Full text: Algebraic decomposition of individual choice behavior

2.2. DECOMPOSITION INTO STRONG COMPONENTS 
47 
and the partition of the associated polynomial expression is illustrated. This 
section also gives an idea how quantitative and qualitative information can 
be extracted from adjacency matrices if they are decomposed into irreducible 
submatrices. 
Note that the following (0,1)-matrix A does not contain artificial data 
but describes the preferences of Subject 1 in Session 1 of Experiment 2B (see 
Chapter 3). 
0 110 1001000 1 
0000000 00 0 0 1 
O10111111101 
110010010001 
0 100000000 0 1 
110110111001 
A = 
110110010001 
0 10 0 1000000 0 
110110110001 
110111111001 
11111111110 
0 0 0 0 0 0 0 1 0 0 0 0 
The index of the rows and columns refers to the number of the twelve lotteries 
as listed in Table B.10. The subject made choices between all possible pairs 
of lotteries in a forced choice pair comparison resulting in a tournament. The 
entry a;; in matrix A is 1 if the subject preferred the Lottery i in row i over 
the Lottery j in columnj and 0 otherwise. By simultaneous permutations of 
the rows and columns according to their row sums, the following Frobenius 
normal form of the tournament can be found. 
1 1 
111111 
1 
10 
1 1 1 1 
1 0 
0 1111 
0 
1 11 1 
0 011111 
1 1 1 1 
0001111 
1 
0 0 0 0 1 1 1 
1 1 1 
0 0 0 0 0 1 1 
1 1 11 
PAPT = 
0 00000 0 
1111 
10000 10 
1 1 11 
0 1 0 11 
0 0 000 0 0 
0 
0 0 1 1 
0 0 0 0 0 0 0 
00 000 0 0 
100 0 
00 00000000 10 
The irreducible submatrices along the diagonal, which correspond to the 
strong components of the tournament, are indicated by square brackets.
	        
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