Trading Card Game

Consider a trading card game $($TCG$)$ with n players. Each player i $=1,2,...,$ n initially receives a different

collectible card. Let us index the cards according to their owners: c$1,$ c$2,...,$ cn$,$ so that player $1$ starts with

card c$1,$ player $2$ starts with card c$2,$ etc. The players want to trade their cards with others, but there is no

money$/$currency$.$ They decide to use the following trading procedure$/$algorithm:

Round $1$: Each player points to the owner of their favorite card. Since there are finitely many players, there

will be at least one cycle. Each player in a cycle is assigned the card of the aplayer he points to and is

removed from the market with his assignment. If there is at least one player left, then proceed to the next

round.

$...$

Round k: Each remaining player points to the owner of their remaining favorite card. Since there are finetly

many players, there will be at least one cycle. Each player in a cycle is assigned the card of the aplayer

he points to and is removed from the market with his assignment. If there is at least one player left, then

proceed to the next round.

The algorithm ends when there are no players left.

$($a$)$ Prove that the algorithm ends in a Pareto Efficient assignment.

$($b$)$ Prove that the final assignment from the algorithm is the only assignment in the Core.