Why is the price of renting an automobile “flat” as a function of its age or odometer value? Specifically, why is it that car rental companies do not offer customers the option of renting older cars at a discount, instead of offering only relatively new cars at full price? We also tackle a related puzzle: why do car rental companies trade-in their vehicles so early? Most US companies purchase brand new rental cars and replace them after 2 years or when their odometer exceeds 34,000 km. That is a very costly strategy due to the well-known by rapid depreciation in used car prices. We show that in a competitive rental market, prices are a declining function of odometer and cars are rented over their full economic lifespan. Our solution to these puzzles is that actual rental markets are not fully competitive and firms may be behaving suboptimally. We provide a case study of a large car rental company that provided us access to its operating data. We develop a model of the company's operations that predicts that the company can significantly increase its profits by keeping its rental cars twice as long as it currently does, discounting the rental prices of older vehicles to induce its customers to rent them. The company undertook an experiment to test our model's predictions. We report initial findings from this experiment which involved over 4500 rentals of over 500 cars in 4 locations over a 5-month period. The results are consistent with the predictions of our model, and suggest that a properly chosen declining rental price function can increase overall revenues. Profits also increase significantly, since doubling the holding period of rental cars cuts discounted replacement costs by nearly 40%.

We define a class of dynamic Markovian games, directional dynamic games (DDG), where directionality is represented by a strategy-independent partial order on the state space. We show that many games are DDGs, yet none of the existing algorithms are guaranteed to find any Markov perfect equilibrium (MPE) of these games, much less all of them. We propose a fast and robust generalization of backward induction we call state recursion that operates on a decomposition of the overall DDG into a finite number of more tractable stage games , which can be solved recursively. We provide conditions under which state recursion finds at least one MPE of the overall DDG and introduce a recursive lexicographic search (RLS) algorithm that systematically and efficiently uses state recursion to find all MPE of the overall game in a finite number of steps. We apply RLS to find all MPE of a dynamic model of Bertrand price competition with cost-reducing investments which we show is a DDG. We provide an exact non-iterative algorithm that finds all MPE of every stage game, and prove there can be only 1, 3, or 5 of them. Using the stage games as building blocks, RLS rapidly finds and enumerates all MPE of the overall game. RLS finds a unique MPE for an alternating move version of the leapfrogging game when technology improves with probability 1, but in other cases, and in any simultaneous move version of the game, it finds a huge multiplicity of MPE that explode exponentially as the number of possible cost states increases.