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 social choice cannot be directly applied to compute a winner. In this paper, we introduce a distributed protocol for collective decision-making in combinatorial domains, which enjoys the following desirable properties: (i) the final decision chosen is guaranteed to be a Smith member; (ii) it enables distributed decision-making and works under incomplete information settings, i.e., the agents are not required to reveal their preferences explicitly; (iii) it significantly reduces the amount of dominance testings (individual 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.
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Notes
A local Condorcet winner is an alternative \(\mathbf {x}\) that beats all its neighbors. And a neighbor of \(\mathbf {x}\) is another alternative that differs only on a single variable from \(\mathbf {x}\).
We call this majority decision tree since the decisions made for the tree expansions are based on the majority rule. That means, a node in the MDTree will be expanded if a majority of agents prefer to.
A leaf node is a node that has zero child nodes.
Note that different agents may propose on different leaf nodes (possibly at different depths) in the same iteration. Therefore, when we say a node \(\varPhi \) receives a majority of agents’ proposals, these proposals may arrive at \(\varPhi \) in different iterations.
An outcome optimization query determines the set of non-dominated outcomes among the feasible outcome space with respect to an agent’s preferences.
A dominance query, given two alternatives \(o\) and \(o'\), asks whether \(o\) is preferred to \(o'\) with respect to an agent’s preferences.
Reasoning about the induced preference ordering over the outcome space may require exponential number of dominance queries.
When the CP tables are represented in a compact way, dominance testing in binary-valued acyclic CP-nets are PSPACE-complete (Goldsmith et al. 2008).
Of course, since the agents’ conditional preference tables are unknown, there is no guarantee about which order will be more efficient.
A CP-net \(\mathcal {N}\) is said to be compatible with a linear order \(\sigma =X_{\sigma _1}>\dots X_{\sigma _m}\) if the dependency graph \(\mathcal {G}_\mathcal {N}\) is compatible with \(\sigma \), i.e., \(\forall X,Y \in \mathbf {V}\) there is an edge \((X,Y)\) in \(\mathcal {G}_\mathcal {N}\) only if \(X >Y\) in \(\sigma \).
Notice that \(Alt_{con}\) is the number of different outcomes each agent encounters when she generates the MDTree. In essence, it is equal to the number of leaf nodes on the MDTree at the end of the decision-making process.
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Li, M., Vo, Q.B. & Kowalczyk, R. A distributed social choice protocol for combinatorial domains. J Heuristics 20, 453–481 (2014). https://doi.org/10.1007/s10732-014-9246-1
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DOI: https://doi.org/10.1007/s10732-014-9246-1