Modelling Multidimensional Search

An individual, based on her/his past experienceand subjective beliefs, forms expectations on potential gain (search gain) from behavioural adjustments along each dimension. The decision to search for a new alternative is based on the interplay of subjective search gain and perceived search cost. Let an agent's generalised cost on the currently used alternative be C. The subjective search gainis based on subjective beliefs,, and defined as the expected improvement in regard to generalised cost savings per trip from an additional search:

(10.4)

where C is actually the minimum of all experienced generalised costs because individuals can select from all tried alternatives in dimension d and pick the one with the lowest costs . We assume all individuals start with a preferred travel pattern. It can be the stabilized travel pattern with an initial generalised cost . Once a policy/ congestion stimulus emerges, travel condition deteriorates. Let us further assume that individuals have the initial beliefs that search and switching to another alternative will lead to a travel condition as good as their original travel condition until they search and experience otherwise. As the search process proceeds, the subjective probability of finding an alternative with after N searches is . Therefore,

Eq. (10.4) can be further simplified as:

(10.5)

While remains universal among all dimensions, , the currently best travel option(s) in dimension d, can differ in each dimension d since the search processes in different dimensions vary and result in diverse outcomes. The subjective search gain evolves and reflects how much value each search can gain based on subjective beliefs. Once is less than or equal to zero, it indicates that search along dimension d is no longer worthwhile and the search process will not initiate. A positive will asymptotically decrease to zero as the number of searches increases and as a better alternative is found ( getting increasingly closer to ).

Furthermore, the theory formulates satisficing behaviour that even with positive gains, individuals may stop search once the gain is lower than the perceived search cost. The search and information acquisition are no longer free as this theory recognises the inconveniences and risks associated with each behaviour adjustment dimension. This impedance is conceptualised as a search cost for each agent and each dimension. Search cost can be perceived and inferred once individuals' searching sequence can be reconstructed using empirical observations collected from survey. The empirical data provides evidence about agents' search and decision processes. Each individual follows her/his own path along the three dimensions in reaching the final behaviour decisions. When it is observed that an individual ends her/his search in dimension d and has searched N times along that dimension for the time being, it infers that the individual satisfices after N rounds of search in d. The search cost must be lower than so that the Nth search is meaningful and rewarding. Meanwhile, the search cost must be higher thanso that the (N+ l)th search does not occur. Let us denote individual f s search cost along dimension d as , which is viewed as an innate personal characteristic for individual i. It can be estimated by using the lower and upper bounds:

(10.6a)

(10.6b)

(10.6c)

Note that for each individual, only one of the multidimensional perceived search costs can be perceived from the empirical data. A subsequent regression analysis for all survey subjects and all dimensions thus needs to be estimated in order to empirically model search cost. We specify that the search cost model in dimension d as:

(10.7)

where C0 is the generalised cost for the originally reported travel experience; distance measures the mileage that the subject travels; Dummy variables include gender (equals to 1 if the subject is female), fixedsch (equals to 1 if the subject has fixed travel schedule), purpose (equals to 1 if the trip purpose is work/school), peak (equals to 1 if the travel is in peak-hour periods) and veh (equals to 1 if household number of vehicles is greater than 2). Different household annual income levels are considered in the model (incomel: less than $50,000; income2: $50,000—100,000; income3: $100,000—150,000; income4: $150,000 and above). In our model, C0 is identified as an instrumental variable (IV) in order to better incorporate the sufficiently high correlation between C0 and other independent variables. We employ the generalised method of moments (GMM) and two-stage least-squares (2SLS) estimator. Denoting the IV as z and the independent variables as x, we can estimate parameters from the population moment conditions:

(10.8)

The estimation result is reported in Table 10.1. The search cost is positively related to the initially experienced generalised cost of the travel. Lower-income agents have higher search costs along the mode dimension. Female agents are more reluctant to search departure times and routes than to search alternative modes. Fixed schedule and travelling during peak-hour increase the search cost for all dimensions. Travel distance has a negative impact on search cost meaning that the longer the travel distance, the more likely she/he will search for alternatives. The coefficients for trip purpose indicate that agents doing commute travels have more incentive to search for alternative modes and departure times. By estimating and applying search cost models, one can make personal/household characteristics endogenous in the search process and model diversified and behaviourally rich multidimensional search. It helps explain why some travellers may adjust routes first while others may adjust departure time first in response to the same stimulus. This feature can potentially provide a rich level of details especially for policy/social equity analysis whence measuring the impacts/benefits by different socio-economic strata of society is of interest.

It is hypothesised that agents will search the most rewarding dimension with the highest search gain/cost ratio. Successive unrewarding searches along a particular

Table 10.1: Multidimensional perceived search cost models (generalised method of moments and instrumental variable).

*,**,*** significant at 90%, 95%, 99% confidence level.

behavioural adjustment dimension (e.g. route) will lead to diminishing subjective search gain for that dimension and at a later point cause the search to shift to a different behaviour dimension (e.g. departure time). Once the ratios for all dimensions drop down below one, the multidimensional search process ceases. Since gdt is monotonically decreasing and converges to zero, the search is guaranteed to reach stability. The interplay of these search gains and costs along all feasible behavioural dimensions defines the bounded rationality embedded in the theory. It collectively determines the prospects for profitable searches over finite horizon and guarantees a convergence of behavioural changes. It quantitatively theorizes when individuals start seeking behavioural changes, how they initially change behaviour, how they switch behavioural adjustment dimensions and when they stop the search.