In the Los Angeles (LA) Metro Rail system and other proof-of-payment transit systems worldwide, passengers are legally required to buy tickets before boarding, but there are no gates or turnstiles. There are, quite literally, no barriers to entry, as illustrated in figure 1. Instead, security personnel are dynamically deployed throughout the transit system, randomly inspecting passenger tickets. This proof-of-payment fare collection method is typically chosen as a more cost-effective alternative to direct fare collection, that is, when the revenue lost to fare evasion is believed to be less than what it would cost to make fare evasion impossible.
For the LA Metro, with approximately 300,000 riders daily, this revenue loss can be significant; the annual cost has been estimated at $5.6 million. (1) The Los Angeles Sheriff's Department (LASD) deploys uniformed patrols onboard trains and at stations for fare checking (and for other purposes such as crime prevention), in order to discourage fare evasion. With limited resources to devote to patrols, it is impossible to cover all locations at all times. The LASD thus requires some mechanism for choosing times and locations for inspections. Any predictable patterns in such a patrol schedule are likely to be observed and exploited by potential fare evaders. The LASD's current approach relies on humans for scheduling the patrols. However, human schedulers are poor at generating unpredictable schedules (Wagenaar 1972, Tambe 2011); furthermore such scheduling for LASD is a tremendous cognitive burden on the human schedulers who must take into account all of the scheduling complexities (for example, train timings, switching time between trains, and schedule lengths). Indeed, the sheer difficulty of even enumerating the trillions of potential patrols makes any simple automated approach--such as a simple dice roll--inapplicable.
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The result of our investigation is a novel application called TRUSTS (tactical randomization for urban security in transit systems) for fare-evasion deterrence in urban transit systems, carried out in collaboration with the LASD. We take a game-theoretic approach, which studies systems with multiple self-interested parties and aims to predict the likely outcomes of the system under rational behavior of the players. In particular, we model this problem as a Stackelberg game with one leader (the LASD) and many followers, in which each metro rider (a follower) takes a fixed route at a fixed time. The leader precommits to a mixed patrol strategy (a probability distribution over all pure strategies), and riders observe this mixed strategy before deciding whether to buy the ticket or not (the decision to ride having already been made) in order to minimize their expected total cost, following for simplicity the classic economic analysis of rational crime (Becket and Landes 1974). Both ticket sales and fines issued for fare evasion translate into revenue to the government. Therefore the optimization objective we choose for the leader is to maximize total revenue (total ticket sales plus penalties).
There are exponentially many possible pure patrol strategies, each subject to both the spatial and temporal constraints of travel within the transit network. Explicitly representing a mixed strategy would be impractical. To remedy this difficulty, TRUSTS uses the transition graph, which captures the spatial as well as temporal structure of the domain, and solves for the optimal (fractional) flow through this graph, using linear programming (LP). Such a flow can be interpreted as a marginal coverage vector. Additionally, we show that a straightforward approach to extracting patrol strategies from the marginals faces important challenges: it can create infeasible patrols that violate the constraint on patrol length, and it can generate patrols that switch too frequently between trains, which can be difficult for patrol personnel to carry out. …