arXiv subjects

Berenice Anne Neumann

Publications and source records attributed to Berenice Anne Neumann.

9 recordsLinked to original sources

arXiv2025

Markovian randomized equilibria for general Markovian Dynkin games in discrete time

We study a general formulation of the classical two-player Dynkin game in a discrete time Markovian setting. We identify an appropriate class of mixed strategies -- \textit{Markovian randomized stopping times} -- in which players stop at any given state with a state-dependent probability. One main result is an explicit characterization of Wald-Bellman-type for Nash equilibria based on this notion of randomization. In particular, we derive a novel characterization of randomized equilibria in zero-sum Dynkin games, which we use to (i) establish the existence and explicit construction of Markovian randomized equilibria, (ii) provide necessary and sufficient conditions for the non-existence of pure strategy equilibria, and (iii) construct an example that admits a unique randomized equilibrium but no pure one. We also provide existence and characterization results in the symmetric version of our game. Finally, we establish existence of a characterizable equilibrium in Markovian randomized stopping times for the general game formulation under the assumption that the state space is countable.

math.PR↗

arXiv2025

An Irreversible Investment Problem with Incomplete Information about Profitability

We analyze an irreversible investment decision for a project which yields a flow of future operating profits given by a geometric Brownian motion with unknown drift. In contrast to similar optimal stopping problems with incomplete information, the agent's payoff now depends directly on the unknown drift and not only indirectly through the underlying dynamics. Hence, many standard arguments are not applicable. Nonetheless, we show that it is optimal to invest in the project if the current profit level exceeds a threshold depending on the current belief for the true state of the unknown drift. These thresholds are described by a boundary function, for which we establish structural properties like monotonicity and continuity. To prove these, we identify a central class of stopping times with useful features. Moreover, we characterize the boundary function as the unique solution of a nonlinear integral equation. Building on this characterization we compute the boundary function numerically and investigate the value of information.

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arXiv2024

Finite State Mean Field Games with Common Shocks

We present a novel framework for mean field games with finite state space and common noise, where the common noise is given through shocks that occur at random times. We first analyze the game for up to $n$ shocks, in which case we are able to characterize mean field equilibria through a system of parameterized and coupled forward-backward equations. We establish existence and uniqueness of solutions to this system for small time horizons. In addition, we show that mean field equilibria for the $n$-shock setting constitute approximate equilibria for the corresponding mean field game with infinitely many common shocks. Our results are illustrated in a corruption detection model with random audits.

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arXiv2023

Characterization of transport optimizers via graphs and applications to Stackelberg-Cournot-Nash equilibria

We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary and sufficient condition for uniqueness of the dual optimizers. Moreover, we provide an algorithm that can efficiently compute the dual optimizer that is the limit, as the regularization parameter goes to zero, of the dual entropic optimizers. Our results find an application in a Stackelberg-Cournot-Nash game, for which we obtain existence and characterization of the equilibria.

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arXiv2021

Competition versus Cooperation: A class of solvable mean field impulse control problems

We discuss a class of explicitly solvable mean field type control problems/mean field games with a clear economic interpretation. More precisely, we consider long term average impulse control problems with underlying general one-dimensional diffusion processes motivated by optimal harvesting problems in natural resource management. We extend the classical stochastic Faustmann models by allowing the prices to depend on the state of the market using a mean field structure. In a competitive market model, we prove that, under natural conditions, there exists an equilibrium strategy of threshold-type and furthermore characterize the threshold explicitly. If the agents cooperate with each other, we are faced with the mean field type control problem. Using a Lagrange-type argument, we prove that the optimizer of this non-standard impulse control problem is of threshold-type as well and characterize the optimal threshold. Furthermore, we compare the solutions and illustrate the findings in an example.

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arXiv2020

Essential Stationary Equilibria of Mean Field Games with Finite State and Action Space

Mean field games allow to describe tractable models of dynamic games with a continuum of players, explicit interaction and heterogeneous states. Thus, these models are of great interest for socio-economic applications. A particular class of these models are games with finite state and action space, for which recently in Neumann (2020a) a semi-explicit representation of all stationary equilibria has been obtained. In this paper we investigate whether these stationary equilibria are stable against model perturbations. We prove that the set of all games with only essential equilibria is residual and obtain two characterization results for essential stationary equilibria.

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arXiv2020

A Myopic Adjustment Process for Mean Field Games with Finite State and Action Space

In this paper, we introduce a natural learning rule for mean field games with finite state and action space, the so-called myopic adjustment process. The main motivation for these considerations are the complex computations necessary to determine dynamic mean-field equilibria, which make it seem questionable whether agents are indeed able to play these equilibria. We prove that the myopic adjustment process converges locally towards stationary equilibria with deterministic equilibrium strategies under rather broad conditions. Moreover, for a two-strategy setting, we also obtain a global convergence result under stronger, yet intuitive conditions.

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arXiv2020

Nonlinear Markov Chains with Finite State Space: Invariant Distributions and Long-Term Behaviour

Nonlinear Markov chains with finite state space have been introduced in Kolokoltsov (2010). The characteristic property of these processes is that the transition probabilities do not only depend on the state, but also on the distribution of the process. In this paper we provide first results regarding their invariant distributions and long-term behaviour. We will show that under a continuity assumption an invariant distribution exists. Moreover, we provide a sufficient criterion for the uniqueness of the invariant distribution that relies on the Brouwer degree. Thereafter, we will present examples of peculiar limit behaviour that cannot occur for classical linear Markov chains. Finally, we present for the case of small state spaces sufficient (and easy-to-verify) criteria for the ergodicity of the process.

math.PR↗

arXiv2019

Stationary Equilibria of Mean Field Games with Finite State and Action Space

Mean field games formalize dynamic games with a continuum of players and explicit interaction where the players can have heterogeneous states. As they additionally yield approximate equilibria of corresponding $N$-player games, they are of great interest for socio-economic applications. However, most techniques used for mean field games rely on assumptions that imply that for each population distribution there is a unique optimizer of the Hamiltonian. For finite action spaces, this will only hold for trivial models. Thus, the techniques used so far are not applicable. We propose a model with finite state and action space, where the dynamics are given by a time-inhomogeneous Markov chain that might depend on the current population distribution. We show existence of stationary mean field equilibria in mixed strategies under mild assumptions and propose techniques to compute all these equilibria. More precisely, our results allow -- given that the generators are irreducible -- to characterize the set of stationary mean field equilibria as the set of all fixed points of a map completely characterized by the transition rates and rewards for deterministic strategies. Additionally, we propose several partial results for the case of non-irreducible generators and we demonstrate the presented techniques on two examples.

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