[Paper Review] Elicitation for preferences single peaked on trees

Palash Dey, Neeldhara Misra|arXiv (Cornell University)|Jul 9, 2016

Game Theory and Voting Systems20 references11 citations

TL;DR

This paper studies preference elicitation for agents whose preferences are single peaked on trees—a generalization of single-peaked preferences. It shows that query complexity depends critically on the number of leaves, path cover number, and distance from path of the tree, not on parameters like diameter or maximum degree, and finds that finding a weak Condorcet winner requires significantly fewer queries than full preference elicitation.

ABSTRACT

Eliciting preferences of a set of agents over a set of items is a problem of fundamental interest in artificial intelligence in general and social choice theory in particular. Prior works on preference elicitation focus on unrestricted domain and the domain of single peaked preferences and show that the preferences in single peaked domain can be elicited by much less number of queries compared to unrestricted domain. We extend this line of research and study preference elicitation for single peaked preferences on trees which is a strict superset of the domain of single peaked preferences. We show that the query complexity crucially depends on the number of leaves, the path cover number, and the distance from path of the underlying single peaked tree, whereas the other natural parameters like maximum degree, diameter, pathwidth do not play any direct role in determining query complexity. We then investigate the query complexity for finding a weak Condorcet winner for preferences single peaked on a tree and show that this task has much less query complexity than preference elicitation. Here again we observe that the number of leaves in the underlying single peaked tree and the path cover number of the tree influence the query complexity of the problem.

Motivation & Objective

• To extend preference elicitation beyond standard single-peaked domains to preferences single peaked on trees.
• To identify which structural properties of the underlying tree influence query complexity in preference elicitation.
• To investigate the query complexity of finding a weak Condorcet winner in the single-peaked-on-tree domain.
• To determine whether the same tree parameters that affect full preference elicitation also govern the complexity of finding a weak Condorcet winner.

Proposed method

• The authors model preferences as single peaked on a tree structure, where each agent's most preferred item is a node and preferences decrease along paths from that node.
• They define and analyze three key tree parameters: number of leaves, path cover number, and distance from path, which measure structural complexity of the tree.
• They derive upper and lower bounds on query complexity for eliciting full preferences using these parameters as central factors.
• They apply similar analysis to the problem of finding a weak Condorcet winner, showing it requires fewer queries than full elicitation.
• They use combinatorial arguments and reductions to show that certain tree parameters are necessary and sufficient for bounding query complexity.
• They demonstrate that standard graph parameters like diameter and maximum degree do not directly influence query complexity in this setting.

Experimental results

Research questions

• RQ1 How does the query complexity of eliciting full preferences depend on the structure of the underlying tree in single-peaked-on-tree domains?
• RQ2 Which tree parameters—such as number of leaves, path cover number, or distance from path—most significantly affect query complexity?
• RQ3 Why do traditional graph parameters like diameter and maximum degree not play a direct role in determining query complexity?
• RQ4 How does the query complexity of finding a weak Condorcet winner compare to that of full preference elicitation in this domain?
• RQ5 To what extent do the same tree parameters govern both preference elicitation and weak Condorcet winner detection?

Key findings

• The number of leaves in the underlying single-peaked tree is a critical determinant of query complexity for preference elicitation.
• The path cover number of the tree also significantly influences the query complexity of eliciting full preferences.
• The distance from path of the tree—measuring how close the tree is to being a path—plays a key role in determining query complexity.
• Query complexity for finding a weak Condorcet winner is substantially lower than that for full preference elicitation.
• The same parameters—number of leaves and path cover number—also govern the query complexity of weak Condorcet winner detection.
• Parameters such as maximum degree, diameter, and pathwidth do not directly affect query complexity in this domain.