$3.(30)$ Matching

$($a$)(4)$ In a matching market with non$-$transferable utility $($NTU$)$ between employees and $$rms$,$

prove that it is not stable to pair every person with their least$-$preferred potential partner

$($i$.$e$.$ for both $$rms to get their least favorite employee, AND both employees to work for

their least favorite $$rm$).$

$2$

$($b$)(8)$ Consider the following matching market, again with NTU:

F $1$ F $2$ F $3$

E$12$; $20$; $40$; $2$

E$23$; $36$; $22$; $6$

E$34$; $67$; $712$; $4$

Run DAA $($deferred acceptance algorithm$)$ with $$rms proposing. What allocation do you

$$nd$?$

$($c$)(8)$ For the matching market in $($b$),$ is there any stable matching allocation in which E$3$

and F$2$ end up together?

$($d$)(10)$ Same market as in $($b$),$ but now assume transferable utility $($TU$)$ and also eliminate

E$3$ and F$3.$ Assume F$2$ o$$ers a job to E$2,$ proposing that they split their $8$ units of surplus

$50-50.$ Find all jobs $($namely surplus shares$)$ that F$1$ could o$$er to E$1$ that will make this

situation stable. $3.(30)$ Matching

$($a$)(4)$ In a matching market with non$-$transferable utility $($NTU$)$ between employees and firms, prove that it is not stable to pair every person with their least$-$preferred potential partner $($i$.$e$.$ for both firms to get their least favorite employee, AND both employees to work for their least favorite firm$).$

$($b$)(8)$ Consider the following matching market, again with NTU:

Run DAA $($deferred acceptance algorithm$)$ with firms proposing. What allocation do you find?

$($c$)(8)$ For the matching market in $($b$),$ is there any stable matching allocation in which E$3$ and F$2$ end up together?

$($d$)(10)$ Same market as in $($b$),$ but now assume transferable utility $($TU$)$ and also eliminate E$3$ and F$3.$ Assume F$2$ offers a job to E$2,$ proposing that they split their $8$ units of surplus $50-50.$ Find all jobs $($namely surplus shares$)$ that F$1$ could offer to E$1$ that will make this situation stable.