Main Answer:
There are 24 possible preference orderings for four alternatives satisfying transitivity.
Step-by-step explanation:
The number of possible preference orderings for four alternatives, ensuring transitivity, can be calculated using the concept of permutations. Each alternative can be ranked in four possible positions, and there are no ties or indifferences in the rankings. Therefore, the total number of permutations is 4 factorial (4!), which equals 24. This accounts for all unique ways the alternatives can be ordered while maintaining the transitive property.
In transitivity, if alternative A is preferred over B, and B is preferred over C, then A must be preferred over C. This implies that once the preferences are established for any three alternatives, the fourth is automatically determined to maintain transitivity. As a result, there are no additional unique orderings when considering the fourth alternative in the sequence. This fact reduces the total number of distinct preference orderings from 4 factorial (4!) to 24.
In summary, the main answer reflects the total number of unique preference orderings for four alternatives, given the constraints of transitivity.