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We study adaptive route choice models that explicitly capture travelers' route choice adjustments according to information on realized network conditions in stochastic time-dependent networks. Two types of adaptive route choice models are explored: an adaptive path model where a sequence of path choice models are applied at intermediate decision nodes; and a routing policy choice model where the alternatives correspond to routing policies rather than paths at the origin. A routing policy in this paper is a decision rule that maps from all possible (node, time) pairs to next links out of the node. A policy-size Logit model is proposed for the routing policy choice, where policy-size is a generalization of path-size in path choice models to take into account the overlapping of routing policies. The specifications of estimating the two adaptive route choice models are established and the feasibility of estimation from path observations is demonstrated on an illustrative network. Prediction results from three models - non-adaptive path model, adaptive path model, and routing policy model - are compared. The routing policy model is shown to better capture the option value of diversion than the adaptive path model. The difference between the two adaptive models and the non-adaptive model is larger in terms of expected travel time, if the network is more stochastic, indicating that the benefit of being adaptive is more significant in a more stochastic network.