CHAPTER 2. THEORY 
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7 
6 
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1 
Figure 2.4: Diagram of Subject 1 in Session 2 (Exp 2B) 
Session 2 and 3 are illustrated in Figures 2.4 and 2.5, respectively. The 
preference structures are more consistent in the second and third session 
than in the first. This improvement is reflected in the decreasing number 
of strong components with size greater than 1 as well as the total number 
of k-dicycles. The partition of% results in the following expression for the 
tournament of Session 2: 
y = 2° + 22' + 323 
(2.14) 
The tournament has seven strong components of order 1 (Aj,A3,..., Ag) and 
one strong component of order 5 (42). This strong component has a single 
directed 5-cycle (2*), two directed 4-cycles (22*), and three directed 3-cycles 
(32°). This example shows how consistency of choices improved over Session 1 
and 2, and were even perfect in the last session. 
With the above decomposition technique at hand tournaments can be 
characterized in great detail on ordinal level. The number of directed cy¬ 
cles of different length as well as the number and size of strong components 
can be assessed at once. With these measures inconsistency of individual 
choice behavior can be studied quantitatively in an exhaustive way. On the 
other hand, the enumeration of all directed cycles in a digraph can be very