Free utility model for explaining the social gravity law
FFree utility model for explaining the social gravity law
Hao Wang, Xiao-Yong Yan,
1, 2, ∗ and Jinshan Wu Institute of Transportation System Science and Engineering,Beijing Jiaotong University, Beijing , China Comple χ Lab, University of Electronic Science and Technology of China, Chengdu , China School of Systems Science, Beijing Normal University, Beijing , China (Dated: September 18, 2020)Social gravity law widely exists in human travel, population migration, commodity trade, in-formation communication, scientific collaboration and so on. Why is there such a simple law inmany complex social systems is an interesting question. Although scientists from fields of statisticalphysics, complex systems, economics and transportation science have explained the social gravitylaw, a theoretical explanation including two dominant mechanisms, namely individual interactionand bounded rationality, is still lacking. Here we present a free utility model from the perspective ofindividual choice behavior to explain the social gravity law. The basic assumption is that boundedrational individuals interacting with each other will trade off the expected utility and information-processing cost to maximize their own utility. The previous explanations of the social gravity lawincluding the maximum entropy model, the free cost model, the Logit model and the destinationchoice game model are all special cases under our model. Further, we extend the free utility modelto the dummy network and real transportation network. This model not only helps us to betterunderstand the underlying mechanisms of spatial interaction patterns in complex social systems,but also provides a new perspective for understanding the potential function in game theory andthe user equilibrium model in transportation science.
I. INTRODUCTION
Predicting the mobility of people, goods and informa-tion between locations is an important problem in fieldsas diverse as sociology [1], economics [2], demography [3],epidemiology [4], transportation science [5] and networkscience [6]. For more than a hundred years, scholars haveproposed a variety of models to predict such mobility [5–12]. These models are named spatial interaction modelsin economics [8] and trip distribution models in trans-portation science [5]. The gravity model is the most in-fluential mobility prediction model and has been appliedin many fields [6]. For example, it is used to predict pop-ulation migration [13], commodity trade [14], commut-ing flows [15] and public transportation flows [15, 16].The gravity model is popular because spatial interactionphenomena in these fields all obey a law known as thesocial gravity law under which the flow between two lo-cations is proportional to the activity (usually quantifiedby population, GDP and so on) of these two locationsand decays with the power function of the distance be-tween them, similar to Newton’s law of universal gravi-tation. As early as 1846, Desart found that the railwaypassenger flow between stations in Belgium obeyed thesocial gravity law [17]. This may be the earliest record ofthe discovery of the social gravity law. Later, Carey [18]and Ravenstein [19] found that population migrations ofthe United States and the United Kingdom, respectively,obeyed the social gravity law. Additionally, Reilly [20]found that retail business drawn from cities obeyed thesocial gravity law. In recent years, with the continuous ∗ [email protected] development of modern electronic and information tech-nology, there have been many approaches (such as GPS,mobile phones and social networking sites) to recordingthe mobility data of people, goods and information overlong periods. By analyzing these data, scientists havefound many phenomena showing obedience to the socialgravity law in various systems [6]. For example, com-muting flow in the United States [21], highway trafficflow in Korea [22], telecommunication flow in Belgium[23], global airline flow [24], global cargo ship movement[25] and even global scientific collaboration [26] followedthe social gravity law.Why does such a simple law apply to so many com-plex social systems? This problem has long excited thecuriosity of many scholars. In the past half century, scien-tists have proposed different explanations of the roots ofthe social gravity law, among which Wilson’s maximumentropy model [27] is the prevailing explanation. He pro-posed an approach to deriving the doubly-constrainedgravity model by maximizing the entropy of the tripdistribution between locations in the transportation sys-tem under the constraints of the total cost, the trip pro-duction volume and attraction volume of each location.However, unlike the total amount of internal energy avail-able to all gas molecules in a system, which is determinedexogenously, the total cost cannot be estimated in a realtransportation system [28]. Tomlin and Tomlin [29] con-structed a free cost model by analogy with the Helmholtzfree energy in physics. This model can lead to the sameresults as Wilson’s maximum entropy model but does notneed the prior constraint of the total cost. However, boththe maximum entropy model and the free cost model aremacroscopic explanations of the social gravity law. Theyprovide only the most probable macroscopic distribution a r X i v : . [ phy s i c s . s o c - ph ] S e p state but do not take into account individual choice be-havior in destination choice [30]. On the other hand,economists have explained the social gravity law by de-scribing individual choice behavior using utility theory,in which the most influential study is that of Domencichand McFadden, who applied random utility theory tomodel individual destination choice behavior [31]. Theyassumed that the traveler always selects the destinationwith the highest utility, but her perception of the desti-nation utility exhibits random error. If these errors fol-low the independent and identical Gumbel distribution,the Logit model [32] can be derived. If the destinationutility consists only of the destination attractiveness andthe travel cost to the destination, the Logit model canderive the singly-constrained gravity model. However,all the aforementioned models neglect individual interac-tion, which is a ubiquitous phenomenon in many complexsystems [33]. For example, in a real transportation sys-tem, a traveler considers not only the constant values ofdestination attractiveness and travel cost but also pos-sible crowding at the destination and congestion on theway. Recently, Yan and Zhou [34] modeled the individ-ual destination choice process as a congestion game, in-cluding interaction among travelers, and further derivedthe singly-constrained gravity model. However, they as-sumed that all travelers are perfectly rational and canaccurately perceive the utilities of all destinations. Inpractice, individual rationality is bounded because of theintractability of the alternative choice problem and lim-ited information-processing resources [35]. However, anexplanation of the social gravity law that simultaneouslyreflects individual interaction and bounded rationality isstill lacking.In this paper, we develop a free utility model fromthe perspective of individual choice behavior to explainthe social gravity law in the context of the destinationchoice problem in transportation science. First, we es-tablish the individual choice model on the basis of howa bounded rational traveler trades off expected utilityand information-processing cost. Then, we extend theindividual choice model to the collective choice model,including infinite noninteractive or interactive travelers,and further derive the gravity model from the collectivemodel. We next extend the collective model of interac-tive travelers, named the free utility model, to a dummynetwork and real transportation network. Finally, wecontrast the similarities and differences between the freeutility model and the free energy in physics and furtherdiscuss the potential application value of the free utilitymodel in different scientific fields. II. MODELA. Individual choice model
Similar to some of the aforementioned studies explain-ing the social gravity law, we conduct our research in the context of individual destination choice behavior inthe transportation system. In this system, there are M origins labeled i ( i = 1 , , . . . , M ) and N destinations la-beled j ( j = 1 , , . . . , N ). O i is the number of trips leav-ing origin i . We initially start with the simplest system,in which there is only one origin ( M = 1) and one trip( O i = 1) made by a traveler who can select N ( N > j for this traveleris u ij , which describes her satisfaction degree concern-ing the attractiveness of destination j and the travel costfrom origin i to destination j . According to the utilitymaximization principle, this traveler will select the desti-nation with the highest utility only if she is perfectly ra-tional [36]. However, it is difficult for a bounded rationaltraveler to exactly perceive the utilities of all destinationsin reality. In this case, her choice is probabilistic, i.e., des-tination j is selected with probability p ij , which dependson u ij [37]. If the traveler is completely uncertain aboutthe utilities of all destinations, she can select destination j only with probability p ij = N , and her expected utility (cid:80) j p ij u ij is the average of the utilities of all destina-tions. If she wants to obtain higher expected utility, shemust acquire knowledge about the utilities of the desti-nations through information processing [38]. A naturalmeasure of information processing is negative informa-tion entropy − H = (cid:80) j p ij ln p ij [38–40]. Assuming thatthe price of unit information is τ , then the information-processing cost is − τ H . If τ → ∞ , which means that theinformation-processing cost is extremely high, the trav-eler does not care about the expected utility and focusesonly on the information-processing cost. Hence, she willmake a totally random choice of destination, that is, auniform distribution over the set of destinations. If τ = 0,which means that information processing has no cost, shewill select only the destination with the highest utility. Ingeneral case τ >
0, she must trade off her expected util-ity and the information-processing cost [41, 42] to achievethe total utility maximization goal, i.e.,max w = (cid:88) j p ij u ij + τ H, s . t . (cid:88) j p ij = 1 , (1)where (cid:80) j p ij u ij + τ H is the objective function subjectedto (cid:80) j p ij = 1. Using the Lagrange multiplier method,we can obtain the solution of Eq. (1) as follows: L ( p ij , λ ) = (cid:88) j p ij u ij + τ H − λ ( (cid:88) j p ij − , (2)where λ is a Lagrange multiplier. According to ∂L∂p ij = 0for all destinations, we can obtain u ij − τ (ln p ij + 1) = λ, (3)which means that all destinations have the same util-ity minus the marginal information-processing cost underthe traveler’s optimal choice strategy. This situation isvery similar to consumer equilibrium in microeconomics[43], in which the marginal utility of each good is equal.Thus, we name u ij − τ (ln p ij + 1) the marginal utility ofdestination j . The total utility of the system is the sum ofthe integrals of the marginal utilities of all destinations.The optimal choice strategy for the traveler is to fol-low the equimarginal principle [43] to select destinationsin order to achieve maximum total utility. CombiningEq. (3) and (cid:80) j p ij = 1, we can derive p ij = e u ij /τ (cid:80) j e u ij /τ , (4)which is the equilibrium solution of Eq. (1). In mathe-matical form, Eq. (4) is the Logit model derived in termsof random utility theory [31]. However, our derivationdoes not need to assume in advance that the destinationutility perception errors follow the Gumbel distribution. B. Collective choice model
We further extend the transportation system to thecase where origin i has infinite homogeneous travelers(i.e., O i (cid:29) j is not affected by the number of trips to j , Eq. (4)can still be applied in this system. On this occasion,the number of trips from i to j is T ij = O i p ij , and thesystem entropy S is the sum of the information entropy H of each trip. Therefore, we can rewrite Eq. (1) asmax W = (cid:88) j T ij u ij + τ S, s . t . (cid:88) j T ij = O i . (5)Similarly, all homogeneous travelers follow theequimarginal principle so that all destinations havethe same marginal utility for them. If the utility ofdestination j is abstractly written as u ij = A j − c ij ,where A j is the constant attractiveness of destination j reflecting the activity opportunities (including variablessuch as retail activity and employment density) [44]available there, and c ij is the constant travel cost from i to j , we can use the Lagrange multiplier method toobtain the equilibrium solution of Eq. (5) as T ij = O i e ( A j − c ij ) /τ (cid:80) j e ( A j − c ij ) /τ , (6)which is the same as the singly-constrained gravity modelderived from the Logit model [31].However, in practice, the utility of destination j is af-fected by the number of trips from i to j [45]; that is, u ij is a function of T ij . For example, the trip num-ber T ij from origin i to destination j will influence thetravel cost from i to j and the attractiveness of desti-nation j , both of which will change the utility of des-tination j . Hence, the utility can be abstractly writtenas u ij ( T ij ) = A j − l j ( T ij ) − c ij − g ij ( T ij ), where l j ( T ij )is the variable attractiveness function at destination j and g ij ( T ij ) is the congestion function on the way from i to j . We already know that each traveler follows theequimarginal principle to make the optimal choice. Inthis context, a traveler’s choice of destination dependson how other travelers are distributed over all destina-tions [46]. The phenomenon of one individual’s behaviorbeing dependent on the behavior of other individuals,known as individual interaction, is widespread in socialsystems. In this interactive system, the optimal choicestrategy for all travelers is still to assign all destinationsthe same marginal utility. Since the total utility of thesystem is the sum of the integrals of the marginal utilitiesof all destinations, the utility maximization model of theinteractive system can be written asmax W = (cid:88) j (cid:90) T ij u ij ( x )d x + τ S, s . t . (cid:88) j T ij = O i . (7)Interestingly, the objective function of Eq. (7) is math-ematically consistent with the Helmholtz free energy inphysics [47]; thus, we name Eq. (7) the free utility model.Analogous to the thermodynamic system, the equilib-rium trip distribution maximizes the free utility functionin the transportation system. In addition, the first termof the free utility function, (cid:80) j (cid:82) T ij u ij ( x )d x , is analogousto internal energy, including potential energy in a ther-modynamic system. Monderer and Shapley [48] thereforenamed this term the potential function . The free utilitymodel with τ = 0 is essentially a potential game with aninfinite number of agents.Here, to generate gravity-like behavior, the utilityneeds a logarithmic dependence on the number of travel-ers from i to j , i.e., u ij ( T ij ) = A j − c ij − γ ln T ij , where γ ln T ij = l j ( T ij ) + g ij ( T ij ) is the simplified cost function,including the travel congestion cost and the destinationvariable attractiveness, and γ is a non-negative parame-ter. Equation (7) can be specifically written asmax W = (cid:88) j (cid:90) T ij ( A j − c ij − γ ln x )d x + τ S, s . t . (cid:88) j T ij = O i . (8)Using the Lagrange multiplier method, we can obtain T ij = O i e ( A j − c ij ) / ( γ + τ ) (cid:80) j e ( A j − c ij ) / ( γ + τ ) . (9)This result is the singly-constrained gravity model [5].Parameter τ reflects the information-processing price forbounded rational travelers, and parameter γ reflects trav-elers’ interaction strength. Figure 1 shows the effect ofchanges in parameters τ and γ on the free utility opti-mal solution in a simple system with one origin and twodestinations. The surface in Fig. 1 describes the opti-mal solution of the free utility model in this simple sys-tem. From Eq. (8), we can see that when γ > τ = 0 (meaning no information-processing cost), the freeutility model is equivalent to the degenerated destinationchoice game model [34] in terms of potential game theory,as shown in the left dashed line of Fig. 1; when γ > τ → ∞ (meaning that the information-processingcost is too high), travelers can only uniformly randomlyselect each destination, as shown in the lower-right solidline of Fig. 1; when τ > γ = 0 (meaning thatthere is no interaction among travelers), the free utilitymodel is equivalent to the Logit model [31], as shown inthe upper-right dotted line of Fig. 1; and when τ = 0and γ = 0, all travelers will select the destination withthe highest constant utility, as shown in the intersectionpoint of the dashed line and dotted line of Fig. 1.Above, we considered only a simple system with asingle origin. However, there is more than one origin( M >
1) in a real transportation system. In such asystem, the utility form of destination j is changed. Itis affected by not only the number of trips from i to j but also the total number of trips attracted to destina-tion j . Hence, the utility of destination j for travelersat origin i can be abstractly written as u ij ( T ij , D j ) = A j − l j ( D j ) − c ij − g ij ( T ij ), where D j is the total numberof trips attracted to destination j , i.e., D j = (cid:80) i T ij . If D j is as constant as O i , that is, not only the total numberof trips emanating from origin i but also the number oftrips destined for any destination is fixed, then, the vari-able attractiveness function l j ( D j ) of destination j is alsoconstant and thus can be merged into the constant at-tractiveness A j of destination j . In other words, only thetravel congestion function g ij ( T ij ) influences the destina-tion choice behavior of travelers. If g ij ( T ij ) = γ ln T ij ,the free utility model of the system can be written asmax W = (cid:88) i (cid:88) j (cid:90) T ij ( A j − c ij − γ ln x )d x + τ (cid:88) i S i , s . t . (cid:88) j T ij = O i , (cid:88) i T ij = D j , (10)where the two constraints (cid:80) j T ij = O i and (cid:80) i T ij = D j are the fixed total number of trips emanating fromorigin i and the fixed total number of trips attracted todestination j , respectively. Using the Lagrange multipliermethod, we can obtain T ij = a i b j O i D j e − c ij / ( γ + τ ) , (11) γ τ ∞ p b FIG. 1.
The effect of parameter changes on the opti-mal solution of the free utility model.
In this simplesystem, there is only one origin, labeled a , and two destina-tions, labeled b and c . The origin has 200 travelers, eachof whom has only one trip. The utilities of b and c are u ab = 20 − γT ab and u ac = 2 − γT ac , respectively, where T ab and T ac are the number of travelers selecting destinations b and c , respectively, and γ is the interaction strength pa-rameter. The information-processing cost for these travelersis − τ S , where τ is the information-processing price. The sur-face of the figure describes probability p b that destination b is selected in the optimal solution of the free utility modelunder different parameter combinations. where a i = 1 / (cid:80) j b j D j e − c ij / ( γ + τ ) and b j =1 / (cid:80) i a i O i e − c ij / ( γ + τ ) . This is the doubly-constrainedgravity model widely used in transportation science [5].This free utility model can be reduced to some classicalmodels under specific parameter combinations. When τ > γ = 0 (meaning that there is no interactionamong travelers), Eq. (10) is equivalent to the free costmodel proposed by Tomlin et al. [29], and its solutionhas the same form as that of Wilson’s maximum entropymodel [27]. When τ = 0 and γ = 0, Eq. (10) is equivalentto the Hitchcock-Koopmans problem [49], which asks, forevery origin i and destination j , how many travelers mustjourney from i to j in order to minimize the total cost (cid:80) i (cid:80) j ( c ij − A j ) T ij . When τ → ∞ (meaning that thefirst term of the free utility function in Eq. (10) is negligi-ble), Eq. (10) is equivalent to the equal priori probabilitySasaki model [29] with the solution T ij ∝ O i D j .In a real transportation system, the travel cost c ij of-ten follows an approximate logarithmic relationship withdistance d ij , i.e., c ij ≈ β ln d ij [50]. Using β ln d ij insteadof c ij in Eq. (11), we can obtain the social gravity law T ij = a i b j O i D j d β/ ( γ + τ ) ij . (12)Thus far, we have achieved the goal of explaining theroot of the social gravity law. The social gravity lawis a macroscopic phenomenon caused by the interactionof bounded rational individuals in destination choice.In contrast, the destination choice game model consid-ers only individual interaction but ignores individualbounded rationality, which is almost universal in prac-tice, while the maximum entropy model, the free costmodel and the Logit model reflect the randomness of in-dividual choice decisions but do not reflect individual in-teraction. C. Network expansion model
The above constraints (cid:80) j T ij = O i and (cid:80) i T ij = D j are essential in the classic four-step travel demand fore-casting models in transportation science since the second-step model predicting trip distribution requires the in-put values of trip production and attraction resultingfrom the first-step model [5]. However, the total num-ber of trips D j attracted to destination j is impossibleto fix beforehand in a real transportation system since D j = (cid:80) i T ij is the product of the individual destinationchoice process. In other words, D j is variable. In thiscase, the free utility model can still describe traveler des-tination choice behavior. For a better understanding, wetransform the destination choice problem in the trans-portation system with multiple origin-destination pairsinto an equivalent route choice problem in a dummy net-work, as shown in Fig. 2. In this network, there are M origin nodes, N destination nodes and one dummy node s . There are M × N dummy links leading from each ori-gin node to each destination node and N dummy linksleading from each destination node to the dummy node.Each origin node has O i travelers whose final destinationin this network is the dummy node s . The traveler jour-neying from i to s has N alternative paths. Each pathis composed of two links, ij and js . The flow along link ij is T ij , and that along js is (cid:80) i T ij . The path utilityconsists of the utility u ij ( T ij ) = − c ij − g ij ( T ij ) of link ij and the utility u j ( (cid:80) i T ij ) = A j − l j ( (cid:80) i T ij ) of link js .Hence, the free utility model for this dummy network canbe written asmax W = (cid:88) i (cid:88) j (cid:90) T ij u ij ( x )d x + (cid:88) j (cid:90) (cid:80) i T ij u j ( x )d x + τ (cid:88) i S i , s . t . (cid:88) j T ij = O i , (13)where S i = − (cid:80) j T ij ln T ij O i . In addition, some pathsmay not be selected since the number of trips O i is fi-nite in practice; thus, constraint T ij ≥ s FIG. 2.
A dummy network to illustrate destinationchoice behavior in a transportation system with mul-tiple origin-destination pairs.
In this network, the greennodes in the bottom layer is are origin nodes, labeled i , andthe orange nodes in the middle layer are destination nodes,labeled j . The solid lines from the green nodes to the orangenodes are dummy links ij , with flow T ij and utility u ij ( T ij ).The blue node in the top layer is the dummy node s . Thedashed lines from the orange nodes to the dummy node aredummy links js , with flow (cid:80) i T ij and utility u j ( (cid:80) i T ij ). In-dividual destination choice behavior is route choice behaviorin this network. The path from i to s is composed of link ij and corresponding link js . fic flow on all links in the network [44]. Although the freeutility model in Eq. (13) cannot provide an explicit ex-pression of T ij , it can better characterize collective desti-nation choice behavior than the doubly-constrained grav-ity model that specifies the number of trips D j attractedto destination j .In the above studies, we assumed that the factorsinfluencing the utility of destination j for travelers atorigin i include the destination variable attractivenessfunction l j ( (cid:80) i T ij ) and the travel congestion function g ij ( T ij ). In practice, the travel congestion cost is deter-mined by the link congestion cost in the transportationnetwork, and the link congestion cost depends travelers’route choice results. Therefore, under the premise of hav-ing transportation network data, a combined destinationand route choice model can predict not only the num-ber of trips between origin and destination pairs but alsothe link flow in the transportation network. Here, wepresent a network expansion method to simultaneouslysolve the destination and route choice problem, as shownin Fig. 3. The bottom layer is the real transportationnetwork, in which each green node labeled i represents alocation (which is the centroid of the traffic analysis zone(TAZ) in transportation science) that can produce andattract trips, the black square nodes represent ordinarynodes such as intersections, and the links labeled a rep-resent segments. Each link is assigned a utility − t a ( x a )summarizing the relationship between link cost t a andlink flow x a . We expand the real transportation networkby adding two types of dummy nodes and two types ofdummy links. First, we add a dummy destination node i (cid:48) for each node i and a dummy link ii (cid:48) from node i tonode i (cid:48) . The flow along link ii (cid:48) is x ii (cid:48) , which is equal tothe trip attraction at the real destination i . The equiva-lent utility of link ii (cid:48) is the negative destination variableattractiveness − l ii (cid:48) ( x ii (cid:48) ). Next, we add a dummy originnode i (cid:48)(cid:48) for each node i and add dummy links from node i (cid:48) to all dummy origin nodes except the correspondingdummy origin i (cid:48)(cid:48) . The flow along link i (cid:48) i (cid:48)(cid:48) is x i (cid:48) i (cid:48)(cid:48) , whichis equal to the number of trips between the correspondingorigin and destination pair. The equivalent utility of link i (cid:48) i (cid:48)(cid:48) is the constant attractiveness A i (cid:48) . In this manner,an expanded network is formed.In this expanded network, the fixed trip volume fromorigin node i to corresponding dummy origin node i (cid:48)(cid:48) is O i , i.e., the number of travelers at origin i . Consequently,the combined destination and route choice problem isconverted into a route choice problem in the expandednetwork. The free utility model for this expanded net-work can be written asmax W = − (cid:88) a (cid:90) x a t a ( x )d x − (cid:88) i (cid:48) (cid:90) x ii (cid:48) l ii (cid:48) ( x )d x + (cid:88) i (cid:48)(cid:48) (cid:88) i (cid:48) (cid:54) = i (cid:48)(cid:48) A i (cid:48) x i (cid:48) i (cid:48)(cid:48) + τ (cid:88) i S i , s . t . (cid:88) k f ki = O i ,f ki ≥ , (14)where S i = (cid:80) k f ki ln f ki O i is the information-processingentropy for bounded rational travelers and f ki is theflow on the k th path from node i to the correspond-ing dummy origin node i (cid:48)(cid:48) . The relationship betweenpath flow f ki and link flow x a in the expanded networkis x a = (cid:80) i (cid:80) k f ki δ a,ki , where δ a,ki is 1 if path k uses link a and 0 otherwise. Similar to Eq. (13), Eq. (14) is aSUE model for traffic assignment on the expanded net-work. By solving it, we can obtain not only the trafficflow on the segment of the real transportation networkbut also the trip volume between TAZs (i.e., flow x i (cid:48) i (cid:48)(cid:48) along dummy link i (cid:48) i (cid:48)(cid:48) ) and the trip volume attracted byeach TAZ (i.e., flow x ii (cid:48) along dummy link ii (cid:48) ). Such amodel, which can simultaneously predict trip distribu-tion and traffic flow, is called a combined distribution-assignment model in transportation science [44]. How-ever, the combined model established by the free utilityframework is more interpretable than most previous com-bined distribution-assignment models. It clearly showsthe interaction of travellers with bounded rationality onthe links and destinations of the real transportation net-work. FIG. 3.
An expanded transportation network.
The bot-tom layer in this network is the real transportation network,in which the green nodes labeled i are the locations that canproduce and attract trips, the black square nodes are the ordi-nary nodes of the transportation network, and the solod linesbetween the nodes are the links of the transportation network.The flow on the link labeled a is x a , and the utility of the linkis − t a ( x a ). The orange nodes in the middle layer are thedummy destination nodes labeled i (cid:48) . The dashed lines fromthe green nodes to the orange nodes are the directed links ii (cid:48) with flow x ii (cid:48) and utility − l ii (cid:48) ( x ii (cid:48) ). The blue nodes in thetop layer are dummy origin nodes labeled i (cid:48)(cid:48) . The dotted linesfrom the orange nodes to the blue nodes are the directed links i (cid:48) i (cid:48)(cid:48) with flow x i (cid:48) i (cid:48)(cid:48) and utility A i (cid:48) . III. DISCUSSION AND CONCLUSIONS
In this paper, we developed a free utility model thatcan explain the social gravity law from the perspec-tive of individual choice behavior. The model makestwo basic assumptions: (1) the individual pursues util-ity maximization, and (2) the individual needs to paythe information-processing cost to acquire more knowl-edge about the utilities of the destinations. The objec-tive function of the free utility model of the destinationchoice system with a single origin (see Eq. (7)) is math-ematically consistent with the Helmholtz free energy inphysics. In other words, this destination choice systemis analogous to the isothermal and isochoric thermody-namic system that consists of several subsystems in ther-mal contact with a large reservoir: the number of travel-ers is analogous to the number of particles; the first termof the free utility model’s objective function is analogousto the thermodynamic system’s internal energy includ-ing potential energy; the information-processing price isanalogous to the temperature of the reservoir; the infor-mation entropy is analogous to the entropy of the ther-modynamic system; the information-processing cost isanalogous to the heat transferred between the thermody-namic system and the reservoir; and the marginal utilityis analogous to the chemical potential of the subsystem.However, the essence of these two systems is different:the thermodynamic system follows the minimum free en-ergy principle to make the system reach the equilibriumstate in which all subsystems have the same chemicalpotential [47], and the maximization free utility of desti-nation choice system is the result of individuals followingthe equimarginal principle to make choices to maximizetheir own utility.Some previous models, including the free cost model[29] and the maximum entropy model [27], have notshown a clear derivation of the gravity model from micro-scopic mechanisms [52]. The free cost model presentedby Tomlin and Tomlin derived the gravity model by anal-ogy with the Helmholtz free energy, but it did not explainthe social gravity law from the perspective of individualchoice behavior. In our opinion, this model is not es-sentially different from the unconstrained gravity modelestablished by direct analogy with Newton’s law of uni-versal gravitation. Similarly, Wilson’s maximum entropymodel on the gravity model can provide only the mostlikely macrostate but cannot describe individual choicebehavior. The maximum entropy model also requires theprior constraint of the total cost, which is actually theresult of individual choice. Additionally, both of thesephysical analogy models ignore the interaction amongindividuals, which is a ubiquitous phenomenon in socialsystems [6]. In comparison, we provide a concise expla-nation for the social gravity law from the perspective ofindividual behavior. It simultaneously reflects two dom-inant mechanisms that are common in social systems,namely, individual interaction and bounded rationality.This line of explanation brings us to the idea that the so-cial gravity law is a phenomenon resulting from boundedrational individuals interacting with each other.The free utility model can explain not only the socialgravity law but also the potential function in game the-ory from the perspective of individual behavior. The freeutility model is a stochastic potential game model [53] inmathematical form. The objective function of the po-tential game model proposed by Monderer and Shapleywas established by analogy with the potential function inphysics [48]. The variation in a player’s individual pay-off due to changes in the player’s strategy is equal to thevariation in the potential function [54, 55]. Every equilib-rium strategy profile of the potential game maximizes thepotential function. Under these circumstances, all play-ers have the same payoff, and they cannot unilaterallychange their strategy to increase their respective pay-off. However, Monderer and Shapley raised a questionabout the interpretation of potential function: “Whatdo the firms try to jointly maximize?” [48]. In fact, thepotential function is not something that players try tojointly maximize. Now we know that players will maxi-mize only their own utility (i.e., the payoff) through themodeling process of the free utility model. A player’soptimal choice strategy is to make the marginal utilityof each alternative equal. When all players follow thisequimarginal principle, the sum of the integral of the marginal utility of each alternative (called total utilityin economics) is naturally maximized. In other words,the maximum potential function is the result of players’optimal choice strategy but is not a goal that the playerspursue jointly.The free utility model can also provide an explanationfor the objective function of the classic traffic assign-ment model in transportation science. From Eq. (13),we can see that the free utility model for the trans-portation network is mathematically consistent with theSUE model [51]. Without considering the information-processing cost, the free utility model is mathematicallyconsistent with the classic user equilibrium (UE) modelestablished by Beckmann [56]. The equilibrium solutionof the UE model, in which all paths used between eachOD pair have equal and minimum costs, is simply theresult of the optimal path choice strategy selected bytravelers following the equimarginal principle. The ob-jective function of the UE model, i.e., the sum of the in-tegral of the marginal utility (negative cost) of each link,is actually the negative free utility without information-processing cost. This provides new insights into UE andSUE models.We know that the free utility model can be used topredict trip distribution and traffic assignment in thefour-step travel forecasting procedure. Moreover, the freeutility model can also be used to predict modal splits inthe four-step procedure [5]. If we regard transportationmodes as alternatives and assign the appropriate utilityto each mode, we can build a free utility model simi-lar to that shown in Eq. (7) to predict the number oftravelers selecting different modes. Compared with theLogit model, which is the model most frequently usedfor mode choice prediction in transportation science, theadvantage of the free utility model is that it can reflectthe interaction among travelers. For example, conges-tion in subways has a negative effect on utility in thismode. However, the Logit model does not consider thecongestion caused by individual interaction [37] that iscommon in actual transportation systems. Furthermore,mode choice can be combined with destination choice androute choice to build a multistep combined model underthe free utility framework. This requires only that thereal transportation network shown in the bottom layerof Fig. 3 be expended into a multimode integrated trans-portation network. By the same token, travel choicebehaviors such as trip frequency choice and departuretime choice can also be modeled in combination with theabove steps under the free utility framework. In sum-mary, such combined models allow us to systematicallystudy the entire travel decision-making process of inter-acting bounded rational travelers and quantitatively an-alyze the complex transportation system under the moreinterpretable analytic framework.Finally, although we use individual destination choicebehavior in the transportation system as background toexplain the social gravity law, such behavior of selectingalternatives exists not only in the transportation systembut also in systems where the spatial interaction patternsfollow the social gravity law. For example, populationmigration involves selecting locations as residences, so-cial interaction involves selecting people as friends andscientific collaboration involves selecting researchers aspartners. In these systems, individuals tend to select al-ternatives with relatively high activity and relatively lowdistance, and changes in the number of individuals whoselect the same alternative will affect the variation in theutility of the alternative. For example, an increase incommodity competitors for the same product will leadto a decrease in the product price, and an increase incollaborators with one scientist will lead to a decrease incooperation intensity between this scientist and her col-laborators. The spatial interaction behavior in these dif-ferent systems can be described by the free utility model.Not only that, the free utility model of the destinationchoice system with a single origin (see Eq. (7)) can beused for generalized problems of human choice, where ho-mogeneous individuals select from multiple alternatives with utility associated with the choices. Moreover, usingthe dummy network method, we can extend the free util-ity model to individuals of different types (i.e., heteroge-neous individuals). The dummy network in Fig. 2 can beregarded as a case in which two types of individuals selectfrom among three alternatives. In this network, the util-ity of the solid line represents the utility component ofthe alternative for the corresponding type of individual,and the utility of the dashed line represents the utilitycomponent of the alternative for both types of individ-uals. In short, the free utility model that explains thesocial gravity law not only helps us deeply understandcollective choice behavior patterns emerging from the in-teraction of bounded rational individuals but also showspotential application in predicting, guiding or even con-trolling human choice behavior in various complex socialsystems.
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