PROBLEM $(5)(34$pts$)($Matching Markets$)$

PART $1($One Sided Matching Markets$)7$ workers $($labeled A to G$)$ in a firm each currently assigned the task $A1,B2,$dots,$G7(7$ tasks in total$)$ are considering a rearrangement of tasks among themselves. The matrix on the left summarizes each worker's preferences on the tasks. $($a$)(8$pts$)$ Derive the TTC $($Top Trading Cycles$)$ matching for this market.

$($b$)(4$pts$)$ Let's try to implement the matching you found in $($a$),$ through a price mechanism. We assign a price for each task, so that a worker's endowed task's price would be her "wealth", and she will be in the market to "buy" the best $($according to her preferences$)$ task that costs

$\backslash$table$[[$Person:$,A,$B$,C,D,E,F,$G$],[1^{st}$ choice,$6,6,1,3,2,3,1],[2^{nd}$ choice,$7,1,6,1,4,6,2],[3^{rd}$ choice,$5,4,2,6,3,1,3],[4^{th}$ choice,$4,7,4,2,7,7,4],[5^{th}$ choice,$3,5,7,7,6,5,5],[6^{th}$ choice,$2,3,5,5,5,4,6],[7^{th}$ choice,$1,2,3,4,1,2,7]]$

less than her wealth. Finally, after market settles with each worker's purchase, they will end up with the matching under the TTC matching in $($a$).$

Calculate the price of each house that would work. Verify numerically that worker B indeed buys the best task she can, among the ones that is priced at most as her own endowed task $2.$

$($c$)(3$pts$)$ Forget about $($a$),($b$)$ and the initial task ownership $($nobody is assigned any task, initially$).$ What matching would serial dictatorship $($with priority order A$-$B$-$C$-..-$G$)$ result in the original market in $($a$)?$

$($d$)(5$pts$)$ What should be the "priority$/$power order" so that when we run the serial dictatorship with that order we would have the TTC outcome matching in $($a$)?$

PART $2($Two Sided Matching Markets$)$ Consider the same situation in Part $1,$ except workers have no current task assignments $($no endowments$/$ initial ownership$)$ this time. Now, the tasks also have "preferences" over workers, in the sense that a task preferring one worker over another is taken to mean that the former worker is better than the other at the task.

$($e$)(10$pts$)$ Find the worker$-$proposing GS mechanism matching.

$($f$)(4$pts$)$ Show that the TTC matching between workers and tasks you have found in $($a$)$ is not a stable matching in the two$-$sided matching market. That is$,$ find a pair of worker and a task, such that each would prefer to be matched to each other, rather than to their match in the TTC matching.

$\backslash$table$[[$task$,1,2,3,4,5,6,7$