Continuous-time Games Research Articles

Non-Zero-Sum Stopping Games in Continuous Time

We consider a two-player non-zero-sum stopping game in which the payoff of each player is revealed when both players stop, instead of it being revealed after the first player's stopping time. Such problems appear in the context of economics and finance, e.g., when two company try to choose good times to enter or leave the market, or when two investors try to maximize their own utilities in the case that each player sells one American option to the other.

Repeated games in continuous time as extensive form games

Extensive form games modeling continuous-time decisions are plagued with problems of nonexistence and nonuniqueness of outcomes. As a “second-best”, the literature has imposed extraneous restrictions on the strategy sets, raising questions on the nature and interpretation of the resulting formal object, and on which restrictions are appropriate. We provide a “first-best” framework, formalizing continuous-time repeated games as extensive form games incorporating natural conditions from the onset. Every strategy profile induces a unique outcome, without any restrictions on the strategy sets. Further, the unrestricted strategy sets are equivalent to a specific class of strategies in a more naïvely specified continuous-time game.

Strongly Symmetric Equilibria in Bandit Games

This paper studies strongly symmetric equilibria (SSE) in continuous-time games of strategic experimentation with Poisson bandits. SSE payoffs can be studied via two functional equations similar to the HJB equation used for Markov equilibria. This is valuable for three reasons. First, these equations retain the tractability of Markov equilibrium, while allowing for punishments and rewards: the best and worst equilibrium payoff are explicitly solved for. Second, they capture behavior of the discrete-time game: as the period length goes to zero in the discretized game, the SSE payoff set converges to their solution. Third, they encompass a large payoff set: there is no perfect Bayesian equilibrium in the discrete-time game with frequent interactions with higher asymptotic efficiency.

Stochastic Games in Continuous Time: Persistent Actions in Long-Run Relationships, Second Version

This paper studies a class of continuous-time stochastic games in which the actions of a long-run player have a persistent effect on payoffs. For example, the quality of a firm's product depends on past as well as current effort, or the level of a policy instrument depends on a government's past choices. The long-run player faces a population of small players, and its actions are imperfectly observed. I establish the existence of Markov equilibria, characterize the Perfect Public Equilibria (PPE) pay-offset as the convex hull of the Markov Equilibria payoff set, and identify conditions for the uniqueness of a Markov equilibrium in the class of all PPE. The existence proof is constructive: it characterizes the explicit form of Markov equilibria payoffs and actions, for any discount rate. Action persistence creates a crucial new channel to generate intertemporal incentives in a setting where traditional channels fail, and can provide permanent non-trivial incentives in many settings. These results offer a novel framework for thinking about reputational dynamics of firms, governments, and other long-run agents.

Differential Neural Networks for Identification and Filtering in Nonlinear Dynamic Games

This paper deals with the problem of identifying and filtering a class of continuous-time nonlinear dynamic games (nonlinear differential games) subject to additive and undesired deterministic perturbations. Moreover, the mathematical model of this class is completely unknown with the exception of the control actions of each player, and even though the deterministic noises are known, their power (or their effect) is not. Therefore, two differential neural networks are designed in order to obtain a feedback (perfect state) information pattern for the mentioned class of games. In this way, the stability conditions for two state identification errors and for a filtering error are established, the upper bounds of these errors are obtained, and two new learning laws for each neural network are suggested. Finally, an illustrating example shows the applicability of this approach.

Dynamic Deception

We characterize the unique equilibrium of a competitive continuous time game between a resource-constrained informed player and a sequence of rivals who partially observe his action intensity. Our game adds noisy monitoring and impatient players to Aumann and Maschler (1966), and also subsumes insider trading models. The intensity bound induces a novel strategic bias and serial mean reversion by uninformed players. We compute the duration of the informed player's informational edge. The uninformed player's value of information is concave if the intensity bound is large enough. Costly obfuscation by the informed player optimally rises in the public deception. (JEL D82, D83, G14)

Tail probabilities for triangular arrays

Different discrete time triangular arrays representing a noisy signal of players' activities can lead to the same limiting diffusion process yet lead to different limit equilibria. Whether the limit equilibria are equilibria of the limiting continuous time game depends on the limit properties of test statistics for whether a player has deviated. We provide an estimate of the tail probabilities along these arrays that allows us to determine the asymptotic behavior of the best test and thus of the best equilibrium.

Is everything just a game? From the discrete to the continuous time modeling of corporate strategy games

Strategy games are a very popular part of today’s management training in education and business. Complex economic interrelationships can be illustrated and interdependencies of economic decisions may be better understood. Most of the business games work with a discrete time model, i.e. decisions are made once for an entire time period which is usually half of a fiscal year. This time modeling is an improper representation of reality and leads to many comprehension problems on the participants’ side, e.g. concerning the effectiveness of decision parameters. This paper presents a strategy game with continuous time modeling in which, in addition to the planning at the beginning of a period, a detailed planning can be done during the period and interventions in the decision process can be made on a daily basis. An illustration of a possible temporary course during a continuous-time game is given. Several scenarios are discussed that have unfavorable results in the usual periodic time modeling. They are then compared to the more realistic modeling in the continuous time strategy game. Maintaining the long-term planning process and the method of debriefing, i.e. the discussion of success parameters and reports at the end of each period, the participants’ comprehension for processes in a company can thus be considerably augmented.

Equilibrium in two-player non-zero-sum Dynkin games in continuous time†

We prove that every two-player non-zero-sum Dynkin game in continuous time admits an ϵ-equilibrium in randomized stopping times. We provide a condition that ensures the existence of an ϵ-equilibrium in non-randomized stopping times.

A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games

We consider the asymptotic value of two person zero-sum repeated games with general evaluations of the stream of stage payoffs. We show existence for incomplete information games, splitting games, and absorbing games. The technique of proof consists of embedding the discrete repeated game into a continuous time game and to use viscosity solution tools.

Game theoretical approaches concerning R&D

We consider two games concerning R&D. One is a deterministic continuous-time nonzero-sum game treating research and development separately. We obtain the Nash equilibrium point and give the explicit solutions of two special cases. Moreover, we consider an extension model. Another is a probabilistic sequential game in which both basic technology level and production technique level are random variables. We obtain its Nash equilibrium point iteratively and give the explicit solution of a special case.

Adaptive dynamic programming for online solution of a zero-sum differential game

This paper will present an approximate/adaptive dynamic programming (ADP) algorithm, that uses the idea of integral reinforcement learning (IRL), to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost. The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation (CT-GARE), which underlies the game problem. We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics. The feasibility of the ADP scheme is demonstrated in simulation for a power system control application. The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.

New optimality conditions for average-payoff continuous-time Markov games in Polish spaces

This paper concerns two-person zero-sum games for a class of average-payoff continuous-time Markov processes in Polish spaces. The underlying processes are determined by transition rates that are allowed to be unbounded, and the payoff function may have neither upper nor lower bounds. We use two optimality inequalities to replace the so-called optimality equation in the previous literature. Under more general conditions, these optimality inequalities yield the existence of the value of the game and of a pair of optimal stationary strategies. Under some additional conditions we further establish the optimality equation itself. Finally, we use several examples to illustrate our results, and also to show the difference between the conditions in this paper and those in the literature. In particular, one of these examples shows that our approach is more general than all of the existing ones because it allows nonergodic Markov processes.

Reputation in Continuous-Time Games

We study reputation dynamics in continuous-time games in which a large player (e.g., government) faces a population of small players (e.g., households) and the large player's actions are imperfectly observable. The major part of our analysis examines the case in which public signals about the large player's actions are distorted by a Brownian motion and the large player is either a normal type, who plays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. We find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type. By contrast, when it is common knowledge that the large player is the normal type, every public equilibrium of the continuous-time game is payoff-equivalent to one in which a static Nash equilibrium is played after every history. Finally, we examine variations of the model with Poisson signals and multiple behavioral types.

Sequential optimizing strategy in multi-dimensional bounded forecasting games

We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001) [10]. By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of a continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.

Identification of Dynamic Discrete Choice Models

Econometric models of dynamic discrete choice processes are applied to a wide variety of economic problems. Recent research on their empirical content has brought important new insights. It has clarified the conditions for their identification from choice and covariate panel data in the absence of dynamic selection on unobservables. It has provided important new identification results for discrete-time models with unobserved heterogeneity and unobserved states. Finally, it has enhanced the attractiveness of continuous-time models, by developing new insights on the identification of continuous-time optimal stopping models. Current developments in the literature promise to shed further light on the specification and identification of models with unobserved state variables, theory-based nonproportional hazard models, continuous-time optimal stopping models with time-varying covariates, and dynamic games in discrete and continuous time.

Multistep Bayesian Strategy in Coin-Tossing Games and Its Application to Asset Trading Games in Continuous Time

We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability of Shafer and Vovk [12]. We show that by a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of nature's moves from independent Bernoulli trials. We then apply our scheme to asset trading games in continuous time and derive the exponential growth rate of the investor's capital when the variation exponent of the asset price path deviates from two.

The Reactive Player for Continuous Time Games

The Reactive Player for Continuous Time Games

A new formulation of asset trading games in continuous time with essential forcing of variation exponent

We introduce a new formulation of asset trading games in continuous time in the framework of the game-theoretic probability established by Shafer and Vovk (Probability and Finance: It’s Only a Game! (2001) Wiley). In our formulation, the market moves continuously, but an investor trades in discrete times, which can depend on the past path of the market. We prove that an investor can essentially force that the asset price path behaves with the variation exponent exactly equal to two. Our proof is based on embedding high-frequency discrete-time games into the continuous-time game and the use of the Bayesian strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008) 1161–1180) for discrete-time coin-tossing games. We also show that the main growth part of the investor’s capital processes is clearly described by the information quantities, which are derived from the Kullback–Leibler information with respect to the empirical fluctuation of the asset price.

Repeated Games with Frequent Signals*

We study repeated games with frequent actions and frequent imperfect public signals, where the signals are the aggregate of many discrete events, such as sales or tasks. The high-frequency limit of the equilibrium set depends on both the probability law governing the discrete events and on how many events are aggregated into a single signal. When the underlying events have a binomial distribution, the limit equilibria correspond to the equilibria of the associated continuous-time game with diffusion signals, but other event processes that aggregate to a diffusion limit can have a different set of limit equilibria. Thus the continuous-time game need not be a good approximation of the high-frequency limit whenever the underlying events have three or more possible values.