Construct a core allocation for the trading problem in which...

Construct a core allocation for the trading problem in which the preferences are given by the following table: $$ \begin{array}{c|ccccccc} & t_{1} & t_{2} & t_{3} & t_{4} & t_{5} & t_{6} & t_{7} \\ \hline t_{1} & 2 & 3 & 1 & 4 & 7 & 5 & 6 \\ t_{2} & 1 & 6 & 4 & 3 & 2 & 7 & 5 \\ t_{3} & 2 & 7 & 3 & 5 & 1 & 4 & 6 \\ t_{4} & 3 & 4 & 2 & 7 & 1 & 6 & 5 \\ t_{5} & 1 & 3 & 4 & 2 & 5 & 7 & 6 \\ t_{6} & 2 & 4 & 1 & 5 & 3 & 7 & 6 \\ t_{7} & 7 & 3 & 4 & 2 & 1 & 6 & 5 \end{array} $$

Construct a core allocation for the trading problem in which the preferences are given by the following table:
$$
\begin{array}{c|ccccccc}
& t_{1} & t_{2} & t_{3} & t_{4} & t_{5} & t_{6} & t_{7} \\
\hline t_{1} & 2 & 3 & 1 & 4 & 7 & 5 & 6 \\
t_{2} & 1 & 6 & 4 & 3 & 2 & 7 & 5 \\
t_{3} & 2 & 7 & 3 & 5 & 1 & 4 & 6 \\
t_{4} & 3 & 4 & 2 & 7 & 1 & 6 & 5 \\
t_{5} & 1 & 3 & 4 & 2 & 5 & 7 & 6 \\
t_{6} & 2 & 4 & 1 & 5 & 3 & 7 & 6 \\
t_{7} & 7 & 3 & 4 & 2 & 1 & 6 & 5
\end{array}
$$

Key Concepts

Preferences

Preferences in trading problems refer to how each agent ranks the available objects or goods. These rankings are crucial for determining the desirability of outcomes in allocation procedures. In core allocation problems, the structure of these preferences ensures that no group of agents can rearrange the objects to make themselves all strictly better off without the cooperation of others.

Top Trading Cycle (TTC) Algorithm

The Top Trading Cycle algorithm is an iterative procedure used to find a core allocation in the Shapley?Scarf housing market. In each iteration, agents point to their most preferred object that is still available, and cycles (or trading loops) are identified. Once a cycle is found, those agents exchange objects among themselves, and are then removed from the market. This process continues until all agents have been assigned an object, resulting in an allocation that lies in the core.

Core Allocation

The core allocation is an outcome where no subset of agents (a coalition) can reassign the objects among themselves in a way that makes every member of the coalition strictly better off. In other words, there is no blocking coalition that can deviate from the allocation to achieve a preferable outcome for all its members, ensuring stability and efficiency in the allocation process.

Trading Problem

A trading problem involves a set of agents, each of whom owns an object and has preferences over the objects available in the market. The problem is to reallocate these objects in such a way that the allocation is stable, meaning that no group of agents would prefer to exchange amongst themselves instead of accepting the proposed allocation. This forms the basis for models of exchange in economics and game theory.

Shapley?Scarf Model

The Shapley?Scarf model is a classic framework in economics for studying exchange when each agent initially owns a single indivisible good. It provides insights into how trade can occur in a market without money, focusing on the behavior of individuals with preferences over the goods. The model underpins many results in matching theory and cooperative game theory, including the existence of a core allocation.

Transcript

00:05 Given the results of a survey taken by a bank about the investment habits of the customers.

00:31 So first we have to find y intersection b.

00:46 So here, y intersection b is all here why intersection b is all.

00:57 Say the younger age between 18 and 29 who are interested investing in bonds will be to why in section b is going to be two now next we have to find m union a m union a that is all the customers who are of middle age between 30 and 49 or who likes to all the customers who likes to have only saving accounts so that will be so all the middle -aged customers will be 14 plus 5 plus 14 that is going to be 33 which is given here and also it includes the customers who prefer having savings savings accounts so that will be 15 plus 14 is already included less 12 this can also be written as 33 plus 41 33 plus 41 minus the common one which is 14 that is going to be 74 minus 14 just 16 now for option c we have to find y intersection s union b why intersection as union b why intersection as union so as union b is going to be all the customers who likes to so it includes all the customers who invest in stocks and also the bonds so stocks or bonds that will be 52 plus 27 now why intersection s union b will be so as union b will be so as union union b will include all these elements, 61432, 2, 5 and 20.

04:42 Now, y intersection s, union b will be 6 and 2.

04:48 So it includes.

04:56 So 6 plus 2, that is 8.

04:58 This represents all the younger age people of 18 to 29 who likes to invest either in stocks or bonds.

05:09 Who invested in stocks or bonds.

05:24 Now the next option, o complement, union, s union a...

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