Sequential Deliberation for Social Choice ALI MOHAMMAD FARAJI - - PowerPoint PPT Presentation

Sequential Deliberation for Social Choice

ALI MOHAMMAD FARAJI MOJTABA FAYAZBAKHSH

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Problem Statement

Protocol for aggregation of Social preferences Difficulties:

• System designer may not be able to enumerate all the outcomes in the decision space
• Minifying the decision space: removing the social optimum
• Agents may not have rankings over the entire decision space
• Difficulty to implement most ordinal voting schemes in continuous spaces

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Problem Statement

Premises:

• No need to formally articulate the entire decision space
• No need for every agent to report his ordinal ranking
• Agents can reason about their preferences
• Small groups can negotiate

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Background

Bargaining Theory:

• Two-person bargaining:
• A game: A disagreement outcome and two agents who must cooperate to reach a decision
• Failure to cooperate: Adoption of the disagreement outcome
• Nash axioms:
• Pareto optimality
• Symmetry between agents
• Invariance with respect to affine transformations of utility
• Independence of irrelevant alternatives

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Background

Bargaining Theory:

• Nash: Solution maximizing the Nash product is the unique solution satisfying the axioms
• If agent u has a bliss point pu, his disutility for an alternative a is d(pu, a)
• Subject to individual rationality:

d(pv, o) ≤ d(pv,a) and d(pu,o) ≤ d(pu,a)

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Background

Social cost: 𝑇𝐷 𝑏 = ෍

𝑣∈𝑂

𝑒(𝑞𝑣, 𝑏) Distortion: 𝐸𝑗𝑡𝑢𝑝𝑠𝑢𝑗𝑝𝑜 𝑏 =

𝑇𝐷(𝑏) 𝑇𝐷(𝑏

∗

)

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Sequential pairwise deliberation

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General Metric Spaces

The Distortion of sequential deliberation is at most 3

• This bound is tight.
• The bound is quite pessimistic.

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Flexibility of Model

Well defined and practical irrespective of an analytical model Generality and high level abstraction Regardless of the underlying decision space or mediator’s understanding of the space

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Median Graphs

A median graph G(S, E)

• unweighted and undirected
• ∀ u, v, w ∈ S × S × S: ∃ a unique point that is common to the shortest paths.
• This point is the unique median of u, v, w.

Trees, points on a line, hypercubes, grid graphs, etc.

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Nash Bargaining on Median Graphs

Nash bargaining will select the median of bliss points of the two agents pu , pv and disagreement alternative a. The median maximizes Nash product and is closest to a. Recall:

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Hypercube Embedding

For any median graph G = (S, E), there is an isometric embedding φ : G → Q

• f G into a hypercube Q

φ(Median(t, u, v)) = Median(φ(t), φ(u), φ(v)) the Distortion of sequential deliberation on G is at most the Distortion of sequential deliberation on φ(G) where each agent’s bliss point is φ(pu).

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Distortion of Sequential Deliberation

As t → ∞, the Distortion of sequential deliberation approaches 1.208 Convergence rate is:

• exponentially fast in t
• independent of |N|, |S|, a1

The Distortion is at most 1.22 in at most 9 steps of deliberation!

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Proof Idea

Hypercube embedding

• Dimension-wise analysis
• optimum social cost
• Median

Defining a Markov chain

• Expected sequential deliberation social cost analysis

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Lower Bounds on Distortion

( I ): Any mechanism constrained to choose outcomes in bliss points has Distortion at least 3.

• Sequential deliberation dominates random dictatorship on every instance for median

graphs.

• Deliberation do play a role in reducing Distortion

( II ): Any mechanism constrained to choose median of three points in bliss points must have Distortion at least 1.316.

• Oligarchy again is not well!

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ANY QUESTIONS?

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