Preference Modelling - Roubens, Marc

. .

Résumé :
The following scheme summarizes the different families introduced in this chapter and the connections between them. Family of interval orders f Row-homogeneous Column-homogeneous Family of family of interval semi orders family of interval orders orders Homogeneous family of i nterva 1 orders Homogeneous family of semi orders Family of weak orders 85 5.13. EXAMPLES We let to the reader the verification of the following assertions. Example 1 is a family of interval orders which is neither row-homogeneous nor column-homogeneous. Example 2 is a column-homogeneous family of interval orders which is not row-homogeneous but where each interval order is a semiorder. Example 3 is an homogeneous family of interval orders which are not semiorders. Example 4 is an homogeneous family of semi orders . . 8 ~ __ --,b ~---i>---_ C a .2 d c Example Example 2 .8 .6 c .5 a 0 a d Example 3 Example 4 5.14. REFERENCES DOIGNON. J.-P ?? Generalizations of interval orders. in E. Degreef and J. Van Buggenhaut (eds). T~ndS in MathematiaaZ PsyahoZogy. Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1984. FISHBURN. P.C., Intransitive indifference with unequal indifference intervals. J. Math. Psyaho.~ 7 (1970) 144-149. FISHBURN. P.C., Binary choice probabilities: on the varieties of stochastic transitivity. J. Math. Psyaho.~ 10 (1973) 327-352.

Sommaire:
1: Binary Relations: Definitions, Representations, Basic Properties.- 1.1. Binary relations.- 1.2. Graph representation of binary relations.- 1.3. Coding the binary relations.- 1.4. Matrix representation of binary relations.- 1.5. Basic properties of binary relations.- 1.6. Particular binary relations.- 1.7. Graph interpretation of the properties.- 1.8. Algebraic interpretation of the properties.- 1.9. References.- 2: The Concept of Preference Structure.- 2.1. Preference, indifference, incomparability.- 2.2. Preference structure.- 2.3. Important agreement.- 2.4. Characteristic relation of a preference structure.- 2.5. Graph representation of a preference structure.- 2.6. Coding the preference structure.- 2.7. Example.- 2.8. References.- 3: Usual Preference Structures.- 3.1. Tournament structure.- 3.2. Total order structure.- 3.3. Weak order structure.- 3.4. Total interval order structure.- 3.5. Total semiorder structure.- 3.6. Partial order structure.- 3.7. Quasi order structure.- 3.8. References.- 4: Two New Preference Structures.- 4.1. Partial interval order structure.- 4.2. Partial semiorder structure.- 4.3. References.- 5: Complete Valued Preference Structures.- 5.1. Definition.- 5.2. Important remark.- 5.3. Particular case.- 5.4. Graph representation.- 5.5. Matridal representation.- 5.6. Particular complete valued preference structures.- 5.7. Binary relations and various properties related to a complete valued preference structure.- 5.8. Characterizations of the families defined in section 5.6...- 5.9. Functional representation of a valued preference structure.- 5.10. Roberts homogeneous families of semiorders.- 5.11. Families of weak orders.- 5.12. Summary.- 5.13. Examples.- 5.14. References.- 6: Complete Two-Valued Preference Structures.- 6.1. Introduction.- 6.2.Two-valued preference structures with constant thresholds.- 6.3. Example.- 6.4. References.