1 RUM vs RRM ◮ Random Utility Maximization (RUM) Systematic Utility Utility U in = V in + ε in = β t x i t n + β c x i c n + ε in ◮ Random Regret Minimization (RRM) Systematic Regret Regret RR in = R in + ε in � J � J = R i ↔ j, t n + R i ↔ j, c n + ε in j � = i j � = i • The notion of regret is characterize by the systematic regret R in . • R in is described in terms of attribute level regret ( R i ↔ j,mn ). 4 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 RUM vs RRM ◮ Random Utility Maximization (RUM) Systematic Utility Utility U in = V in + ε in = β t x i t n + β c x i c n + ε in ◮ Random Regret Minimization (RRM) Systematic Regret Regret RR in = R in + ε in � J � J = R i ↔ j, t n + R i ↔ j, c n + ε in j � = i j � = i • The notion of regret is characterize by the systematic regret R in . • R in is described in terms of attribute level regret ( R i ↔ j,mn ). 4 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 The Attribute level regret R i ↔ j,mn Attribute \ Route 1 2 3 Travel Time 23 min. 27 min. 35 min. Travel Cost 6 euros 4 euros 3 euros ◮ R i ↔ j,mn describes the pairwise combinations of regret derived from alternatives. 5 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 The Attribute level regret R i ↔ j,mn Attribute \ Route 1 2 3 Travel Time 23 min. 27 min. 35 min. Travel Cost 6 euros 4 euros 3 euros ◮ R i ↔ j,mn describes the pairwise combinations of regret derived from alternatives. ( x jm − x i m ) Attribute \ Route j = 1 j = 2 j = 3 ( x jm − x 1 t ) Travel Time 0 4 12 ( x jm − x 1 c ) Travel Cost 0 -2 -3 ( x jm − x 2 t ) Travel Time -4 0 8 ( x jm − x 2 c ) Travel Cost 2 0 -1 ( x jm − x 3 t ) Travel Time -12 -8 0 ( x jm − x 3 c ) Travel Cost 3 1 0 5 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 The Attribute level regret R i ↔ j,mn Attribute \ Route 1 2 3 Travel Time 23 min. 27 min. 35 min. Travel Cost 6 euros 4 euros 3 euros ◮ R i ↔ j,mn describes the pairwise combinations of regret derived from alternatives. ( x jm − x i m ) Attribute \ Route j = 1 j = 2 j = 3 ( x jm − x 1 t ) Travel Time 0 4 12 ( x jm − x 1 c ) Travel Cost 0 -2 -3 ( x jm − x 2 t ) Travel Time -4 0 8 ( x jm − x 2 c ) Travel Cost 2 0 -1 ( x jm − x 3 t ) Travel Time -12 -8 0 ( x jm − x 3 c ) Travel Cost 3 1 0 ◮ Takeaway: We will define R i ↔ j,mn in terms of the attribute differences. 5 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) ◮ (Chorus, 2010) proposed the following attribute level regret: R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] ◮ R i ↔ j,mn compares alternative i with alternative j in attribute m . 6 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) ◮ (Chorus, 2010) proposed the following attribute level regret: R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] ◮ R i ↔ j,mn compares alternative i with alternative j in attribute m . ◮ � j � = i R i ↔ j,mn is the equivalent to x imn · β m in an utilitarian model. 6 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) ◮ (Chorus, 2010) proposed the following attribute level regret: R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] ◮ R i ↔ j,mn compares alternative i with alternative j in attribute m . ◮ � j � = i R i ↔ j,mn is the equivalent to x imn · β m in an utilitarian model. ◮ β m is the taste parameter of attribute m . 6 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: Attribute level regret. J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: Attribute level regret. J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i Linear sum of all attribute level regret. 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: Attribute level regret. J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i Linear sum of all attribute level regret. • From our example: M = { t, c } , J = 3 . 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: Attribute level regret. J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i Linear sum of all attribute level regret. • From our example: M = { t, c } , J = 3 . • Regret of alternative 1 ( R 1 ) will be described by: 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010) • (Chorus, 2010) proposed the following systematic regret: Attribute level regret. J M J M � � � � R in = R i ↔ j,mn = ln [1 + exp { β m · ( x jmn − x imn ) } ] (1) m =1 m =1 j � = i j � = i Linear sum of all attribute level regret. • From our example: M = { t, c } , J = 3 . • Regret of alternative 1 ( R 1 ) will be described by: � 3 � R 1 = ln [1 + exp { β m ( x jm − x im ) } ] j � = i m ∈M = ln [1 + exp { β t ( x 2 t − x 1 t ) } ] + ln [1 + exp { β c ( x 2 c − x 1 c ) } ] + ln [1 + exp { β t ( x 3 t − x 1 t ) } ] + ln [1 + exp { β c ( x 3 c − x 1 c ) } ] 7 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010):Towards the log-likelihood. Defining RR in = R in + ε in , where ε in is a type I Extreme Value i.i.d. 1 error. 8 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010):Towards the log-likelihood. Defining RR in = R in + ε in , where ε in is a type I Extreme Value i.i.d. 1 error. Acknowledging that the minimization of the random regret is 2 mathematically equivalent to maximizing the negative of the regret. 8 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010):Towards the log-likelihood. Defining RR in = R in + ε in , where ε in is a type I Extreme Value i.i.d. 1 error. Acknowledging that the minimization of the random regret is 2 mathematically equivalent to maximizing the negative of the regret. Hence, the probabilities may be derived using the Multinomial Logit: 3 exp ( − R in ) P in = for i = 1 , . . . , J (2) � J j =1 exp ( − R jn ) 8 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
1 Classical RRM (Chorus, 2010):Towards the log-likelihood. Defining RR in = R in + ε in , where ε in is a type I Extreme Value i.i.d. 1 error. Acknowledging that the minimization of the random regret is 2 mathematically equivalent to maximizing the negative of the regret. Hence, the probabilities may be derived using the Multinomial Logit: 3 exp ( − R in ) P in = for i = 1 , . . . , J (2) � J j =1 exp ( − R jn ) 4 Consequently, the log-likelihood will be described by: � N � J ln L = y in ln ( P in ) n =1 i =1 � N � J � N � J � J = − y in R in − y in ln exp ( − R jn ) (3) n =1 i =1 n =1 i =1 j =1 8 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Outline 1 Introduction 2 Differences between RUM and RRM models. Taste Parameter Interpretation in RRM models Semi-compensatory Behavior and the Compromise Effect 3 Extensions of the Classical RRM model 4 Relationships among the different models 5 Implementation 6 Download 9 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata 7 Bibliography
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. • For instance: � β m > 0 suggests that regret increases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. • For instance: � β m > 0 suggests that regret increases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Comfortable level). 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. • For instance: � β m > 0 suggests that regret increases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Comfortable level). • For instance: � β m < 0 suggests that regret decreases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. • For instance: � β m > 0 suggests that regret increases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Comfortable level). • For instance: � β m < 0 suggests that regret decreases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Total Time). 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Taste Parameter Interpretation in RRM models ◮ RUM: parameters are interpreted as the change in utility caused by an increase of a particular attribute level. ◮ RRM: parameters represent the potential change in regret associated with comparing a considered alternative with another alternative in terms of the attribute, caused by one unit change in a particular attribute level. • For instance: � β m > 0 suggests that regret increases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Comfortable level). • For instance: � β m < 0 suggests that regret decreases as the level of that attribute increases in a non-chosen alternative , in comparison to the level of the same attribute in the chosen alternative (e.g: Total Time). ◮ All in all. the parameters in RUM and RRM, are expected to have the same sign, even though their interpretation is dramatically different. 10 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Semi-compensatory Behavior and the Compromise Effect r 2 . 5 2 1 . 5 ( B ) Regret domain 1 0 . 5 ( A ) Rejoice domain − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ Attribute level regret R i ↔ j,mn with β m = 1 . 11 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Semi-compensatory Behavior and the Compromise Effect r 2 . 5 2 1 . 5 ( B ) Regret domain 1 0 . 5 ( A ) Rejoice domain − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ Attribute level regret R i ↔ j,mn with β m = 1 . ◮ ( A ) = rejoice and ( B ) = regret on an equal difference of attribute level. 11 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Semi-compensatory Behavior and the Compromise Effect r 2 . 5 2 1 . 5 ( B ) Regret domain 1 0 . 5 ( A ) Rejoice domain − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ Attribute level regret R i ↔ j,mn with β m = 1 . ◮ ( A ) = rejoice and ( B ) = regret on an equal difference of attribute level. ◮ For an equal difference of the attribute levels ⇒ regret >>> rejoice 11 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Semi-compensatory Behavior and the Compromise Effect r 2 . 5 2 1 . 5 ( B ) Regret domain 1 0 . 5 ( A ) Rejoice domain − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ Attribute level regret R i ↔ j,mn with β m = 1 . ◮ ( A ) = rejoice and ( B ) = regret on an equal difference of attribute level. ◮ For an equal difference of the attribute levels ⇒ regret >>> rejoice ◮ Linear RUM models ⇒ fully-compensatory model. 11 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
2 Semi-compensatory Behavior and the Compromise Effect r 2 . 5 2 1 . 5 ( B ) Regret domain 1 0 . 5 ( A ) Rejoice domain − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ Attribute level regret R i ↔ j,mn with β m = 1 . ◮ ( A ) = rejoice and ( B ) = regret on an equal difference of attribute level. ◮ For an equal difference of the attribute levels ⇒ regret >>> rejoice ◮ Linear RUM models ⇒ fully-compensatory model. ◮ Compromise Effect : Alternatives with “balanced” performance in all attributes are more attractive than alternatives with a severe poor performance in one attribute. 11 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Outline 1 Introduction 2 Differences between RUM and RRM models. 3 Extensions of the Classical RRM model Generalized RRM (Chorus, 2014) µ RRM (van Cranenburgh et al., 2015) Pure RRM (van Cranenburgh et al., 2015) 4 Relationships among the different models 5 Implementation 6 Download 12 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Extensions of the Classical RRM model The extensions of the classical regret model (Chorus, 2010) are derived using modified versions of the attribute level regret R i ↔ j,mn . 13 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Extensions of the Classical RRM model The extensions of the classical regret model (Chorus, 2010) are derived using modified versions of the attribute level regret R i ↔ j,mn . Attribute Level Comparison Regret Function R ∗ R ∗ ( x jm − x im ) i ↔ j,mn in Attribute Level Regret 13 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Extensions of the Classical RRM model The extensions of the classical regret model (Chorus, 2010) are derived using modified versions of the attribute level regret R i ↔ j,mn . Attribute Level Comparison Regret Function R ∗ R ∗ ( x jm − x im ) i ↔ j,mn in Attribute Level Regret ◮ ⇒ all the steps described in order to obtain the log-likelihood of the model remain constant. 13 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Extensions of the Classical RRM model The extensions of the classical regret model (Chorus, 2010) are derived using modified versions of the attribute level regret R i ↔ j,mn . Attribute Level Comparison Regret Function R ∗ R ∗ ( x jm − x im ) i ↔ j,mn in Attribute Level Regret ◮ ⇒ all the steps described in order to obtain the log-likelihood of the model remain constant. ◮ All we need to do is replace the new attribute level regret from the extended model to compute the new log-likelihood. 13 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Generalized RRM (Chorus, 2014) ◮ (Chorus, 2014) proposed a new attribute level regret : � J � M � J � M R GRRM R GRRM = i ↔ j,mn = ln [ γ + exp { β m ( x jmn − x imn ) } ] in j � = i m =1 j � = i m =1 (4) 14 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Generalized RRM (Chorus, 2014) ◮ (Chorus, 2014) proposed a new attribute level regret : New parameter! � J � M � J � M R GRRM R GRRM = i ↔ j,mn = ln [ γ + exp { β m ( x jmn − x imn ) } ] in j � = i m =1 j � = i m =1 (4) ◮ The regret function ( R GRRM ) (again) is just the sum of those attribute in level regret ( R GRRM i ↔ j,mn ) across attributes. 14 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Generalized RRM (Chorus, 2014) ◮ (Chorus, 2014) proposed a new attribute level regret : New parameter! � J � M � J � M R GRRM R GRRM = i ↔ j,mn = ln [ γ + exp { β m ( x jmn − x imn ) } ] in j � = i m =1 j � = i m =1 (4) ◮ The regret function ( R GRRM ) (again) is just the sum of those attribute in level regret ( R GRRM i ↔ j,mn ) across attributes. ◮ The new parameter ( γ ) alters the shape of the regret, and the degree of asymmetries between regret and rejoice. 14 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Generalized RRM (Chorus, 2014) ◮ (Chorus, 2014) proposed a new attribute level regret : New parameter! � J � M � J � M R GRRM R GRRM = i ↔ j,mn = ln [ γ + exp { β m ( x jmn − x imn ) } ] in j � = i m =1 j � = i m =1 (4) ◮ The regret function ( R GRRM ) (again) is just the sum of those attribute in level regret ( R GRRM i ↔ j,mn ) across attributes. ◮ The new parameter ( γ ) alters the shape of the regret, and the degree of asymmetries between regret and rejoice. ◮ Model generalized the original RRM model and also the RUM model! (how?) 14 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R GRRM 3 i ↔ j,mn at different values of γ conditional on β m = 1 . r 2 1 . 5 Regret domain 1 0 . 5 Rejoice domain γ = 1 0 0 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 0 . 5 γ = 0 . 5 ( x jmn − x imn ) 1 γ = 0 . 25 1 . 5 γ = 0 . 1 2 2 . 5 3 γ = 0 . 01 3 . 5 γ = 0 ◮ γ = 1 ⇒ Classic RRM. 15 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R GRRM 3 i ↔ j,mn at different values of γ conditional on β m = 1 . r 2 1 . 5 Regret domain 1 0 . 5 Rejoice domain γ = 1 0 0 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 0 . 5 γ = 0 . 5 ( x jmn − x imn ) 1 γ = 0 . 25 1 . 5 γ = 0 . 1 2 2 . 5 3 γ = 0 . 01 3 . 5 γ = 0 ◮ γ = 1 ⇒ Classic RRM. ◮ γ ∈ ]0 , 1[ asymmetries are present but smaller than with γ = 1 . 15 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R GRRM 3 i ↔ j,mn at different values of γ conditional on β m = 1 . r 2 1 . 5 Regret domain 1 0 . 5 Rejoice domain γ = 1 0 0 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 0 . 5 γ = 0 . 5 ( x jmn − x imn ) 1 γ = 0 . 25 1 . 5 γ = 0 . 1 2 2 . 5 3 γ = 0 . 01 3 . 5 γ = 0 ◮ γ = 1 ⇒ Classic RRM. ◮ γ ∈ ]0 , 1[ asymmetries are present but smaller than with γ = 1 . ◮ γ = 0 , no convexity ⇒ fully compensatory behavior (RUM!). 15 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) New parameter... 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) New parameter... ...the scale parameter � 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) New parameter... ...the scale parameter � ◮ The scale parameter is not identified in the RUM context. 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) New parameter... ...the scale parameter � ◮ The scale parameter is not identified in the RUM context. ◮ However, RRM models can describe a semi-compensatory behavior ⇒ identification of the µ parameter. 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 µ RRM (van Cranenburgh et al., 2015) • (van Cranenburgh et al., 2015) proposed the following systematic regret: J M J M � � � � R µ RRM µ · R µ RRM = i ↔ j,mn = µ · ln [1 + exp { ( β m /µ ) ( x jmn − x imn ) } ] in m =1 m =1 j � = i j � = i (5) New parameter... ...the scale parameter � ◮ The scale parameter is not identified in the RUM context. ◮ However, RRM models can describe a semi-compensatory behavior ⇒ identification of the µ parameter. ◮ µ is informative of the degree of regret imposed by the model, stated otherwise, how much semi-compensatory behavior we are observing in the decision makers choice behavior . 16 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ µ = 1 ⇒ Classic RRM model. 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ µ = 1 ⇒ Classic RRM model. ◮ µ → ∞ ⇒ the smaller the ratio ( β m /µ ) ⇒ the smaller the asymmetries. 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ µ = 1 ⇒ Classic RRM model. ◮ µ → ∞ ⇒ the smaller the ratio ( β m /µ ) ⇒ the smaller the asymmetries. • The model collapses into a RUM model. 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ µ = 1 ⇒ Classic RRM model. ◮ µ → ∞ ⇒ the smaller the ratio ( β m /µ ) ⇒ the smaller the asymmetries. • The model collapses into a RUM model. ◮ µ → 0 ⇒ the higher the ratio ( β m /µ ) , ⇒ the higher the asymmetries. 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
R µ RRM 3 i ↔ j,mn at different values of µ conditional on β m = 1 r µ = 0 . 05 µ = 0 . 5 2 . 5 µ = 1 2 µ = 2 1 . 5 1 µ = 15 0 . 5 − 4 − 3 . 5 − 3 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 ( x jmn − x imn ) ◮ µ = 1 ⇒ Classic RRM model. ◮ µ → ∞ ⇒ the smaller the ratio ( β m /µ ) ⇒ the smaller the asymmetries. • The model collapses into a RUM model. ◮ µ → 0 ⇒ the higher the ratio ( β m /µ ) , ⇒ the higher the asymmetries. • The model collapses into a new model: Pure RRM. 17 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in M � R PRRM β m x PRRM = (6) in imn m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 ...with transformed attributes 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn for ”positive” attributes m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 ...with transformed attributes 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn for ”positive” attributes m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 for ”negative” attributes ...with transformed attributes 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn for ”positive” attributes m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 for ”negative” attributes ...with transformed attributes ◮ We need to know the sign of the attributes a priori ! 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn for ”positive” attributes m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 for ”negative” attributes ...with transformed attributes ◮ We need to know the sign of the attributes a priori ! ◮ In some situations, this requisite is not very restrictive (e.g. price, cost). 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
3 Pure RRM (van Cranenburgh et al., 2015) µ → 0 R µ RRM ◮ For arbitrary small values of µ : lim i ↔ j,mn = R PRRM in ...linear specification! M � R PRRM β m x PRRM = (6) in imn for ”positive” attributes m =1 �� J j � = i max { 0 , x jmn − x imn } if β m > 0 x PRRM = � J (7) imn j � = i min { 0 , x jmn − x imn } if β m < 0 for ”negative” attributes ...with transformed attributes ◮ We need to know the sign of the attributes a priori ! ◮ In some situations, this requisite is not very restrictive (e.g. price, cost). ◮ This model yields the strongest semi-compensatory behavior among all the RRM family 18 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
4 Outline 1 Introduction 2 Differences between RUM and RRM models. 3 Extensions of the Classical RRM model 4 Relationships among the different models 5 Implementation 6 Download 7 Bibliography 19 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
4 Relationships among the different models GRRM �� M � R GRRM = � J m =1 ln { γ + exp [ β m · ( x jmn − x imn )] } in j � = i γ = 1 RRM �� M � R in = � J m =1 ln { 1 + exp [ β m · ( x jmn − x imn )] } j � = i µ = 1 γ = 0 µ RRM �� M � R µ RRM = � J m =1 ln { 1 + exp [( β m /µ ) · ( x jmn − x imn )] } in j � = i µ → ∞ µ → 0 RUM PRRM U in = � M = � M m =1 β m · x imn R PRRM m =1 β m · x PRRM in imn Figure: Interrelationship among the models based on parameters 20 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
4 Relationships among the different models Table: LR test for model comparison. Models Hypothesis LR statistic Distribution under H 0 � � H 0 : γ = 1 ℓ ( � θ GRRM ) − ℓ ( � RRM v.s GRRM 2 θ RRM ) 0 . 5( χ 2 0 + χ 2 1 ) H 1 : γ < 1 � � H 0 : γ = 0 ℓ ( � θ GRRM ) − ℓ ( � RUM v.s GRRM 2 θ RUM ) 0 . 5( χ 2 0 + χ 2 1 ) H 1 : γ > 0 � � H 0 : µ = 1 ℓ ( � θ µ RRM ) − ℓ ( � χ 2 RRM v.s µ RRM 2 θ RRM ) 1 H 1 : µ � = 1 ◮ ℓ ( . ) represents the loglikelihood of the model, and � θ RRM , � θ GRRM , � θ µ RRM , � θ RUM represent the full set of parameters of the classical RRM, GRRM, µ RRM and linear RUM model, respectively. 21 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
4 Relationships among the different models Table: LR test for model comparison. Models Hypothesis LR statistic Distribution under H 0 � � H 0 : γ = 1 ℓ ( � θ GRRM ) − ℓ ( � RRM v.s GRRM 2 θ RRM ) 0 . 5( χ 2 0 + χ 2 1 ) H 1 : γ < 1 � � H 0 : γ = 0 ℓ ( � θ GRRM ) − ℓ ( � RUM v.s GRRM 2 θ RUM ) 0 . 5( χ 2 0 + χ 2 1 ) H 1 : γ > 0 � � H 0 : µ = 1 ℓ ( � θ µ RRM ) − ℓ ( � χ 2 RRM v.s µ RRM 2 θ RRM ) 1 H 1 : µ � = 1 ◮ ℓ ( . ) represents the loglikelihood of the model, and � θ RRM , � θ GRRM , � θ µ RRM , � θ RUM represent the full set of parameters of the classical RRM, GRRM, µ RRM and linear RUM model, respectively. ◮ The fact that the two first hypotheses follow a different distribution from the traditional χ 2 1 , is because we are testing a null hypothesis on the boundary of the parametric space of γ (Gutierrez et al., 2001). 21 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 Outline 1 Introduction 2 Differences between RUM and RRM models. 3 Extensions of the Classical RRM model 4 Relationships among the different models 5 Implementation Syntax Outputs 6 Download 22 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata 7 Bibliography
5 Syntax randregret is implemented as a Mata-based d0 ml evaluator. The command allows to implement four different regret functions in logit form. � � � � � � randregret depvar indepvars if in group(varname) � alternative(varname) rrmfn(string) , basealternative( string ) noconstant uppermu( # ) negative( varlist ) positive( varlist ) show notrl initgamma initmu robust cluster( varname ) level( # ) � maximize options 23 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 Syntax randregret is implemented as a Mata-based d0 ml evaluator. The command allows to implement four different regret functions in logit form. � � � � � � randregret depvar indepvars if in group(varname) � alternative(varname) rrmfn(string) , basealternative( string ) noconstant uppermu( # ) negative( varlist ) positive( varlist ) show notrl initgamma initmu robust cluster( varname ) level( # ) � maximize options The command randregretpred can be used following randregret to obtain predicted choice probabilities. It is also possible to recover the linear prediction of the systematic regret from equations (1), (4) (5) or (6). � � � � randregretpred newvar if in group( varname ) � � alternatives( varname ) , proba xb 23 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 The Data ◮ Data from van Cranenburgh (2018): Stated Choice (SC) experiment. 24 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 The Data ◮ Data from van Cranenburgh (2018): Stated Choice (SC) experiment. . list obs altern choice id tt tc in 1/6, sepby(obs) obs altern choice id tt tc 1. 1 First 0 1 23 6 2. 1 Second 0 1 27 4 3. 1 Third 1 1 35 3 4. 2 First 0 1 27 5 5. 2 Second 1 1 35 4 6. 2 Third 0 1 23 6 24 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 The Data ◮ Data from van Cranenburgh (2018): Stated Choice (SC) experiment. . list obs altern choice id tt tc in 1/6, sepby(obs) obs altern choice id tt tc 1. 1 First 0 1 23 6 2. 1 Second 0 1 27 4 3. 1 Third 1 1 35 3 4. 2 First 0 1 27 5 5. 2 Second 1 1 35 4 6. 2 Third 0 1 23 6 • Three unlabeled route alternatives ( J = 3 ). 24 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
5 The Data ◮ Data from van Cranenburgh (2018): Stated Choice (SC) experiment. . list obs altern choice id tt tc in 1/6, sepby(obs) obs altern choice id tt tc 1. 1 First 0 1 23 6 2. 1 Second 0 1 27 4 3. 1 Third 1 1 35 3 4. 2 First 0 1 27 5 5. 2 Second 1 1 35 4 6. 2 Third 0 1 23 6 • Three unlabeled route alternatives ( J = 3 ). • Described by Travel Cost ( tc ) and Travel Time ( tt ) ( M = 2 ). 24 Guti´ errez, Meulders & Vandebroek: Random regret minimization models using Stata
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