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A Distributed Protocol for Collective Decision-Making in Combinatorial Domains

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A Distributed Protocol for Collective Decision-Making in Combinatorial Domains Swinburne Research Bank http://researchbank.swinburne.edu.au Author: Li, Minyi; Vo, Quoc Bao; Kowalczyk, Ryszard Title: A distributed protocol for collective decision- making in combinatorial domains Editor: Takayuki Ito, Catholijn Jonker, Maria Gini, and Onn Shehory Conference name: 12th international conference on Autonomous agents and multi-agent systems (AAMAS 2013) Conference location: Saint Paul, Minnesota, United States Conference dates: 06-10 May 2013 Series title and volume: Vol. 2 Publisher: International Foundation for Autonomous Agents and Multiagent Systems Year: 2013 Pages: 1117-1118 URL: http://hdl.handle.net/1959.3/378574 Copyright: Copyright © 2013. International Foundation for Autonomous Agents and Multiagent Systems. All rights reserved.This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Proceedings of the 2013 international conference on Autonomous agents and multi-agent systems, vol. 2, 2013, http ://dl.acm.org/citation.cfm?id=2484920.2485099&c oll=DL&dl=ACM This is the author’s version of the work, posted here with the permission of the publisher for your personal use. No further distribution is permitted. You may also be able to access the published version from your library. The definitive version is available at: http://dl.acm.org/citation.cfm?id=2484920.24850 99&coll=DL&dl=ACM Swinburne University of Technology | CRICOS Provider 00111D | swinburne.edu.au Powered by TCPDF (www.tcpdf.org) A distributed protocol for collective decision-making in combinatorial domains Minyi Li, Quoc Bao Vo, and Ryszard Kowalczyk Swinburne University of Technology Abstract. In this paper, we study the problem of collective decision- making over combinatorial domains, where the set of possible alternatives is a Cartesian product of (finite) domain values for each of a given set of variables, and these variables are not preferentially independent. Due to the large alternative space, most common rules for preference aggregation cannot be directly applied to compute a winner. In this paper, we focus on a particular rule, namely Smith/Minimax. We introduce a distributed pro- tocol for collective decision-making, which enjoys the following desirable properties: i) the final decision chosen is guaranteed to be a Smith mem- ber, and is sufficiently close to the Smith/Minimax candidate; ii) it enables distributed decision-making and works under incomplete information settings; iii) it significantly reduces the amount of dominance testings (in- dividual outcome comparisons) that each agent needs to conduct, as well as the number of pairwise comparisons; iv) it is sufficiently general and does not restrict the choice of preference representation languages. 1 Introduction In this paper, we focus on the problem of collective decision-making in combina- torial domains, where the set of candidates is a Cartesian product of finite value domains for each of a given set of possibly inter-dependent variables. For in- stance, a research group plan to order several personal computers (PCs) and the group members need to decide on a common group PC configuration. Consider for example that the common PC configuration to be agreed on is composed of the processor, hard disk, RAM and operating system, with a choice of six possi- ble options for each, which makes 64 = 1296 candidates. The agents’ preferences on variables and the collective decisions are not independent, because, perhaps, the preferred operating systems may depend on the given processor type. The inter-dependency between variables and the exponential alternative space in such domains lead to a big challenge of computing the social aggregation rules. When the domain is combinatorial, a straightforward way to compute a social decision rule might be using issue-by-issue (a.k.a. seat-by-seat) sequential election. However, as soon as the variables are not preferentially independent, it is very likely that deciding on the issues separately will lead to suboptimal choices [4]. The only way to avoid inefficiency is ”deciding on combinations [of variable values]” [4]. Nonetheless, most common aggregation methods need a number of operations at least linear (sometimes even quadratic or exponential) 2 Minyi Li, Quoc Bao Vo, Ryszard Kowalczyk in the number of possible alternatives [9]. When the set of alternatives has a combinatorial structure and the agents’ preferences are described in a compact representation language, generating the whole relation from those inputs and directly applying such rules to compute a collective decision is impractical. Collective decision-making in combinatorial domains has been studied in a number of papers. Many of them consider logical or structured preferences, and mainly focus on analysing negative complexity results or finding reason- able restrictions on the preference relations such that sequential winners and direct winners always coincide, e.g., [10, 13, 12]. Some later works [12, 11] relax those restrictions and introduce the notion of local Condorcet winners 1. They also propose (and implement) algorithms for computing them. Nevertheless, there is still another problem that has not been addressed: with a limited number of agents and a large alternative space, collective preference cycles (i.e., para- doxes) generally occur. Therefore, investigating techniques to generate a “good” alternative in the presence of paradoxes is certainly of interest in the field. In this paper, we address the aforementioned problems by proposing a proto- col that enables the agents to identify efficient collective decisions. The proposed protocol enjoys the following desirable properties: i) Regarding the social prop- erties, the final chosen collective decision is guaranteed to be a Smith member. Moreover, we show experimentally that the collective decision chosen by the proposed protocol is sufficiently close to the Smith/Minimax candidate i.e., the Minimax candidate in the Smith set. ii) From the perspective of computation and communication, the proposed protocol enables the agents to make propos- als on partial assignments (i.e., an assignment of a subset of variables) until all variables are assigned. As a result, it does not require counting candidates as employed in a majority of simple aggregation mechanisms [8]. It reduces the total number of alternatives that each agent needs to consider in practice, and requires significantly fewer outcome comparisons compared to an exhaustive search in most decision-making instances. iii) Last but not least, the proposed protocol supports real-world applications. It works under incomplete informa- tion settings and does not require the agents to submit their preferences; and thus, it can be implemented in a distributed manner. Moreover, the proposed protocol is sufficiently general and applicable to most preference representation languages in combinatorial domains. There are no restrictions on the agents’ preferences: their preference orders could be complete or partial, and they can extend certain structural properties, e.g., CP-nets, soft constraints, etc. 2 Preliminaries 2.1 Combinatorial domains and preferences Let V = X1;:::; Xm be a set of m variables in a combinatorial domain, where each variablef X takesg values in a local variable domain D(X). A variable X is 1 A local Condorcet winner is an alternative that beats all its neighbours. And a neigh- bour of an alternative is another alternative that differs only on a single variable from that alternative. Collective decision-making in combinatorial domains 3 binary if D (X) = x; x¯ . An alternative is uniquely identified by the combination of variable values.f Hence,g the possible alternatives (outcomes) are D(X ) 1 × · · · × D(Xm), denoted by O. If X = X ;:::; X and X V, with σ < < σ then DX f σ1 σ` g ⊆ 1 ··· ` denotes DX DX and x denotes an assignment of variable values to X, σ1 × · · · × σ` i.e., x DX. If X = V, x is a complete assignment (corresponds to an alternative); otherwise2 x is called a partial assignment. We allow concatenations of disjoint sets of variable values: for instance, let X = A; B , Y = C; D , x DX and x = ab¯, f g f g 2 y DY and y = cd¯, then xye¯ denotes an assignment abc¯ d¯e¯. If X Y = V, we call xy2a completion of assignment x. We denote by Comp(x) the set[ of completions of x. A preference order is a reflexive, transitive and antisymmetric relation on (recall that is antisymmetric iff x; y , x y and y x implies x = y). X denotes the strict relation induced8 from2 X: x y iff x yand not y x. An order is complete (linear order) iff , eitherx y or y xholds for all x; y ; otherwise, it is a partial order, i.e., there existx; y , such that neither x2 Xy 2 X nor y x holds, written as x Z y (incomparable). Since direct assessment of the preference relations in combinatorial domains is usually infeasible, several compact languages have been proposed to avoid the exponential blow up. Some of these languages are cardinal, e.g., utility net- works [1] and soft constraints [7, 5]; some others are purely qualitative, e.g., CP- nets [2] and CI-nets [3]. Some representation models are based on conditional preferences, for instance CP-nets [2] and its variants; some others are based on propositional logic (or possibly a fragment of it), for instance prioritized goals, distance-based goals, weighted goals, bidding languages for combinato- rial auctions. For brevity, we don’t present an exhaustive list here. The reader is referred to Lang ([9]) for a detailed survey of compact preference representation languages in combinatorial domains. In the following, we only briefly describe CP-nets, a widely studied compact graphical language for representing quali- tative preferences in combinatorial domains. A Conditional Preference Network (CP-net) [2] over a set of variables V is an annotated directed graph , in which nodesN stand for the variables. Each node X is annotated with aG conditionalN preference table CPT(X), which associates a strict total order over DX given every combination values u of parents Pa(X) in .
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