CHAPTER 2. THEORY 
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2.5 Summary 
In this chapter we defined intransitive choice as k-dicycles and discussed the 
general algebraic decomposition of digraphs into strong components which is 
related to the identification of all k-dicycles. The decomposition into strong 
components can be achieved by matrix operations and corresponds to the 
factorization or partition of associated polynomials. The partition has the 
advantage that it completely characterizes the intransitivities within a di- 
graph. The coefficients of the partitioned polynomial y equal the number of 
dicycles in each strong component. Associated with the problem of identi¬ 
fying a minimal set of critical arcs that are responsible for all dicycles in a 
digraph is the so called acyclic subgraph problem. For tournaments a simple 
matrix technique was suggested and examples of decompositions into strong 
components were presented. 
In a more specific model the ear decomposition was introduced. This 
technique is known to be efficient, and has a minimal solution. The ear 
decomposition by sequence finds a unique set of dicycles that constitutes 
a directed ear basis in a suitable space of incidence vectors. By using the 
sequence of intransitive choices in a pair comparison a unique basis can be 
identified. 
In a digression related to the ear decomposition by sequence the comple¬ 
tion by cuts on the sequence of choice-trials was suggested. This is a simple 
technique which can be performed on subchains associated with any family of 
intransitive dicycles. It leads to subsets of choice-trials which may be respon¬ 
sible for most intransitivities. Some aspects of the completion by cuts were 
discussed which may have implications on the detection of critical choices. 
One might presume that we are now well-equipped with empirically test- 
able assumptions which can be derived from the theoretical models. At this 
point, however, a few cautious words may be appropriate. The algebraic de¬ 
composition as outlined in Section 2.2 is not a model of choice but an exhaus¬ 
tive descriptive characterization of individual choice behavior. It therefore 
offers limited opportunity for testing except for comparative purposes. As 
mentioned before, the completion by cuts introduced in Section 2.4 has no 
computer-based implementation and was not applied. The ear decomposition 
is based on the sequence of intransitive choice-trials. Consequently, it should 
be sensitive to systematic changes of the sequence of choice-trials in a pair 
comparison. This specific assumption was tested in the next chapter. The 
completion by cuts makes a slightly stronger assumption about the sequence