On the two-armed bandit problem with continuous time parameter and discounted rewards

Explicit formulae are obtained for the value and a stationary optimal policy in some cases of the continuous-time two-armed bandit problem with expected discounted reward.     

基于双曲折现的跨期消费和投资组合的一种解析

通过在默顿(1969年,1971年)的经典模型中引入Harris和Laibson(2013年)的随机双曲偏好,研究得到了针对常绝对风险厌恶效用函数的最优消费和投资组合的解析解.与默顿的结果相比,发现消费与财富尽管仍有线性关系,但其比例再也不是一个常数.投资于风险资产的比例也非固定常数,但投资于风险资产的总价值保持不变.     

Exploring the trade-off between immediate gratification and delayed network externalities in the consumption of information goods

Our motivation for this paper originates from recent advances in consumption and payment technologies available for mobile commerce. Ubiquitous computing is enabling consumers worldwide to reach digital content and services whenever and wherever they request it. This capability further fuels “impatience” in consumption of such information goods. We model such consumer behavior using a hyperbolic discounting approach. A subset of these products, especially software, also inherit delayed network externalities as part of their consumption characteristics. This builds a tension between decision to consume now or to expedite consumption. We build a stylized model to assess the impact of immediate gratification on the profit maximizing behavior of a monopolist firm which produces an information good with network externalities. We find that serving “impatient” consumers is always profitable for a monopolist. For lower levels of network externalities, the monopolist can increase first period and decrease second period prices in equilibrium. As network externalities effect increases, prices converge to the traditional market (with exponential discounters) levels.     

Some comments on a theorem of Hardy and Littlewood

In this note, we reconstruct a proof of a classical result due to Hardy and Littlewood. While this result has played an important role in the modern theories of Markov decision processes and stochastic games, it is not that easy to find its proof in the literature in the format in which it has been applied. Furthermore, we supply either examples or complete citations for the other related cases which are not covered by the Hardy-Littlewood theorem.     

Strong consistency of maximum likelihood estimators for a discrete-time random field HJM-type interest rate model

We consider a discrete time Heath–Jarrow–Morton-type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. Strong consistency of maximum likelihood estimators is proved for stable and unstable no-arbitrage models containing a simple stochastic discounting factor. This research was supported by the Hungarian Scientific Research Fund under Grant No. OTKA–T048544/2005.     

Maximizing the length of a success run for many-armed bandits

One of a number of Bernoulli processes is selected at each of a number of stages. A success at stage i is worth α_i and the problem is to maximize the expected payoff before the first failure. Results of Berry and Viscusi (1981) are generalized. In particular, we show that there is always an optimal strategy that uses a single process exclusively and indefinitely whenever the arms are independent and the discount sequence (α₁, α₂,…) is superregular. There is not always a similar reduction in the number of strategies when the discount sequence is not superregular.     

Minimization of risks in defined benefit pension plan with time‐inconsistent preferences

In this paper, we investigate the defined benefit pension plan, where the object of the manager is to minimise the contribution rate risk and the solvency risk by considering a quadratic performance criterion. To incorporate some well‐documented behavioural features of human beings, we consider the situation where the discounting is non‐exponential. It leads to a time‐inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, we assume that the benefit outgo is constant, and the pension fund can be invested in a risk‐free asset and a risky asset whose return follows a geometric Brownian motion. We characterise the time‐consistent strategies and value function in terms of the solution of a system of integral equations. The existence and uniqueness of the solution is verified, and the approximation of the solution is obtained. Some numerical results of the equilibrium contribution rate and equilibrium investment policy are presented for three types of discount functions. Copyright © 2015 John Wiley & Sons, Ltd.     

Omnichannel operations with ship-from-store

While embracing omnichannel operations, firms are rolling out ship-from-store (SFS). This paper considers a newsvendor-like retailer operating both online and offline channels, and analyzes how SFS affects the seller’s profits. In our model, consumers strategically choose a channel to optimize their utility, which is investigated by applying the theory of hyperbolic discounting. The results show that, interestingly, the firm can be worse off implementing SFS, when the customer-base expansion thanks to SFS is not significant.     

An agreeable collusive equilibrium in differential games with asymmetric players

We study a class of collusive equilibria in differential games with asymmetric players discounting the future at different rates. For such equilibria, at each moment, weights of players can depend on the state of the system. To fix them, we propose using a bargaining procedure according to which players can bargain again at every future moment. By choosing as threat point the feedback noncooperative outcome, the corresponding solution, if it exists, is agreeable. An exhaustible resource game illustrates the results.     

Optimal equilibrium barrier strategies for time-inconsistent dividend problems in discrete time

This paper studies a time-inconsistent dividend problem in discrete time with nonexponential discounting. Motivated by the decreasing impatience in behaviour economics, a general discount function is used and assumed to be log sub-additive. Using a game-theoretic approach equilibrium barrier strategies are considered. It is shown that in the case of multiple equilibria, there exists an optimal one that pointwisely dominates all the other equilibria. Case studies are conducted where there is no equilibrium, multiple equilibria, and a unique equilibrium.