Gernot Tragler

Optimal Dynamic Allocation of Treatment and Enforcement in Illicit Drug Control

There has been considerable debate about what share of drug control resources should be allocated to treatment vs. enforcement. Most of the debate has presumed that there is one answer to that question, but it seems plausible that the mix of interventions should vary as the size of the problem changes. We formulate the choice between treatment and enforcement as an optimal control problem and reach the following conclusions. If initiation into drug use is an increasing function of the current number of users and control begins early, then it is optimal to use very large amounts of both enforcement and treatment to cut short the epidemic. Otherwise the optimal policy is not to stop the growth of the epidemic, but rather to moderate it. Initially this should be done primarily with enforcement. Over time, enforcement spending should increase, but not nearly so fast as treatment spending. Hence, treatment should receive a larger share of control resources when a drug problem is mature than when it is first growing. If initiation rates subsequently decline, enforcements budget share should drop further in the ensuing declining stage of the epidemic.

Why present-oriented societies undergo cycles of drug epidemics

Musto (The American Disease, Yale University Press, New Homen, 1987) hypothesizes that cycles of drug use arise when the current generation of youth no longer remembers the adverse experiences of their forebears. This paper underlines this hypothesis through the results of an optimal control model of drug epidemics that incorporates an endogenous initiation function, models the reputation of a drug as being determined by memories of past use, and finds the optimal drug treatment strategy. Interestingly, unless the societal discount rate is quite low, if memories of past users decay moderately quickly, the optimal strategy is cyclic. Hence, not only do we find that ‘those who forget the past are condemned to repeat it’, but also that ‘for those who forget the past and over-value the present it may even be optimal to have their future recreate the past’. These findings are illustrated by numerical analysis based on empirical data from the current US cocaine epidemic.

An age-structured single-state drug initiation model--cycles of drug epidemics and optimal prevention programs

Abstract This paper introduces a model for drug initiation that extends traditional dynamic models by considering explicitly the age distribution of the users. On the basis of a 2-groups model in which the population is split into a user and a non-user group the advantage of a continuous age distribution is shown by considering more details and by yielding new results. Neglecting death rates reduces the model to a single-state (1-group) descriptive model which can still simulate some of the complex behavior of drug epidemics such as repeated cycles. Furthermore, prevention programs—especially school-based programs—can be targeted to certain age classes. So in order to discover how best to allocate resources to prevention programs over different age classes we formulate and solve optimal control models.

Estimating the relative efficiency of various forms of prevention at different stages of a drug epidemic

Drug use and problems change dramatically over time in ways that are often described as reflecting an “epidemic cycle”. We use simulation of a model of drug epidemics to investigate how the relative effectiveness of different types of prevention varies over the course of such an epidemic. Specifically we use the so-called LHY model (see, Discussion Paper No. 251 of the Institute of Econometrics, OR, and Systems Theory, Vienna University of Technology, Vienna, Austria, 2000) which includes both “contagious” spread of initiation (a positive feedback) and memory of past use (a negative feedback), which dampens initiation and, hence, future use. The analysis confirms the common sense intuition that prevention is more highly leveraged early in an epidemic, although the extent to which this is true in this model is striking, particularly for campaigns designed to preserve or amplify awareness of the drugs dangers. The findings also suggest that the design of “secondary” prevention programs should change over the course of an epidemic.

Optimal control of terrorism and global reputation: A case study with novel threshold behavior

A control model is presented which studies optimal spending for the fight against terrorism. Under the assumptions that economic damages are larger the greater the number of terrorists and that the success of counter terror operations depends on public opinion, it is demonstrated that a so-called DNSS threshold may exist, separating the basin of attraction of optimal paths.

Managing the reputation of an award to motivate performance

Abstract.Managers wish to motivate workers to exert effort. There is large literature on the use of wages and monetary incentives for this purpose, but in practice the “honor” or “prestige” of an award can be a significant motivator as well, unless the award is given so often that its prestige is diluted. The model here focuses on management of the reputation of an award that may or may not have a fixed monetary component. The model is an optimal dynamic control model, so its solution suggests how to manage the award over time. The analysis is interesting because of a “false” steady state that is adjacent to but outside the admissible region and which otherwise has the qualitative properties of a steady state; there are (infinitely many) trajectories converging to it and (infinitely many) trajectories starting arbitrarily close to it. For all initial conditions there are infinitely many candidates for the optimal solution that cannot be evaluated in the standard way. We resolve that problem by proving a new proposition concerning the value of the utility functional when the limit of the Hamiltonian is non-zero. Managerial implications are derived.

Optimal dynamic law enforcement

Abstract In this paper we present an intertemporal extension of Beckers [Journal of Political Economy 76 (1968) 169] static economic approach to crime and punishment. For a dynamic supply of offenders we determine the optimal dynamic trade-off between damages caused by offenders, law enforcement expenditures and cost of imprisonment. By using Pontryagins maximum principle we obtain interesting insight into the dynamical structure of optimal law enforcement policies. It is found that inherently multiple steady states are generated which can be saddle points, unstable points and boundary solutions. As in other non-linear control models there exists a threshold (a so-called Skiba point) which makes the optimal enforcement policy dependent on the initial conditions. It turns out that above the Skiba point the optimal trade-off between social costs implies a steady state with a high level of offences, while below the threshold the optimal law enforcement should eradicate crime.

Incentive Stackelberg Strategies for a Dynamic Game on Terrorism

This paper presents a dynamic game model of international terrorism. The time horizon is finite, about the size of one presidency, or infinite. Quantitative and qualitative analyses of incentive Stackelberg strategies for both decisionmakers of the game (“theWest” and “International Terror Organization”) allow statements about the possibilities and limitations of terror control interventions. Recurrent behavior is excluded with monotonic variation in the frequency of terror attacks whose direction depends on when the terror organization launches its terror war. Even optimal pacing of terror control operations does not greatly alter the equilibrium of the infinite horizon game, but outcomes from theWest’s perspective can be greatly improved if the game is only “played” for brief periods of time and if certain parameters can be influenced, notably those pertaining to the terror organization’s ability to recruit replacements.

Keeping Options Open: an Optimal Control Model with Trajectories That Reach a DNSS Point in Positive Time

The so-called DNSS points of indifference are of interest because they give decision makers in optimal control problems a choice between following either of two or more trajectories while still achieving optimality. Usually they are described in terms of initial conditions, so that if the system starts at a DNSS point, the decision maker can proceed in either of two or more directions. Here we present a model that has an entire curve of indifference points away from which the decision maker can move in only one direction but does so by choosing either of two trajectories that initially coincide in the state space but later diverge, approaching different long-run steady states.

Price-raising drug enforcement and property crime: a dynamic model

Price-raising drug enforcement suppresses drug use, but it is expensive and may increase property crime. This has led to contradictory recommendations concerning how drug enforcement should or should not be used. We reconcile these recommendations by incorporating the enforcements effects on both drug use and on property crime within an optimal-control model that recognizes whether convicted drug-involved property offenders are merely incarcerated or whether they receive some form of drug treatment.