A game-theoretic model of car ownership and household time allocation

Transportation Research Part B 104 (2017) 667–685

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

A game-theoretic model of car ownership and household time allocation Mingzhu Yao a, Donggen Wang a,∗, Hai Yang b a

Department of Geography, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong, PR China Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China

b

a r t i c l e

i n f o

Article history: Received 5 May 2016 Revised 29 May 2017 Accepted 30 May 2017 Available online 9 June 2017 Keywords: Car ownership Time allocation Game-theoretic approach Household interaction Vehicle usage rationing policy

a b s t r a c t The explosive growth of private cars in China and other developing countries has attracted a great deal of renewed research interest in car ownership. This paper investigates households’ car ownership decision-making process from the perspective of household time allocation. Applying the game-theoretic approach to capturing household members’ interactive decision-making mechanism, we propose a two-stage model that links household members’ short-term time allocation decisions to long-term car ownership decisions. The first stage models the bargaining of household members (e.g., husband and wife) over the car ownership decision, taking into consideration of government policies for regulating car ownership; and the second stage is a generalized Nash equilibrium model for activitytravel pattern analysis incorporating individuals’ interactions concerning activity participation. The existence and uniqueness of the generalized Nash equilibrium solution is examined, and a heuristic procedure that combines backwards induction and method of exhaustion is adopted to solve the two-stage game. The proposed model is applied to an empirical case study in Beijing, which demonstrates the applicability of the model in predicting car ownership and examining interactions between car ownership and household time allocation. The empirical model is applied to assess the impacts of plate-number-based vehicle usage rationing policies on car ownership and time allocation to travel and daily activities. Results show that the model can be applied to evaluate the car ownership impacts of car usage rationing policies. © 2017 Published by Elsevier Ltd.

1. Introduction 1.1. Background and motivation The fast economic growth has greatly contributed to the rapid pace of motorization in developing countries in recent years. Taking China as an example, it experienced an exponential growth in total number of private cars in less than a decade, from 18.5 million in 2005 to 88.4 million in 2012. In the same period, the number of private cars in Beijing, the capital of China, nearly tripled from 1.5 million (14.1% household car ownership) to 4.1 million (42.3% household car ownership) .1 The fast growth in vehicle ownership has resulted in serious traffic congestion and air pollution problems. As a ∗

1

Corresponding author. E-mail address: [email protected] (D. Wang). Information source: National Bureau of Statistics of China. http://www.stats.gov.cn/.

http://dx.doi.org/10.1016/j.trb.2017.05.015 0191-2615/© 2017 Published by Elsevier Ltd.

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response, since 2008, the transportation authority in Beijing has been implementing the plate-number-based vehicle usage rationing policy with the objectives of mitigating traffic congestion as well as discouraging car ownership. The policy stipulates the usage of cars with certain plate numbers on specific workdays.2 This policy intervention may have helped in restraining people’s desire to buy cars, but it may also turn out to motivate financially well-off people to buy additional cars in order to maintain access to cars (Goddard, 1999). Therefore, research efforts are needed to investigate the real effects of policy measures such as the vehicle usage rationing policy on car ownership. Further, the interrelationship between car usage and time allocation have been revealed in the literature (e.g., Golob et al., 1995; Ding et al., 2014), it is thus worth noting that car ownership plays an important role in facilitating household members’ engagement in daily activities and associated travel (Li et al., 2010). Although previous studies have made significant contributions to acknowledging the influence of car usage on time allocation patterns and investigating the decision mechanism of household car ownership, they can hardly tackle questions regarding how household members make tradeoffs regarding spending money on cars to save travel time for activity participation, to what extent car usage influences the daily time allocation to activities, and how the vehicle usage rationing policies impact car ownership decision and the resultant time allocation decision, etc. 1.2. Previous studies on car ownership With respect to static modelling, there are in general two approaches to modelling car or auto ownership in the existing studies. The first approach applies the computationally efficient aggregate models that predict car ownership at zonal, regional or national level (see Jong et al., 2004 for a detailed review); the second approach makes use of the disaggregate models (often at household level), which treat single household as a decision making unit and examine the determinants of household car ownership such as household social-economic variables and built environment attributes (Bhat and Pulugurta, 1998). By dealing with individual households separately, some researchers argued that the disaggregate modelling approach was demonstrated to be more appropriate and preferable to car ownership modeling in terms of reducing aggregation bias, estimating high precision model parameters, and addressing human behavior (Potoglou and Kanaroglou, 2008). When addressing household car ownership problem itself or the interrelation between household car ownership and other long-term/short-term choices at a disaggregate level, disaggregate car ownership models in literature usually take the form of discrete choice because car ownership is a categorical variable (Li et al., 2010; Pinjari et al., 2011). Depending on whether the ordinal nature of car numbers is utilized in the modeling mechanism or not, the discrete choice over car ownership alternatives can be further divided into ordered-response models represented by ordered-response logit and ordered-response probit (Golob and Van Wissen, 1989; Bhat and Koppelman, 1993; Kim and Kim, 2004), and unorderedresponse models including multinomial logit (MNL) and multinomial probit (MNP) models. Ordered-response structure has the advantage of discerning unequal differences between ordinal categories in a dependent variable, but Bhat and Pulugurta (1998) empirically demonstrated that unordered-response structure could more closely represent car ownership decision-making behavior. Within unordered-response models, multinomial logit was most frequently used for car ownership study (e.g. Lerman, 1976; Purvis, 1994; Ryan and Han, 1999; Potoglou and Kanaroglou, 2008), and it has been applied to different geographical areas since its first introduction by Lerman and Ben-Akiva (1976). As an extension structure of the MNL model, the nested logit model was also used for multidimensional cases when jointly dealing with car ownership choice and other possible choices such as mode (Train, 1980) and vehicle type (Mannering and Winston, 1985). Besides, the MNL model was combined with group decision theoretic approaches to examine interactions among household members on car size choice (Zhang et al., 2009). Another representative type of unordered-response model, the multinomial probit model was also applied by Bunch and Kitamura (1989) to auto ownership prediction. But as they stated, MNP is less commonly used due to its various computational difficulties associated with parameter estimation. Apart from these static models, researchers have begun to model car ownership from a dynamic perspective recently. Roorda et al. (2009) proposed an integrated model of dynamic vehicle transactions (change car ownership by purchasing, disposing or replacing vehicles) and activity scheduling/mode choice, where intra-household interactions of vehicle allocation, ridesharing, and drop-off/pick-up of household members were considered. More choice dimensions were addressed by embedding a discrete-continuous choice model into a dynamic programming framework, which allowed a joint modelling of transaction type, annual driving distance, fuel type, car ownership status and car state relative to each car in a household’s fleet (Glerum et al., 2013). The existing dynamic discrete choice models based on pure dynamic programming perspective were improved by Cirillo et al. (2015) in the aspects that the optimal time of purchase must be decided and quality of different vehicles types changed stochastically over time. However, the above-mentioned methodologies of disaggregate modeling employed in car ownership research generally ignore the individual differences and interactions within a household in reaching an agreement on car ownership decision (in terms of number of cars to own). They usually treat a household as if it were an individual, despite the fact that intra-household interactions in decision making have gained research attention in recent decades (Bhat and Pendyala, 2005; Timmermans and Zhang, 2009; De Palma et al., 2016). Indeed, male head, female head, children and other household members with different car ownership or car usage preferences may interact with each other by means like bargaining to reach a final agreement. Roorda et al. (2009) and Meister et al. (2005) dealt with intrahousehold interactions of vehicle allocation

2

Information source: Official website of Beijing Traffic Management Bureau. http://www.bjjtgl.gov.cn/jgj/95332/127211/index.html

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and usage; however, the interactions among household members with divergent preferences in reaching an agreement on car ownership level were not explicitly considered. Furthermore, existing studies mostly examine car ownership decision as an isolated household long-term decision, but car ownership and usage are demonstrated to have significant impact on the time allocation patterns of household members (e.g. Zhang and Fujiwara, 2006; Zhang et al., 2007; Wang and Li, 2009; Bee, 2016); therefore, it is worthwhile to investigate car ownership decision from the viewpoint of time allocation.

1.3. Time allocation study incorporating interactions Originated from the economic theories of family that study households’ resource allocation problem (Becker, 1965), considerable research attention has been devoted to the development, operationalization and refinement of time allocation analysis over the past decades (e.g. Kitamura, 1984; Kitamura et al., 1996; Bhat and Misra, 1999). These early unitary models typically assume individual level time allocation decision-making. But later it has been recognized and emphasized that in practical situations of time allocation, household members do not make isolated decisions over activity participation, and they are likely to interact with each other in many manners (e.g. Golob and McNally, 1997; Bhat and Pendyala, 2005). The possible single- or multi-facet interactions among individuals are normally through activity participation and resource allocation, which may include sharing of household responsibilities (e.g. maintenance responsibilities), joint engagement in travel and non-travel activities, and sharing of limited household resources (e.g. available vehicles in multi-driver households) (Timmermans, 2009). In order to accommodate these different types of interactions, a variety of modeling methodologies have been adopted. The first approach used structural equations modeling methodology to study the interrelationships between household members in allocating time to out-of-home activities (Van Wissen and Meurs, 1989), identify potential interconnection between household heads in activity participation and travel (Golob and McNally, 1997), and analyze individuals’ joint activity engagement with family members and non-family members (Fuji et al., 1999). Structural equations methods offer an approach to analyzing the interrelationship between household members’ activity participation; however, they are unable to accommodate unordered multinomial discrete choice variables (Pendyala, 2009). Consequently, the second approach using discrete choice models derived from random utility theory was applied to further explain the decision-making processes (e.g. Gliebe and Koppelman, 20 01; ;20 02 Srinivasan and Athuru, 2005 ; Scott and Kanaroglou, 2002). For example, Gliebe and Koppelman (2002) proposed a nested choice structure that incorporated all household members’ joint participation alternatives at the upper level and individual’s independent participation alternatives at the lower level, so as to capture interactions at both household and individual level. Because the discrete choice models cannot differentiate the relative influence of members in joint decision-making process, the third approach was developed by Zhang et al. (2002) to explore household group decision-making mechanism in a more behaviorally-oriented way. This approach adopted group utility functions (GUF) rooted from group decision theory to incorporate household members’ cooperative interactions in time allocation decision. Two classes of group utility functions— the multilinear group utility function and isoelastic group utility function—were introduced (Zhang et al., 20 02; 20 05) for modeling household task allocation and time use. In essence, both these two classes are composite utilities that combine family members’ individual utilities in certain ways to reflect interaction mechanism. A special case of multilinear group utility function, the additive-type utility function has been frequently used (e.g. Wang and Li, 2009), which is a weighted summation of all members’ utilities that reflects their relative importance. The additive-type household utility function form is similar to the collective model (Chiappori, 1992), but the latter requires Pareto efficiency based on hypothesis of collective rationality, as can be seen in the collective model employed by Kato and Matsumoto (2009) for joint time allocation analysis. The group decision-making mechanism is favorable to representing the cooperative interactions among household members in time allocation decision process. But they require binding forces that integrate household members’ utilities in some manner into a group utility, which represents their common interests. Household members make time allocation decisions to jointly maximize the group utility. For further exploration and prediction of individuals’ interactions in activity participation and travel behavior analysis, researchers have argued for the necessity of incorporating new techniques such as game theory approaches in group behavior modeling (Zhang and Daly, 2009). Nevertheless, most of the above time allocation studies did not devote efforts towards separating travel time from time allocated to non-travel activity participation. The line of research pioneered by Train and McFadden (1978) and developed by other researchers have tried to make a separation to evaluate travel time saving effects, which addressed travel mode choice and activity time assignment in a common microeconomic framework (e.g. Jara-Díaz and Farah, 1987; Jara-Diaz and Guevara, 2003; Jara-Díaz and Guerra, 2003; Munizaga et al. 2008). However, car ownership was exogenously given in this line of model system that dealt with short-term time assignment to activity participation and travel. The model system assumed that car mode would be available to drivers as long as there was a car in the household, without explicit consideration of car deficiency in multi-driver households. Besides, the impacts of household/individual level socio-demographics and socioeconomics on time assignment was left out of consideration. In view of these previous research limitations, it is necessary to integrate the endogenous choice of car ownership into the short-term time assignment and travel model system and explore the allocation of deficient car resource within multi-driver households; because the short-term time assignment and travel feedback information about car resource needs and may exert an influence on car ownership choice. Moreover,

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intra-household interactions need to be taken into account in this line of research to maintain consistency with practical situations. 1.4. Study objectives In view of the above discussion, this study aims to propose a modeling framework that incorporates game-theoretic approaches into travel behavior modeling to address the aforementioned issues. Specifically, (1) we propose a Nash bargaining model for disaggregate car ownership choice that explicitly accounts for individual differences and intra-household interactions in car ownership decision, and introduce the split of car usage among household members. The car ownership decision is also investigated from the perspective of time allocation, which is reflected by a sequential decision mechanism drawn from game theory. (2) We employ a Generalized Nash equilibrium (GNE) model to incorporate household members’ interactions through maintenance time allocation. Unlike those cooperative time allocation models that require outside binding forces, GNE is self-enforcing. (3) We calibrate the model with real world data and use the calibrated model to examine the impacts of transport policies such as the vehicle usage rationing policies. The rest of this paper is organized as follows. Section 2 develops a two-stage model framework based on the gametheoretic approach that explicitly incorporates various intra-household interactions, and proposes a solution approach. The econometric model and estimation method are presented in Section 3. Section 4 shows the calibration results of the theoretical model and its application for prediction and impact analysis. Section 5 concludes the study by summarizing the major research findings and pointing out future research directions. 2. Model formulation Car ownership decision is usually regarded as a joint decision due to its long-lasting impact on family members (Zhang et al., 2009), which makes it reasonable to examine from the perspective of household group decision. Considering that children in many cases neither have the capacity nor the power to influence household decisions and following the normal practice of studies on household decision problems in previous literature (such as bargaining over car allocation (Anggraini et al., 2008; Anggraini et al., 2012), bargaining over consumption of private goods and public goods (Lundberg and Pollak, 1993), etc.), we assume that only husband and wife are involved in the household decision making process concerning car ownership to simplify the problem. A direct extension can be made to include a third stakeholder (e.g., children with driving license) in the bargaining over car ownership, which will make the two-stage model more complicated but yet still tractable; it will not fundamentally change the model structure. To facilitate the presentation of the modeling framework, we only consider the two household heads, which should not be at the cost of generalizability of the model. As a decision that exerts long-term influence, car ownership is highly likely to have direct or indirect implication for household members’ daily time allocation decisions, especially through its impact on travel time. In order to capture the link between long-term vehicle ownership decision and short-term time allocation decision, a two-stage decision making framework is proposed. The framework is outlined in Fig. 1. As shown in Fig. 1, the whole decision process is initiated by household members’ desire to improve their utilities by obtaining an optimal household car ownership level, which is formulated as the first stage model. In the second stage, they make daily time allocation decisions, conditional on car ownership strategy from the first stage. In turn, when making car ownership decision in the first stage, players realize and take into account the responsive time allocation strategies in the second stage. The two-stage model facilitates the realization of this sequential decision-making mechanism, and the decision on car ownership and time allocation are still modeled as joint decisions because they come from the common modeling framework.

Fig. 1. Two-stage game-theoretic framework.

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2.1. First stage model: car ownership decision In previous study of household bargaining and group decision, efforts were devoted to examining intra-household decision making on fertility (Eswaran, 2002; Rasul, 2008), goods consumption (Lundberg and Pollak, 1993), time allocation (Zhang et al., 20 02; 20 05), residential location (Timmermans et al., 1992; Borgers and Timmermans, 1993) and so on, with applications to stated/revealed preference data. The approaches adopted in these studies include either cooperative gametheoretic approach (represented by Nash bargaining solution) or group decision-theoretic approach (e.g. the commonly used additive group utility function and multilinear group utility function). In Nash bargaining solution, individuals are motivated by proportionate cooperation. That is, each individual wants to increase the joint gains, but only to the extent that his/her own share in it also increases (MacCrimmon and Messick, 1976; Corfman and Gupta, 1993). As long as there is one individual prefers his/her status-quo to an alternative chosen by others, the latter alternative will not be the group’s choice. Effectively, each individual has the power to veto the desires of all others, but there is no such veto power in the decision-theoretic models. In terms of the predicative accuracy, Eliashberg et al., (1986) demonstrated that in comparison with group decision-theoretic approach which used group utility functions (including additive form and multilinear form), Nash bargaining approach performed better in accurately predicting outcome of a marketing channel laboratory simulation. Therefore, Nash bargaining approach is adopted in the proposed model. Bargaining between the wife and the husband forms the basis of household decisions (Iyigun and Walsh, 2007). In the process of reaching an agreement on car ownership level, husband and wife may have imbalanced bargaining powers on decision-making. This effect is supposed to be captured by an asymmetric/generalized Nash bargaining model (Harsanyi and Selten, 1972; Chen and Woolley, 2001). In the context of discrete choice over possible car ownership alternatives, the point with the largest Nash gain will be chosen (Ott, 1992). The Nash bargaining solution of car ownership level n is characterized by the following maximization problem,



max unh − u0h n

α1 

unw − u0w

1−α1

n = 0, 1, 2, . . . , N

(1)

The product in this formula is termed as Nash product.3 unh and unw are the respective utilities of husband (denoted by h) and wife (denoted by w) under car ownership level n. u0h and u0w are threat point utilities of husband and wife in the case that they fail in consensual decision-making, which indicates that 0 car will be bought. Parameter α 1 ∈ [0, 1] measures the relative bargaining power of husband in car purchasing decision. If α 1 = 0.5, then it is the special case that both parties are equal in power. Otherwise, they have imbalanced bargaining powers. If α 1 = 0 or α 1 = 1 then it turns into extreme cases that the decision is made by a single spouse. N is set as the upper limit of cars in a family, because it is unlikely for a household to purchase excessive cars. To represent individuals’ tradeoff between time expenditure and money expenditure, the goods/leisure framework proposed by Train and McFadden (1978) is adopted here. Cobb-Douglas utility function with preference parameter α i is selected from many possible functional forms to represent the car ownership utility uni that spouse i(i= h, w) wants to maximize by choosing car ownership level n (Amador and Cherchi, 2011),



uni = S · Tir,n

αi  n 1−αi Gi

∀i = h, w

Subject to monetary constraints,



(2)



Gnh + h · n · pcars = S · wh · Thw,n +  · ww · Tww,n + Ih +  · Iw Gnw + w · n · pcars = S ·



  · wh · Thw,n + ww · Tww,n +  · Ih + Iw

(3) (4)

where α i measure spouse i’s output elasticity of time. A larger value of α i means a higher value for spouse i’s leisure time surplus. Correspondingly, 1 − α i measure spouse i’s output elasticity of money surplus for goods consumption. Tir,n and Tiw,n are spouse i’s time allocation for recreation and subsistence activities under car ownership level n, which will be obtained from the later second stage time allocation model. Gni represents spouse i’s money surplus for goods consumption. Service life of cars S and price of cars pcars both take an average value. Wage rates of the couple are given by wh and ww respectively. Except wage income, other income sources are captured by Ih and Iw . The management of family finance may vary across households; thus we introduce parameters  , ε h and  w to dictate the degree of independence or dependence of family finance. If family finance is independent, then  = 0 and  h +  w = 1; if family finance is pooled, then  = 1,  h = 1 and  w = 1. 2.2. Second stage model: time allocation decision Under given car ownership level from the first stage, we can proceed to calculate the daily travel time that husband and wife take to reach their destinations. They may either engage in solo or joint travel. If the car ownership level is zero, 3 Nash product can be regarded as the deterministic part (without adding the stochastic error term) of a type of meta-utility (Zhang et al., 2009). The concept of meta-utility was initially proposed by Swait et al. (2004) for relating utilities from various temporal states, while Zhang et al. (2009) also applied such concept in relating household members’ utilities. In general, meta-utility relates utilities from various temporal states, different individuals and so on together, and the unobservable aspects associated with the meta-utility function are assumed to be independent over time or across different household members (Swait et al. 2004; Zhang et al., 2009).

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whether they travel together or not makes no difference to their travel time,

Tit,0 =

Di

vother

∀i = h, w

(5)

Dividing spouse i’s total travel distance Di by the average travel speed of all other modes vother (except auto), his/her average travel time Tit,0 under 0 car ownership can be obtained. Daily total travel distance of an individual is assumed to be given exogenously to make a differentiation between travelling by car and by other modes and evaluate the time saving effect of purchasing cars, that is how household members would make tradeoffs regarding spending money on cars to save travel time for activity participation. This assumption can be relaxed in future research by incorporating the benefits of possible longer travel distance brought by car ownership, and the tradeoff between travel time and travel distance. If car ownership level is non-zero (n > 0), both parties’ access to car usage for joint travel can be guaranteed. But ind denote their respective car husband and wife may have to bargain over car usage for independent travel. Let βhind and βw usage probability for independent travel, the travel time of spouse i under car ownership n can be derived as follows,

Tit,n = βiind

 (1 − τi )Di τi Di (1 − τi )Di  + 1 − βiind + ∀i = h, w vcar vother vcar

(6)

The travel time is comprised of three components: independent travel time by car, independent travel time by other modes and joint travel time by car, as sequentially expressed in Eq. (6). βiind is spouse i’s car usage probability for independent travel. vcar represents the average travel speed by car. τ i is spouse i’s joint travel percentage satisfying an implicit relationship τ h Dh = τ w Dw . To obtain βiind , we consider four possible scenarios based on different feasible combinations of the number of cars available (denoted by n · r) and whether spouse i holds a driver’s license (δ i ). The parameter r represents car usage rationing policy (Han et al., 2010), which indicates the percentage of days in the week that private cars are allowed to use for travel in the city. δ i indicates whether spouse i has driving license or not. It equals 1 if spouse i has driving license and 0 otherwise. (1) If δ h + δ w = 2 (equivalent to δ h = 1 and δ w = 1) and n · r < 2, then husband and wife have to bargain over the limited usage of n · r available cars. According to the ‘Split-The-Difference Rule’ (Muthoo, 1999), let α 2 denote husband’s relative bargaining power for car usage decision, then the partition of n · r available cars is supposed to be α 2 · n · r for husband and (1 − α 2 ) · n · r for wife. However, from the perspective of efficient resource allocation in a household, the car resource for husband or wife should not exceed one. Thus, we assume the excessive car resource is automatically transferable between husband and wife. That is to say, if one spouse’s bargained car resource exceeds one, it will be transferred to the other spouse. This effect is reflected in the following equations designed to calculate ind for independent travel, the car usage probability of husband βhind and wife βw

βhind = min[1, α2 · n · r] + max[0, (1 − α2 ) · n · r − 1]

(7)

βwind = min[1, (1 − α2 ) · n · r] + max[0, α2 · n · r − 1]

(8)

(2) If δ h + δ w = 2 (equivalent to δ h = 1 and δ w = 1) and n · r ≥ 2, which signifies the case that the number of available cars equals or exceeds the number of drivers, then no bargaining is needed for access to cars. In this case, βhind = 1,

βwind = 1. (3) If δ h + δ w = 1, implying there is only one driver within the household, then bargaining is unnecessary for access to cars,

βhind = min[1, δh · n · r]

(9)

βwind = min[1, δw · n · r] (4) If δ h + δ w = 0, then there is no need to buy cars. Hence,

(10)

βhind

= 0,

βwind

= 0.

Travel time derived in the way described above incorporates interactions between husband and wife concerning household vehicles allocation and joint engagement in travel, and will be an input variable for the decisions concerning time allocation to activities. In terms of modeling methodologies for time allocation decisions, Nash equilibrium model, the leading non-cooperative model in game theory is adopted. Why we adopt Nash equilibrium model is based on the assumption that unlike the longterm car ownership decision formulated as a cooperative Nash bargaining model in the first stage, which requires binding forces for commitment, household members may not bother or be able to negotiate binding commitments, or cannot adhere to possible commitments on a daily basis, so the non-cooperative Nash equilibrium model (self-enforcing) seems to be more suitable for capturing their daily time allocation patterns, where household members act separately to maximize their own utilities but their actions will influence each other through the share of maintenance responsibility. Mathematically speaking, the second stage model can be formulated as a generalized Nash equilibrium problem (GNEP) because the feasible strategy profiles of husband and wife are interdependent, which is reflected specifically in the coupled maintenance time constraint (Eq. (15)) (see Facchinei and Kanzow, 2007 for details of GNEP).

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The literature suggests that the utility of an activity increases and the marginal utility diminishes as the amount of time allocated to it increases, and this relationship may be represented by a logarithmic function (Kitamura, 1984; Kitamura et al., 1996; Bhat and Misra 1999). By adopting logarithmic utility function for activity participation, the GNEP under given car ownership level from the first stage can be formulated as,













max

w,n m,n uw + 1 + um + 1 + urh ln Thr,n + 1 h ln Th h ln Th

(11)

max

w,n m,n uw + 1 ) + um + 1 ) + urw ln (Twr,n + 1 ) w ln (Tw w ln (Tw

(12)

Thw,n , Thm,n

Tww,n , Twm,n

Thw,n + Thm,n + Thr,n + Tht,n = Th

(13)

Tww,n + Twm,n + Twr,n + Twt,n = Tw

(14)

Thm,n + Twm,n = M

(15)

j

where ui (i = h, w; j = w, m, r ) is household member i’s baseline utility for engaging in activity j. Eqs. (13) and (14) are the respective time budget for husband and wife. We assume that each household has certain maintenance task that needs the amount of time M to accomplish and this task can be shared by the two spouses, with maintenance time input Thm,n and Twm,n respectively. Substitute Eq. (13) and Eq. (14) into the objective functions respectively and we can derive KKT (Karush– Kuhn–Tucker) conditions for the above GNEP,

⎧ uw urh h ⎪ − =0 ⎪ w,n ∗ ⎪ Th +1 Th − Thw,n∗ − Thm,n∗ − Tht,n + 1 ⎪ ⎪ ⎪ m r ⎪ uh ⎪ uh ⎨ − − λh = 0 m,n∗ w,n∗ m,n∗ t,n Th

+1

Th − Th

− Th

− Th + 1

Tw

+1

Tw − Tw

− Tw

− Tw + 1

urw uw ⎪ w ⎪ − =0 ⎪ w,n∗ w,n∗ ⎪ Tw + 1 ⎪ Tw − Tw − Twm,n∗ − Twt,n + 1 ⎪ m r ⎪ uw ⎪ ⎩ m,nu∗w − − λw = 0 w,n∗ m,n∗ t,n

(16)

where Lagrangian multipliers λh and λw are the marginal utilities of time use for maintenance activities for male and female heads respectively, as can be noted in the second and fourth component of equation set (16). It is well known that GNEP is likely to have multiple equilibrium points. However, as proved in Appendix A, the proposed GNEP is a diagonally strictly concave game, indicating that there is a one-to-one correspondence between the share of maintenance responsibility between husband and wife and the equilibrium point (Rosen, 1965). Once such share is fixed, only one corresponding equilibrium point could be identified. Like previous studies with GNEP applications, such as Krawczyk (2005) and Boucekkine et al. (2010), we assume that husband and wife have equal share of maintenance responsibility, which leads to λh = λw as shown in Appendix A. That is to say, husband and wife will experience the same degree of penalty if anyone of them fails to accomplish his/her maintenance responsibility. Nevertheless. the equal share assumption can be relaxed. As long as the share of maintenance responsibility is given, one and only one corresponding equilibrium point can be determined. 2.3. Solution approach Because payoffs of the first stage rely on results of the second stage in the proposed sequential decision game, backwards induction as a common approach to computing subgame perfect equilibria is adopted to obtain the solution (Gibbons, 1992). Furthermore, since n is a non-negative discrete variable, the procedure of solving the two-stage game could be a combination of backwards induction and method of exhaustion, and the optimal outcome of this two-stage game of complete information will be a sub-game perfect outcome. The procedure works as follows, Step 0: Initialization. Set car ownership level n = 0. Step 1: Solve the second-stage model under car ownership level n from the first stage. Firstly determine travel time Tit,n from Eqs. (5)–(10). Next, apply Newton-Raphson algorithm to solve the KKT conditions expressed by equation set (16) to obtain the optimal time allocation pattern Tiw,n∗ and Tim,n∗ for subsistence and maintenance activity. Finally, obtain the optimal time for recreation activity Tir,n∗ by substituting Tit,n , Tiw,n∗ and Tim,n∗ into time constraint (13) and (14) respectively. Step 2: If Gnh < 0 or Gnw < 0, assign a very small negative value to the Nash product n ; otherwise, substitute Tit,n and j,n∗

Ti (i = h, w j = w, m, r ) into the first-stage model to obtain the corresponding Nash product under car ownership level n, namely n = (unh − u0h )α1 (unw − u0w )1−α1 . Step 3: If n < N, let n = n + 1, and return to step 1. Otherwise, end the iteration process.

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Step 4: Find the maximal value from these Nash products n (n = 0, 1, 2..N), then the corresponding car ownership level j,n∗ n∗ in conjunction with the corresponding time allocation pattern Ti constitute the outcome of this two-stage game.

3. Model estimation We assume two sources of error terms exist in the proposed two-stage model; one is the stochastic error terms in car ownership model as shown in Eq. (27), and the other is the stochastic error terms in baseline utility of time allocation model in Eq. (17). Theoretically, there might be correlation between these two sources of error terms. But the econometric calibration would be intractable if the correlation is considered. Due to this econometric challenge, we assume these two sources are independent, so that parameters in car ownership model and time allocation model could be sequentially estimated by a “bottom up” procedure, starting with the second stage estimation, and then input parameters estimated at the second stage to estimate the first stage model. Similar simplification was also made by other researchers, and it was supposed not to impact the validity of the model (e.g. Jara-Díaz and Guevara, 2003). Jara-Díaz and Guevara (2003) assumed independence between error terms for mode choice decision and work time decision, which come from the same microeconomic framework. Nevertheless, this assumption can be relaxed in future research by exerting more efforts to explore the linkage of these two sources of error terms for simultaneous estimation (Munizaga et al., 2008).

3.1. Parameter estimation for second stage j

Assume an additive error term exists in baseline utility ui ,





uij = vij + εij = fij β˜ij , x˜ij + εij

∀ i = h, w j = w, m, r

(17)

j j j j where ui consists of the systematic component vi and the error term εi . x˜i is a vector of socio-economic variables and j β˜i is the vector of coefficients associated with these variables through function fij . We further assume the error terms are independently and identically standard normal distributed, so the following joint density function can be obtained,



g

2 2 2  (εhw ) +(εhm ) +(εhr ) +(εww )2 +(εwm )2 +(εwr )2 1 2 εhw , εhm , εhr , εww , εwm , εwr = √ 6 e− 2π

(18)

m , φ = ε r . From Equation set Eq. (16), Eq. (17) and the condition λ = λ we may obtain, Let φ1 = εhr , φ2 = εw w 3 h w

⎧  w,k∗   w,k∗  Th +1 Th +1 ⎪ ⎪ w ⎪ εh = φ1 + vrh − vwh ⎪ ⎪ Th − Thw,k∗ − Thm,k∗ − Tht,k + 1 Th − Thw,k∗ − Thm,k∗ − Tht,k + 1 ⎪ ⎪   m,k∗  ⎪ ⎪ Thm,k∗ + 1 Thm,k∗ + 1 Th +1 ⎪ ⎪ m ⎪ ⎨εh = T − T w,k∗ − T m,k∗ − T t,k + 1 φ1 + M − T m,k∗  + 1 φ2 − T − T w,k∗ − M − T m,k∗  − T t,k + 1 φ3 w h h w w h h  m,k∗ h  h Thm,k∗ + 1 Th +1 Thm,k∗ + 1 ⎪ r m ⎪ ⎪    + vh +  vw − vrw − vm ⎪ h w,k∗ m,k∗ t,k m,k∗ w,k∗ m,k∗ t,k ⎪ T − T − T − T + 1 M − T + 1 T − T − M − T − T + 1 ⎪ w h w h h h  h h ⎪   w,kw∗  ⎪ w,k ∗ ⎪ Tw + 1 Tw + 1 ⎪ ⎪ w ⎪     φ3 + vrw − vww ⎩εw = w,k∗ m,k∗ t,k w,k∗ m,k∗ t,k Tw − Tw

− M − Th

− Tw + 1

Tw − Tw

− M − Th

(19)

− Tw + 1

where k refers to the observed household car ownership level in data. Substitute equation set (19) into Eq. (18), the density function of the random vector (Thw,k∗ , Thm,k∗ , Tww,k∗ , φ1 , φ2 , φ3 ) may be derived as,



f Thw,k∗ , Thm,k∗ , Tww,k∗ , φ1 , φ2 , φ3 ; β˜ij



⎧ ⎡

2  w,k∗  ⎨ Th +1 1 ⎣ = √ 6 exp − · φ1 + C1 ⎩ 2 Th − Thw,k∗ − Thm,k∗ − Tht,k + 1 2π 1



+







Thm,k∗ + 1

Th − Thw,k∗ − Thm,k∗ − Tht,k + 1



+ ( φ1 ) + 2

φ1 + 

 w,k∗  Tw + 1   m,k∗

Tw − Tww,k∗ − M − Th



Thm,k∗ + 1



M − Thm,k∗ + 1

− Twt,k + 1

φ2 −

φ3 + C3

2

 m,k∗  Th +1   m,k∗

Tw − Tww,k∗ − M − Th

⎤⎫ ⎬ 2 2 + (φ2 ) + (φ3 ) ⎦ |J | ⎭

2 − Twt,k + 1

φ3 + C2

(20)

M. Yao et al. / Transportation Research Part B 104 (2017) 667–685

where the Jacobian determinant is

⎡

∂εhw ∂ Thw,k∗ ⎢ ∂εm ⎢ w,kh ∗ ⎢ ∂ Th ⎢ ∂εhr ⎢ ∂ T w,k∗ J=⎢ ⎢ ∂εh ww ⎢ ∂ Thw,k∗ ⎢ ∂εm ⎢ w,kw ∗ ⎣ ∂ Th r ∂εw ∂ Thw,k∗

∂εhw ∂ Thm,k∗ ∂εhm ∂ Thm,k∗ ∂εhr ∂ Thm,k∗ ∂εww ∂ Thm,k∗ ∂εwm ∂ Thm,k∗ ∂εwr ∂ Thm,k∗

∂εhw ∂ Tww,k∗ ∂εhm ∂ Tww,k∗ ∂εhr ∂ Tww,k∗ ∂εww ∂ Tww,k∗ ∂εwm ∂ Tww,k∗ ∂εwr ∂ Tww,k∗

(Th −Thm,k∗ −Tht,k +2)(φ1 +vrh ) 2 (Th −Thw,k∗ −Thm,k∗ −Tht,k +1)



∂εhw ∂ φ1 ∂εhm ∂ φ1 ∂εhr ∂ φ1 ∂εww ∂ φ1 ∂εwm ∂ φ1 ∂εwr ∂ φ1

675

⎤

∂εhw ∂ φ2 ∂εhm ∂ φ2 ∂εhr ∂ φ2 ∂εww ∂ φ2 ∂εwm ∂ φ2 ∂εwr ∂ φ2

∂εhw ∂ φ3 ∂εhm ⎥ ⎥ ∂ φ3 ⎥ r ⎥ ∂εh ∂ φ3 ⎥ ⎥ ∂εww ⎥ ∂ φ3 ⎥ ∂εwm ⎥ ⎥ ∂ φ3 ⎦ ∂εwr ∂ φ3

(21)

(Th −Thw,k∗ −Tht,k +2)(φ1 +vrh ) + (M+2)(φ2 +vmw ) − (Tw −Tww,k∗ −M−Twt,k )(φ3 +vrw ) 2 2 2 [(M−Thm,k∗ )+1] [Tw −Tww,k∗ −(M−Thm,k∗ )−Twt,k +1] (Th −Thw,k∗ −Thm,k∗ −Tht,k +1)   [Tw −(M−Thm,k∗ )−Twt,k +2](φ3 +vrw ) (Thw,k∗ +1 )(Thm,k∗ +1 )(φ1 +vrh ) (Thm,k∗ +1 )(Tww,k∗ +1 )(φ3 +vrw )2 − · − 2 2 4 [Tw −Tww,k∗ −(M−Thm,k∗ )−Twt,k +1] [Tw −Tww,k∗ −(M−Thm,k∗ )−Twt,k +1] (Th −Thw,k∗ −Thm,k∗ −Tht,k +1) (Th −Thm,k∗ −Tht,k +2)  w,k∗ 

=−

C1 =

Th −

 C2 =

C3 =

+1

Th

Thw,k∗

Thm,k∗

−

− Tht,k + 1

vrh − vwh



(22)



Thm,k∗ + 1

Th − Thw,k∗ − Thm,k∗ − Tht,k + 1

 w,k∗  Tw + 1   m,k∗

Tw − Tww,k∗ − M − Th

v + r h



Thm,k∗ + 1



M − Thm,k∗ + 1

− Twt,k + 1

m w

v −

 m,k∗  Th +1   m,k∗

Tw − Tww,k∗ − M − Th

− Twt,k + 1

vrw − vww

vrw − vm h

(23)

(24)

Then the marginal density function of the random vector (Thw∗ , Thm∗ , Tww∗ ) may be derived by performing the following triple integration,

f



Thw,k∗ , Thm,k∗ , Tww,k∗ ;



β˜ij =

+∞





f Thw,k∗ , Thm,k∗ , Tww,k∗ , φ1 , φ2 , φ3 ; β˜ij dφ1 dφ2 dφ3

(25)

−∞

(See Appendix B for detailed calculation) j Introducing index g for household, the log-form maximum likelihood function for estimation of β˜i can be written as follows,

 

L β˜ij =





ln f Thw,k∗ , Thm,k∗ , Tww,k∗ ; β˜ij



(26)

g

j j j j where β˜i = (βi0 , βi1 , . . . , βiH ) is a vector of parameters associated with socio-economic variables.

3.2. Parameter estimation for first stage To reflect stochastic characteristics in households’ decision making, error term ε˜gk is added to the Nash product in Eq. (1), and household g’s utility Ugk (meta-utility) for choosing alternative car level k can be expressed as follows:



Ugk = Vgk + ε˜gk = ukh − u0h

α1 

ukw − u0w

1−α1

+ ε˜gk

(27)

where Vgk is the non-stochastic term and ε˜gk is the stochastic/error term with respect to household g and car ownership alternative k. Assume the error term follows identical and independent Gumbel distribution across alternatives, then the logit model for discrete group decision can be adopted. Based on random utility maximization theory, the probability that household g chooses alternative k from possible car ownership level n is, k

Pgk =

k

eVg +

eVg 

n=k

n

eVg

(28)

To calculate the non-stochastic term Vgn , time allocation pattern for household g under car ownership alternative n needs to be inputted into the first stage model. For each household g with car level n, both travel time and non-travel time can be obtained according to the step 1 in Section 2.3.

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When using maximum likelihood method to estimate the preference parameters α h and α w in Cobb-Douglas functions uh and uw , the following log-likelihood function could be obtained,

L(αh , αw ) =





Vgk



Vgk

− ln e

g

+





Vgn

e

(29)

n=k

4. Empirical application 4.1. Data source and sample formation The data used for the empirical application are from an activity-travel diary survey conducted in Beijing from November 2011 to June 2012. In total, 1243 individuals from 467 households sampled from 12 urban and suburban districts of Beijing had successfully completed the questionnaire survey through face-to-face interview. This study only included households with two heads so that the interactions between male and female heads can be analyzed. After the elimination of singlehead household and other cases with missing or inconsistent data, the sample for this study involves 342 households. The dataset includes information on individual and household socio-demographic and socio-economic characteristics, and activity-travel diaries that provide details about travel and non-travel activities. Household level information such as auto ownership and housing tenure were also collected. Table 1 summarizes the important individual and household level characteristics that are employed in the proposed model framework. Within the 342 households, it is worth noting that only a very small portion of households own more than one car. 4.2. Specification of variables Some inputs required for model calibration are not readily available from the dataset. Originally, there are 33 types of non-travel activities. These non-travel activities are aggregated into three main categories comprised of subsistence (work related activities), maintenance (housework, shopping, etc.) and recreation (entertainment, social gathering, etc.). As for travel related activities, they are classified into 12 sub-categories. By assigning an average travel speed to each sub-category, travel distance as a product of travel time and travel speed can be obtained. If households have ever bought cars, car price can be extracted directly from the data. However, for households that have never bought cars, an average car price will be given. Hourly wage rate is also needed for the estimation, but the data cannot provide direct information in this regard. It can be derived by splitting the available monthly household income to husband and wife at a ratio of 1 to 0.825 (figures Table 1 Sample profile of individual and household level characteristics. Variable

Classification

Individual characteristics Male head

Age

Driving license Subsistence time Maintenance time Recreation time Travel time

Car ownership

Housing tenure Household size

Monthly household income

Presence of child under 11

19–39 40–59 >59 Yes No Daily average Daily average Daily average Daily average

0 1 2 3 Owner Renter and others 2 3 >=4 <= 5999 60 0 0–9999 10,0 0 0–19,999 >= 20,0 0 0 Yes No

Female head

Frequency

Percentage

Frequency

Percentage

147 183 12 255 87 6.34(h) 4.26(h) 3.78(h) 1.45(h) Household characteristics

43.0% 53.5% 3.5% 74.6% 25.4%

168 170 4 126 216 5.95(h) 5.45(h) 3.83(h) 1.22(h)

49.1% 49.7% 1.2% 36.8% 63.2%

Frequency 209 125 7 1 212 130 117 66 159 34 89 172 47 90 252

Percentage 61.1% 36.5% 2.0% 0.3% 62.0% 38.0% 34.2% 19.3% 46.5% 9.9% 26.0% 50.3% 13.8% 26.3% 73.7%

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677

from national program of Chinese Family Panel Studies conducted by Peking University, see Heshmati and Su, 2015), and then dividing the partitioned income by monthly working hours. Because the plate-number based car usage rationing policy have been implemented in Beijing during the survey period, with a usage limitation of two out of ten end-numbers every workday, a car rationing ratio 0.8 is derived for the calibration. j To define the baseline utility vi , an exponential form is adopted to guarantee its positivity. The definition consists of five socio-economic variables including housing tenure (differentiated by private housing and public housing), household size (number of household members), presence of child under 11, household income and age of household heads. These variables are selected by referring to the results from earlier studies (e.g. Pinjari et al., 2009; Wang and Li, 2009) and based on a systematic process of eliminating variables that are statistically insignificant at the 5% level. The baseline utility is expressed as,



v = fi β j i

j

˜ j , x˜ j i i





= exp

β + j i0



β

j j x ih ih

, i = h, w j = w, m, r

(30)

h

To obtain the car ownership utility of non-chosen alternative (i.e., alternative level of car ownership), travel time for non-chosen alternative needs to be calculated first. The travel time is determined by the probability of car usage under the non-chosen car ownership alternative and can be computed according to Eqs. (5) to (10). After getting the travel time for non-chosen car ownership alternative, the travel time will be input to the second stage time allocation model (Eq. (11) to Eq. (15)) to calculate the optimal time allocation pattern for non-travel activities under the non-chosen car ownership alternative. Finally, input the travel time and optimal time allocation pattern to the first stage model to obtain the car ownership utility of the non-chosen alternative. 4.3. Results of model estimation The maximum likelihood estimation (Eq. (26)) results for parameters in baseline utility function (Eq. (30)) are presented in Table 2. The significance and goodness-of-fit of the proposed model is assessed by likelihood-ratio test, where the proposed model is compared with null (simple) model that assumes all the parameters to be 0. The value of likelihood ratio LR=−2[L(0)-L(βˆ )] follows a χ 2 distribution with 36 degrees of freedom. For a significance level of α = 0.05, the critical value of chi-square distributed statistic χr2 (α ) is 50.999. As can be seen in Table 2, the likelihood ratio (LR) is significantly larger than this critical value, which demonstrates that the proposed model fits the data significantly better than the null(simple) model. Besides, likelihood ratio index ρ 2 is between 0.2 to 0.4, indicating that the goodness-of-fit is satisfactory (McFadden, 1977). Table 2 Baseline parameter estimates in time allocation model. Variables

Male head

Female head

Coefficients

t

Coefficients

t

Subsistence activity Constant Housing tenure Household size Presence of child under 11 Household income Age of household head

1.0299 0.2079 0.0157 0.5051 0.0105 −0.0192

4.1626∗ ∗ 2.1383∗ ∗ 0.3505 5.2556∗ ∗ 1.6992∗ −4.4753∗ ∗

0.6597 0.2016 −0.0361 0.3845 0.0190 −0.0186

2.5448∗ ∗ 2.0644∗ ∗ −0.7532 3.7836∗ ∗ 3.0024∗ ∗ −4.0885∗ ∗

Maintenance activity Constant Housing tenure Household size Presence of child under 11 Household income Age of household head

1.3635 −0.2058 −0.0369 −0.0286 0.0259 −0.0070

6.6411∗ ∗ −2.0283∗ ∗ −0.6736 −0.2491 3.9587∗ ∗ −1.9208∗

1.1953 0.0209 −0.0302 −0.0378 0.0198 0.0 0 09

5.5609∗ ∗ 0.1966 −0.5831 −0.2956 3.1916∗ ∗ 0.2432

Recreation activity Constant Housing tenure Household size Presence of child under 11 Household income Age of household head

−0.5280 0.2680 0.1124 0.0358 −0.0358 0.0140

−1.4301 2.2759∗ ∗ 2.2438∗ ∗ 0.2807 −2.8700∗ ∗ 2.1287∗ ∗

−0.5359 0.2792 0.0110 −0.3183 −0.0231 0.0245

−1.6678∗ 2.6909∗ ∗ 0.2298 −2.3567∗ ∗ −2.3338∗ ∗ 4.3477∗ ∗

Summary statistics: number of observations = 342; L(0) = −1743.7; L(βˆ ) =− 1366.5; LR = −2[L(0)-L(βˆ )] = 754.4; ρ 2 = 0.2163 Note: ∗ ∗ 5% significant level; ∗ 10% significant level j j j j Table 2 also lists the estimated parameters β˜i = (βi0 , βi1 , . . . , βi5 ) that represent the influence of the aforementioned socio-economic variables on the baseline preference for each activity type. Most estimates of male head and head in undertaking these activities are significant at the 5% level (marked by two asterisks). Housing tenure (‘1’ for ownership and ‘0’

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for rental or others) tend to positively influence time expenditure on subsistence activities for both heads. It might be the reason that they both have to earn more to pay housing mortgage if a house is purchased. A possible explanation for the negative relationship between household size and time allocated for maintenance activity is that if there are more household members to share the maintenance responsibility, the maintenance activity performed per person can be reduced. When there are children under 11 in a family, both heads tend to spend more time on subsistence activities. One reason could be that they need extra money for the financial expenditure of raising children. It is worth noting that presence of child under 11 has negative influence on the recreation time of the female head, because she might need to spare more time for child-care responsibilities. As can be expected, more household income will be generated if both heads input more time on subsistence activities, and their time input for recreation activities could be reduced as a result. As both heads get older, they tend to significantly reduce their time expenditure on subsistence activities, and input much more time to recreation activities. The maximum likelihood estimation (Eq. (29)) results for preference parameter α i (i= h, w) of Cobb-Douglas function (Eq. (2)) in the first stage model are shown in Table 3. For both heads, a higher preference parameter indicates greater individual fondness for time surplus than for money surplus. The result is in consistency with the preliminary findings of Train and McFadden (1978) on preference parameter of leisure in goods/leisure framework, which indicated that the value of time preference parameter was between 0.7 and 1.0. Table 3 Preference parameters in Cobb–Douglas utility function.

Male head Female head

Preference parameter

t

0.8782 0.8957

147.7702 196.5210

4.4. Prediction and impact analysis The proposed theoretical model framework in conjunction with the solution approach and estimated parameters in previous sections can be used for prediction and sensitivity analysis to demonstrate its application and predictive power. Table 4 shows the prediction results regarding ownership of cars within households. As mentioned in Section 4.1, households with more than one car only accounts for a tiny percentage; therefore, it is unnecessary to classify those non-zero car ownership households into one car, two cars or more than three cars. Out of the 209 households observed with zero car ownership, 170 cases are predicted correctly, indicating a precision rate of 81.3%. In comparison, out of the 133 households that actually bought cars, 78 (58.6%) cases are predicted with precision. Overall, 72.5% correctness is achieved in predicting the sample data, suggesting a high accuracy rate for sample size of 342 households. Table 4 Observed and predicted case that households own cars or not. Observed

Predicted Without cars

With cars

Percentage correct (%)

Without cars With cars Overall % correct

170 55

39 78

81.3% 58.6% 72.5%

Additionally, the developed model could be employed to analyze the impacts of relevant vehicle rationing policies. As displayed in Fig. 2, when vehicle rationing degree deepens from 1 (no restriction) to 0 (full restriction), the average car ownership level per household increases at first and then decreases. The increase under mild restriction is because some financially well-off households tend to buy additional cars to ensure their access, although few households may also be demotivated to give up their car purchase plan, the effect on the latter is less significant. However, when restriction degree gets severe, making it inefficient to buy additional cars or first cars, the drop of average car level becomes dramatic. The maximal point of average car ownership level is achieved at a rationing degree of around 0.7. In order to reveal how the varying rationing degrees affect households’ time allocation patterns, Fig. 3 plots the percentage change of time allocation for subsistence, maintenance, recreation and travel activities with respect to rationing degree. Generally speaking, the influence of rationing on travel time is more significant than other activities. As can be noted, the travel time of both heads rise remarkably in response to increasingly strict restriction on car usage. This is because they have to switch to other transportation modes like buses when cars are unavailable for travel, which tends to be more timeconsuming. Besides, male heads seem to be more sensitive to changing rationing degrees than their female counterparts. This might be the reason that males are more prone to drive cars in daily travel. To some extent, this is consistent with the fact that much more male heads possess driving license in comparison with female heads, as noticed in Table 1. Fig. 3 also exhibits that, for both heads, time allocated for subsistence and recreation activities will decrease when rationing degree deepens, but at much smaller scale than that of travel time. In comparison with subsistence activities, which are mostly

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679

Fig. 2. Effects of car rationing degree on household car ownership level.

Fig. 3. Effects of car rationing degree on household time allocation.

out-of-home activities necessitating travel to workplaces, some of the recreation activities are performed at home; therefore, recreation activities are slightly less sensitive to car rationing intensification than subsistence activities. The relative invariance for maintenance activities is because a certain amount of maintenance tasks is required to be finished by the two household heads. 5. Conclusions This paper developed a game-theoretic model to study auto ownership decisions from the point of view of household time allocation. The model explicitly considered interactions between spouses in long-term household car ownership de-

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cision and short-term time allocation to daily activities by adopting a two-stage framework. Car ownership decision was formulated as a Nash bargaining problem in the upper level with husband and wife involved in decision-making, while time allocation decision was considered as a generalized Nash equilibrium problem in the lower level incorporating spouses’ interactions concerning activity participation. The proposed two-stage model was solved by a heuristic procedure, which is a combination of backwards induction and method of exhaustion. To demonstrate the applicability of the proposed model in car ownership prediction and investigation of interactions between household long-term and short-term time decisions, empirical data collected from Beijing in 2011–2012 were used to calibrate the model and assess the impacts of plate-numberbased vehicle usage rationing policies on car ownership and daily time allocation as a case study. Results showed that when car rationing intensified from free to full restriction, the total car ownership within the sampled area increased slightly at first and then decreased continuously after reaching a peak point. Another finding was that the changing rationing degree had a significant impact on household members’ travel time, minor impact on time allocated to subsistence and recreation activities, and almost no impact on maintenance activity participation. By providing insights on how household members decide car ownership level and the resultant time allocation patterns, the proposed model and findings can serve as effective guidance to transport planners and policy makers. For future research, some directions can be explored. First, since household members are usually subject to income constraints, it is worthwhile to treat car ownership as a household expenditure commitment competing with other important long-term decisions, such as housing related decisions. Besides, short-term expenditures on various activity participation could also be included to analyze diversified monetary expenditure tradeoffs, and the proposed model can be easily modified to allow for non-negative restrictions on time and money expenditures. Second, for working activity income, it could be extended by assuming different wage patterns, such as a combination of fixed salary for mandatory working hours and incremental salary for extra working hours. Third, for the proposed game-theoretic formulation, generalized Nash equilibrium (GNE) as a solution to non-cooperative game may not always be Pareto efficient (Chen and Woolley, 2001), which makes it possible for household members to form a cooperation in the repeated daily time allocation so as to make everyone better off. In this scenario, the second stage game can be formulated as a bargaining problem as well, with GNE as its internal threat point. Moreover, a direct extension can be made to the proposed model framework: in addition to the two household heads, a third stakeholder can be included as well (e.g., children with driving license). Fourth, with regard to model estimation, efforts can be exerted to explore the linkages between the stochastic error terms in car ownership model and the stochastic error terms in baseline utility of time allocation model, so as to address the econometric challenge of simultaneous estimation. Fifth, the timing choice of car purchase can be incorporated to account for purchases at different time points, and the implicit consideration of travel mode choice can be bettered by an explicit consideration with more mode alternatives available. For travel time, it is worthwhile to consider the reciprocal interactions between travel time and car usage probability, as well as between travel time and time allocated for non-travel activities.

Acknowledgements The authors wish to express their thanks to four anonymous reviewers for their useful comments on an earlier version of the paper. This research is sponsored by a General Research Fund grant from the Hong Kong Research Council: No. HKBU247813.

Appendix Appendix A. Proof of proposition 1 An important theoretical foundation for the following proof is the theorem of existence and uniqueness defined by Rosen (1965). Existence According to Rosen’s existence theorem for concave game, an equilibrium exists for every concave 2-person game. In other words, concavity is the sufficient condition for existence of an equilibrium. In the proposed generalized Nash equilibrium problem (11)-(15), let Uin (i = h, w) denotes player i’s utility for time allocation under car ownership n, and substitute Eq. (13) and Eq. (14) into the objective functions respectively, we can have













w,n m,n Uin = uw + 1 + um + 1 + uri ln Ti − Tiw,n − Tim,n − Tit,n + 1 i ln Ti i ln Ti

(A.1)

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681

It is obvious that Uin is continuous and differentiable with respect to the strategy set. The Hessian matrix of Uin is

 n

⎡

⎤

∂ 2Uin ∂ (Tiw,n )2 ⎣ = ∂ 2Uin ∂ Tim,n ∂ Tiw,n

 m,n

∇ 2Ui Tiw,n , Ti

⎡

−

⎣ (

=

−

(

∂ 2Uin ∂ Tiw,n ∂ Tim,n ⎦ ∂ 2Uin m,n 2 ∂ (Ti )

uw i 2 w,n Ti +1

uri 2 w,n Ti −Ti −Tim,n −Tit,n +1 um uri i 2 2 Ti −Tiw,n −Tim,n −Tit,n +1 Tim,n +1

)

−

(

−

)

−

)

(

−

)

uri 2 w,n Ti −Ti −Tim,n −Tit,n +1 uri 2 Ti −Tiw,n −Tim,n −Tit,n +1

⎤

(

) ⎦

(

)

(A.2)

Because the Hessian matrix is negative definite, so the payoff function is concave with respect to player i’s strategies while keeping the other player’s strategies fixed. In addition, the linear constraints (Eq. (15)) in conjunction with the nonnegative constraints of allocated time for every activity constitute a compact convex set. Therefore, it is a concave game and the existence of a Nash equilibrium solution can be guaranteed. Uniqueness Let σ (T, r) denotes a weighted nonnegative sum of husband’s and wife’s utility functions expressed in equation A.1, where their weights are rh and rw respectively, so we can have

  σ (T , r ) = rhUhn Thw,n , Thm,n + rwUwn (Tww,n , Twm,n ), rh , rw ≥ 0

where



T w,n T = hm,n Th

Tww,n Twm,n

(A.3)

 (A.4)

The pseudogradient of σ (T, r) is defined by



g (T , r ) =

∂U n

∂U n

h rh ∂ T w,n

h rh ∂ T m,n

w rw ∂ T w,n

w rw ∂ T m,n

h



(A.5)

h

∂U n

∂U n

w

w

A sufficient condition that σ (T, r) be diagonally strictly concave is that the symmetric matrix [G(T, r) + GT (T,r)] be negative definite, where G(T, r) is the Jacobian matrix with respect to T of g(T, r). Specifically, the Jacobian matrix G(T, r) is

⎡

∂ 2U n

rh ∂ T w,n ∂hT w,n

∂ 2U n

∂ 2U n

rh ∂ T m,n ∂hT w,n

rh ∂ T w,n ∂hT w,n

∂ 2U n

rh ∂ T m,n ∂hT w,n

⎤

⎢ ∂ 2Uhn ⎢ rh ∂ T w,n ∂ Thm,n ⎢ h G (T , r ) = ⎢ ∂ 2Uwn ⎣rw ∂ Tww,n ∂ Thw,n

rh ∂ T m,n ∂hT m,n

rh ∂ T w,n ∂ hT m,n

rh ∂ T m,n ∂hT m,n ⎥

rw ∂ T m,n ∂wT w,n

rw ∂

∂

rw ∂

rw ∂ T w,n ∂wT m,n

rw ∂ T m,n ∂wT m,n

rw ∂

∂

rw ∂

h

h

∂ 2U n

w

Substitute function

Ai = Ti −

Uhn

Tiw,n

−

h

h

∂ 2U n

h

h

h

∂ 2U n

w

and

Tim,n

− Tit,n + 1

∂ 2U n

∂ 2U n

w

Uwn

h

h

h

w

w h 2 n Uw w,n Tw Tww,n 2 n Uw Tww,n Twm,n

∂ ∂

h

∂ 2U n

w

w h 2 n Uw m,n Tw Tww,n 2 n Uw Twm,n Twm,n

∂ ∂

∂

⎥ ⎥ ⎥ ⎦

(A.6)

∂

(see Eq. (A.1)) into the matrix, and let

∀i = h, w

(A.7)

We can obtain the following symmetric matrix





G ( T , r ) + GT ( T , r ) =

⎡



⎢−2rh ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

uw h

(Thw,n +1 )

2

+

ur

−2rh Ah2

urh A2h

  −2rh

h

0

h

um h

(Thm,n +1 )

2

0

0

⎤

ur

−2rh Ah2

0

+

urh A2h

0

  −2rw

0

0 uw w

(Tww,n +1 )

2

−2rw

0

 urw A2w

+

urw A2w

ur

 −2rw

−2rw Aw2

w

(

um w 2 Twm,n +1

)

+

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ urw

(A.8)

A2w

Because [G(T, r) + GT (T ,r)] is negatively definite, so σ (T, r) is diagonally strictly concave for every r ∈ Q, where Q is a convex subset of the positive orthant of En . Thus, for each r ∈ Q there is a unique normalized equilibrium point based on Rosen’s uniqueness theorem for a concave game with coupled constraint set. In Rosen’s definition for normalized equilibrium point, λh and λw are given by,

λh =

λ rh

,

λw =

λ rw

(A.9)

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M. Yao et al. / Transportation Research Part B 104 (2017) 667–685

The weights rh and rw tell how the burden of satisfying the maintenance constraint is to be distributed between husband and wife. Because we assume husband and wife equally share the burden of household maintenance work, an implied condition from this assumption is that rh = rw = 1 (Krawczyk, 2005), indicating λh = λw as well. In this situation, the proposed problem has a unique equilibrium point called normalized equilibrium, which is a special kind of generalized equilibrium. Appendix B: Computation of triple integration The density function of the random vector (Thw,k∗ , Thm,k∗ , Tww,k∗ ) may be derived by conducting the triple integration in Eq. (25) by the following logic,

f



Thw,k∗ , Thm,k∗ , Tww,k∗ ;



β˜ij =

+∞





f Thw,k∗ , Thm,k∗ , Tww,k∗ , φ1 , φ2 , φ3 ; β˜ij dφ2 dφ1 dφ3

−∞

 +∞ =



+∞



f Thw,k∗ , Thm,k∗ , Tww,k∗ , φ1 , φ3 ; β˜ij dφ1 dφ3 =

−∞





f Thw,k∗ , Thm,k∗ , Tww,k∗ , φ3 ; β˜ij dφ3

−∞

According to Eq. (18) and equation set (19), it can be further written as



f



Thw,k∗ , Thm,k∗ , Tww,k∗ ;



β˜ij =

 +∞ B DC − E −

√

2π

−∞

+∞ = −∞

5

1 1 N M2 e− 2 + 8L √ √ 4 2 L 2π (C13 ) + 1

DC6 F C13 (C13 )2 +1



A+

e

−

2

(C13 )2 + 1  ! " 2 H M 1+ L 4L

where

 C4 =

C6 =

C8 =



M − Th

 



C7 =

8S3

X3

3+



−

I·M + K d φ3 2L











+

(S2 )2

+

 2

Th − Thw,k∗ − Thm,k∗ − Tht,k + 1

Tw − Tww,k∗ − M − Twt,k







2

Tw − Tww,k∗ − M − Thm,k∗ − Twt,k + 1







Th − Thm,k∗ − Tht,k + 2



Tw − M − Thm,k∗ − Twt,k + 2



( S 2 )2

Th − Thw,k∗ − Tht,k + 2

Thw,k∗ + 1 Thm,k∗ + 1



2

Tw − Tww,k∗ − M − Thm,k∗ − Twt,k + 1





Thm,k∗ + 1 Tww,k∗ + 1





4

Tw − Tww,k∗ − M − Thm,k∗ − Twt,k + 1



Thw,k∗ + 1

Th − Thw,k∗ − Thm,k∗ − Tht,k + 1



( S 2 )2

C5 =

2

2 



C13 =

2

+1



C11 =

S1 +



Th − Thw,k∗ − Thm,k∗ − Tht,k + 1



C10 =

e



Th − Thm,k∗ − Tht,k + 2

(M + 2 )  m,k∗

 

1

Th − Thw,k∗ − Thm,k∗ − Tht,k + 1



C9 =

1



d φ1 d φ3

 (S2 )4 2 S3 16(S3 )2 ( S 3 )2 (C13 )2 + 1 S3      S2 1 1 S2 (S2 )2 (S2 )2 r r r − (X3 + vrw X4 ) 1+ + + X + v X 1 + − X + v X + v X ( 2 ( 1 w 3) w 2) w 1 S3 2S3 S3 S3 8(S3 )2 4(S3 )2

1 1 =  √ 3 √ L 2π



F2

(C13 )2 +1

C12 =



Thm,k∗ + 1



M − Thm,k∗ + 1

C14 =



Th − Thw,k∗ − Thm,k∗ − Tht,k + 1

 m,k∗  Th +1   m,k∗

Tw − Tww,k∗ − M − Th

A = (C11 φ1 + C1 ) + (φ1 ) + (C15 φ3 + C3 ) + (φ3 ) 2



2



Thm,k∗ + 1

2

2

− Twt,k + 1

B = C4



φ1 + vrh

C15 =

 w,k∗  Tw + 1   m,k∗

Tw − Tww,k∗ − M − Th



 r D = C9 φ3 + C9 vrw φ1 + vrh + C6 vm w − C7 (φ3 + vw )   DC4C6C12C13 2 2 E = C10 (φ3 ) + 2C10 vrw φ3 + C10 (vrw ) F = C12 φ1 − C14 φ3 + C2 H = DC4 (C5 − C8 ) − (C13 )2 + 1

C = (C5 − C8 )

− Twt,k + 1

M. Yao et al. / Transportation Research Part B 104 (2017) 667–685

I = DC4 (C5 − C8 )vrh −

DC4C6C12C13

(C13 )2 + 1



vrh +

r D(C5 − C8 )vrh + DC6 vm w − DC7 (φ3 + vw ) − E −

D(C2 − C14 φ3 )C6C13

 K =

D(C5 − C8 )v + r h

v − DC7 (φ3 + v ) − E −

DC6 m w

r w



2C2C12

(C13 )2 + 1

P1 = C9 C4 (C5 − C8 ) −

O2 =

C4C6C12C13

 C4

(C13 )2 + 1 D(C2 − C14 φ3 )C6C13

(C13 )2 + 1 2 2C12 (C2 − C14 φ3 ) (C12 ) L = (C11 )2 + 1 + M = 2C11C1 + 2 (C13 ) + 1 (C13 )2 + 1 (C2 − C14 φ3 )2 2 2 N = (C1 )2 + (C15 φ3 + C3 ) + (φ3 ) + (C13 )2 + 1 O1 = 2C11C1 +

683

 C4 vrh

2C12C14

(C13 )2 + 1 

(C13 )2 + 1

C4C6C9C13C14

− C4 (C7C9 + C10 ) (C13 )2 + 1   C2C6C9C13 C6C9C13C14 vrw r m r P3 = C4 C9 vh (C5 − C8 ) + C6C9 vw − 2(C7C9 + C10 )vw − + (C13 )2 + 1 (C13 )2 + 1   C6C9C13C2 vrw r r 2 P4 = C4 C9 vrh (C5 − C8 )vrw + C6C9 vm v − C C + C v − ( ) ( ) 7 9 10 w w w (C13 )2 + 1 P2 =

S1 = (C1 )2 + (C3 )2 +

(C2 )2 (O1 )2 − 2 4L (C13 ) + 1

S2 = 2C3C15 −

2C2C14

(C13 ) + 1 2

+

O1 O2 2L

X1 =

P1 (O2 )2 P2 O2 + 2L 4L2

X3 =

P1 vrw vrh O2 + P4 O2 O1 P1 vrh P1 vrw O1 O2 O1 P3 P1 P1 (O1 )2 + − − − + + vrh P3 L 2L 2L 2L 4L2 2L2

X4 =

O1 P1 vrw vrh + O1 P4 P1 vrw P1 vrw (O1 )2 + − + vrh P4 L 2L 4L2

X2 =

S3 = (C15 )2 + 1 +

(C14 )2 (O2 )2 − 2 4L (C13 ) + 1

P1 vrh O2 P1 O1 O2 O1 P2 P1 vrw (O2 )2 P3 O2 − + − + + vrh P2 2 2 2L 2L 2L 4L 2L

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