Prove that, in a trading problem, there is a core allocation...

Prove that, in a trading problem, there is a core allocation in which every trader gets the item he ranks number 1 if and only if the digraph $D^{1}$ constructed in the proof of Theorem $13.1 .9$ consists of directed cycles, no two of which have a vertex in common.

Key Concepts

Core Allocation

A core allocation is an outcome of a trading problem where no coalition of traders can reallocate their items among themselves in a way that makes everyone in the coalition strictly better off. It represents a stable allocation where no group can deviate to improve their individual positions, ensuring that the allocation is resilient against any blocking coalition.

Trading Problem

A trading problem involves a set of traders or agents, each endowed with an item and having preferences over the items possessed by others. The objective is to reassign these items among the traders in a manner that maximizes their individual satisfaction while ensuring no subgroup of traders can rearrange the allocation to all benefit, thereby achieving stability.

Directed Graph Representation

In analyzing trading problems, a directed graph (digraph) is used where vertices represent traders and directed edges indicate preferences, typically pointing from a trader to the trader holding his most preferred item. This graphical representation helps in visualizing and identifying possible trade cycles that can resolve the allocation problem.

Directed Cycles and Disjoint Cycles

A directed cycle in this context is a sequence in which each trader prefers the item held by the next trader in the cycle, eventually looping back to the starting trader. When such cycles are disjoint—meaning no trader appears in more than one cycle—they guarantee that each trader in these cycles can obtain his top choice without conflicts. This structure is crucial for proving that there exists a core allocation where every trader receives his most preferred item.

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