Content

1. Concepts �: Basic Model of Exchange Problem

​

​

• Model with Externalities �: Assumptions for Externalities and Some Literatures

​

​

• Project

​

1. Concepts

Concepts

1.Housing Market� (Simple Model)

Concepts

2. Core

Concepts

3. Trading cycle

Concepts

4. TTC algorithm

Developed by David Gale, introduced in Shapely and Scarf(1974)

​

Step k: Each remaining agents points to the agent who follows him in a trading cycle he prefers most and check whether each cycle formed is a TTC for the remaining agents.

①If there is at least one, trade according to the TTCs and remove the agents. If there remains at least one agents, proceed to the next step. Otherwise, stop

②If there’s no TTC, it fails

​

Concepts

5. An Example of TTC algorithm

Step1

Step2

2

5

4

3

1

5

4

TTC: (1,2,3)

TTC: (4,5)

Concepts

6. Preferences

Only cares about the trading cycle he belongs to

Same trading cycle => Feels same

Cares about his own allotment prior to others

Standard

1^st: Agents in the trading cycle he belongs to

2^nd: Other agents

Same house, same preference

Hedonic

Egocentric

Trading-cycle-lexicographic

Demand-lexicographic

(Supply)

1^st: House(Agent), 2^nd: Agent(House)

Separable

Separate demand and supply side preferences by applying each order as is when the other one is same

Concepts

4. Preferences

Standard

Separable

Trading cycle lexicographic

Hedonic

Demand Lexicographic

Egocentric

2. Model with externalities

Model

Housing Market with externalities

• For example, temporal exchange problems(e.g. vacation house exchange program), kidney exchange
• In the case of house exchanging, who uses my house is also important in decision making
• In the case of kidney exchanging, suitability of the recipient is also important to doners

Literature

Shapley and Scarf 1974 [On Cores And Indivisibility]

Model

Results

An exchange economy with finite N agents in which each agents initially owns an indivisible commodity (ex. House) and has use for 1 and only 1 house

The market in question has a non-empty core

In the basic exchange model,

if there aren’t any externality, then nonempty core always exist

Literature

Klaus and Meo 2021

Model

Results

Demand-lexicographic

(Supply)

Primarily cares about the house he receives and only secondarily about who receives his house and vice versa in supply-lexicographic

House is acceptable if the agent prefers others’ house to his

Agent is acceptable if the house owner prefers other agents to himself

Acceptable

Separable preference

Order of preference b/w houses keeps same in preference b/w house-agent pairs

5. Project

Question

First, try to find an appropriate variation of TTC

1. Apply TTC to demand side preference and supply side preference each and check if they are same
2. Apply TTC according to the preference b/w the house-agent pairs
3. Apply a variation algorithm

Approach Using TTC

Demand side TTC and supply side TTC yield same core in set 1

1. Apply TTC to demand side preference and supply side preference each

Demand side TTC and supply side TTC could yield different results in set 2

To address the problem,

Let the agents point as a pair during the algorithm

Also same core in set 1

2) Apply TTC according to the preference b/w the house-agent pairs

Can fail in set 2

So, we need to narrow down the preference domain

Assume Demand(or supply)-lexicographic preference

But then, the results are same as 1) so that some standards of choosing b/w D-TTC and S-TTC are needed

3) Apply a variation algorithm

An algorithm

[Step 1]

• Run D-TTC and S-TTC as usual

[Step 2]

• Check if all agents in the D-TTC(S-TTC) primarily cares about the houses(agents) assigned in the TTC

[Step 3]

• If there are D-TTC and S-TTC that overlap with each other, choose and implement one of them arbitrarily
• If there are D-TTC and S-TTC that coincides with, implement trade like that

​

​

Let the domain of preferences be the preference profiles that yields a matching in the above algorithm

​

Now checking the coverage of above algorithm

2

3

1

4

Formal Def of preferences

Standard

Hedonic

Egocentric

Formal Def of preferences

Demand-

lexicographic

Separable