Innovations in Travel Demand Modeling, Volume 2: Papers (2008)

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ties, and expected values of attributes of the transporta- tion and land use system. The location choice set consists of all locations known by the individual. “Known” in this context means that the agent knows not only the physical location but also the attributes that are potentially relevant for evaluating utility values for all potential activities. Nevertheless, location choice sets are dynamic. Changes follow from processes of knowledge decay, reinforcement, and explo- ration (Arentze and Timmermans 2005b, 2006). The strength of a memory trace of a particular item in the choice set is modeled as follows: (6) where Wi t = strength of the memory trace (awareness) of location i at time t; Ii t = 1, if the location was chosen at time t, and = 0, otherwise; Ui t = utility attributed to location i; 0  γ  1 = parameter representing a recency weight; and 0  λ  1 = parameter representing the retention rate. The coefficients γ and λ determine the size of reinforce- ment and memory retention, respectively, and are param - eters of the system. Exploration, in contrast, is a process by which new elements can enter the choice set. The probability that a certain location i is added to the choice set in a given time step is modeled as (7) where P(Gt) is the probability that the individual decides to explore and P(Hi t | Gt) is the probability that location i is discovered during exploration and tried on a next choice occasion. Whereas the former probability is a parameter of the system to be set by the modeler, the lat- ter probability is modeled as a function of attractiveness of the location based on the Boltzman model (Sutton and Barton 1998): (8) where Vi t is the utility of location i according to some measure and τ is a parameter determining the degree of randomness in the selection of new locations but which can also be interpreted as the degree of agent uncer- tainty (Han and Timmermans 2006). The higher the τ parameter is, the more evenly probabilities are distrib- uted across alternatives and, hence, the higher the ran- domness and vice versa. More than one location may be added to the choice set in a given time step. A new loca- tion has priority over known locations in location choice and cannot be removed from the choice set before it has been tried once. Once tried, the new loca- tion receives a memory-trace strength and is subject to the same reinforcement and decay processes that hold for memory traces in general. As a consequence of these mechanisms, higher-utility locations have a higher prob- ability of being chosen, for three reasons: (a) they have a higher probability of being discovered; (b) if discov- ered, they have a higher probability of being chosen, and (c) if chosen, they are more strongly reinforced. At the same time, they are not guaranteed of staying in the choice set because of two other mechanisms: (a) if the utility decreases due to nonstationarity in the system (e.g., the locations do not longer fit in changed sched- ules), the decay process will ensure that they vanish from the choice set, and (b) if more attractive locations are discovered, the original locations will be outper- formed and, therefore, will decay. Finally, learning involves updating default settings of activities, such as duration, start time, transport mode, and location. For this updating, each agent keeps a record of the probability distribution across each choice set. For start time and duration, which are continuous variables, a reasonable subrange is identified and subdi- vided into n rounded values. For each choice facet, the following Bayesian method of updating is used: (9) (10) where Pi t = probability of choice i at time t, M = weighted count of the number of times the choice has been made in the past, Ii t = indication of whether iwas chosen at time t, and 0  α  1 = retention rate of past cases. As implied by Equation 9, more recent cases have a higher weight in the update (if α < 1), to account for pos- sible nonstationarity in the agent’s choice behavior. With the probability distribution of each choice facet at the current time step defined, the default is simply identified as the option having the highest probability across the choice set. 74 INNOVATIONS IN TRAVEL DEMAND MODELING, VOLUME 2 W W U I W i t i t i t i t i t + = + =⎧⎨⎪ ⎩⎪ 1 1γ λ if otherwise P H P G P H Gi t t i t t( ) ( ) ( | )= P H G V V i t t i t i t i ( | ) exp( / ) exp( / ) = ∑ τ τ P P M M I P M M i t i t t t i t i t t t + = + + = + 1 1 1 1 1 if otherwise ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ M Mt t+ = +1 1α