Markov Perfect Equilibria in Multi-Mode Differential Games with Endogenous Timing of Mode Transitions

We study Markov perfect equilibria (MPE) of two-player multi-mode differential games with controlled state dynamics, where one player controls the transition between modes. Different types of MPE are characterized distinguishing between delay equilibria, inducing for some initial conditions mode switches after a positive finite delay, and now or never equilibria, under which, depending on the initial condition, a mode switch occurs immediately or never. These results are applied to analyze the MPE of a game capturing the dynamic interaction between two incumbent firms among which one has to decide when to extend its product range by introducing a new product. The market appeal of the new product can be influenced over time by both firms through costly investments. Under a wide range of market introduction costs a now or never equilibrium coexists with a continuum of delay equilibria, each of them inducing a different time of product introduction.

Generalisations of Quasi-Hyperbolic Discounting

<p>This paper proposes several time preference specifications that generalise quasi-hyperbolic discounting, while retaining its analytical tractability. We define their discount functions and provide a recursive formulation of the implied lifetime payoffs. A calibration exercise demonstrates that these specifications deliver better approximations to true hyperbolic discounting. We characterise the Markov-perfect equilibrium of a general intra-personal game of agents with various time preferences. When applied to specific economic examples, our proposals yield policies that are close to those of true hyperbolic discounters. Furthermore, these approximations can be used in settings where an exact solution for hyperbolic agents is not available. Finally, we suggest further generalisations which would provide an even better fit.</p>

Generalisations of Quasi-Hyperbolic Discounting

<p>This paper proposes several time preference specifications that generalise quasi-hyperbolic discounting, while retaining its analytical tractability. We define their discount functions and provide a recursive formulation of the implied lifetime payoffs. A calibration exercise demonstrates that these specifications deliver better approximations to true hyperbolic discounting. We characterise the Markov-perfect equilibrium of a general intra-personal game of agents with various time preferences. When applied to specific economic examples, our proposals yield policies that are close to those of true hyperbolic discounters. Furthermore, these approximations can be used in settings where an exact solution for hyperbolic agents is not available. Finally, we suggest further generalisations which would provide an even better fit.</p>

On a special case of non-symmetric resource extraction games with unbounded payoffs

The game of resource extraction/capital accumulation is a stochastic infinite-horizon game, which models a joint utilization of a productive asset over time. The paper complements the available results on pure Markov perfect equilibrium existence in the non-symmetric game setting with an arbitrary number of agents. Moreover, we allow that the players have unbounded utilities and relax the assumption that the stochastic kernels of the transition probability must depend only on the amount of resource before consumption. This class of the game has not been examined beforehand. However, we could prove the Markov perfect equilibrium existence only in the specific case of interest. Namely, when the players have constant relative risk aversion (CRRA) power utilities and the transition law follows a geometric random walk in relation to the joint investment. The setup with the chosen characteristics is motivated by economic considerations, which makes it relevant to a certain range of real-word problems.

The commons problem in the presence of negative externalities

PurposeWhile the commons problem and the issues related to the negative externalities of harvesting have been studied extensively, there remains a need to bridge these two streams of studies to comprehensively investigate the implications of the strategic interactions among resource harvesters in the presence of such negative externalities. This paper aims to fill this gap.Design/methodology/approachThe authors study a common-pool harvest problem when the extractive activities leave behind negative externalities which affect the resource growth rate and reduce the stock beyond the extracted levels. Markov perfect noncooperative and optimal solutions are presented under different scenarios regarding considerations of negative externalities into harvest decisions.FindingsResults of the study suggest that, in the presence of such externalities, all parties must scale down their extraction in accordance with their externalities. The resource can be preserved by implementation of such harvest rule. However, failure to incorporate the externalities exacerbates the commons problem and can even lead to exhaustion of the biomass even if countries manage to cooperate and coordinate their harvest. Suggesting that if such externalities are large enough – which empirical literature suggests they are – then recognition and consideration of these externalities in the harvest decisions is as crucial as cooperation.Originality/valueThis paper provides a framework that is capable of incorporating the negative externalities of harvest activities into a bioeconomic game theoretic model and thereby providing a more real-world representation of the state of the common-pool resource management. While, the authors extend a well-known simple model, the model of this research study has the capacity to explain the widespread incidences of resource collapses. Therefore, the important policy implication is that agents should rigorously work together to understand the extent of the negative externalities of their harvests on the resources.

A Model of Multiproduct Firm Growth

This paper studies optimal growth strategies of a multiproduct firm that invests in the qualities of different products, which have persistent effects on future payoffs and are modeled as a state variable of a stochastic game. We derive a unique Markov perfect equilibrium under a monotonicity condition. At the early stage, the firm focuses on the product with higher quality, and may switch its specialization. If the quality of the specialized good is high enough, the firm diversifies to capture demands for all products. However, the firm may lose its focus on either product and get no demand, due to a moral hazard problem.

Markov decision processes with quasi-hyperbolic discounting

We study Markov decision processes with Borel state spaces under quasi-hyperbolic discounting. This type of discounting nicely models human behaviour, which is time-inconsistent in the long run. The decision maker has preferences changing in time. Therefore, the standard approach based on the Bellman optimality principle fails. Within a dynamic game-theoretic framework, we prove the existence of randomised stationary Markov perfect equilibria for a large class of Markov decision processes with transitions having a density function. We also show that randomisation can be restricted to two actions in every state of the process. Moreover, we prove that under some conditions, this equilibrium can be replaced by a deterministic one. For models with countable state spaces, we establish the existence of deterministic Markov perfect equilibria. Many examples are given to illustrate our results, including a portfolio selection model with quasi-hyperbolic discounting.