔′0 𝒔′′0
𝒔′′′0
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Desiderata for voting models
• Theoretic criteria
(Rationality, equilibrium)
• Behavioral criteria
(voters’ beliefs and
capabilities)
• Scientific criteria:
(Robustness, Simplicity,
consistent with data,
Discriminative power)
Local-
Dominance
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Related work
• Voting experiments – VoteLib.org [Tal, M., Gal ‘15]
• Voting under strict uncertainty: – [Conitzer, Walsh, Xia ‘11] (dominance with information sets)
– [Reijngoud, Endriss ‘12] (∏-manipulation)
– [van Ditmarsch, Lang, Saffidine ‘13] (de re knowledge)
• Regret minimization [M.‘15]
• Lazy/truth-biased voters […]
• Coordination in polls [Reyhani, Wilson, Khazaei ’12]
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What next?
• Doodle scheduling
• Uncertainty and Modal Logic
• Proof sketch for Plurality convergence
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Doodle Scheduling
Scheduling
Questions:
• Do people strategize when
seeing previous responses?
• How?
Based on analyzing > 340,000 real Doodle polls
[Zou, M., Parkes, CSCW’15]
Findings for open polls:
1. More correlation with
previous responses
2. Availability 35% higher
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• Where are the extra available slots?
The probability that the 11th
responder approves the slot
Number of previous responders
who approved
“Popular slots” “Unpopular slots”
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• Where are the extra available slots?
Be more
cooperative
?
Respondents strategically mark additional unpopular slots.
Want to appear cooperative!
“Popular slots” “Unpopular slots”
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Uncertainty in scheduling
Time1 Time2 Time3 Time4
Scheduling
I should answer next.
I want Thu. 10am.
ok ok
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Uncertainty in scheduling
Time1 Time2 Time3 Time4
Scheduling
Not “possible winners.”
Safe for strategic vote
ok I should answer next.
I want Thu. 10am.
ok ok ok
skip
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Uncertainty and modal logic
s
𝑎 dominates 𝑏 if :
◻ (𝑓 𝒔, 𝑎 ≽𝑖 𝑓 𝒔, 𝑏 ) ⋄(𝑓(𝒔, 𝑎) ≻𝑖 𝑓(𝒔, 𝑏))
What is the set of
states accessible
from 𝒔 ?
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Uncertainty and modal logic
Possible states under the
S5 axioms – a partition 𝑃
s
𝑃(𝒔)
“If I am in 𝒔, then I know
I am in 𝑃(𝒔)”
𝑎 dominates 𝑏 if :
◻ (𝑓 𝒔, 𝑎 ≽𝑖 𝑓 𝒔, 𝑏 ) ⋄(𝑓(𝒔, 𝑎) ≻𝑖 𝑓(𝒔, 𝑏))
What is the set of
states accessible
from 𝒔 ?
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Uncertainty and modal logic
Possible states under the
S5 axioms – a partition 𝑃
s S 𝒔, 𝑟
s
𝑃(𝒔)
Possible states under the
distance-based uncertainty
“If I am in 𝒔, then I know
I am in 𝑃(𝒔)” “If I am in 𝒔, then I know
I am close to 𝒔”
45
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Uncertainty and modal logic
Possible states under the
S5 axioms – a partition 𝑃
s S 𝒔, 2𝑟
s
𝑃(𝒔)
Possible states under the
distance-based uncertainty
“If I am in 𝒔, then I know
I am in 𝑃(𝒔)” “If I am in 𝒔, then I know
I am close to 𝒔”
46
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Uncertainty and modal logic
Possible states under the
S5 axioms – a partition 𝑃
s S 𝒔, 𝑟
s
𝑃(𝒔) 𝒔′
S 𝒔′, 𝑟
Possible states under the
distance-based uncertainty
“If I am in 𝒔, then I know
I am in 𝑃(𝒔)” “If I am in 𝒔, then I know
I am close to 𝒔”
Violates transitivity
𝒔′
𝑃(𝒔’)
47 Doodle
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Recipe for general games
𝑠1 𝑠2 𝑠3 𝑠4
r
States are all strategy profiles
“possible states” S 𝒔, 𝑟
are all states close to 𝒔 𝒔
𝒔 is the prospective state, induced by
the current strategies
• Avoid dominated actions
• Minimize worst-case cost
• Minimize worst-case regret
• Other?
Epistemic model
Behavioral model
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Recipe for general games
𝑠1 𝑠2 𝑠3 𝑠4
r
States are all strategy profiles
“possible states” S 𝒔, 𝑟
are all states close to 𝒔 𝒔
𝒔 is the prospective state, induced by
the current strategies
• Avoid dominated actions
• Minimize worst-case cost
• Minimize worst-case regret
• Other?
Epistemic model
Behavioral model
Example: Congestion Games
with strict uncertainty
[M. & Parkes, ‘15]
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Summary
Epistemic model
Distance-based uncertainty
No probabilities
Behavioral
model
Local-
dominance
Results
Behavioral
model
Mark ``safe’’
slots
Results
Plurality
voting
Online
scheduling ?
Behavioral
model
?
Results
![Page 38: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/38.jpg)
Omer Lev,
HUJI
David Parkes,
Harvard
James Zou,
Harvard &
MSR
Jeff Rosenschein,
HUJI
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![Page 39: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/39.jpg)
The slides are based on the
following papers: • A Local-Dominance Theory of Voting Equilibria. Reshef Meir,
Omer Lev, and Jeffrey S. Rosenschein. EC’14.
• Plurality Voting under Uncertainty, Reshef Meir. AAAI’15.
• Strategic Voting Behavior in Doodle Polls, James Zou, Reshef
Meir, and David Parkes. CSCW ’15.
Other related papers:
• Convergence to Equilibria of Plurality Voting, Reshef Meir,
Maria Polukarov, Jeffrey S. Rosenschein and Nicholas R. Jennings.
AAAI’10.
• A Study of Human Behavior in Voting Systems, Maor
Tal, Reshef Meir, and Kobi Gal. AAMAS’15.
• Congestion Games with Distance-Based Strict
Uncertainty, Reshef Meir and David Parkes.
![Page 41: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/41.jpg)
Uniform uncertainty (𝑟𝑖 = 𝑟):
Existence + Convergence if start by voting truthfully
[M., Lev, Rosenschein, EC’14]
Proof intuition:
Results
𝑇 𝒔, 𝑟
![Page 42: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/42.jpg)
Uniform uncertainty (𝑟𝑖 = 𝑟):
Existence + Convergence if start by voting truthfully
[M., Lev, Rosenschein, EC’14]
Results
𝑇 𝒔, 𝑟
![Page 43: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/43.jpg)
Uniform uncertainty (𝑟𝑖 = 𝑟):
Existence + Convergence if start by voting truthfully
[M., Lev, Rosenschein, EC’14]
Results
𝑇 𝒔, 𝑟1
𝑇 𝒔, 𝑟2
𝑇 𝒔, 𝑟3
![Page 44: Reshef Meir - Bilgi · 2015-11-10 · Desiderata for voting models •Theoretic criteria (Rationality, equilibrium) •Behavioral criteria (voters’ beliefs and capabilities) •Scientific](https://reader034.vdocuments.net/reader034/viewer/2022042613/5f9068b4e8e65826e006aa86/html5/thumbnails/44.jpg)
Uniform uncertainty (𝑟𝑖 = 𝑟):
Existence + Convergence if start by voting truthfully
[M., Lev, Rosenschein, EC’14]
Results
𝑇 𝒔, 𝑟1
𝑇 𝒔, 𝑟2
𝑇 𝒔, 𝑟3
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