Residence exchange wanted: A stable residence exchange problem

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 90 (1996)536-546

Theory and Methodology

Residence exchange wanted: A stable residence exchange problem Yufei Yuan Michael G. DeGroote School of Business, McMaster University, Hamilton, Ont., Canada L8S 4M4

ReceivedJune 1994;revisedOctober 1994

Abstract

This paper introduces a special matching problem, the stable residence exchange problem, originated from the need for residence exchange in China. The problem involves families wishing to exchange their residences based on their preferences. The stability of exchange is defined and the problem can be formed as a special case of trading indivisible goods with preferences. The solution method and the properties of the solution are discussed and the factors that may affect the performance of the residence exchange are investigated through computer simulation. Finally, the application considerations are addressed. Keywords: Matching; Multi-person game; Assignment problem; Trading; Public service

1. Introduction

A variety of problems can be classified as matching problems where pairing is made between match participants on the basis of their ordinal preferences (see Gusfield and Irving, 1989, for a general discussion). Three related matching problems have been introduced by Gale and Shapley (1962). Among them, the simplest and best known is the stable marriage problem. This problem involves pairing men and women according to their preferences towards the marriage. The unique feature which makes the problem different from the classic personnel job assignment problem is the requirement of stability for the assignment. A matching assignment is stable if there do not exist a man and a woman who prefer each other to their assigned partners.

Gale and Shapley prove that a stable solution always exists and can be found through a 'deferred-acceptance' procedure. A polygamous version of the stable marriage problem is the stable college admission problem where the matching is made between the students and college programs with a targeted number of enrolments. The stable assignment algorithm for the problem has been successfully applied to the National Resident Matching Program (NRMP) for matching medical graduates to hospital internship positions in the United States and Canada (see Graettinger and Perason 1981, and Roth, 1984). The third matching problem is the stable roommates problem where students are paired to be roommates based on their preferences and the stability is defined in a similar way. The major

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Y. Yuan / European Journal of Operational Research 90 (1996) 536-546

difference between the stable roommates problem and the stable marriage problem is that the roommate matching is formed within the same group rather than between two disjoint groups. Irving (1985) shows that a stable solution for the roommates problem does not always exist. He develops an algorithm to check the existence of a stable solution and to find it when it does exist. In this paper we study another kind of matching problem called the stable residence exchange problem. The problem involves voluntary exchange of rented residences among families based on their ordinal preferences on move-in residences. This problem originates from the needs for residence exchange in China. In China the majority of urban people do not own houses. They live in public housing with very low rent subsidized by the government. Due to the high population growth and inadequate housing development, there is serious housing shortage, especially in large cities. To apply for an accommodation from the government is very difficult, often involves many years of waiting. With no other choices, people have to search for residence exchange (actually, exchange the right of renting, not the ownership of the residence) in order to meet their special needs. For instance, they exchange residences in order to move to a location closer to their workplace or to their parents, to trade space for better facilities such as a private bathroom or a kitchen equipped with gas stoves, to trade one living place for two separate living places or vice versa because of marriage or divorce, etc. The demand for residence exchange is very high. However, looking for residence exchange is very difficult and often very frustrating. There are two major problems. One problem is that there is no efficient information channel to facilitate information exchange. To search for potential exchange, people have to depend on personal contact such as friends or relatives. They may also attend residence exchange fairs occasionally organized by government housing agencies. In such residence exchange fair, people hang a board indicating what kind of residence they have and what kind of residence they want. They look around and talk with each other hoping to find a

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exchange partner. Some residence exchange fairs were quite large. For instance, a residence exchange fair held in Beijing in May 1991 opened for 5 days and attracted 80000 person-attendances. The labour hours wasted however, were also considerable. Another problem is that mutually preferred exchange pairs seldom exist. One family may like another family's residence but not vice versa. A pair-wise exchange therefore cannot be arranged. However, a potential cycle exchange among several families may lead a good result. For instance, a cycle exchange among A, B, C (A to B, B to C, and C to A) may make everyone happy. There was a case that a cycle exchange was made among 9 families through great effort of searching and negotiation. The problem is that such a cycle exchange is very difficult to explore through individual contact. Even if a cycle exchange is possible, the parties involved may still hesitate due to the fear of losing a potentially better opportunity later on. The question therefore is to determine what is the best residence exchange and how to find it. The main objective of this paper, therefore, is to define and solve the residence exchange problem. The rest of the paper is organized as follows. In Section 2, exchange stability is defined as the criterion for residence exchange and the stable residence exchange problem is formulated as a special case of a trading problem with indivisible goods. The solution methods is discussed in Section 3, and the properties of the solution are discussed in Section 4. In Section 5, computer simulation is used to investigate which factors may affect the performance of the exchange. The application considerations of residence exchange are addressed in Section 6. To conclude, future researches are discussed in Section 7.

2. The stable residence exchange problem Assume that there are n families in a city who wish to exchange their residences for a variety of reasons. Each family is allowed to submit a preference list consisting of up to n choices with the last choice being its own residence without

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change. A residence exchange may be arranged to reallocate n residences to n families in accordance with families' preferences. If the residence exchange is restricted to be pair-wise only, the problem can be formulated as the stable r o o m m a t e s problem (Irving, 1985). A stable solution for the r o o m m a t e s problem, however, does not always exist, and the restriction of pair-wise exchanged may prevent better results for some families through cycle exchange. Without restriction of pair-wise exchange, the residence exchange can be any feasible reallocation as long as each residence is occupied by only one family and each family moves to only one of the residences on its preference list. A reallocation can always be implemented through disjoint exchange cycles where each family moves to the next one's residence in a cycle. A question therefore is to define what is the desirable or the best residence exchange. One approach, which is quite common in management science, assumes the existence of a single authority or decision m a k e r who takes care of all families' interests. For each residence reallocation, a utility is specified according to the families' preferences. The objective therefore is to maximize the total utility for the residence reallocation assignment. T h e problem can be formed as an integer programming problem with multiplicative utility converted from families' preferences (see Keeney and Raiffa, 1976, and Yuan, Mehrez and Gafni, 1992). The problem with this approach however, is that it is difficult to impose a single authority on a mass of families with conflict interests. Considering families' interests as the constraints not the objectives, a housing reallocation problem is formulated by Wright (1975) where tenants are allowed to change housing from one category to another category. The objective of reallocation, from a housing administrator's point of view, is to minimize the length of the exchange cycle. A network best path algorithm is applied to solve the problem. The model however does not allow tenants to have more than one choice and the reallocation criterion is different from the one studied here. Another approach, which is taken in this paper, treads the residence exchange as a market or

multi-player game where each family acts as an independent decision m a k e r to cooperate or compete with each other. As a desired criterion, the stability of a residence exchange can be defined as follows. Definition 1. Given families' ordinal preferences on move-in residences, a residence exchange arrangement is stable if there does not exist any family subset consisting of more than one family, in which the rearrangement of residence exchange makes at least one family better off but none worse off than their current move-in arrangement in terms of preferences. The stability indicates that no coalition among families is capable to improve their move-in choices further more regardless other families outside of the coalition. To illustrate the concept, let us look at a simple example where four families wish to exchange their residences. Assume that residence i is currently occupied by family i. The families' residence exchange preference is shown in Example A in Table 1 where the preferred residences to move in for each family are

Table 1 Examples of the residence exchange problem a Family Move-in residence preference choices 1st 2nd 3rd 4th Example A: 1

2

3

1

2 3 4

3 1 1

1 2 2

4 3 3

An instable exchange: 1 ~ 3, 3 --*1, 2 --, 4, 4 ~ 2. Stable cycle exchange: 1 ~ 2, 2 ~ 3, 3 ~ 1, 4 ~ 4. Stable pair-wise exchange: does not exist. Example B: 1 2 3 4

2 3 1 3

1 1 4 4

2 3

Stable cycle exchange: 1 ~ 2, 2 ~ 3, 3 ~ 1, 4 --*4. Stable pair-wise exchange: 1 ---,2, 2 --*1, 3 ~ 4, 4 ~ 3. The table lists each family's ordered choices on move-in residences. The numbers in the table are the family (and corresponding residence) numbers. a

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listed in the ranked order from most preferred to least preferred. For instance, on the first line, family l's first choice is residence 2, second choice is residence 3, and third choice is its own residence, i.e. no exchange. One possible residence exchange, say M', can be 1 --->3, 3--, 1, 2 ~ 4 , and 4 ~ 2 , where 1 ~ 3 means that family 1 moves to residence 3 and so on. This exchange is unstable since there exists a family subset {1 2} consisting of family 1 and 2, in which the residence exchange 1 ~ 2 and 2 ~ 1 can make both families better off (in this subset, family 1 can move to residence 2, its first choice, which is better than residence 3, its second choice, under M', and family 2 can move to residence 1, its second choice, which is more preferred than residence 4, its third choice, under M'). The exchange solution M, in which 1 ~ 2, 2 --* 3, 3 -~ 1 and 4 ~ 4, is stable. To verify the stability of this solution we can check all possible subsets of families to verify if any exchange within that subset can make further improvement. For instance, let us consider a subset {1, 2, 4} in which 1 ~ 2, 2 ~ 4, and 4 -~ 1. Although in this subset family 4 will be better off, family 2 will be worse off therefore there is no motivation for family 2 to participate in this coalition. In fact family 4 cannot join any subgroup to improve its situation without making other members in that subgroup worse off. This is the reason why in solution M family 4 cannot make any residence exchange with other families. This however does not violate the definition of stability. It is interesting to see that in this simple example no stable pair-wise exchange exists. There are 9 possible pair-wise exchange assignments, but none are stable. It should be mentioned that we should not conclude that every family should be better off or at least not worse off under stable cycle exchange than under stable pair-wise change. A case is shown in Example B in Table 1. The stable cycle exchange is 1 ~ 2, 2 -~ 3, 3 1, 4 --* 4 and the stable pair-wise exchange is 1 2, 2 -~ 1, 3 ~ 4, 4 ~ 3. Families 2, 3 are better off and family 4 is worse off in stable cycle exchange than in stable pair-wise exchange. We prefer stable cycle exchange to stable pair-wise exchange not becauase it would make every family better

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off but because it is more fair in terms of free competition. Pair-wise should not be used as a restriction to prevent some families to be better off through fair competition. We should also not conclude that every family should be better off or at least not worse off under stable cycle exchange than under any instable cycle exchange. The stable resident exchange problem defined here, can be considered as a special case of the trading model defined by Shapley and Scarf (1974) as follows. A market with n traders, each with an indivisible good to offer in trade (e.g., a house). Each trader has purely ordinal preference over the goods, and no trader has any use for more than one item. An allocation defines a one-to-one mapping between traders and items after trading. An allocation will be said to be a core allocation if there is no submarket that could have done better for all its members. Although Shapley and Scarf mentioned a house as an example of indivisible good, people in the Western society usually do not trade houses directly based on preferences as described by their model. The residence exchange in China, however, does fit into Shapley and Scarf's model well. Interpreting the families as traders, the rental rights as indivisible goods, and the residence exchange as an allocation, the stable solution of residence exchange is exactly the core allocation.

3. The forward chaining algorithm An algorithm called the Top Trading Cycle M e t h o d is suggested by Gale to find the core allocation (Shapley and Scarf, 1974). This procedure is dearly stated in Roth (1982). To make the terminology more relevant to our problem, the algorithm is renamed Forward Chaining Algorithm and is briefly explained here. Specific notations are used to describe the procedure. A notation i ~ j indicates a tentative move of family i to family j's residence. A chain, e.g. i ~ j ~ k ~ h, represents subsequential tentative moves. A cycle represents a cyclical chain. For example, in addition to the above chain, if h ~ j , then a cycle denoted as (j, k, h ) is formed. Once a cycle is

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formed, the actual residence exchange can be arranged according to the cycle. A waiting list is used to represent all the families whose move-in residences have not been arranged yet, and an arranged list represents all the families whose 'move-in' residences have been finally arranged. The forward chaining works as follows. At the beginning, all families are put into the waiting list and the arranged list is empty. Starting from one family arbitrary picked (the order is irrelevant because the solution is unique) from the waiting list, a chain extends when each preceding family in the chain tends to move to a succeeding family's residence which is its best choice from its preference list. T h e chain will continue to grow until the last family in the chain tends to move to one already in the chain. In this case a cycle is formed. T h a t cycle will be the final exchange arrangement for the m e m b e r s of the cycle. These families will be removed from the waiting list and put into the arranged list and their residence will be removed from the residence preference list of all other families in the waiting list. The cycle then will also be removed from the chain. If no family is left in the chain, a new chain should start with a family picked from the waiting list. If the rest of the chain is not empty, the last family in the remaining chain will try to move to the best choice in its remaining preference list. Since the n u m b e r of families and the n u m b e r of choices are limited, before the choices exhausted, all families will eventually be assigned to a residence, either someone else's residence or their own residence.

Table 2 Example of the Forward Chaining Algorithm a Family Move-in residence preference choices 1st 2nd 3rd 4th 5th 1

6

5

4

1

2 3 4 5 6

6 1 6 3 4

5 2 2 2 3

4 4 3 1 6

1 5 4 6

step 2

Fig. 1. An illustration of the forward chaining process.

T h e pseudo-code of the algorithm is shown in the Appendix. We illustrate the algorithm through a simple example shown in Table 2 and Fig. 1. In Fig. 1, a chain starts from family 1. Family 1 proposes to its best choice family 6 (i.e. makes a tentative move to family 6's residence), family 6 proposes to its best choice family 4, and so on. The chain is formed as 1 ~ 6 ~ 4. Since family 4 proposes to family 6 which is already in the chain, the cycle (6, 4) is formed and removed from the chain. It also indicates that the family l's previous proposal to family 6 failed. Continuing forward the remaining chain from the last m e m b e r family 1, we have 1 ~ 5 ~ 3 ~ 1, and another cycle, (1, 5, 3), is formed. After removing (1, 5, 3), the chain is empty. We have only family 2 left to start a new chain, which has no other choice but 2 ~ 2. The corresponding cycle then is (2). The residence exchange now consists of three cycles: (6, 4), (1, 5, 3) and (2). The reader can verify from this sample that no matter when and which family initiates the chaining, the solution is always the same.

6th

2 6

4. Properties o f the s o l u t i o n

5

The solution generated from the Forward Chaining Algorithm has following properties as described in T h e o r e m s 1 to 5. Most of these theorems have been proved in the literature based on game theory (see Shapley and Scarf, 1974,

a The table lists each family's ordered choices on move-in residences. The numbers in the table are the family (and corresponding residence) numbers.

Y. Yuan/ EuropeanJournalof OperationalResearch 90 (1996)536-546 Roth and Postlewaite, 1976 and Roth, 1982). However, the proofs are rather difficult to understand for those people without the background of game theory. H e r e we provide rather simple and intuitive proofs. Theorem 1. The solution generated from the For-

ward Chaining Algorithm is stable. Proof. The theorem can be easily proved by contradiction. Assume that the solution generated from the Forward Chaining Algorithm is not stable. According to the definition of stability, there must exist a family subset consisting of more than one family in which the reallocation of the residences within the subset makes at least one better off and none worse off than the currently assigned one. Since any residence reallocation can be represented as a ring, or a set of partitioned rings. With the assumptions we have made there must exist a ring which is different from the rings generated from the Forward Chaining Algorithm and every family in the ring is better than or at least the same as the assigned one. If a family can move into a better residence, it should be able to propose (i.e. make a tentative move) to that residence with success. However, based on our algorithm, every family starts proposal to its best choice, the family must have proposed to that residence before its current assigned residence and must have failed. This contradicts with our assumption. A proof based on game theory is given by Shapley and Scarf (1974).

T h e o r e m 1 leads to the following corollary. Corollary 1.1. For any instance of the residence exchange problem, a stable solution always exists.

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both solutions. Since the two solutions are not the same, the set of remaining families is not empty. There must exists a family, say i, assigned to residence j in one solution and residence k in another solution. Start from family i; each remaining family proposes to the best residence between the two residences assigned to that family in two solutions. Continue the proposal chain; a ring must be formed and every member of the ring will be better than or at least the same as the existing two solutions. Since the ring must be different from one of the two solutions, the stability of that solution will be violated. A proof based on game theory is given by Roth and Postlewaite (1976). Theorem 3. No family can be better off by misrep-

resenting its true preferences, assuming other families keep their preferences unchanged. Even when several families try to collude by misrepresenting their true preferences, it is impossible to make at least one better off and none worse off among themselves. Proof. The proof for a single family misrepresentation case is given by Roth (1982). Notice that although there are n!-I different ways to misrepresent a family's true preference, it is sufficient to examine n ways of misrepresentation with only one choice listed each time. Obviously, to be not worse off, a family should only list a choice not worse than the one it had received with true preferences. However, those choices had already been explored during the forward chaining process. To apply this argument to multiple families, the second part of the theorem then can be proved. Theorem 4. A family will not be worse off by

Theorem 2. For any instance of the residence

exchange problem, the stable solution is unique. Proof. The theorem can be easily proved by contradiction. Assume that there are two different stable solutions. We first remove all the exchange rings which are the same for both solutions. All the remaining families must have been assigned to the residences of the remaining families from

adding more choices into its preference list, while assuming other families keep their preferences unchanged. Proof. Assume that a family i adds a new choice, residence j, into its current preference list without changing the relative order of existing choices. There are two possible outcomes. If residence j is less preferred by family i to its assigned move-in

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residence say k, under the stable solution with its current preferences list, adding i will not effect the forward chaining process and the result will be the same. If j is m o r e preferred than k, family i must propose to j before proposing to k. If the ring including i and j is successfully formed, family j will be better off. If the proposal to j failed, the solution will be the same as before since other families did not change their preferences and the rings will be formed the same as before.

Theorem 5. A stable solution may not exist if indifference in families' preference list is allowed. Proof. This can be proved by giving an example. Assume there are three families. Both family 1 and family 2 prefer family 3's residence but family 3 has no difference in terms of move into family l's or 2's residence. T h e r e are two solutions: (1, 3) (2) and (2, 3) (1). None of them is stable. Family 3 will be forced to make a choice when both family 1 and 2 want to exchange residence with family 3. T h e o r e m s 1 and 2 imply that the sequence of selecting a family to initiate the forward chaining will not affect the solution. T h e o r e m 3 shows that there is no incentive to misrepresent true preferences. T h e o r e m 4 shows that there is no risk of submitting more choices. T h e o r e m 5 shows that indifference should be avoided. It is clear that the optimal strategy for each family therefore is to submit its true and strict preferences and to list more choices as long as they are acceptable. To explore a family's potential success in the residence exchange, a concept of accessibility is defined as follows.

Definition 2. For given exchange preferences of all families, a residence j is accessible by family i, if family i can move into residence j by adjusting its preferences, while assuming other families keep their preferences unchanged. Denote by A ( i ) the set of accessible residences for family i; each family can be classified as either belongs to or does not belong to A(i). To find A(i), make family i's preference list empty, then start the forward chaining procedure. All the residences on the chains which terminate at i belong to A(i). The concept of accessibility is useful in helping families to assess their strengths in competition for adjusting their preferences. One can expect to move into one of its accessible residence but will not be able to move into an unaccessible one.

5. Factors that may affect exchange results In practice it is often desirable to know what factors will affect the exchange result. These factors could be the pool size (number of families that participate in the residence exchange), the n u m b e r of choices listed by each family, etc. The performance of resulting exchange can be measured by the success rate (percentage of families being arranged for exchange) and the choice preference obtained. C o m p u t e r simulations are conducted to investigate possible relations between these variables. It is assumed in the simulation that each family's preferences are independent of other families' preferences and that every residence has an equal chance of being ranked by any family in any order.

Table 3 Simulation results: Exchange success rate with different number of choices a Success Number of choices listed by each family rate 2 3 4 5 6

10

20

50

Mean STD

0.878 0.043

0.951 0.028

0.990 0.010

0.173 0.083

0.483 0.086

0.630 0.077

0.722 0.062

0.775 0.060

a The table lists the mean and standard deviation of success rate for different scenarios where 50 families participated in residence exchange but the number of choices listed by each family is varied between different scenarios. Simulation runs for 1000 times for each scenario.

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Table 4 Simulation results: received choice distribution (%) among families with different numbers of choices listed a Move-in choice received

Number of choices listed by each family 2 3 4 5

1st 2nd 3rd 4th 5th 6-10th > 10th No change

17.3

37.6 10.7

44.2 12.9 6.2

82.7

51.7

37.0

6

10

20

50

47.4 14.1 6.6 4.0

48.7 14.7 7.0 4.1 2.8

51.0 15.8 7.9 4.6 3.0 5.7

27.8

22.5

12.2

51.5 16.5 8.2 4.8 3.2 6.3 4.5 4.9

52.1 16.4 8.2 5.0 3.3 6.5 7.4 1.0

The table lists the distribution of move-in choice received for different scenarios where 50 families participated in residence exchange but the number of choices listed by each family is varied between different scenarios. Simulation runs for 1000 times for each scenario.

T h e s i m u l a t i o n s are d o n e in two different settings. I n the first setting, the pool size is fixed at 50. T h e n u m b e r of choices that each family c a n s u b m i t is the s a m e in each s c e n a r i o b u t varied in different scenarios from 2 to 50. T h e s i m u l a t i o n r u n s for 1000 times for each scenario a n d the m e a n a n d s t a n d a r d d e v i a t i o n of the success rate are shown in T a b l e 3. It is clear from T a b l e 3 that the m o r e choices are listed, the higher the success rate. W h e n each family lists only 2 choices ( i n c l u d i n g the s e c o n d choice as n o change), the success rate is only 0.173. By a d d i n g 2 choices, the success rate j u m p s to 0.630. W h e n m o r e choices are a d d e d , the m a r g i n a l gain of success rate diminishes. However, to list m o r e choices r e q u i r e s families to do

m o r e intensive searching a n d / o r to lower the level of acceptance. T a b l e 4 shows the choice p r e f e r e n c e received w h e n different n u m b e r s of choices are listed. A l t h o u g h , as i n d i c a t e d in T h e o r e m 4, a family will n o t b e worse off by listing m o r e choices, the side effect o n o t h e r families c a n n o t be determ i n e d . A d d i n g a choice by o n e family may m a k e some families b e t t e r off a n d some families worse off. B e c a u s e of the existence of b o t h positive a n d negative effects, it is difficult to p r e d i c t the overall effect even w h e n only o n e family has a d d e d o n e choice. It might be expected that w h e n m o r e choices are listed by m a n y families, the overall choice p r e f e r e n c e received will be worse d u e to the i n t e n s i f i e d c o m p e t i t i o n . T h e simulation, how-

Table 5 Simulation results: Cycle length distribution (%) with different numbers of choices listed by each family a Exchange cycle length

Number of choices listed by each family 2 3 4 5

6

10

20

50

1 2-5 6-10 10-15 > 15

82.7 7.7 6.8 1.9 0.9

22.5 34.0 32.1 9.5 1.9

12.2 40.8 35.1 10.1 1.8

4.9 48.0 35.1 10.6 1.4

1.0 51.2 36.2 10.0 1.6

51.7 20.2 19.7 7.3 1.1

37.0 26.7 26.8 8.5 1.0

27.8 31.1 30.4 9.4 1.3

The table lists the distribution of exchange cycle length for different scenarios where 50 families participated in residence exchange but the number of choices listed by each family is varied between different scenarios. Simulation runs for 1000 times for each scenario. a

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ever, shows the opposite direction. When the number of choices increases, the percentage of families that received the first choice increases most rapidly, followed by groups with less preferred choices. Since the success rate increases significantly, it seems that the dominant effect of adding choices is to enable families to form exchange cycles that could not be formed otherwise. The positive effect seems to offset the negative effect on those families that were able to make exchanges before but lost the battle due to intensified competition. The cycle length distribution with different numbers of choices listed per family is shown in Table 5. It shows that when the number of choices listed per family increases, the largest portion of families are still making residence exchanges with short cycles ranging from length 2 to 5. Considering the cost and benefit involved in deciding the number of choices listed, about 5 to 6 choices in our case seem reasonable and satisfactory. In the second setting, simulation is carried out for scenarios with different pool sizes ranging from 6 to 1000. In all situations, each family lists 6 choices. For each scenario the simulation runs 1000 times. Table 6 shows that the success rate decreases when the pool size increases. The result seems unexpected. For small pools with size 6 or 10, the high success rate is somehow artificial because families usually cannot list 6 choices within such a small pool. When the pool size increases from 50 to 1000, more alternatives will be available for families to choose from, but the choices among families can spread widely. The

Table 6 Simulation results: Exchange success rate with different pool sizes a Success

Pool size

rate

6

10

50

200

1000

Mean STD

0.932 0.082

0.863 0.097

0.776 0.059

0.741 0.034

0.722 0.015

a T h e table lists the m e a n and standard deviation of success rate for different scenarios where 6 choices are listed by each family but the pool size is varied between different scenarios. Simulation runs for 1000 times for each scenario.

Table 7 Simulation results: Choice preference distribution (%) with different pool sizes a Move-in choice received

Pool size 6

10

50

200

1000

1st 2nd 3rd 4th 5th

64.2 13.4 7.8 4.6 3.2

58.0 14.3 7.4 4.0 2.5

49.2 14.5 7.1 4.2 2.7

46.7 14.2 6.8 3.9 2.5

45.8 13.9 6.5 3.7 2.3

a The table lists the distribution of move-in choice received for different scenarios where 6 choices are listed by each family but the pool size is varied between different scenarios. Simulation runs for 1000 times for each scenario.

fixed number of choices causes the success rate to decrease slightly for large pools. The distribution of choice preference received with different pool sizes is shown in Table 7. The percentage of families that receive the first choice for residence exchange decreases most significantly when the pool size increases from 6 to 50. The percentage decrease diminishes when the pool size further increases from 50 to 1000. However, it does not necessarily mean that the quality of the exchange will be lower since the rank order only represents relative preference. Choices with the same rank selected from a large pool usually should be better than that from a small pool if more intensive searching is possible. In summary, the simulation shows that listing more choices will improve both the success rate and the overall satisfaction level of residence exchange. Although large pool size does not directly contribute to the success rate and the average rank received, it allows families to make more and better choices and therefore indirectly improve the performance of residence exchange.

6. Application considerations To facilitate residence exchange, a residence exchange service center needs to be established. The service center should provide following basic

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functions: registration, searching, preference submission, and exchange assignment. A computerized information retrieval system with a large database can be used to register the requests for residence exchange and to facilitate the search for desirable residences. The types and features of the residences for exchange should be well recorded and represented. The use of information technology can significantly increase the size of exchange pool therefore increase the chance of finding suitable exchange partners. The computer search will be more efficient than manually browsing registration cards or spending days looking around and reading posts in residence exchange fairs. For easy access, computer network and user-friendly interface are essential. Helping people to search for right exchange candidates and make reasonable choices is the key to success. A scoring system should be created so that each residence can be evaluated based on the size, the location, or other related features. The search for exchange can be made among residences at the similar level. Visiting the residences of interests should be restricted and authorized, and charged for fees to avoid unnecessary visit. After careful consideration and consultation with service agencies, a family may submit a preference list. Once all families' preference lists are initially submitted, an accessible set can be calculated at the center for each family. Comparing the accessible set with each one's submitted preference list, it may reveal that some families may have ignored some good alternatives, while others (naturally, the majority) might have had unrealistic expectations. Some kind of feedback information can then be provided to help families to make preference adjustment. After one or several runs of preference adjustment, each family will be required to confirm its final preference list and make a legal commitment in accepting the exchange assignment within its preference list. This legal procedure is very important because if one family breaks its promise, all other families in the assigned exchange cycle will be affected. A computerized forward chaining algorithm can then be used to generate the stable solution for residence exchange. The exchange assignment can be run

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periodically to arrange exchanges for participating families. With this approach, the quality of residence exchange can be significantly improved. It should be mentioned that the implementation of a computerized residence exchange service is not simply a technical issue. The political, economical, organizational, and cultural issues, however, are beyond the scope of this paper.

7. Conclusion

In this paper we have introduced the residence exchange problem and formulated it as a special case of trading indivisible goods with preferences. By using the Forward Chaining Algorithm, stable residence exchange can be made through computerized exchange services. The exchange model can also be applied to other settings such as the exchange of rooms in a student dormitory, the rotation of people among different jobs or working places within the military, governments, or any large organization. In the definition of the stable residence exchange problem we did not consider the situation where some families want to divide (change from) one residence into two separate residences because of divorce or a child's marriage, while other families want to combine (change from) two separate residences into one in order to take care of elderly parents or grandchildren. If we restrict to the case where a 'splitting' family moves only to the residences of one 'combining' family, the model will still be valid since the preferences can still be referred to each family. For the case where a 'splitting' family moves to two residences of two unrelated families or where two unrelated families move to two residences of a 'combining' family, the preference presentation will be more complex. The residence exchange problem, in turn, will be more complicated. While it can be formulated in a more general trading model, a core may not exist (see a counterexample given by Shapley and Scarf, 1974). Further research is needed to define and solve more general residence exchange problems. Another situation is that some families may come from or move to other cities so that they

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need to be assigned to new accommodation or vacate their current residences. If a city housing administration office is responsible in both assisting residence exchange and assigning new residences, the residence exchange should be considered jointly with residence assignment. This problem is also worthy of further research.

Appendix. The forward chaining algorithm put all families in a waiting list W; while the waiting list W is not empty do begin {the beginning of a forward chaining} clean chain C; let s = a family picked from the waiting list W; put s as the first member of the chain C; while chain C is not empty do

be~n {the beginning of one forward step} let t = the last member of the chain C; let u = the best choice from t's remaining preference list P; if u = any member v of chain C then

bel~n {form a ring} clean ring R; put chain members start from v to t into ring R; put ring R into arranged list M; remove members of ring R from the waiting list W; remove members of ring R from the chain C; remove members of ring R from the preference list P of all families in the waiting list W; end; else {extend the chain} put u as the last member of chain C; end; {the end of one forward step} end; {the end of a forward chaining} output the stable exchange solution stored in the arranged list M.

Acknowledgements Financial support for this research was provided by the Natural Sciences and Engineering Research Council of Canada and the YorkNankai Exchange Program sponsored by the Canadian International Development Agency. Special thanks go to Mr. Yuhua He for introducing the application problem and to Mr. Xinde Wang for computer programming. The author is grateful to the anonymous referees for their valuable comments and suggestions on improving the manuscript.

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