Conclusion

Our endeavour to provide a systematic view of the utility within a threshold maximization paradigm and its relations with preference and choice models has been completed. Throughout this study we considered the rational choice paradigm defined by maximization of a utility within a threshold, i.e., presented in the form 6.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGdbGaaiikamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hfayjaa % cMcacqGH9aqpcaGG7bGae8xEaKNaeyicI4SaamiwaiaacYhatuuDJX % wAK1uy0HwmaeXbfv3ySLgzG0uy0Hgip5wzaGGbaiab+rGiZJqaaiaa % 9bcacqWF4baEcqGHiiIZcaWGybWdaiaayEW7iuaacqaFZbWCcqaF1b % qDcqaFJbWycqaFObaAcaqFGaGaeWhDaqNaeWhAaGMaeWxyaeMaeWhD % aqNaaG5bV-qacqWF1bqDcaGGOaGae8hEaGNaaiykaiabgkHiTiab-v % ha1jaacIcacqWF5bqEcaGGPaGaeyOpa4JaeqyTduMaaiyFaaaa!70B2! $$C(X) = \{ y \in X|\nexists x \in X{\mkern 1mu} such that{\mkern 1mu} u(x) - u(y) > \varepsilon \} $$ where the threshold ε is given in the forms % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacM % cacaaMc8UaeqyTduMaeyypa0JaaGimaiaacYcaaaa!3D27!$$ a)\,\varepsilon = 0,$$ % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacM % cacaaMc8UaeqyTduMaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaa % dshacaWGHbGaamOBaiaadshacaGGSaaaaa!4400! $$ b)\,\varepsilon = constant, $$ % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaacM % cacaaMc8UaeqyTduMaeyypa0JaeqyTduMaaiikaiaadIhacaGGPaGa % aiilaaaa!406C! $$c)\,\varepsilon = \varepsilon (x), $$ % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaacM % cacaaMc8UaeqyTduMaeyypa0JaeqyTduMaaiikaiaadIhacaGGSaGa % amyEaiaacMcacaGGSaaaaa!421B! $$ d)\,\varepsilon = \varepsilon (x,y), $$ % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiaacM % cacaaMc8UaeqyTduMaeyypa0JaeqyTduMaaiikaiaadIfacaGGPaGa % aiilaaaa!404E! $$e)\,\varepsilon = \varepsilon (X), $$ % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacM % cacaaMc8UaeqyTduMaeyypa0JaeqyTduMaaiikaiaadIhacaGGSaGa % amyEaiaacYcacaWGybGaaiykaiaacYcaaaa!43AA! $$f)\,\varepsilon = \varepsilon (x,y,X), $$.