Suppose that, in a trading problem, some trader ranks his...

Suppose that, in a trading problem, some trader ranks his own item number $k$. Prove that, in each core allocation, that player obtains an item he ranks no lower than $k$. (Thus, a player never leaves with an item that he values less than the item he brought to trade.)

Key Concepts

Core Allocation

Core allocation is a concept from cooperative game theory in which an assignment of items to players is stable against any group deviations. In other words, no coalition of players can reassign the items among themselves so that every member of the coalition is strictly better off. This concept ensures that every player’s allocation is robust to potential improvements by any subgroup of players.

Trading Problems

Trading problems, often illustrated by models such as the Shapley?Scarf housing market, involve players who each initially own an object and have preferences over all objects. The fundamental idea is to allow mutually beneficial trades among the players. In these problems, stability notions such as the core are used to evaluate if an allocation is immune to deviations by groups of players.

Individual Rationality

Individual rationality in this context means that each player should receive an object that they find at least as desirable as the object they initially owned. If a player values his own item at a certain rank, then in any stable or core allocation, he is guaranteed to receive an object that is no worse than his own endowment. This ensures that no player is made worse off by the trading process.

Blocking Coalition

A blocking coalition is a group of players who can reassign the items among themselves so that every member of the group strictly prefers the new assignment over the current allocation. In demonstrating that a player who ranks his own item at least k will not be worse off, one typically argues by contradiction: if the player were to receive an item worse than his own, then he and possibly other players could form a blocking coalition by trading their own items to achieve a mutually better outcome.

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