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The problem of fair division of surplus development rights in redevelopment of urban areas: Can the Shapley value help?
Land Use Policy xxx (xxxx) xxxx
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The problem of fair division of surplus development rights in redevelopment of urban areas: Can the Shapley value help? K. Mert Cubukcu Professor of City and Regional Planning, Dokuz Eylul University, Izmir, Turkey
ARTICLE INFO
ABSTRACT
Keywords: Game theory Cooperative games The Shapley value Urban redevelopment Fair division Landownership
Many more people are expected to live in urban areas in the coming decades, and much of the physical transformation will take place in the built-up areas of cities. Landownership is a key factor in urban regeneration, and the fair division of benefits is a major obstacle to redeveloping urban land in a cooperative context. This paper aims to show that the Shapley value, a fair division scheme developed within the cooperative game theory framework, can be used to achieve a fair division of the surplus development rights among landowners remolding smaller and irregular parcels into bigger and regular ones, resulting in higher aggregate development rights. The methodology is illustrated by a case study of three parcels selected from the Karabaglar district in Izmir, Turkey, where surplus development rights are available for landowners cooperating for land amalgamation. The findings show that the Shapley values for the landowners satisfy the core conditions of the game and landowners can benefit from the highest possible development rights and share the surplus fairly. However, the current institutional setting has to be restructured to apply such division schemes.
1. Introduction The share of people living in urban areas has surpassed the share in rural areas for the first time in 2009 (United Nations, 2010). By 2050, 68% of the world population is expected to live in cities (United Nations, 2018), and the major share of this urban growth is expected to take place in developing and poor countries, which will bring additional responsibilities to local and municipal governments (Montgomery, 2008). The problems are related to both the expanding boundaries of the city and the redevelopment of the existing parts. Traditional city centers and their surrounding ring of old neighborhoods started losing their attraction as early as the 1950s. Rural areas at the fringe of the cities have been transformed into mass housing projects through master plans (Barnett, 2018). This process has accelerated in the 1980s, giving rise to the concentration of shopping, business, and entertainment activities outside the urban core and decentralizing some of the main functions of the traditional city centers to the so-called “edge cities” (Garreau, 1991; Henderson and Mitra, 1996). The separation between old and new city became obvious, and something had to be done in the old city. As a response, the notion of urban regeneration evolved in stages: (1) reconstruction (1950s), (2) revitalization (1960s), (3) renewal (1970s), (4) redevelopment (1980s), (5) regeneration (1990s), (6) regeneration in recession (2000s). Although the contents of these stages somehow differ and the key actors
and spatial levels of activity vary, there are no clear-cut boundaries among them (Li et al., 2014; Roberts et al., 2016). The main motivation has been to redevelop the urban structure to avoid abandonment and prevent economic, social, and physical deterioration. Although urban renewal policies have been implemented to deal with a variety of urban problems including low building quality, increasing social segregation and poverty, insufficient infrastructure, and the need for public spaces, most policies have been oriented towards improving living conditions in residential areas (Kleinhans, 2004). Landownership is one of the key factors in the urban regeneration process, and ownership constraints are among the major obstacles. Fragmented or multiple ownership may be a significant barrier to coordinated redevelopment (Louw, 2008). The real estate market has three actors: (1) the developer, (2) the landowner and (3) an authority (local and/or central government) (Mu and Ma, 2007). The landowners are the suppliers of property or development rights, and developers seek land for redevelopment (Li and Li, 2007). In the absence of a governmental authority during the land assembly or acquisition phase of the redevelopment process, there are different options for the landowners and the developers. The developer may purchase the property rights from the property owners, replace old buildings with new ones, and sell them for profit. However, resettlement or displacement is not generally the desired result for many landowners, even with adequate compensation. Thus, landowners and developer may adopt a bilateral
E-mail address: [email protected]. https://doi.org/10.1016/j.landusepol.2019.104320 Received 25 March 2019; Received in revised form 20 September 2019; Accepted 20 October 2019 0264-8377/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: K. Mert Cubukcu, Land Use Policy, https://doi.org/10.1016/j.landusepol.2019.104320
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structure (Li and Li, 2007), where land readjustment is the most common approach. In this approach, the ownership of multiple irregular parcels is first pooled and then subdivided into regular urban parcels for redevelopment (Sorensen, 1999). Remolding of smaller and irregular parcels into bigger and regular ones may result in higher aggregate development rights, as bigger parcels are sometimes assigned higher development rights in development plans. The question then becomes: How to divide the surplus development rights (Li and Li, 2007)? The issue of fair division is a major obstacle to redevelop urban land in a cooperative context. Fair division is defined as the process of dividing a set of divisible goods or resources among more than one person (Brams and Taylor, 1996), and one of the most common fair division problems is the ownership allocation problem (Moulin, 1992). When the players have conflicting preferences over division alternatives for a collective entity, a division problem emerges. In the classical study of the fair division problem, the number of players is fixed (Thomson, 1983) and Game Theory provides a variety of tools to cope with it. This paper shows that the Shapley value, a fair division scheme developed by Lloyd Shapley (1953) within the cooperative game theory framework, can be used to achieve a fair division of the surplus development rights among landowners remolding smaller and irregular parcels into bigger and regular ones, resulting in higher aggregate development rights. The remainder of the paper is organized as follows. Section 2 consists in a review of the literature. The methodology is described in Section 3. A case study is presented in Section 4, and public policy implications are assessed in Section 5. Section 6 concludes the paper.
allocation problem where every player gets at most one object and monetary compensation is not possible. Abdulkadiroglu and Sonmez (1998, 1999), Svensson (1999); Bogomolnaia and Moulin (2001); Ehlers (2002); Ergin (2000); Ehlers and Klaus (2004), and Sonmez and Unver (2010) deal with the problem of allocating indivisible objects among players. Most of the literature is concerned with property rights. However, land and development rights are divisible and monetary compensation is possible. Aside from studies on house allocation, the issues regarding urban development have rarely been assessed through a cooperative game theory-based approach. Hui and Bao (2013) assess conflicts between local governments and farmer regarding legal land acquisition and illegal land conversion. They develop game theorybased analytic models to explain the negotiation process between the stakeholders. They conclude that farmers tend to cooperate if the land acquisition is for the public interest. However, if the local government negotiates with farmers for individual interests, conflict emanates, resulting in additional compensation. Samsura et al. (2010) analyze the decisions-making process in land and property development projects taking into consideration the interactions between the decision makers and their payoffs. They apply their approach to a theoretical residential development project in the Netherlands with four players (the municipality, a property developer who acquired the land, a property developer who did not acquire the land, and the landowner) under two scenarios (old regulation and new regulation). Wang et al. (2007) proposes a cooperative game theory-based approach to support negotiations on fair allocations of costs and benefits in abandoned and underused industrial and commercial sites, or so-called “brownfields”. The nucleolus and the Shapley value solutions are presented for a hypothetical example with the players, the current landowners, the developer, and the government (Wang et al., 2007).
2. Literature review Game Theory is the study of decision problems including more than one person (Gibbons, 1992). Its main objective is to determine the strategies that players choose to maximize their utilities. Thus, game theory-based approaches are normative in nature (Colman, 2003a,b). Game theory models are mostly abstract representations, but their abstract nature allows them to be used in a variety of real-life situations (Osborne and Rubinstein, 1994). N-person games theory, or later called cooperative game (coalitional) theory, has been introduced by von Neumann and Morgenstern (1944). Cooperative game theory deals with the formation of coalitions by the players and the analysis of the various coalitions (Nash, 1951). Gillies presents the notion of “the core” as the unimprovable set of feasible allocations in n-person games in the early 1950s (published in 1959), and “the core” is later developed by Shapley as a solution concept in lectures at Princeton University in 1953 (Shapley and Shubik, 1963). A valuation function for each player in n-player games was first defined by Shapley (1953). The Shapley value has been applied to a wide range of fields including cost allocation, social networks, water allocation, waste management, pollution reduction, airport management, biology, reliability theory, and belief formation (Littlechild and Owen, 1973; Petrosjan and Zaccour, 2003; Moretti and Patrone, 2008; Jørgensen, 2010). Fig. 1 shows Google ngram viewer results of the case-insensitive search for “the Shapley value”. Ngram is a statistical analysis of text to derive the share of frequency of a word or a phrase in a given text. Fig. 1 shows that the share of references to this particular technique has constantly increased since its very first introduction in the early 1940s. The Shapley value has been recently applied to more complex phenomena, such as network problems. Niyato and Hossain (2006, 2008), for example, use the Shapley value for bandwidth allocation in wireless networks. Michalak et al. (2013) examines the computational properties of the Shapley value as a measure of centrality in networks and conclude that the solutions outperform Monte Carlo simulations. Despite the wide range of fields where cooperative game theory has been used, its applications to urban development and redevelopment have been limited. Shapley and Scarf (1974) introduce the house
3. Methodology The allocation problem is considered in two main classes of games within the game theory framework: (1) bargaining games and (2) coalition games (Thomson, 2007). Although building and managing a coalition is also an important theme (Guajardo and Rönnqvist, 2016), sharing the surplus of the coalition has been the primary subject of cooperative game theory (Moulin, 1987). In this regard, cooperative game theory provides efficient and unique solutions when players are allowed to act independently or cooperatively (Dinar et al., 1992). There are a number of allocation methods developed within the cooperative game theory framework, but the Shapley value and the nucleolus developed by Shapley (1953) and Schmeidler (1969) respectively have received more theoretical attention. The other available solutions include the stable sets developed by von Neumann and Morgenstern (1944), the kernel by Davis and Maschler (1965), and the bargaining set by Aumann and Maschler (1964). The Shapley value has been a focus of cooperative game theory as it introduces a distinct approach to analyze strategic behavior based on the concepts of the core and the stable set (Roth, 1988). It has many attractive normative properties and meets certain criteria (Moulin, 1992; Michalak ea., 2013). A cooperative game can be defined by a finite set of players N = {1, 2, 3, 4, …, n} and a real-valued characteristic function v, or “the coalitional form”:
v: 2 N
R
with v ( ) = 0 . The game is then denoted by (N , v ) . The players of N are allowed to participate in a coalition S. The Shapley value includes the probabilistic interpretations of the marginal contributions of the players, and it can be interpreted as giving players their expected marginal contributions (Schotter and Schwödiauer, 1980). The formulation for the Shapley value for player i, i (v ), is computed as follows: 2
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Fig. 1. Case-insensitive Google ngram viewer results for “the Shapley value”. i (v )
(S
= S N \ {i}
1) ! (N N!
S )!
(v (S )
v (S
Second, population solidarity requires that when a new player is introduced into a coalition, all existing players are affected (benefit or cost) to a minor degree. Third, the stand-alone axiom requires that the total of the players’ individual utilities without a coalition cannot exceed the total utility of a coalition (Moulin, 1992). Finally, the resource monotonicity axiom requires that the utility of each player does not diminish when the total utility to be shared increases (Moulin, 1992). Other division schemes including the nucleolus, the bargaining set, and the kernel are proven to be non-monotonic (Ortmann, 2000). There are some drawbacks related to the Shapley solution. For example, the Shapley value may not be in the core when the game is nonconvex (Shapley, 1971). The core may also be empty or quite large (Saad et al., 2009). Also, when the number of players increases, the complexity of computations increases significantly (van Campen et al., 2018). Finally, the additivity axiom limits the range of the games that the Shapley solution can be used. Nonetheless, there are still a number of other good reasons for selecting the Shapley value as a division scheme (Krajewska et al., 2008). It provides a unique solution for each player and leaves no room for a bargaining process. It is fairly easy to implement when the number of players is limited.
i ))
where S N is a coalition of size |S|, and |N| is the number of players. The expected sum of payoffs for the players participating coalition S, or the worth of coalition S, is denoted by v (S ) . v (S i ) is the value of the coalition in the absence of player i (Colman, 2003a,b). Thus, (v (S ) v (S i)) is a measure of the marginal contribution of player i in coalition S, where all coalitions of size S with player i is taken into consideration. The Shapley value meets four desirable criteria: (1) efficiency, (2) symmetry, (3) null player, and (4) additivity. Efficiency requires that the total utility (or benefit) is distributed among the players participating in the coalition implying i N i (v ) = v (N ) . Symmetry requires that the utilities assigned to each player is not related to the player’s identity, implying v (S {i}) = v (S {j }) where i and j are equivalent players in coalition S. Null player, also called as the dummy player, requires that the players with no marginal contributions receive no utility implying i (v ) = 0 if i is a null player. Finally, additivity requires that the value obtained from the sum of two different games (v + w ) is equal to the sum of the values obtained from these two games individually (v and w ) . That is to say (v ) + (w ) = (v + w ) (Roth, 1988). The Shapley value is characterized by fairness. As stated by van den Brink (2002) a solution that satisfies symmetry and additivity, also satisfies fairness. However, defining fairness in a cooperative environment is not an easy task (Chen et al., 2013), and has been a major conceptual problem in game theory (Deng and Papadimitriou, 1994). In its simplest form, fairness has two major components: proportionality and envy-freeness (Chen et al., 2013). Fairness as a concept in cooperative games has been initially introduced by Myerson (1977). Myerson (1977) argues that players have limited communication opportunities in forming coalitions, and when a communication between two players is deleted, the payoffs of both players are affected in the same way. In his pioneer work incorporating fairness into game theory, Rabin (1993) asserts that people are willing to give away their own benefits in favor of those who help them and to punish those who are unkind. Although the majority of economic models assume that decision-makers seek to maximize their own utility, there is strong empirical evidence that fairness is an important factor in decision making (Fehr and Schmidt, 1999). The Shapley value formula holds four other desirable axioms for efficient allocation of goods: (1) individual rationality, (2) population solidarity, (3) stand-alone test, and (4) resource monotonicity. First, the individual rationality axiom requires that the utility of each player in the coalition should not be less than the player’s benefit without a collation.
4. Case study Along with industrialization, most cities in Turkey have experienced a massive migration from rural areas, resulting in rapid urbanization and unexpected population growth, starting in the 1950s (TurkerDevecigil, 2010). Insufficient supply of affordable housing led to the development of squatter settlements (Uzun et al., 2010). The government was not opposed to urbanization, and the emergence of squatter settlements seemed like its inevitable result. The number of squatter houses, or so-called gecekondus, increased from 100.000 to 1.25 million between 1950 and 1983 (Özler, 2000). In the 1980s, with the enactment of the new amnesty law (Law No. 2981), parcel owners or current occupiers of public land were given development rights through improvement plans. The gecekondus thus have been transformed into low-quality apartments (Turker-Devecigil, 2010; Türkün, 2011). This “populist” approach shifted to a “neo-liberal” one in the early 2000s (Kuyucu and Ünsal, 2010) and urban regeneration has been one of the top priorities of the ruling party in Turkey since then. The year 2011 was declared as “the year of transformation” by the Mass Housing Development Administration, the predominant governmental institution regarding mass housing and mass urban renewal (Karaman, 2013). The era replacing the rational comprehensive planning approach by large scale development projects 3
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Fig. 2. Total numbers of parcel-level redevelopment applications and rebuilt buildings in Turkey (2012–2018).
Fig. 3. Total numbers of parcel-level redevelopment applications and rebuilt buildings in Izmir, Turkey (2013–2018).
had already begun (Kuyucu and Ünsal, 2010). A great portion of the laws regarding the physical transformation of the urban setting has been altered and the roles of key institutions, including the Mass Housing Development Administration (TOKI) and Privatization Administration have been strengthened to enable the realization of large scale urban renewal and regeneration projects (Türkün, 2011). However, the law requires a minimum area of five hectares to declare an urban area as “urban transformation zone”, and this legislative constraint limits large scale transformation in the built-up areas of the cities. As a result, the government has started up new initiatives to trigger smaller scale transformations. Through these initiatives, landowners and developers benefit from a variety of subsidies, including tax cuts and rent subsidy. This process has been eased by the enactment of the Law on the Regeneration of Areas Under the Risk of Disasters (Law no. 6306) in 2012 to give urban regeneration some acceleration at the parcel level (Güzey, 2016). According to Law no. 6306, the redevelopment process is straightforward at the parcel level. The dwellers of the building apply for inspection, and the inspection procedure is held by licensed companies. Then, the redevelopment phase is pursued through a contract between the landowners and the developer. The landowners are free to use land readjustments, that is they can cooperate with neighboring landowners for amalgamation to benefit from higher development rights if available. However, parcel-level transformations have been limited and have not reached the desired level. The number of applications peaked in 2016 with 41,343 applications and has declined dramatically in the following years. Similarly, a sharp decrease in the number of total rebuilt buildings has been observed since the year 2015 (Fig. 2). One of the main problems seems to be land reallocation, as surplus development rights would not be possible for most parcels due to the lack of cooperation between the landowners. A major dispute among the landowners is on the fair share of the extra development rights that are to be achieved through the amalgamation of two or more parcels. The Shapley value can be used as a fair division scheme in such situations. A real-life example from the Karabaglar district, Izmir, Turkey is presented. Since ancient times, Izmir has been one of the largest seaport cities on the Mediterranean coast (Freely, 2004). Izmir has become a city that reflects the characteristics of the Republic of Turkey in the 20th century (Goffman et al., 1999) and reached a population of 4,279,677 by 2017. The very first squatter houses in Izmir were recorded in the early 1930s, and they were mostly located around the city center (Karadag and Miraglu, 2011). The first comprehensive Izmir plan of the 2000s has
been completed in 2007 and revised in 2009. In this 1/25000 scale land-use plan, 15 urban regeneration and renewal sites were delineated covering a total area of 4328 ha. These sites cover 13.7% of the total planning area and 39.0% of the residential areas in Izmir (Izmir Metropolitan Municipality, 2009). However, as in the case in Turkey, parcel-level transformations have been limited. The number of rebuilt buildings peaked in 2016, with 3783 buildings, followed by a sharp decrease in the following years. The number of applications has been 17,869 since the enactment of the Law on the Regeneration of Areas Under the Risk of Disasters (Law No. 6306) in 2012, and the number of rebuilt buildings 14,405 (Fig. 3). A case study is selected from the Karabaglar district in Izmir (Fig. 4) for several reasons. First, the Karabaglar district is the most populous district in Izmir, and one of the most populous in Turkey, with a population of 480,790 in 2017. Secondly, a large share of the regeneration and renewal sites in the 1/25000 scale land-use plan of Izmir are located in the Karadaglar district. Finally, the development plan of the Karabaglar district offers surplus development rights for larger parcels, and thus allows surplus development rights for landowners cooperating for land amalgamation. The development block no. 31044 is selected for further analysis. There are 3 parcels in this block, namely 4, 5 and 6 (Fig. 5). The small number of parcels make the calculations of the Shapley values simple and comprehensible as an example. The zoning decision for the block in the development plan is “residential”, and a maximum of 4 floors is permitted. The floor area ratio is 0.40, the front yard setback 5 m, and the side and rear yard setbacks 3 m (Fig. 6). However, if the total land area is over 1000 square meters, the number of permitted floors increases to 5.5, with the floor area ratio at 0.40 and the front, side and rear setbacks are 5 m. These planning decisions result in a variety of development rights for each parcel in case they are developed individually or in different land amalgamation combinations (Table 1). The Shapley value approach assumes that all combinations are possible and equally likely (Heaney and Dickinson, 1982). There are five possible outcomes in this case, called scenarios. The setbacks and the building floor areas are presented in Fig. 7 and Table 1. For example, in scenario A, all of the three parcels are redeveloped individually. In scenario B, parcels 5 and 6 are amalgamed for redevelopment, excluding parcel 4. Finally, scenario E is the grand coalition, where all three parcels are developed together. Note that the setbacks may limit the footprint of the building, thus the floor area ratio (0.40) may not be fulfilled. For example, in scenarios A and C, the buildable parcel area after setbacks (d) is less than
4
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Fig. 4. Location of the example case in Karabaglar district, Izmir, Turkey.
the maximum floor area (c), which is found by multiplying the parcel area (a) by the floor area ratio (b). Thus, the building floor area (e) is less than the maximum buildable floor area (c), resulting in underuse of the development rights for the parcels. The gross floor area (g) is calculated by multiplying the building floor area (e) by the number of floors (f) (Table 1, Fig. 8). Here, the gross benefits are considered, but the cost of building, taxes and other costs are ignored. This is a plausible approach since these costs are directly proportional to the land size in practice and have a limited effect on the results of the division scheme.
The benefits (measured as the gross floor areas) for the individual parcels and the coalitions are:
v ( ) = 0, v ({4}) = 1504.0, v ({4, 5}) = 2950.0, v ({4, 5, 6}) = 3882.5 v ({5}) = 546.3, v ({4, 6}) = 3000.5 v ({6}) = 649.4, v ({5, 6}) = 1319.7 n
The game is essential. That is v (N ) > i = 1 v ({i}) , implying that the benefits of the grand coalition are greater than the sum of the benefits of the individual players:
v ({4, 5, 6})
v ({4}) + v ({5}) + v ({6}).
Fig. 5. Aerial photo for the parcels in development block no. 31044. 5
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Fig. 6. Development plan decisions for the development block no. 31044.
Also, the game is superadditive, with v (S T ) v (S ) + v (T ) S, T N and v (S ) v (T ) = , where S and T are two disjoint subsets of N:
v ({4, v ({4, v ({5, v ({4, v ({4, v ({4,
v (S T ) + v (S T ) lowing coalitions: v ({4, v ({4, v ({5, v ({4, v ({4, v ({4, v ({4, v ({4,
5}) v ({4}) + v ({5}), 6}) v ({4}) + v ({6}), 6}) v ({5}) + v ({6}), 5, 6}) v ({4, 5}) + v ({6}), 5, 6}) v ({4, 6}) + v ({5}), 5, 6}) v ({5, 6}) + v ({4}).
On the other hand, the game is non-convex. The condition that
v ( S ) + v (T )
5}) v ({4}) + v ({5}), 6}) v ({4}) + v ({6}), 6}) v ({5}) + v ({6}), 5, 6}) v ({4}) + v ({5, 5, 6}) v ({5}) + v ({4, 5, 6}) v ({6}) + v ({4, 5, 6}) + v ({4}) v ({4, 5, 6}) + v ({6}) v ({4,
S, T
N , holds for the fol-
6}), 6}), 6}), 5}) + v ({4, 6}), 6}) + v ({5, 6}),
Table 1 The development rights for the three parcels under all possible scenarios. Scenario
Singleton or Coalition
Parcel Area (m2) (a)
Floor Area Ratio (b)
Maximum Buildable Floor Area (m2) (c) = (a) x (b)
Buildable Parcel Area After Setbacks (m2) (d)
Building Floor Area (m2) (e)=Min{(c), (d)}
Number of Floors (f)
Gross Floor Area (m2) (g) = (e) x (f)
A
{4} {5} {6} {4} {5, 6} {4, 6} {5} {4, 5} {6} {4, 5, 6}
940.0 400.9 423.9 940.0 824.8 1363.9 400.9 1340.9 423.9 1764.8
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
376.0 160.4 169.6 376.0 329.9 545.6 160.4 536.4 169.6 705.9
464.8 136.6 162.3 464.8 386.8 645.6 136.6 558.9 162.3 1034.0
376.0 136.6 162.3 376.0 329.9 545.6 136.6 536.4 162.3 705.9
4 4 4 4 4 5.5 4 5.5 4 5.5
1504.0 546.3 649.4 1504.0 1319.7 3000.5 546.3 2950.0 649.4 3882.5
B C D E
6
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Fig. 7. The setbacks and the building floor areas in each parcel under five different scenarios.
Fig. 8. 3D models regarding the development rights in each parcel under five different scenarios.
but not for the coalition:
Similarly, the Shapley value for Parcel 5,
5 (v ) ,
is:
v ({4, 5, 6}) + v ({5}) v ({4, 5}) + v ({5, 6}). Thus, the Shapley solution may not be in the core, and the core conditions have to be checked Lemaire (1984). When calculating the Shapley value, marginal contributions of the players, (v (S ) v (S i)), are taken into consideration. For example, there are a number of alternatives for the landowner of Parcel 4. The landowner may decide to redevelop the parcel individually (S = 1) or participate in a group of two (S = 2) or three landowners (S = 3). If the landowner decides to participate in a group of two (S = 2), there are two alternatives: cooperating with the landowner of Parcel 5 or of Parcel 6, where the probability of cooperating with either of the other landowners is the same. The Shapley value for Parcel 4, 4 (v ) , is then calculated considering every possible cooperation: S=1
(1
S=2
(2 (2
S=3
(3
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) !
(v ({4})
2) !
(v ({4, 5})
v ({5})=
2) !
(v ({4, 6})
v ({6})=
3) !
1 (1504.0 3
v ({ })=
(v ({4, 5, 6})
v ({5, 6})=
1 (2950.0 6 1 (3000.5 6 1 (3882.5 3
S=1
(1
S=2
(2 (2
S=3
(3
5 (v )
=
S N {4}
|S | ! (| N| |S| |N | !
1) !
(v (S )
v (S
i))=
(v ({5})
2) !
(v ({4, 5})
v ({4})=
2) !
(v ({5, 6})
v ({6})=
3) !
(v ({4, 5, 6})
v ({ })=
v ({4, 6})=
1 (546.3 3 1 (2950.0 6 1 (1319.7 6 1 (3882.5 3
0)=
182.1
1504.0)=
241.0
649.4)=
111.7
3000.5)=
294.0
S N {5}
|S | ! (| N | |S| |N | !
1) !
(v (S )
v (S
828.8
i))=
182.1 + 241.0 + 111.7 + 294.0 =
Finally, the Shapley value for Parcel 6,
0)=
501.3
546.3)=
400.6
S=1
(1
649.4)=
391.9
S=2
(2
1319.7)=
854.3
(2
(3
Thus,
=
1) !
Thus,
S=3
4 (v )
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) !
(v ({6})
2) !
(v ({4, 6})
v ({4})=
2) !
(v ({5, 6})
v ({5})=
3) !
(v ({4, 5, 6})
v ({ })=
v ({4, 5})=
6 (v ) ,
is:
1 (649.4 3 1 (3000.5 6 1 (1319.7 6 1 (3882.5 3
0)=
216.5
1504.0)=
249.4
546.3)=
128.9
2950.0)=
310.8
Thus,
2148.1
501.3 + 400.6 + 391.9 + 854.3 =
6 (v )
=
S N {6}
|S | ! (| N| |S| |N | !
1) !
(v (S )
216.5 + 249.4 + 128.9 + 310.8 =
7
v (S
i ))=
905.6
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as:
Thus, the Shapley Values for the parcels 4, 5, and 6 can be written
4 (v )
= 2148.1
5 (v )
= 828.8
6 (v )
= 905.6
Since the game is non-convex, it is necessary to assess whether the core is non-empty Lemaire (1984). The core of the game is non-empty since the following inequality holds: 4 (v )
+
5 (v )
+
v ({4, 5}) v ({4, 6}) + 2 2 v ({5, 6}) + v ({4, 5, 6}) 2
6 (v )
2148.1 + 828.8 + 905.6
2950 3000.5 1319.7 + + 2 2 2
3882.5
Although the core of the game is non-empty, the Shapley values may fall outside of the core for non-convex games and this is considered to be a very undesirable outcome (Heaney and Dickinson, 1982). The core conditions in the present case can be written as: 4 (v )
1504.0,
5 (v )
546.3,
4 (v )
4 (v ) +
+
6 (v )
3000.5
6 (v )
649.4,
5 (v )
6 (v )
1319.7,
+
5 (v )
2950.0,
4 (v )
+
5 (v )
+
6 (v )
= 3882.5
It is clear that the Shapley values satisfy these conditions. The Shapley value can also be easily calculated, and core conditions can be checked by using the R Package “CoopGame” developed by Staudacher et al. (2019). Fig. 9 shows the core for the game and the Shapley solution belongs to the core of the game as a CoopGame output (see Appendix for the code). As a result, the landowners benefit from the highest possible development rights, which are 3882.5 square meters of gross floor area, and are able to share the surplus development rights fairly: landowner of Parcel 4 gets 2148.1 m2, landowner of Parcel 5828.8 m2, and landowner of Parcel 6905.6 m2. 5. Conclusions As many more people are expected to live in urban areas in the coming decades, much of the physical transformation will take place in the built-up areas of the existing cities. Local and municipal governments have to put more effort in reshaping the cityscape and struggling with the landownership disputes. Thus, new and innovative ways of collaborations should be sought. This paper shows that a fair sharing of the surplus development rights through the utilization of the Shapley value solution may provide landowners the highest development rights, which can be considered as a crucial incentive for small scale redevelopment in the built-up parts of the cities. However, such fair division schemes are only possible with a proper institutional setting. Currently, the redevelopment process is governed by a contract between the landowners and the developer. The present institutional setting allows landowners of neighboring parcels to cooperate for land amalgamation but does not provide any insights on the fair sharing of the resulting surplus development rights. An institutional setting providing such fair division instruments to the stakeholders may ease the parcel-level transformation in the built-up areas, allowing landowners benefit from maximum development rights. A proper and well-functioning institutional setting is also crucial in the sense that alternative solutions, which are also in the core of the game, can be put forward or asserted by different players in the game. As shown by Nash (1953), the same fair solution may also be obtained through a bargaining process within the non-cooperative game theory framework, but, an
Fig. 9. The core for the game and the Shapley solution.
institutional mechanism must be established to govern the interactions among the players, even if these players have limited or no information on their preferences over outcomes (Serrano, 2004). However, in the case of redevelopment in urban areas, an agreement between the stakeholders may not be achieved through a negotiation process, especially when the number of players is high. Thus, an institutional setting empowering with fair division solutions based on cooperative game theory can be the long-awaited solution to the sharing of surplus development rights among landowners. The main incentive for landowners to act in collaboration is the planning decision allowing higher development rights for larger parcels. This incentive is a plausible approach to achieve better living conditions in bigger and legally built buildings, as most of the parcels in urban regeneration and renewal sites are relatively small and subject to individual illegal building or modification process. However, there is an upper limit for higher development rights, as urban amenities have to be planned for the increasing population caused by the additional development rights. Thus, the regulator making planning decisions has to consider the balance between the incentives for redevelopment and the share of land allocated for urban amenities, which is, in fact, another allocation problem relating to a fair sharing of land between public and private stakeholders. 8
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Acknowledgments
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I would like to thank Jean-Michel Guldmann for reviewing the final draft of this paper, and the two anonymous reviewers for their valuable comments. However, any mistakes that remain are my own. Appendix A The following code can be used with the “CoopGame” package in R to derive the results obtained in the study. The outputs are provided under the command lines. # To define the characteristic functions by using the payoffs for the players/coalitions of {4}, {5}, {6}, {4,5}, {4,6}, {5,6}, and {5,6,7} from Table 1 > characteristic_function < - c(1504.0, 546.3, 649.4, 2950.0, 3000.5, 1319.7, 3882.5) # To check if the game is essential > isEssentialGame(characteristic_function) [1] TRUE # To check if the game is superadditive > isSuperadditiveGame(characteristic_function) [1] TRUE # To check if the game is convex > isConvexGame(characteristic_function) [1] FALSE # to derive the Shapley values for each player > shapleyValue(characteristic_function) [1] 2148.0667 828.8167 905.6167 # to check if the Shapley solution is in the core of the game > belongsToCore((shapleyValue(characteristic_function)), characteristic_function) [1] TRUE # To draw the core of the game > drawCore(characteristic_function) # To draw the Shaply solution on core of the game > drawShapleyValue(characteristic_function, holdOn = TRUE) References Abdulkadiroglu, A., Sonmez, T., 1998. Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66, 689–701. Abdulkadiroglu, A., Sonmez, T., 1999. House allocation with existing tenants. J. Econ. Theory 88, 233–260. Aumann, R.J., Maschler, M., 1964. The bargaining set for cooperative games. In: Dresher, M., Shapley, L.S., Tucker, A.W. (Eds.), Advances in Game Theory (Annals of Mathematics Studies, 52). Princeton University Press, Princeton, pp. 443–476. Barnett, J., 2018. The Fractured Metropolis: Improving the New City, Restoring the Old City, Reshaping the Region. Routledge. Bogomolnaia, A., Moulin, H., 2001. A new solution to the random assignment problem. J. Econ. Theory 100, 295–328. Brams, S.J., Taylor, A.D., 1996. Fair Division: From Cake-cutting to Dispute Resolution. Cambridge University Press. Chen, Y., Lai, J.K., Parkes, D.C., Procaccia, A.D., 2013. Truth, justice, and cake cutting. Games Econ. Behav. 77 (1), 284–297. Colman, A.M., 2003a. Cooperation, psychological game theory, and limitations of rationality in social interaction. Behav. Brain Sci. 26 (2), 139–153. Colman, A.M., 2003b. Game Theory and Its Applications: in the Social and Biological Sciences. Routledge. Deng, X., Papadimitriou, C.H., 1994. On the complexity of cooperative solution concepts. Math. Oper. Res. 19 (2), 257–266. Dinar, A., Ratner, A., Yaron, D., 1992. Evaluating cooperative game theory in water resources. Theory Decis. 32 (1), 1–20. Ehlers, L., 2002. Coalitional strategy-proof house allocation. J. Econ. Theory 105, 298–317. Ehlers, L., Klaus, B., 2004. Resource-monotonicity for house allocation problems. Int. J. Game Theory 32 (4), 545–560. Ergin, H.I., 2000. Consistency in house allocation problems. J. Math. Econ. 34, 77–97. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition, and cooperation. Q. J. Econ. 114, 817–868. Freely, J., 2004. The Western Shores of Turkey: Discovering the Aegean and Mediterranean Coasts. Tauris Parke., London. Garreau, J., 1991. Edge City: Life on the New Frontier. Anchor Books. Gibbons, R., 1992. A Primer in Game Theory. Harvester Wheatsheaf. Goffman, D., 1999. Izmir: from village to colonial port city. In: Eldem, E., Goffman, D.,
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K.M. Cubukcu Staudacher, J., Anwander, J., Tiukkel, A., Maerz, M., Mueller, F., Gebele, D., Merkle, A., Tokay, F., Tokay, K., Cyl, N., 2019. CoopGame: Important Concepts of Cooperative Game Theory. https://CRAN.R-project.org/package=CoopGame. Svensson, L.-G., 1999. Strategy-proof allocation of indivisible goods. Soc. Choice Welfare 16, 557–567. Thomson, W., 1983. The fair division of a fixed supply among a growing population. Math. Oper. Res. 8 (3), 319–326. Thomson, W., 2007. Cost allocation and airport problems. Rochester Center Econ. Res Working Paper, 538. Türkün, A., 2011. Urban regeneration and hegemonic power relationships. Int. Plan. Stud. 16 (1), 61–72. United Nations, 2010. World Urbanization Prospects: the 2009 Revision. Department of Economic and Social Affairs Population Division, New York. United Nations, 2018. 2018 Revision of World Urbanization Prospects. Department of
Economic and Social Affairs Population Division, New York. Uzun, B., Çete, M., Palancıoğlu, H.M., 2010. Legalizing and upgrading illegal settlements in Turkey. Habitat Int. 34 (2), 204–209. van Campen, T., Hamers, H., Husslage, B., Lindelauf, R., 2018. A new approximation method for the Shapley value applied to the WTC 9/11 terrorist attack. Soc. Netw. Anal. Min. 8 (1), 3. van den Brink, R., 2002. An axiomatization of the Shapley value using a fairness property. Int. J. Game Theory 30 (3), 309–319. Von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior. Princeton University Press, pp. 1944. Wang, L., Fang, L., Hipel, K.W., 2007. A game-theoretic approach to brownfield redevelopment: negotiation on cost and benefit allocation. In: Proceedings of the 2007 IEEE International Conference on Systems, Man, and Cybernetics. Montreal, Canada. pp. 1800–1805.
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The problem of fair division of surplus development rights in redevelopment of urban areas: Can the Shapley value help? K. Mert Cubukcu Professor of City and Regional Planning, Dokuz Eylul University, Izmir, Turkey
ARTICLE INFO
ABSTRACT
Keywords: Game theory Cooperative games The Shapley value Urban redevelopment Fair division Landownership
Many more people are expected to live in urban areas in the coming decades, and much of the physical transformation will take place in the built-up areas of cities. Landownership is a key factor in urban regeneration, and the fair division of benefits is a major obstacle to redeveloping urban land in a cooperative context. This paper aims to show that the Shapley value, a fair division scheme developed within the cooperative game theory framework, can be used to achieve a fair division of the surplus development rights among landowners remolding smaller and irregular parcels into bigger and regular ones, resulting in higher aggregate development rights. The methodology is illustrated by a case study of three parcels selected from the Karabaglar district in Izmir, Turkey, where surplus development rights are available for landowners cooperating for land amalgamation. The findings show that the Shapley values for the landowners satisfy the core conditions of the game and landowners can benefit from the highest possible development rights and share the surplus fairly. However, the current institutional setting has to be restructured to apply such division schemes.
1. Introduction The share of people living in urban areas has surpassed the share in rural areas for the first time in 2009 (United Nations, 2010). By 2050, 68% of the world population is expected to live in cities (United Nations, 2018), and the major share of this urban growth is expected to take place in developing and poor countries, which will bring additional responsibilities to local and municipal governments (Montgomery, 2008). The problems are related to both the expanding boundaries of the city and the redevelopment of the existing parts. Traditional city centers and their surrounding ring of old neighborhoods started losing their attraction as early as the 1950s. Rural areas at the fringe of the cities have been transformed into mass housing projects through master plans (Barnett, 2018). This process has accelerated in the 1980s, giving rise to the concentration of shopping, business, and entertainment activities outside the urban core and decentralizing some of the main functions of the traditional city centers to the so-called “edge cities” (Garreau, 1991; Henderson and Mitra, 1996). The separation between old and new city became obvious, and something had to be done in the old city. As a response, the notion of urban regeneration evolved in stages: (1) reconstruction (1950s), (2) revitalization (1960s), (3) renewal (1970s), (4) redevelopment (1980s), (5) regeneration (1990s), (6) regeneration in recession (2000s). Although the contents of these stages somehow differ and the key actors
and spatial levels of activity vary, there are no clear-cut boundaries among them (Li et al., 2014; Roberts et al., 2016). The main motivation has been to redevelop the urban structure to avoid abandonment and prevent economic, social, and physical deterioration. Although urban renewal policies have been implemented to deal with a variety of urban problems including low building quality, increasing social segregation and poverty, insufficient infrastructure, and the need for public spaces, most policies have been oriented towards improving living conditions in residential areas (Kleinhans, 2004). Landownership is one of the key factors in the urban regeneration process, and ownership constraints are among the major obstacles. Fragmented or multiple ownership may be a significant barrier to coordinated redevelopment (Louw, 2008). The real estate market has three actors: (1) the developer, (2) the landowner and (3) an authority (local and/or central government) (Mu and Ma, 2007). The landowners are the suppliers of property or development rights, and developers seek land for redevelopment (Li and Li, 2007). In the absence of a governmental authority during the land assembly or acquisition phase of the redevelopment process, there are different options for the landowners and the developers. The developer may purchase the property rights from the property owners, replace old buildings with new ones, and sell them for profit. However, resettlement or displacement is not generally the desired result for many landowners, even with adequate compensation. Thus, landowners and developer may adopt a bilateral
E-mail address: [email protected]. https://doi.org/10.1016/j.landusepol.2019.104320 Received 25 March 2019; Received in revised form 20 September 2019; Accepted 20 October 2019 0264-8377/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: K. Mert Cubukcu, Land Use Policy, https://doi.org/10.1016/j.landusepol.2019.104320
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structure (Li and Li, 2007), where land readjustment is the most common approach. In this approach, the ownership of multiple irregular parcels is first pooled and then subdivided into regular urban parcels for redevelopment (Sorensen, 1999). Remolding of smaller and irregular parcels into bigger and regular ones may result in higher aggregate development rights, as bigger parcels are sometimes assigned higher development rights in development plans. The question then becomes: How to divide the surplus development rights (Li and Li, 2007)? The issue of fair division is a major obstacle to redevelop urban land in a cooperative context. Fair division is defined as the process of dividing a set of divisible goods or resources among more than one person (Brams and Taylor, 1996), and one of the most common fair division problems is the ownership allocation problem (Moulin, 1992). When the players have conflicting preferences over division alternatives for a collective entity, a division problem emerges. In the classical study of the fair division problem, the number of players is fixed (Thomson, 1983) and Game Theory provides a variety of tools to cope with it. This paper shows that the Shapley value, a fair division scheme developed by Lloyd Shapley (1953) within the cooperative game theory framework, can be used to achieve a fair division of the surplus development rights among landowners remolding smaller and irregular parcels into bigger and regular ones, resulting in higher aggregate development rights. The remainder of the paper is organized as follows. Section 2 consists in a review of the literature. The methodology is described in Section 3. A case study is presented in Section 4, and public policy implications are assessed in Section 5. Section 6 concludes the paper.
allocation problem where every player gets at most one object and monetary compensation is not possible. Abdulkadiroglu and Sonmez (1998, 1999), Svensson (1999); Bogomolnaia and Moulin (2001); Ehlers (2002); Ergin (2000); Ehlers and Klaus (2004), and Sonmez and Unver (2010) deal with the problem of allocating indivisible objects among players. Most of the literature is concerned with property rights. However, land and development rights are divisible and monetary compensation is possible. Aside from studies on house allocation, the issues regarding urban development have rarely been assessed through a cooperative game theory-based approach. Hui and Bao (2013) assess conflicts between local governments and farmer regarding legal land acquisition and illegal land conversion. They develop game theorybased analytic models to explain the negotiation process between the stakeholders. They conclude that farmers tend to cooperate if the land acquisition is for the public interest. However, if the local government negotiates with farmers for individual interests, conflict emanates, resulting in additional compensation. Samsura et al. (2010) analyze the decisions-making process in land and property development projects taking into consideration the interactions between the decision makers and their payoffs. They apply their approach to a theoretical residential development project in the Netherlands with four players (the municipality, a property developer who acquired the land, a property developer who did not acquire the land, and the landowner) under two scenarios (old regulation and new regulation). Wang et al. (2007) proposes a cooperative game theory-based approach to support negotiations on fair allocations of costs and benefits in abandoned and underused industrial and commercial sites, or so-called “brownfields”. The nucleolus and the Shapley value solutions are presented for a hypothetical example with the players, the current landowners, the developer, and the government (Wang et al., 2007).
2. Literature review Game Theory is the study of decision problems including more than one person (Gibbons, 1992). Its main objective is to determine the strategies that players choose to maximize their utilities. Thus, game theory-based approaches are normative in nature (Colman, 2003a,b). Game theory models are mostly abstract representations, but their abstract nature allows them to be used in a variety of real-life situations (Osborne and Rubinstein, 1994). N-person games theory, or later called cooperative game (coalitional) theory, has been introduced by von Neumann and Morgenstern (1944). Cooperative game theory deals with the formation of coalitions by the players and the analysis of the various coalitions (Nash, 1951). Gillies presents the notion of “the core” as the unimprovable set of feasible allocations in n-person games in the early 1950s (published in 1959), and “the core” is later developed by Shapley as a solution concept in lectures at Princeton University in 1953 (Shapley and Shubik, 1963). A valuation function for each player in n-player games was first defined by Shapley (1953). The Shapley value has been applied to a wide range of fields including cost allocation, social networks, water allocation, waste management, pollution reduction, airport management, biology, reliability theory, and belief formation (Littlechild and Owen, 1973; Petrosjan and Zaccour, 2003; Moretti and Patrone, 2008; Jørgensen, 2010). Fig. 1 shows Google ngram viewer results of the case-insensitive search for “the Shapley value”. Ngram is a statistical analysis of text to derive the share of frequency of a word or a phrase in a given text. Fig. 1 shows that the share of references to this particular technique has constantly increased since its very first introduction in the early 1940s. The Shapley value has been recently applied to more complex phenomena, such as network problems. Niyato and Hossain (2006, 2008), for example, use the Shapley value for bandwidth allocation in wireless networks. Michalak et al. (2013) examines the computational properties of the Shapley value as a measure of centrality in networks and conclude that the solutions outperform Monte Carlo simulations. Despite the wide range of fields where cooperative game theory has been used, its applications to urban development and redevelopment have been limited. Shapley and Scarf (1974) introduce the house
3. Methodology The allocation problem is considered in two main classes of games within the game theory framework: (1) bargaining games and (2) coalition games (Thomson, 2007). Although building and managing a coalition is also an important theme (Guajardo and Rönnqvist, 2016), sharing the surplus of the coalition has been the primary subject of cooperative game theory (Moulin, 1987). In this regard, cooperative game theory provides efficient and unique solutions when players are allowed to act independently or cooperatively (Dinar et al., 1992). There are a number of allocation methods developed within the cooperative game theory framework, but the Shapley value and the nucleolus developed by Shapley (1953) and Schmeidler (1969) respectively have received more theoretical attention. The other available solutions include the stable sets developed by von Neumann and Morgenstern (1944), the kernel by Davis and Maschler (1965), and the bargaining set by Aumann and Maschler (1964). The Shapley value has been a focus of cooperative game theory as it introduces a distinct approach to analyze strategic behavior based on the concepts of the core and the stable set (Roth, 1988). It has many attractive normative properties and meets certain criteria (Moulin, 1992; Michalak ea., 2013). A cooperative game can be defined by a finite set of players N = {1, 2, 3, 4, …, n} and a real-valued characteristic function v, or “the coalitional form”:
v: 2 N
R
with v ( ) = 0 . The game is then denoted by (N , v ) . The players of N are allowed to participate in a coalition S. The Shapley value includes the probabilistic interpretations of the marginal contributions of the players, and it can be interpreted as giving players their expected marginal contributions (Schotter and Schwödiauer, 1980). The formulation for the Shapley value for player i, i (v ), is computed as follows: 2
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Fig. 1. Case-insensitive Google ngram viewer results for “the Shapley value”. i (v )
(S
= S N \ {i}
1) ! (N N!
S )!
(v (S )
v (S
Second, population solidarity requires that when a new player is introduced into a coalition, all existing players are affected (benefit or cost) to a minor degree. Third, the stand-alone axiom requires that the total of the players’ individual utilities without a coalition cannot exceed the total utility of a coalition (Moulin, 1992). Finally, the resource monotonicity axiom requires that the utility of each player does not diminish when the total utility to be shared increases (Moulin, 1992). Other division schemes including the nucleolus, the bargaining set, and the kernel are proven to be non-monotonic (Ortmann, 2000). There are some drawbacks related to the Shapley solution. For example, the Shapley value may not be in the core when the game is nonconvex (Shapley, 1971). The core may also be empty or quite large (Saad et al., 2009). Also, when the number of players increases, the complexity of computations increases significantly (van Campen et al., 2018). Finally, the additivity axiom limits the range of the games that the Shapley solution can be used. Nonetheless, there are still a number of other good reasons for selecting the Shapley value as a division scheme (Krajewska et al., 2008). It provides a unique solution for each player and leaves no room for a bargaining process. It is fairly easy to implement when the number of players is limited.
i ))
where S N is a coalition of size |S|, and |N| is the number of players. The expected sum of payoffs for the players participating coalition S, or the worth of coalition S, is denoted by v (S ) . v (S i ) is the value of the coalition in the absence of player i (Colman, 2003a,b). Thus, (v (S ) v (S i)) is a measure of the marginal contribution of player i in coalition S, where all coalitions of size S with player i is taken into consideration. The Shapley value meets four desirable criteria: (1) efficiency, (2) symmetry, (3) null player, and (4) additivity. Efficiency requires that the total utility (or benefit) is distributed among the players participating in the coalition implying i N i (v ) = v (N ) . Symmetry requires that the utilities assigned to each player is not related to the player’s identity, implying v (S {i}) = v (S {j }) where i and j are equivalent players in coalition S. Null player, also called as the dummy player, requires that the players with no marginal contributions receive no utility implying i (v ) = 0 if i is a null player. Finally, additivity requires that the value obtained from the sum of two different games (v + w ) is equal to the sum of the values obtained from these two games individually (v and w ) . That is to say (v ) + (w ) = (v + w ) (Roth, 1988). The Shapley value is characterized by fairness. As stated by van den Brink (2002) a solution that satisfies symmetry and additivity, also satisfies fairness. However, defining fairness in a cooperative environment is not an easy task (Chen et al., 2013), and has been a major conceptual problem in game theory (Deng and Papadimitriou, 1994). In its simplest form, fairness has two major components: proportionality and envy-freeness (Chen et al., 2013). Fairness as a concept in cooperative games has been initially introduced by Myerson (1977). Myerson (1977) argues that players have limited communication opportunities in forming coalitions, and when a communication between two players is deleted, the payoffs of both players are affected in the same way. In his pioneer work incorporating fairness into game theory, Rabin (1993) asserts that people are willing to give away their own benefits in favor of those who help them and to punish those who are unkind. Although the majority of economic models assume that decision-makers seek to maximize their own utility, there is strong empirical evidence that fairness is an important factor in decision making (Fehr and Schmidt, 1999). The Shapley value formula holds four other desirable axioms for efficient allocation of goods: (1) individual rationality, (2) population solidarity, (3) stand-alone test, and (4) resource monotonicity. First, the individual rationality axiom requires that the utility of each player in the coalition should not be less than the player’s benefit without a collation.
4. Case study Along with industrialization, most cities in Turkey have experienced a massive migration from rural areas, resulting in rapid urbanization and unexpected population growth, starting in the 1950s (TurkerDevecigil, 2010). Insufficient supply of affordable housing led to the development of squatter settlements (Uzun et al., 2010). The government was not opposed to urbanization, and the emergence of squatter settlements seemed like its inevitable result. The number of squatter houses, or so-called gecekondus, increased from 100.000 to 1.25 million between 1950 and 1983 (Özler, 2000). In the 1980s, with the enactment of the new amnesty law (Law No. 2981), parcel owners or current occupiers of public land were given development rights through improvement plans. The gecekondus thus have been transformed into low-quality apartments (Turker-Devecigil, 2010; Türkün, 2011). This “populist” approach shifted to a “neo-liberal” one in the early 2000s (Kuyucu and Ünsal, 2010) and urban regeneration has been one of the top priorities of the ruling party in Turkey since then. The year 2011 was declared as “the year of transformation” by the Mass Housing Development Administration, the predominant governmental institution regarding mass housing and mass urban renewal (Karaman, 2013). The era replacing the rational comprehensive planning approach by large scale development projects 3
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Fig. 2. Total numbers of parcel-level redevelopment applications and rebuilt buildings in Turkey (2012–2018).
Fig. 3. Total numbers of parcel-level redevelopment applications and rebuilt buildings in Izmir, Turkey (2013–2018).
had already begun (Kuyucu and Ünsal, 2010). A great portion of the laws regarding the physical transformation of the urban setting has been altered and the roles of key institutions, including the Mass Housing Development Administration (TOKI) and Privatization Administration have been strengthened to enable the realization of large scale urban renewal and regeneration projects (Türkün, 2011). However, the law requires a minimum area of five hectares to declare an urban area as “urban transformation zone”, and this legislative constraint limits large scale transformation in the built-up areas of the cities. As a result, the government has started up new initiatives to trigger smaller scale transformations. Through these initiatives, landowners and developers benefit from a variety of subsidies, including tax cuts and rent subsidy. This process has been eased by the enactment of the Law on the Regeneration of Areas Under the Risk of Disasters (Law no. 6306) in 2012 to give urban regeneration some acceleration at the parcel level (Güzey, 2016). According to Law no. 6306, the redevelopment process is straightforward at the parcel level. The dwellers of the building apply for inspection, and the inspection procedure is held by licensed companies. Then, the redevelopment phase is pursued through a contract between the landowners and the developer. The landowners are free to use land readjustments, that is they can cooperate with neighboring landowners for amalgamation to benefit from higher development rights if available. However, parcel-level transformations have been limited and have not reached the desired level. The number of applications peaked in 2016 with 41,343 applications and has declined dramatically in the following years. Similarly, a sharp decrease in the number of total rebuilt buildings has been observed since the year 2015 (Fig. 2). One of the main problems seems to be land reallocation, as surplus development rights would not be possible for most parcels due to the lack of cooperation between the landowners. A major dispute among the landowners is on the fair share of the extra development rights that are to be achieved through the amalgamation of two or more parcels. The Shapley value can be used as a fair division scheme in such situations. A real-life example from the Karabaglar district, Izmir, Turkey is presented. Since ancient times, Izmir has been one of the largest seaport cities on the Mediterranean coast (Freely, 2004). Izmir has become a city that reflects the characteristics of the Republic of Turkey in the 20th century (Goffman et al., 1999) and reached a population of 4,279,677 by 2017. The very first squatter houses in Izmir were recorded in the early 1930s, and they were mostly located around the city center (Karadag and Miraglu, 2011). The first comprehensive Izmir plan of the 2000s has
been completed in 2007 and revised in 2009. In this 1/25000 scale land-use plan, 15 urban regeneration and renewal sites were delineated covering a total area of 4328 ha. These sites cover 13.7% of the total planning area and 39.0% of the residential areas in Izmir (Izmir Metropolitan Municipality, 2009). However, as in the case in Turkey, parcel-level transformations have been limited. The number of rebuilt buildings peaked in 2016, with 3783 buildings, followed by a sharp decrease in the following years. The number of applications has been 17,869 since the enactment of the Law on the Regeneration of Areas Under the Risk of Disasters (Law No. 6306) in 2012, and the number of rebuilt buildings 14,405 (Fig. 3). A case study is selected from the Karabaglar district in Izmir (Fig. 4) for several reasons. First, the Karabaglar district is the most populous district in Izmir, and one of the most populous in Turkey, with a population of 480,790 in 2017. Secondly, a large share of the regeneration and renewal sites in the 1/25000 scale land-use plan of Izmir are located in the Karadaglar district. Finally, the development plan of the Karabaglar district offers surplus development rights for larger parcels, and thus allows surplus development rights for landowners cooperating for land amalgamation. The development block no. 31044 is selected for further analysis. There are 3 parcels in this block, namely 4, 5 and 6 (Fig. 5). The small number of parcels make the calculations of the Shapley values simple and comprehensible as an example. The zoning decision for the block in the development plan is “residential”, and a maximum of 4 floors is permitted. The floor area ratio is 0.40, the front yard setback 5 m, and the side and rear yard setbacks 3 m (Fig. 6). However, if the total land area is over 1000 square meters, the number of permitted floors increases to 5.5, with the floor area ratio at 0.40 and the front, side and rear setbacks are 5 m. These planning decisions result in a variety of development rights for each parcel in case they are developed individually or in different land amalgamation combinations (Table 1). The Shapley value approach assumes that all combinations are possible and equally likely (Heaney and Dickinson, 1982). There are five possible outcomes in this case, called scenarios. The setbacks and the building floor areas are presented in Fig. 7 and Table 1. For example, in scenario A, all of the three parcels are redeveloped individually. In scenario B, parcels 5 and 6 are amalgamed for redevelopment, excluding parcel 4. Finally, scenario E is the grand coalition, where all three parcels are developed together. Note that the setbacks may limit the footprint of the building, thus the floor area ratio (0.40) may not be fulfilled. For example, in scenarios A and C, the buildable parcel area after setbacks (d) is less than
4
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Fig. 4. Location of the example case in Karabaglar district, Izmir, Turkey.
the maximum floor area (c), which is found by multiplying the parcel area (a) by the floor area ratio (b). Thus, the building floor area (e) is less than the maximum buildable floor area (c), resulting in underuse of the development rights for the parcels. The gross floor area (g) is calculated by multiplying the building floor area (e) by the number of floors (f) (Table 1, Fig. 8). Here, the gross benefits are considered, but the cost of building, taxes and other costs are ignored. This is a plausible approach since these costs are directly proportional to the land size in practice and have a limited effect on the results of the division scheme.
The benefits (measured as the gross floor areas) for the individual parcels and the coalitions are:
v ( ) = 0, v ({4}) = 1504.0, v ({4, 5}) = 2950.0, v ({4, 5, 6}) = 3882.5 v ({5}) = 546.3, v ({4, 6}) = 3000.5 v ({6}) = 649.4, v ({5, 6}) = 1319.7 n
The game is essential. That is v (N ) > i = 1 v ({i}) , implying that the benefits of the grand coalition are greater than the sum of the benefits of the individual players:
v ({4, 5, 6})
v ({4}) + v ({5}) + v ({6}).
Fig. 5. Aerial photo for the parcels in development block no. 31044. 5
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Fig. 6. Development plan decisions for the development block no. 31044.
Also, the game is superadditive, with v (S T ) v (S ) + v (T ) S, T N and v (S ) v (T ) = , where S and T are two disjoint subsets of N:
v ({4, v ({4, v ({5, v ({4, v ({4, v ({4,
v (S T ) + v (S T ) lowing coalitions: v ({4, v ({4, v ({5, v ({4, v ({4, v ({4, v ({4, v ({4,
5}) v ({4}) + v ({5}), 6}) v ({4}) + v ({6}), 6}) v ({5}) + v ({6}), 5, 6}) v ({4, 5}) + v ({6}), 5, 6}) v ({4, 6}) + v ({5}), 5, 6}) v ({5, 6}) + v ({4}).
On the other hand, the game is non-convex. The condition that
v ( S ) + v (T )
5}) v ({4}) + v ({5}), 6}) v ({4}) + v ({6}), 6}) v ({5}) + v ({6}), 5, 6}) v ({4}) + v ({5, 5, 6}) v ({5}) + v ({4, 5, 6}) v ({6}) + v ({4, 5, 6}) + v ({4}) v ({4, 5, 6}) + v ({6}) v ({4,
S, T
N , holds for the fol-
6}), 6}), 6}), 5}) + v ({4, 6}), 6}) + v ({5, 6}),
Table 1 The development rights for the three parcels under all possible scenarios. Scenario
Singleton or Coalition
Parcel Area (m2) (a)
Floor Area Ratio (b)
Maximum Buildable Floor Area (m2) (c) = (a) x (b)
Buildable Parcel Area After Setbacks (m2) (d)
Building Floor Area (m2) (e)=Min{(c), (d)}
Number of Floors (f)
Gross Floor Area (m2) (g) = (e) x (f)
A
{4} {5} {6} {4} {5, 6} {4, 6} {5} {4, 5} {6} {4, 5, 6}
940.0 400.9 423.9 940.0 824.8 1363.9 400.9 1340.9 423.9 1764.8
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
376.0 160.4 169.6 376.0 329.9 545.6 160.4 536.4 169.6 705.9
464.8 136.6 162.3 464.8 386.8 645.6 136.6 558.9 162.3 1034.0
376.0 136.6 162.3 376.0 329.9 545.6 136.6 536.4 162.3 705.9
4 4 4 4 4 5.5 4 5.5 4 5.5
1504.0 546.3 649.4 1504.0 1319.7 3000.5 546.3 2950.0 649.4 3882.5
B C D E
6
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Fig. 7. The setbacks and the building floor areas in each parcel under five different scenarios.
Fig. 8. 3D models regarding the development rights in each parcel under five different scenarios.
but not for the coalition:
Similarly, the Shapley value for Parcel 5,
5 (v ) ,
is:
v ({4, 5, 6}) + v ({5}) v ({4, 5}) + v ({5, 6}). Thus, the Shapley solution may not be in the core, and the core conditions have to be checked Lemaire (1984). When calculating the Shapley value, marginal contributions of the players, (v (S ) v (S i)), are taken into consideration. For example, there are a number of alternatives for the landowner of Parcel 4. The landowner may decide to redevelop the parcel individually (S = 1) or participate in a group of two (S = 2) or three landowners (S = 3). If the landowner decides to participate in a group of two (S = 2), there are two alternatives: cooperating with the landowner of Parcel 5 or of Parcel 6, where the probability of cooperating with either of the other landowners is the same. The Shapley value for Parcel 4, 4 (v ) , is then calculated considering every possible cooperation: S=1
(1
S=2
(2 (2
S=3
(3
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) !
(v ({4})
2) !
(v ({4, 5})
v ({5})=
2) !
(v ({4, 6})
v ({6})=
3) !
1 (1504.0 3
v ({ })=
(v ({4, 5, 6})
v ({5, 6})=
1 (2950.0 6 1 (3000.5 6 1 (3882.5 3
S=1
(1
S=2
(2 (2
S=3
(3
5 (v )
=
S N {4}
|S | ! (| N| |S| |N | !
1) !
(v (S )
v (S
i))=
(v ({5})
2) !
(v ({4, 5})
v ({4})=
2) !
(v ({5, 6})
v ({6})=
3) !
(v ({4, 5, 6})
v ({ })=
v ({4, 6})=
1 (546.3 3 1 (2950.0 6 1 (1319.7 6 1 (3882.5 3
0)=
182.1
1504.0)=
241.0
649.4)=
111.7
3000.5)=
294.0
S N {5}
|S | ! (| N | |S| |N | !
1) !
(v (S )
v (S
828.8
i))=
182.1 + 241.0 + 111.7 + 294.0 =
Finally, the Shapley value for Parcel 6,
0)=
501.3
546.3)=
400.6
S=1
(1
649.4)=
391.9
S=2
(2
1319.7)=
854.3
(2
(3
Thus,
=
1) !
Thus,
S=3
4 (v )
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) ! (3 3! 1) ! (3 3! 1) ! (3 3! 1) ! (3 3!
1) !
(v ({6})
2) !
(v ({4, 6})
v ({4})=
2) !
(v ({5, 6})
v ({5})=
3) !
(v ({4, 5, 6})
v ({ })=
v ({4, 5})=
6 (v ) ,
is:
1 (649.4 3 1 (3000.5 6 1 (1319.7 6 1 (3882.5 3
0)=
216.5
1504.0)=
249.4
546.3)=
128.9
2950.0)=
310.8
Thus,
2148.1
501.3 + 400.6 + 391.9 + 854.3 =
6 (v )
=
S N {6}
|S | ! (| N| |S| |N | !
1) !
(v (S )
216.5 + 249.4 + 128.9 + 310.8 =
7
v (S
i ))=
905.6
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as:
Thus, the Shapley Values for the parcels 4, 5, and 6 can be written
4 (v )
= 2148.1
5 (v )
= 828.8
6 (v )
= 905.6
Since the game is non-convex, it is necessary to assess whether the core is non-empty Lemaire (1984). The core of the game is non-empty since the following inequality holds: 4 (v )
+
5 (v )
+
v ({4, 5}) v ({4, 6}) + 2 2 v ({5, 6}) + v ({4, 5, 6}) 2
6 (v )
2148.1 + 828.8 + 905.6
2950 3000.5 1319.7 + + 2 2 2
3882.5
Although the core of the game is non-empty, the Shapley values may fall outside of the core for non-convex games and this is considered to be a very undesirable outcome (Heaney and Dickinson, 1982). The core conditions in the present case can be written as: 4 (v )
1504.0,
5 (v )
546.3,
4 (v )
4 (v ) +
+
6 (v )
3000.5
6 (v )
649.4,
5 (v )
6 (v )
1319.7,
+
5 (v )
2950.0,
4 (v )
+
5 (v )
+
6 (v )
= 3882.5
It is clear that the Shapley values satisfy these conditions. The Shapley value can also be easily calculated, and core conditions can be checked by using the R Package “CoopGame” developed by Staudacher et al. (2019). Fig. 9 shows the core for the game and the Shapley solution belongs to the core of the game as a CoopGame output (see Appendix for the code). As a result, the landowners benefit from the highest possible development rights, which are 3882.5 square meters of gross floor area, and are able to share the surplus development rights fairly: landowner of Parcel 4 gets 2148.1 m2, landowner of Parcel 5828.8 m2, and landowner of Parcel 6905.6 m2. 5. Conclusions As many more people are expected to live in urban areas in the coming decades, much of the physical transformation will take place in the built-up areas of the existing cities. Local and municipal governments have to put more effort in reshaping the cityscape and struggling with the landownership disputes. Thus, new and innovative ways of collaborations should be sought. This paper shows that a fair sharing of the surplus development rights through the utilization of the Shapley value solution may provide landowners the highest development rights, which can be considered as a crucial incentive for small scale redevelopment in the built-up parts of the cities. However, such fair division schemes are only possible with a proper institutional setting. Currently, the redevelopment process is governed by a contract between the landowners and the developer. The present institutional setting allows landowners of neighboring parcels to cooperate for land amalgamation but does not provide any insights on the fair sharing of the resulting surplus development rights. An institutional setting providing such fair division instruments to the stakeholders may ease the parcel-level transformation in the built-up areas, allowing landowners benefit from maximum development rights. A proper and well-functioning institutional setting is also crucial in the sense that alternative solutions, which are also in the core of the game, can be put forward or asserted by different players in the game. As shown by Nash (1953), the same fair solution may also be obtained through a bargaining process within the non-cooperative game theory framework, but, an
Fig. 9. The core for the game and the Shapley solution.
institutional mechanism must be established to govern the interactions among the players, even if these players have limited or no information on their preferences over outcomes (Serrano, 2004). However, in the case of redevelopment in urban areas, an agreement between the stakeholders may not be achieved through a negotiation process, especially when the number of players is high. Thus, an institutional setting empowering with fair division solutions based on cooperative game theory can be the long-awaited solution to the sharing of surplus development rights among landowners. The main incentive for landowners to act in collaboration is the planning decision allowing higher development rights for larger parcels. This incentive is a plausible approach to achieve better living conditions in bigger and legally built buildings, as most of the parcels in urban regeneration and renewal sites are relatively small and subject to individual illegal building or modification process. However, there is an upper limit for higher development rights, as urban amenities have to be planned for the increasing population caused by the additional development rights. Thus, the regulator making planning decisions has to consider the balance between the incentives for redevelopment and the share of land allocated for urban amenities, which is, in fact, another allocation problem relating to a fair sharing of land between public and private stakeholders. 8
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Acknowledgments
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I would like to thank Jean-Michel Guldmann for reviewing the final draft of this paper, and the two anonymous reviewers for their valuable comments. However, any mistakes that remain are my own. Appendix A The following code can be used with the “CoopGame” package in R to derive the results obtained in the study. The outputs are provided under the command lines. # To define the characteristic functions by using the payoffs for the players/coalitions of {4}, {5}, {6}, {4,5}, {4,6}, {5,6}, and {5,6,7} from Table 1 > characteristic_function < - c(1504.0, 546.3, 649.4, 2950.0, 3000.5, 1319.7, 3882.5) # To check if the game is essential > isEssentialGame(characteristic_function) [1] TRUE # To check if the game is superadditive > isSuperadditiveGame(characteristic_function) [1] TRUE # To check if the game is convex > isConvexGame(characteristic_function) [1] FALSE # to derive the Shapley values for each player > shapleyValue(characteristic_function) [1] 2148.0667 828.8167 905.6167 # to check if the Shapley solution is in the core of the game > belongsToCore((shapleyValue(characteristic_function)), characteristic_function) [1] TRUE # To draw the core of the game > drawCore(characteristic_function) # To draw the Shaply solution on core of the game > drawShapleyValue(characteristic_function, holdOn = TRUE) References Abdulkadiroglu, A., Sonmez, T., 1998. Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66, 689–701. Abdulkadiroglu, A., Sonmez, T., 1999. House allocation with existing tenants. J. Econ. Theory 88, 233–260. Aumann, R.J., Maschler, M., 1964. The bargaining set for cooperative games. In: Dresher, M., Shapley, L.S., Tucker, A.W. (Eds.), Advances in Game Theory (Annals of Mathematics Studies, 52). Princeton University Press, Princeton, pp. 443–476. Barnett, J., 2018. The Fractured Metropolis: Improving the New City, Restoring the Old City, Reshaping the Region. Routledge. Bogomolnaia, A., Moulin, H., 2001. A new solution to the random assignment problem. J. Econ. Theory 100, 295–328. Brams, S.J., Taylor, A.D., 1996. Fair Division: From Cake-cutting to Dispute Resolution. Cambridge University Press. Chen, Y., Lai, J.K., Parkes, D.C., Procaccia, A.D., 2013. Truth, justice, and cake cutting. Games Econ. Behav. 77 (1), 284–297. Colman, A.M., 2003a. Cooperation, psychological game theory, and limitations of rationality in social interaction. Behav. Brain Sci. 26 (2), 139–153. Colman, A.M., 2003b. Game Theory and Its Applications: in the Social and Biological Sciences. Routledge. Deng, X., Papadimitriou, C.H., 1994. On the complexity of cooperative solution concepts. Math. Oper. Res. 19 (2), 257–266. Dinar, A., Ratner, A., Yaron, D., 1992. Evaluating cooperative game theory in water resources. Theory Decis. 32 (1), 1–20. Ehlers, L., 2002. Coalitional strategy-proof house allocation. J. Econ. Theory 105, 298–317. Ehlers, L., Klaus, B., 2004. Resource-monotonicity for house allocation problems. Int. J. Game Theory 32 (4), 545–560. Ergin, H.I., 2000. Consistency in house allocation problems. J. Math. Econ. 34, 77–97. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition, and cooperation. Q. J. Econ. 114, 817–868. Freely, J., 2004. The Western Shores of Turkey: Discovering the Aegean and Mediterranean Coasts. Tauris Parke., London. Garreau, J., 1991. Edge City: Life on the New Frontier. Anchor Books. Gibbons, R., 1992. A Primer in Game Theory. Harvester Wheatsheaf. Goffman, D., 1999. Izmir: from village to colonial port city. In: Eldem, E., Goffman, D.,
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