Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities

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1 Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities Miho Hong and Jaeok Park January 29, 2018 Yonsei University 1

2 Introduction

Introduction of Housing Markets Housing Markets Shapely and Scarf(1974) introduces an exchange economy in which each agent owns an indivisible object, such as a house, can own at most one object, and

3 Introduction of Housing Markets Housing Markets Shapely and Scarf(1974) introduces an exchange economy in which each agent owns an indivisible object, such as a house, can own at most one object, and seeks to trade with others. Such a market is referred to as a housing market. This model has served as a basic model for indivisible goods allocation problems and one-sided matching problems. Figure 1: Illustration of a Housing Market 2

4 Existing Results on the Core and the TTC Solution Concept Core: the set of allocations immune to coalitional deviations, which is a central solution concept for cooperative games. The key results based on previous works include: Shapley and Scarf(1974) The core of a housing market is nonempty and there exists an algorithm called the top trading cycle(tcc) algorithm that produces an allocation in the core. Roth and Postlewaite(1977) The core of a housing market is a singleton. Roth(1982) The TTC algorithm as a mechanism is strategy-proof. Ma(1994) A mechanism is individually rational, Pareto efficient, and strategy-proof if and only if it is the TTC mechanism. 3

The Top Trading Cycle(TTC) algorithm The TTC algorithm without externalities (one cares only about the house he/she was assigned(egoistic preferences)).

5 The Top Trading Cycle(TTC) algorithm The TTC algorithm without externalities (one cares only about the house he/she was assigned(egoistic preferences)). Step 1: Each agent points to the owner of the house he prefers most. There must exist a cycle of agents pointing to one another. Each agent in a cycle is assigned the endowment of the agent he pointed to and removed from the market. (a) TTC:List of Preferences (b) TTC: Step 1 4

Each agent in a cycle is assigned the endowment of the agent he points to and removed from

6 The Top Trading Cycle(TTC) algorithm Step k: Each remaining agent points to the owner of the house he prefers most among the remaining houses. Each agent in a cycle is assigned the endowment of the agent he points to and removed from the market. If there is at least one remaining agent, proceed to the next step. Otherwise, stop. (c) TTC: Step 2 (d) Resulting allocation 5

7 Motivation: The Presence of Externalities In real life, externalities are common in housing markets. For example, when choosing the house to live in, we consider not only the quality of the house but also the surrounding neighborhoods of each house. In other words, one s preference is defined over the overall allocation of houses. Therefore, it is natural to introduce externalities in housing markets to generalize the base model. Baccara et al. (2012) shows in a field experiment that there exist network externalities between faculty members assigned to different offices. Figure 2: Presence of Externalities 6

8 Failure of Existing Properties with Externalities Mumcu and Saglam(2007) give examples of when: 1. the core is empty with externalities. 2. the core is not singleton with externalities. 3. TTC mechanism cannot be applied to the market with externalities. We consider restrictions of preferences and introduce solution concepts to render the existing results robust. 7

9 Model

10 Model N: a finite set of agents, H: a finite set of houses. ( N = H 3) An allocation a A describes the house assigned to each agent and is represented by a one-to-one function from N onto H. For any allocation a A, a(i) denotes the house received by agent i, and we call it the allotment of agent i at allocation a. Each agents preferences are defined over the set of allocations instead of houses. For each agent i N, R i denotes his preference relation, which is a complete and transitive binary relation on A. A profile of agents preference relations is denoted by R = (R i ) i N. The initial endowment allocation is denoted by e A. A housing market (with externalities) is a 4-tuple N, H, R, e. 8

11 Preference Restrictions Definition We call a preference relation R i egoistic if it satisfies a I i b for any a and b such that a(i) = b(i). Definition We call a preference relation egocentric if it satisfies: each agent is not indifferent between any two allocations where he receives different houses. each agent cares about his own allotment prior to others(the size of externalities is small relative to that of utility from one s own allotment). An advantage of focusing on egocentric preferences is that we can still apply the TTC algorithm. We investigate the properties of allocations generated by the TTC algorithm. 9

12 Behavior of the Outside Agents When there are externalities, when a coalition plans to deviate, it matters how the residual agents react to the deviation. We consider two kinds of behavior of the residual agents from Hart and Kurz(1983). γ-model: the outside agents stay put with their endowments. δ-model: the outside agents trade their endowments as before whenever possible. Figure 3: Illustration of a deviation in the γ-model and the δ-model From these assumptions, we derive γ(δ)-domination and γ(δ)-core concepts. 10

13 The γ-core γ-domination An allocation b A γ-dominates an allocation a A via coalition T (or, coalition T γ-blocks a via b) in the market N, H, R, e if (i) b(t ) = e(t ), (ii) b(i) = e(i) for all i / T, (iii) br i a for all i T, and (iv) bp i a for some i T. The γ-core The γ-core of the market N, H, R, e is the set of allocations that are not γ-dominated by any allocation in N, H, R, e. 11

14 Refinement of Solution Concepts Irreversible γ-core We show that with the γ-core, the core still may be empty. Therefore, the existence of the core requires a minor weakening of the γ-core, which we call the irreversible γ-core. Consider the deviation(d) of a coalition of agents, from allocation µ to allocation µ. It may be the case that the outside agents can deviate(d ) from µ back to µ, the original allocation. We prohibit deviations like d. Stability With externalities, it may also be the case that the agents have an incentive to trade their received allotments even from an allocation of the irreversible γ-core, which is not desirable. Therefore, we additionally require allocations to be stable. Stability with externalities corresponds to Pareto Efficiency without externalities. 12

15 Our Main Results in Comparison with the Existing Results (1) Theorem I: Consider a housing market < N, H, R, e > with egocentric preferences. The TTC allocation of < N, H, R, e > is stable in < N, H, R > and is in the irreversible γ-core of < N, H, R, e >. If R has aligned interests among externality creators, then the TTC allocation of < N, H, R, e > is the unique allocation that is stable, and is in the irreversible γ-core. Existing Results Combined: Consider a housing market < N, H, R, e > with egoistic preferences. The TTC allocation of < N, H, R, e > is Pareto Efficient in < N, H, R > and is the unique allocation in the core of < N, H, R, e >. 13

16 Our Main Results in Comparison with the Existing Results (2) Proposition I Consider a house allocation problem N, H, R with egocentric preferences. An allocation a A is stable(pareto efficient) in N, H, R if and only if it is the TTC allocation of N, H, R, e for some endowment allocation e A. Proposition II The TTC mechanism ϕ T is coalitionally strategy-proof. Proposition III A mechanism ϕ is individually rational, stable, and strategy-proof if and only if it is the TTC mechanism ϕ T. 14

17 The Contribution of this Paper In two-sided matching markets, there exist a significant amount of literature studying the presence of externalities, and generalizing the existing results (see Bando et al., 2016). This has not been done in one-sided matching markets. In this paper, with reasonable restrictions on the domain of preferences and applications of new solution concepts, we generalized the existing results of the TTC allocation in house allocation problems. 15

Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities

Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities Miho Hong Jaeok Park August 2, 2018 Abstract In this paper, we incorporate externalities into Shapley-Scarf housing markets.