an efficient distributed protocol for collective decision-making in combinatorial domains

An efficient distributed protocol for collective decision-making in combinatorial domains

CMSS Feb. 20-21, 2012

Minyi Li

Intelligent Agent Technology Group

Centre for Computing and Engineering Software Systems (SUCCESS)

Swinburne University of Technology

Melbourne, Australia

> Society

– Voters (agents)

> Alternatives

– Domain

– Candidates

> Problem

– Choice of one among several Candidates

– Selection criterion

Motivation and problem settingMotivation and problem setting

An efficient distributed protocol for collective decision-making in combinatorial domains CMSS 2012

> Majority rule

– Majority rule for elections with only two candidates which the candidate preferred by more than half the agents is the winner.

> Condorcet’s Method

– The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more than half of the agents (winner in pair-wise comparison).

> Condorcet’s Paradox

Selection criteriaSelection criteria

a b

c

3 candidates: {a, b, c}3 agents: {1,2,3}1 : a > b > c 2 : b > c > a3 : c > a > b An efficient distributed protocol for collective decision-making in combinatorial domains CMSS 2012

> What to do when there is no (weak) Condorcet winner

– Smith set: the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pair-wise election.

The Smith set provides one standard of optimal choice for an election outcome.

– Minimax (Simpson): selects the candidate for whom the greatest pair-wise score for another candidate against him is the least such score among all candidates.

• May possible choose a Condorcet loser (is a candidate who can be defeated in pair-wise competition against each other candidate).

– Smith/minimax: chooses a minimax candidate from the Simith set.

Selection criteriaSelection criteria


> When the domain is huge, and variables are interdependent

– Unrealistic to assume linear orders are given by the individual agents

– Exhaustive pair-wise comparison in the entire alternative space becomes impractical.

– Decompose rules or issue by issue voting may produce undesirable outcomes, and might be impossible.

– Individual outcome comparison might be very difficult.

Combinatorial VoteCombinatorial Vote


> Possible solution

– To find reasonable restrictions on the preference structures so that sequential voting still preserves desirable properties;

– Design good algorithms to compute the winner for the combinatorial vote problem, by:

• Utilizing the structures of the preferences; work on the preference language directly.

• Avoiding pair-wise comparison as much as possible

• Choose from a small subset of alternatives (it might be easy to detect that some alternatives are not socially optimal)

Combinatorial VoteCombinatorial Vote


The proposed protocol: The proposed protocol: MDTreeS/MMDTreeS/M

> Basic principles:Basic principles:

– Allow partial vote with optimistic strategy

– A partial vote is a proposal on an assignment over a subset of variables in the domain

– In each iteration of the protocol, different agents may vote for assignments on different subset of variables

– A partial vote can be extend only if it received more than half of the agents’ proposal

– The agents continue to propose and extend partial assignment until all variables are assigned

– Allow regret of partial choice

– The agent is not required to stick with and extend one partial assignment (or even a complete assignment) if there is something else better!


The frameworkThe framework

> Majority decision tree (MDTree):Majority decision tree (MDTree):

– A k-ary tree (k is maximum domain size) with depth m (m is the number of variables)

– Root begin at depth 0 with an empty assignment

– Each level assigns possible values to a single variable; the order following which the variables

will be assigned is randomly chosen.

– Each node at level p (p<=m) in a MDTree represents a unique assignment to a subset of variables

> Open node: Open node: 1) a leaf node at level 1) a leaf node at level p (p<m); 2) a majority of agents have proposed on

> Open node will be expanded automatically for further consideration.

> Winning node: 1) a leaf node at level m; a majority of agents have proposed on

> Winning node will no longer be feasible for making proposal on.

21... XX

} ..., ,{1 p

XX


> Best possible alternativeBest possible alternative

– At each node of the negotiation tree (represents a partial At each node of the negotiation tree (represents a partial assignment), each agent has a best possible alternative (BPA), assignment), each agent has a best possible alternative (BPA), i.e., an optimal outcome with the variable values in the partial i.e., an optimal outcome with the variable values in the partial assignment being fixed.assignment being fixed.

e.g. e.g.

> BATCD Strategy:BATCD Strategy: to proposed on a node that optimistically can give to proposed on a node that optimistically can give him the best! him the best!

– Feasible nodes for an agent Feasible nodes for an agent ii: 1) a leaf node; 2) not a winning : 1) a leaf node; 2) not a winning node; 3) agent hasn’t proposed on that node.node; 3) agent hasn’t proposed on that node.



cbaBPAbat

abcBPAa

,assignmen node

,assginment node

The process of negotiationThe process of negotiation

Step 1: Each agent proposes on a node, unless:Step 1: Each agent proposes on a node, unless:

– He has proposed on every node.He has proposed on every node.

– Nothing will be better for him than the chosen Nothing will be better for him than the chosen

If all the agents stop making offers, the process ends and returns the If all the agents stop making offers, the process ends and returns the current chosen current chosen

Step 2: Expanding open nodes.Step 2: Expanding open nodes.

If there exists no winning node, go back to Step 1.If there exists no winning node, go back to Step 1.

Step 3: Choose a Smith/Minimax solution among the winning nodes. Step 3: Choose a Smith/Minimax solution among the winning nodes.

Go back to Step 1. Go back to Step 1.

> From a broad view, during the negotiation process, the agents first try to From a broad view, during the negotiation process, the agents first try to identify a socially prefer winning node by iteratively making proposals. identify a socially prefer winning node by iteratively making proposals. Then, they try to converge to that node unless there is something better Then, they try to converge to that node unless there is something better (socially more preferred)! (socially more preferred)!

> The detailed process:The detailed process:


An exampleAn example

1 :

2 :

3 :

1：

2：

3：

2, 3

1, 31, 2


1 :

2 :

3 :

Experiment and evaluationExperiment and evaluation

> Experiment Setting:Experiment Setting:

– 2-12 variables; 5 and 15 agents2-12 variables; 5 and 15 agents

– Preferences: SLO-SCP-net (Soft constraint Lexicographic Ordering)Preferences: SLO-SCP-net (Soft constraint Lexicographic Ordering)

[Carmel Domshlak, Steven David Prestwich, Francesca Rossi, Kristen Brent Venable, and [Carmel Domshlak, Steven David Prestwich, Francesca Rossi, Kristen Brent Venable, and Toby Walsh. Hard and soft constraints for reasoning about qualitative conditional Toby Walsh. Hard and soft constraints for reasoning about qualitative conditional preferences. J. Heuristics, 12(4-5):263–285, 2006.]preferences. J. Heuristics, 12(4-5):263–285, 2006.]

: the average number of alternatives that each agent needs to consider.

: the average number of alternatives that require pair-wise comparison.

: the average number of dominance queries that each agent needs to answer.

In these experiments, for each number of agents and variables, in more than 99:8% cases, the final decision chosen by MDTree-S/M protocol is the Smith/Minimax solution.


> Desirable properties:Desirable properties:

– Satisfies Smith criterion;Satisfies Smith criterion;

– The final decision chosen is sufficiently close to the Smith/Minimax The final decision chosen is sufficiently close to the Smith/Minimax solution (experimentally evaluated);solution (experimentally evaluated);

– Enables distributed decision-making and works under incomplete Enables distributed decision-making and works under incomplete information setting; information setting;

– The amount of dominance testing needed from each agent, as well as The amount of dominance testing needed from each agent, as well as the number of pair wise comparisons required is significantly less;the number of pair wise comparisons required is significantly less;

– It is sufficiently general that it is applicable to most preference It is sufficiently general that it is applicable to most preference representation languages in combinatorial domainrepresentation languages in combinatorial domain

> Future WorkFuture Work

– Other voting rules, Copeland, range voting;Other voting rules, Copeland, range voting;

– Develop more accurate algorithms.Develop more accurate algorithms.

Property of the protocolProperty of the protocol


Thank you & Questions?


an efficient distributed protocol for collective decision-making in combinatorial domains

Documents

collective decisionmaking

combinatorialdomains

combinatorialdomains

minimax candidate

proposed protocol

pairwise election

set of candidates

simith set