Reshef Meir Tehcnion-Israel Institute of Technology Based on joint - PowerPoint PPT Presentation

Reshef Meir Tehcnion-Israel Institute of Technology Based on joint work(s) with Omer Lev, David Parkes, Jeff Rosenschein, and James Zou

Plurality voting - example “strategic” ?? 90 votes 85 “truthful” 45 20 … Can theory explain/predict voting behavior?

Desiderata for voting models (arguable) “Leader rule” Expected utility [Laslier’09] [MW’93,MP’02,...] • Theoretic criteria (Rationality, equilibrium) Bounded • Behavioral criteria rationality (voters’ beliefs and capabilities) • Scientific criteria: (Robustness, Simplicity, consistent with data, Discriminative power)

Our contribution Formal and empirical results Behavioral model (for limited capabilities) Epistemic model (for limited information)

Epistemic model 𝒕 = 90,20,85,45 Prospective scores 𝒕 𝜗 ℝ |𝒟| • E.g. from a poll • “world state” Uncertainty level 𝑠 𝑗 ≥ 0 𝒟 = { } [Simon’57]: “...the state of information may as well be regarded as a characteristic of the decision- maker as a characteristic of his environment”

Epistemic model 𝒕 = (90,20,85,45) Prospective scores 𝒕 90 + 𝑠 𝑗 • E.g. from a poll 90 − 𝑠 𝑗 • “world state” Uncertainty level 𝑠 𝑗 ≥ 0 Voter 𝑗 considers as “possible” all states close enough to 𝒕 . 𝑇 𝒕, 𝑠 𝑗 = {𝒕 ′ : 𝒕 ′ − 𝒕 ≤ 𝑠 𝑗 } – Example I: “ 𝑏𝑒𝑒𝑗𝑢𝑗𝑤𝑓 𝑣𝑜𝑑𝑓𝑠𝑢𝑏𝑗𝑜𝑢𝑧"

Epistemic model 𝒕 = (90,20,85,45) Prospective scores 𝒕 90 (1 + 𝑠 𝑗 ) • E.g. from a poll 90/(1 + 𝑠 𝑗 ) • “world state” Uncertainty level 𝑠 𝑗 ≥ 0 Voter 𝑗 considers as “possible” all states close enough to 𝒕 . 𝑇 𝒕, 𝑠 𝑗 = {𝒕 ′ : 𝒕 ′ − 𝒕 ≤ 𝑠 𝑗 } – Example I : “ 𝑏𝑒𝑒𝑗𝑢𝑗𝑤𝑓 𝑣𝑜𝑑𝑓𝑠𝑢𝑏𝑗𝑜𝑢𝑧" – Example II: “ 𝑛𝑣𝑚𝑢𝑗𝑞𝑚𝑗𝑑𝑏𝑢𝑗𝑤𝑓 𝑣𝑜𝑑𝑓𝑠𝑢𝑏𝑗𝑜𝑢𝑧"

Behavioral model Rational agents 𝒕 = (90,20,85,45) avoid dominated strategies! 𝒕 𝜗 ℝ | 𝒟 | : state (scores) 𝑇 = 𝑇 𝒕, 𝑠 𝑗 : possible states Def. I ( Local dominance ): A candidate 𝑑 ′ S-dominates candidate 𝑑 if it is always weakly better for 𝑗 to vote for 𝑑 ′ . in every state 𝒕 ′ ∈ S

Behavioral model Rational agents avoid dominated strategies! 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. Local dominance move

Strategic voting (one shot)

Lemma: All dominance relations in state 𝒕 are characterized by a Strategic voting (one shot) 𝑗 : (depends on winner’s score) single threshold 𝑈 𝒕, 𝑠 𝑑 is dominated iff below the threshold or least preferred .* 2𝑠 𝑗 𝑈 𝒕, 𝑠 𝑗 A B C D E F 𝑠 𝑗 A B C D E F

Strategic voting (iterative) Local dominance move A → B 𝑑′ 𝑈 𝒕, 𝑠 𝑗 𝑡[𝑗] A B C D E F 𝑡 𝑗 ∈ 𝒟 : the vote of voter 𝑗 in state 𝒕 A B C D E F

Strategic voting (iterative) 𝑑′ 𝑈 𝒕, 𝑠 𝑗 𝑡[𝑗] A B C D E F 𝑡 𝑗 ∈ 𝒟 : the vote of voter 𝑗 in state 𝒕 A B C D E F

Strategic voting (iterative) 𝑈 𝒕, 𝑠 𝑗 𝑡[𝑗] A B C D E F 𝑡 𝑗 ∈ 𝒟 : the vote of voter 𝑗 in state 𝒕 A B C D E F

Strategic voting (iterative) 𝒕 0 (initial state) 𝑡 0 [𝑗] A B C D E F 𝑡 𝑢 𝑗 ∈ 𝒟 : the vote of voter 𝑗 in state 𝒕 𝑢 A B C D E F

Strategic voting (iterative) 𝑡 2 [𝑗] Def. II ( voting equilibrium ): A state 𝒕 where for every voter 𝑗 , the 𝒕 0 (initial state) 𝒕 1 candidate 𝑡[𝑗] is not 𝑇(𝒕, 𝑠 𝑗 ) -dominated. 𝑡 1 [𝑗] 𝒕 2 𝒕 3 … A B C D E F 𝑡 𝑢 𝑗 ∈ 𝒟 : the vote of voter 𝑗 in state 𝒕 𝑢 A B C D E F

Def. II ( voting equilibrium ): A state 𝒕 where for every voter 𝑗 , the candidate 𝑡[𝑗] is not 𝑇(𝒕, 𝑠 𝑗 ) -dominated. • Existence? Independent of voting order Prop. [M., Polukarov, Rosenschein, • Convergence? Jennings , AAAI’10]: “ best-response in voting converges to a Nash equilibrium .” • Properties?

Results Main Theorem [M. AAAI’15] : Any sequence 𝒕 0 → 𝒕 1 → 𝒕 2 → ⋯ of Local- dominance moves is acyclic (must converge). In particular, a voting equilibrium always exists. • From any initial state 𝒕 0 • Uncertainty levels 𝑠 𝑗 may be diverse • Arbitrary order of players • For a nonatomic model : Also holds under simultaneous moves

Results Main Theorem [M. AAAI’15] : Any sequence 𝒕 0 → 𝒕 1 → 𝒕 2 → ⋯ of Local- dominance moves is acyclic (must converge). In particular, a voting equilibrium always exists. Prop. [M., Polukarov, Rosenschein, Jennings , AAAI’10 ]: “best - response in voting converges to a Nash equilibrium.” Follows as a special case! ⇒ 𝑗 ) = {𝒕} 𝑗 = 0 for all 𝑗 Proof sketch: 𝑇(𝒕, 𝑠 𝑠 ⇒ Local- dominance ≡ Best response ⇒ Voting equilibrium ≡ Nash equilibrium

Results Def. II ( voting equilibrium ): A state 𝒕 where for every voter 𝑗 , the candidate 𝑡[𝑗] is not 𝑇(𝒕, 𝑠 𝑗 ) -dominated. • Existence? • Convergence? Extensive computer simulations: • Properties? >100 distributions of preferences >10K profiles in total >1M simulations

Results (computer simulations) 𝒕 0 𝒕′′′ 0 • Decisiveness 𝒕′′ 0 𝒕′ 0 • Duverger Law • Participation More uncertainty • Welfare [M., Lev, Rosenschein , EC’14]

Desiderata for voting models Local- Dominance • Theoretic criteria (Rationality, equilibrium) • Behavioral criteria (voters’ beliefs and capabilities) • Scientific criteria: (Robustness, Simplicity, consistent with data, Discriminative power)

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]

What next? • Doodle scheduling • Uncertainty and Modal Logic • Proof sketch for Plurality convergence

Doodle Scheduling Scheduling Questions: • Do people strategize when seeing previous responses? • How? [Zou, M., Parkes , CSCW’15] Findings for open polls: 1. More correlation with previous responses 2. Availability 35% higher Based on analyzing > 340,000 real Doodle polls

• Where are the extra available slots? “Unpopular slots” “Popular slots” The probability that the 11 th Number of previous responders responder approves the slot who approved

• Where are the extra available slots? ? Be more cooperative “Unpopular slots” “Popular slots” Respondents strategically mark additional unpopular slots. Want to appear cooperative!

Uncertainty in scheduling Scheduling Time1 Time2 Time3 Time4 ok ok I should answer next. I want Thu. 10am.

Uncertainty in scheduling Scheduling Time1 Time2 Time3 Time4 ok ok ok ok I should answer next. I want Thu. 10am. Not “possible winners.” skip Safe for strategic vote

Uncertainty and modal logic 𝑏 dominates 𝑐 if : ◻ (𝑔 𝒕, 𝑏 ≽ 𝑗 𝑔 𝒕, 𝑐 ) ⋄(𝑔(𝒕, 𝑏) ≻ 𝑗 𝑔(𝒕, 𝑐)) s What is the set of states accessible from 𝒕 ?

Uncertainty and modal logic Possible states under the S5 axioms – a partition 𝑄 𝑏 dominates 𝑐 if : ◻ (𝑔 𝒕, 𝑏 ≽ 𝑗 𝑔 𝒕, 𝑐 ) ⋄(𝑔(𝒕, 𝑏) ≻ 𝑗 𝑔(𝒕, 𝑐)) 𝑄(𝒕) s What is the set of states accessible from 𝒕 ? “If I am in 𝒕 , then I know I am in 𝑄(𝒕) ”

Uncertainty and modal logic Possible states under the Possible states under the distance-based uncertainty S5 axioms – a partition 𝑄 𝑄(𝒕) s s S 𝒕, 𝑠 “If I am in 𝒕 , then I know “If I am in 𝒕 , then I know I am in 𝑄(𝒕) ” I am close to 𝒕 ” 45

Uncertainty and modal logic Possible states under the Possible states under the distance-based uncertainty S5 axioms – a partition 𝑄 𝑄(𝒕) s s S 𝒕, 2𝑠 “If I am in 𝒕 , then I know “If I am in 𝒕 , then I know I am in 𝑄(𝒕) ” I am close to 𝒕 ” 46

Uncertainty and modal logic Possible states under the Possible states under the distance-based uncertainty S5 axioms – a partition 𝑄 𝑄(𝒕) 𝒕′ 𝑄(𝒕’) s S 𝒕′, 𝑠 𝒕′ s S 𝒕, 𝑠 “If I am in 𝒕 , then I know “If I am in 𝒕 , then I know I am in 𝑄(𝒕) ” I am close to 𝒕 ” Violates transitivity 47 Doodle