Payoff Rates Research Articles

Risk-sensitive zero-sum games for continuous-time jump processes with unbounded rates and Borel spaces

This paper studies two-person zero-sum risk-sensitive stochastic games for continuous time jump processes with the following features: (1) the payoff and transition rates are unbounded and time-dependent, (2) the state and action spaces are general Borel sets, (3) the discount factors can vary in time. Differing from those with the risk-sensitive parameter as a differential variable in previous works, we introduce a new type of Shapley equation (NSE) with the time as a differential variable. Under some technical assumptions, we derive a representation of a solution to the NSE, which in turn is used to establish the existence of a solution to the NSE by iterations and approximation. With the help of the NSE, we prove the existence of the value of the game and a saddle-point equilibrium depending on the time, which explicitly shows the nonstationarity of the risk-sensitive discounted equilibria. We illustrate our results by two examples with unbounded transition and payoff rates.

Continuous-time zero-sum stochastic game with stopping and control

We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.

Small-Dollar Installment Loans: An Empirical Analysis

Small-dollar credit is a form of unsecured consumer credit primarily characterized by the low dollar amounts of loans. There has been increasing debate about the benefit and harm to consumers from small-dollar loans, along with recent discussion of greater regulation. Determining the need for and appropriate form of regulation requires an understanding of the current state of the small-dollar credit industry based on actual industry data. One significant change is the shift from single-payment payday loans to multiple-payment loans or installment loans. This paper is the first systematic study of small-dollar installment loans.Our main findings are as follows. A typical installment loan is for $900 and is scheduled to be repaid in 12 biweekly installments over six months. Less than a quarter of borrowers take another loan within 14 days of ending a previous loan. Most borrowers do not keep their loan until maturity. Even repeat borrowers generally either pay off the loan or have it charged off before the maturity date of their original loan. Those who borrow repeatedly are more likely to repay their loans on average and are offered lower interest rates, indicating that at least some of these repeat borrowers are utilizing the opportunity to borrow again based on their past track record of payments. Payment-to-income ratio alone is a poor metric for predicting whether the loan will be paid off or not. Regulation imposing an upper limit on payment-to-income ratio is likely to result in reduced access to credit for a large majority of current borrowers, without a large improvement in loan payoff rates. A payment-to-income limit of 5% would reduce the volume of credit between 55.1% and 92.6%. It would only increase the loan payoff rate by 0.7% (from 72.9% to 73.6%). A payment-to-income limit of 10% will result in reduction of credit between 26.5% and 67.8% without any increase in loan payoff rate.A regulation that prohibits lending based on simple affordability criteria risks substantial reductions in credit availability to a population that often has few available alternatives. A cost-benefit analysis of the proposed regulation requires weighing the cost of reduced access to a financially underserved segment of the population against the benefit of a higher loan payoff rate and lower incidence of indebtedness. Limiting the payment-to-income ratio to 5% would benefit fewer than 1% of borrowers by reducing the incidence of loans that are not paid off, but it would impose costs on 86% of current borrowers, who could not be offered the same credit on the same terms that they now obtain. Raising the permitted ratio can reduce these costs, but it also reduces the benefits, because payment-to-income ratio alone is poor predictor of the likelihood of repayment.

Nonzero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper attempts to study two-person nonzero-sum games for denumerable continuous-time Markov chains determined by transition rates, with an expected average criterion. The transition rates are allowed to be unbounded, and the payoff functions may be unbounded from above and from below. We give suitable conditions under which the existence of a Nash equilibrium is ensured. More precisely, using the socalled “vanishing discount” approach, a Nash equilibrium for the average criterion is obtained as a limit point of a sequence of equilibrium strategies for the discounted criterion as the discount factors tend to zero. Our results are illustrated with a birth-and-death game.

NEW EVIDENCE ON CREDIT CARD BORROWING AND REPAYMENT PATTERNS

A new dataset allows researchers to examine patterns of credit card use for both borrowing and payoff. This article addresses changes in these behaviors for different birth cohorts by estimating cohort‐adjusted age profiles for debt and payoff rates based on a time series of cross sections. Younger consumers are found to be borrowing more heavily and repaying at lower rates than older generations. The accumulation of credit card debt is found to continue over the lifecycle. This has implications for recent changes in laws governing the credit card industry. Increases in minimum required payment rates are examined and are found to increase actual payoff rates more than proportionately. (JEL D12)

Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to beunbounded, and the payoff rates may haveneither upper nor lower bounds. We give conditions on the game'sprimitive dataunder which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using theextended infinitesimal operatorassociated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide arecursiveway of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game's primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates

This paper is concerned with two-person zero-sum games for continuous-time Markov chains, with possibly unbounded payoff and transition rate functions, under the discounted payoff criterion. We give conditions under which the existence of the value of the game and a pair of optimal stationary strategies is ensured by using the optimality (or Shapley) equation. We prove the convergence of the value iteration scheme to the game's value and to a pair of optimal stationary strategies. Moreover, when the transition rates are bounded we further show that the convergence of value iteration is exponential. Our results are illustrated with a controlled queueing system with unbounded transition and reward rates.

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

Motivational bases in mixed-motive interactions: The effects of comparison levels

The relationship between reward level and the basic motives which underlie strategic choices (competitive and noncompetitive) in a mixed-motive game was examined from two theoretical perspectives. The “regret” interpretation proposed by McClintock and McNeel was compared to an approach based upon Thibout and Kelley's concept of comparison level (CL). Two independent variables, CL and Reward, were manipulated by varying the payoff rates in two playing series of the MDG. Initially, 50 iterations were played for high (4¢ per point) or low ( 1 5¢ per point) incentive to provide subjects with outcomes upon which to base their CL's. The Reward manipulation was introduced in 150 subsequent trials of the game with subjects playing for 8¢ or 1¢ per point. The results show that the absolute magnitude of reward did not influence the extent of cooperative or competitive behavior. On the other hand, the magnitude of reward relative to CL was significantly associated with the degree of cooperative behavior—supra-CL outcomes yielding a higher frequency of cooperative behavior than infra-CL outcomes.

Efficient allocation of public resources in manpower training

This paper utilizes the linear programming technique for developing a general model to solve choice-involving questions in the area of manpower training policy. The model can be used to bring about a more efficient allocation of budgetary and non-financial resources among given training activities. It was applied in an empirical evaluation of on-the-job and institutional training activities in the Province of Ontario, Canada. Some of the findings are reported in the last section of the paper. Theoretically, the objective function is set up to maximize the contribution of government-sponsored training to social real income subject to constraints. Empirically, this contribution is measured as the present value of the cumulated stream of additional earnings of placed graduate-trainees; externalities are ignored due to measurement difficulties. Three resource constraints are used: (1) trainee allowances paid during the course of training, (2) cost of instruction, and (3) government administration expense, which reflect actual budgetary conditions of the training activities evaluated. Also there is an upper boundary for each training activity showing the maximum volume of demand for training by employers and trainees. The empirical evaluation refers to the fiscal year 1966/67. A total of six linear programming models were constructed for purposes of evaluation. The main empirical results were as follows : 1. There was considerable sub-optimality in the conduct of on-the-job activities, chiefly for two reasons; (i) much training was carried in relatively low-yielding industries, (ii) excess administration capacity existed in the handling of these training activities. 2. Training of the under-employed was virtually as efficient as that of the unemployed. 3. Institutional training had comparative advantage in academic upgrading courses. The operation of institutional training activities was sub-optimal, but less so than on-the-job activities, because of (i) the provision of a large volume of commercial training for which the pay-off per unit of cost was relatively low, and (ii) low demand for skill development courses in institutions in spite of fairly high pay-off rates. 4. While some economies of integration of institutional and on-the-job activities could be realized, it would be virtually as efficient to maintain the existing departmental division of responsibilities provided that each were run optimally. In that event, the most efficient choice would be to devote institutional training exclusively to academic upgrading courses and on-the-job training to occupational skill development.