Non-exponential Discounting Research Articles

Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting

Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting

Lifecycle consumption and welfare with nonexponential discounting in continuous time

In a continuous-time lifecycle model with log utility and a general time-inconsistent discount function, we establish necessary and sufficient conditions under which commitment to the initial plan will increase the realized objective function for all future selves. We also establish necessary and sufficient conditions under which the log consumption profile over the lifecycle is locally concave. Empirically, the lifecycle profile of average household consumption is hump-shaped and thus concave at the peak, so this result is useful for identifying what discount functions are consistent with data. We express these conditions in terms of what we call the future weighting factor, which measures the deviation of the discount function from an (arbitrarily chosen) exponential discount function. Both the welfare and concavity conditions depend on how the marginal future weighting factor compares to a weighted average of marginal future weighting factors. If the marginal future weighting factor is sufficiently high at a given delay, i.e. if the discount function decays sufficiently more slowly than the chosen exponential at that delay, this implies that log consumption is concave at a point on the lifecycle profile and there will be a positive contribution to commitment utility relative to the realized utility for one of the selves.

Non-exponential Reward Discounting in Reinforcement Learning

Reinforcement learning methods typically discount future rewards using an exponential scheme to achieve theoretical convergence guarantees. Studies from neuroscience, psychology, and economics suggest that human and animal behavior is better captured by the hyperbolic discounting model. Hyperbolic discounting has recently been studied in deep reinforcement learning and has shown promising results. However, this area of research is seemingly understudied, with most extant and continuing research using the standard exponential discounting formulation. My dissertation examines the effects of non-exponential discounting functions (such as hyperbolic) on an agent's learning and aims to investigate their impact on multi-agent systems and generalization tasks. A key objective of this study is to link the discounting rate to an agent's approximation of the underlying hazard rate of its environment through survival analysis.

Stability of Equilibria in Time-Inconsistent Stopping Problems

We investigate the stability of equilibrium-induced optimal values with respect to reward functions and transition kernels for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of and equipped with total variation distance, we show that the optimal value is semicontinuous with respect to . We provide examples showing that continuity may fail in general, and the convergence for in total variation cannot be replaced by weak convergence. Next we show that, with the uniform convergence of and , the optimal value is continuous with respect to when we consider a relaxed limit over -equilibria. We also provide an example showing that for such continuity the uniform convergence of cannot be replaced by locally uniform convergence.

Renegotiation and dynamic inconsistency: Contracting with non-exponential discounting

This paper studies a continuous-time, finite-horizon contracting problem with renegotiation and dynamic inconsistency arising from non-exponential discounting. The problem is formulated as a dynamic game played among the agent, the principal and their respective future “selves”, each with their own discount function. We identify the principal optimal renegotiation-proof contract as a Markov perfect equilibrium (MPE) of the game, prove that such an MPE exists, and characterize the optimal contract via an extended Hamilton-Jacobi-Bellman system. We solve the optimal contract in closed-form when discounting is a function of the time-difference only and demonstrate the applicability of the results in several different settings.

Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

<p style='text-indent:20px;'>This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.</p>

A time-inconsistent Dynkin game: from intra-personal to inter-personal equilibria

This paper studies a nonzero-sum Dynkin game in discrete time under non-exponential discounting. For both players, there are two intertwined levels of game-theoretic reasoning. First, each player looks for an intra-personal equilibrium among her current and future selves, so as to resolve time inconsistency triggered by non-exponential discounting. Next, given the other player’s chosen stopping policy, each player selects a best response among her intra-personal equilibria. A resulting inter-personal equilibrium is then a Nash equilibrium between the two players, each of whom employs her best intra-personal equilibrium with respect to the other player’s stopping policy. Under appropriate conditions, we show that an inter-personal equilibrium exists, based on concrete iterative procedures along with Zorn’s lemma. To illustrate our theoretical results, we investigate a two-player real options valuation problem where two firms negotiate a deal of cooperation to initiate a project jointly. By deriving inter-personal equilibria explicitly, we find that coercive power in negotiation depends crucially on the impatience levels of the two firms.

Time-Consistent Investment and Consumption Strategies under a General Discount Function

In the present paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that gives rise to time-inconsistency of the decision-maker. We consider equilibrium policies within the class of open-loop controls that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

Controlled Markov chains with non-exponential discounting and distribution-dependent costs

This paper deals with a controlled Markov chain in continuous time with a non-exponential discounting and distribution-dependent cost functional. A definition of closed-loop equilibrium is given and its existence and uniqueness are established. Due to the time-inconsistency brought by the non-exponential discounting and distribution dependence, it is proved that the equilibrium is locally optimal in some appropriate sense. Moreover, it is shown that our problem is equivalent to a mean-field game for infinite-many symmetric players with a non-exponential discounting cost.

Optimal Equilibria for Multidimensional Time-Inconsistent Stopping Problems

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.

Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms “strong equilibria” and “weak equilibria” are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.

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.

Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes

<p style='text-indent:20px;'>In the dual risk model, we study the periodic dividend problem with a non-exponential discount function which results in a time-inconsistent control problem. Viewing it within the game theoretic framework, we extend the Hamilton-Jacobi-Bellman (HJB) system of equations from the fixed terminal to the time of ruin and derive the verification theorem, and we generalize the theory of classical optimal periodic dividend. Under two special non-exponential discount functions, we obtain the closed-form expressions of equilibrium strategy and the corresponding equilibrium value function in a compound Poisson dual model. Finally, some numerical examples are presented to illustrate the impact of some parameters.

Optimal equilibria for time‐inconsistent stopping problems in continuous time

For an infinite-horizon continuous-time optimal stopping problem under non-exponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log sub-additive and the state process is one-dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. While there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.

Observational Implications of Non-Exponential Discounting

Dans cette note, nous donnons un exemple élémentaire avec une fonction d’utilité concave où la règle d’épargne essentiellement unique et consistante est discontinue ; une telle règle d’épargne ne peut pas survenir avec une fonction d’escompte exponentielle.

Non-exponential discounting portfolio management with habit formation

<p style='text-indent:20px;'>This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky assets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash subgame perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person's consumption and habit level are initially lower than those of a naive person but are higher in later periods.

Optimal equilibria for time‐inconsistent stopping problems in continuous time

AbstractFor an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.

Contracting with Non-Exponential Discounting: Moral Hazard and Dynamic Inconsistency

This paper studies a continuous-time, nite-horizon contracting problem with renegotiation and dynamic inconsistency arising from non-exponential discounting. The problem is formulated as a dynamic game played among the agent, the principal and their respective future selves, each with their own discount function. We identify the principal optimal renegotiation-proof contract as a Markov Perfect Equilibrium (MPE) of the game, prove such a MPE exists, and characterize the optimal contract via an extended Hamilton-Jacobi-Bellman system. We solve the optimal contract in closed form when the discount functions of the selves are related by time di erence, a property that is satis ed by common forms of non-exponential discounting such as quasi-hyperbolic discounting and anticipatory utility. In particular, quasi-hyperbolic discounting leads to a U-shaped action path and anticipatory utility leads to a humshaped path, both are qualitatively di erent from the monotonic action path that would arise under exponential discounting.

The Optimal Equilibrium for Time-Inconsistent Stopping Problems---The Discrete-Time Case

We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibria. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.

Equilibrium strategies for time-inconsistent stochastic switching systems

An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation which is constructed via a sequence of multi-person differential games. A verification theorem is proved and, under proper conditions, the well-posedness of the equilibrium HJB equation is established as well.