Mining consensus preference graphs from users' ranking data

Abstract

The group ranking problem consists of constructing coherent aggregated results from preference data provided by decision makers. Traditionally, the output of a group ranking problem can be classified into ranking lists and maximum consensus sequences. In this study, we propose a consensus preference graph approach to represent the coherent aggregated results of users' preferences. The advantages of our approach are that (1) the graph is built based on users' consensuses, (2) the graph can be understood intuitively, and (3) the relationships between items can be easily seen. An algorithm is developed to construct the consensus preference graph from users' total ranking data. Finally, extensive experiments are carried out using synthetic and real data sets. The experimental results indicate that the proposed method is computationally efficient, and can effectively identify consensus graphs.

Introduction

The group ranking decision process involves aggregating individual rankings to obtain a representative group ranking. In other words, the group ranking algorithm generates consolidated ranking results that represent the group will, preference, or decision based on decision makers' preference data. In recent decades, the group ranking problem has become an important and interesting issue in decision making [11], [15], machine learning [14], web search strategies [2], [7] and others. The essence of this problem is how to consolidate and aggregate decision makers' rankings to obtain a group ranking that represents “better coherent” ordering in regards to the decision makers' rankings.

Generally, the traditional group ranking problem can be classified using three aspects: the completeness of the user-provided preference information, the input format used to express users' preferences, and the type of compromised output results. The group ranking problem can be roughly classified into two major approaches based on the completeness of the decision maker's preference information: the total ranking approach [7], [17], [20], [21], [22], [23] and the partial ranking approach [3], [4], [8], [9], [10], [18], [19]. The former requires individuals to appraise all items (called alternatives), while the latter appraises only a subset of items. There are three typical input formats decision makers can use express their item preferences: weighting models, pair-wise comparisons, and ranking lists. All three formats have been used in previous studies to express individuals' input preferences. These formats may not be perfect, but they express user preferences reasonably well in most practical situations. Depending on the input format adopted, users are asked to rank (in the ranking list model), rate (in the weighting model), or compare (in the pair-wise comparison model) the items. After all preference data have been collected, an algorithm is applied to generate the consolidated output results. In previous research, output results could be divided into two main types. One is a total ranking list, which is an ordering list of all items that represent the achieved consensus. The other is a maximum consensus sequence, which gives the longest ranking lists of items that agree with the majority and disagree with the minority.

Unfortunately, both output formats have their own weaknesses. Most previous ranking list approaches attempted to minimize the total disagreement between multiple input rankings in order to obtain an overall ranking list that represented the achieved consensus. This disregards the fact that user opinions may be discordant and have no consensus, forcing a complete ranking result even if there is no consensus or only a slight consensus. In such a situation, what we obtain is merely the algorithm output, since different algorithms derive different ranking results due to their different designs. To overcome this weakness, Chen and Cheng [5], [6] proposed the maximum consensus approach, which generates only those maximum sequences on which users have consensus, meaning they are agreed upon by a majority of users and disagreed with by a minority of users. However, this approach may generate many maximum consensus sequences, making the results fragmented and difficult to understand and use.

Therefore, we propose a method that finds consensus preferences and represents these relationships as a graph. This is called a preference graph, where the relationships are agreed upon by majority of users and disagreed with by only a minority of users. Accordingly, we develop algorithms to discover preference graphs from users' ranking lists, and use the graphs to present the preferences of all users.

Example 1

Suppose we have the three ranking lists shown in Table 1. We will show their consolidated results in the ranked order, maximum consensus sequence, and preference graph formats.

Using a total ranking list, we may get the result {A ≥ C > B ≥ D > E}, which represents a coherent ranking of all items. There is no consensus, however, on the rankings of A and C in the preference data. The reason they are arranged that way is simply because we are forced into a complete ranking.

Using maximum consensus sequences, the longest patterns of coherent item rankings are {A ≥ D > E} and {C > B > E}. The problem is that it may output many consensus sequences that need to be checked. Additionally, the preference between items A and C is unknown.

Using our approach, the result may look like Fig. 1. Items in the same cluster are similarly preferred by users. Therefore, items A and C are similarly preferred, and B and D are similarly preferred. Additionally, G₁ is preferred more than G₂ and G₃, and G₂ is preferred more than G₃. There are several advantages to this approach. First, the graph is built based on users' consensuses. Second, the graph can be intuitively understood. Third, relationships between items can be observed from the graph.

This paper is divided into six sections. Our motivations are discussed in Section 1. Section 2 reviews related works. Section 3 defines the problem of mining consensus preference graphs and provides definitions. Section 4 introduces the preference graph mining algorithm. Experimental results are presented in Section 5. Finally, we draw conclusions in Section 6.