Suppose that an agent$$s$2-($Incompleteness and Status Quo Bias$)$ Suppose that an agent's

choice context is defined by a pair $(B,e)$ where $B$ is a menu and einB is the

default option in the menu. She possesses a transitive but potentially incomplete

preference. Her choice in context $(B,e),$ written as $C(B,e),$ is determined by

a two step procedure. In the first step, she determines the set of undominated

alternatives in the menu $($recall that ainB is undominated in $B$ if there is no bin

$B$ such that $b>-a.$ Denote these by $C_1(B).$ In the second step, she chooses the

default $e$ uniquely if $e$ is undominanted, otherwise she picks any undominated

alternative: $C(B,e)=\{e\}$ if $ein{C}_1(B),$ otherwise $C(B,e)=C_1(B).$ That

is$,$ she picks the default uniquely whenever it is undominated, otherwise her

choice just follows the undominated choice hypothesis. Intuitively, she deals

with incompleteness by choosing the default option $($when it is undominated$).$

$($i$)$ Prove that this model satisfies Sen's alpha.

$($ii$)$ Show that this model exhibits a default effect$/$status quo bias in the

following sense: for any $a,$einB with $ae,$

ainC$(B,e)LongrightarrowC(B,a)=\{a\}.$

This says that an alternative becomes more "special" when it becomes a default

option, in that if $a$ is picked in a menu when it is not a default option, then it

is uniquely picked when it is made the default option.

$($iii$)$ Does this model serve as a satisfactory model of choice overload?

choice context is defined by a pair $($B$,$ e$)$ where B is a menu and e in B is the

default option in the menu. She possesses a transitive but potentially incomplete

preference. Her choice in context $($B$,$ e$),$ written as C$($B$,$ e$),$ is determined by

a two step procedure. In the first step, she determines the set of undominated

alternatives in the menu $($recall that a in B is undominated in B if there is no b in

B such that b a$).$ Denote these by C$1($B$).$ In the second step, she chooses the

default e uniquely if e is undominanted, otherwise she picks any undominated

alternative: C$($B$,$ e$)=\{$e$\}$ if e in C$1($B$),$ otherwise C$($B$,$ e$)=$ C$1($B$).$ That

is$,$ she picks the default uniquely whenever it is undominated, otherwise her

choice just follows the undominated choice hypothesis. Intuitively, she deals

with incompleteness by choosing the default option $($when it is undominated$).$

$($i$)$ Prove that this model satisfies Sen$$s alpha.

$($ii$)$ Show that this model exhibits a default effect$/$status quo bias in the

following sense: for any a$,$ e in B with a $6=$ e$,$

a in C$($B$,$ e$)=$ C$($B$,$ a$)=\{$a$\}.$

This says that an alternative becomes more $$special$$ when it becomes a default

option, in that if a is picked in a menu when it is not a default option, then it

is uniquely picked when it is made the default option.

$($iii$)$ Does this model serve as a satisfactory model of choice overload?