Quantitative convergence for displacement monotone mean field games with controlled volatility

We study the convergence problem for mean field games with common noise and controlled volatility. We adopt the strategy recently put forth by Lauri\`ere and the second author, using the maximum principle to recast the convergence problem as a question of ``forward-backward propagation of chaos", i.e (conditional) propagation of chaos for systems of particles evolving forward and backward in time. Our main results show that displacement monotonicity can be used to obtain this propagation of chaos, which leads to quantitative convergence results for open-loop Nash equilibria for a class of mean field games. Our results seem to be the first (quantitative or qualitative) which apply to games in which the common noise is controlled. The proofs are relatively simple, and rely on a well-known technique for proving well-posedness of FBSDEs which is combined with displacement monotonicity in a novel way. To demonstrate the flexibility of the approach, we also use the same arguments to obtain convergence results for a class of infinite horizon discounted mean field games

Numerical Analysis of Markov-Perfect Equilibria with Multiple Stable Steady States: A Duopoly Application with Innovative Firms

Dawid H, Keoula M, Kort PM. Numerical Analysis of Markov-Perfect Equilibria with Multiple Stable Steady States: A Duopoly Application with Innovative Firms. Dynamic Games and Applications. 2017;7(4):555-577

Present-Biased Lobbyists in Linear Quadratic Stochastic Differential Games

We investigate a linear quadratic stochastic zero-sum game where two players lobby a political representative to invest in a wind turbine farm. Players are time-inconsistent because they discount performance with a non-constant rate. Our objective is to identify a consistent planning equilibrium in which the players are aware of their inconsistency and cannot commit to a lobbying policy. We analyze the equilibrium behavior in both single player and two-player cases, and compare the behavior of the game under constant and non-constant discount rates. The equilibrium behavior is provided in closed-loop form, either analytically or via numerical approximation. Our numerical analysis of the equilibrium reveals that strategic behavior leads to more intense lobbying without resulting in overshooting

Tacit collusion in a dynamic duopoly with indivisible production and cumulative capacity constraints

This paper studies a dynamic, quantity setting duopoly game characterized as follows: Each firm produces an indivisible output over a potentially infinite horizon, facing the constraint that its cumulative production cannot exceed an initially given bound. The environment is otherwise stationary; the remaining productive capacities of the firms at any moment are common knowledge; the firms choose production plans contingent on these capacities which are mutual best responses in every contingency. The resulting Markov Perfect Equilibria are analyzed using a two-dimensional backward induction, and compared with the equilibria which emerge when precommitment to time paths of output is possible. It is shown that the ability to precommit can be disadvantageous; that collusion in Markov Equilibrium is facilitated by the symmetrical placement of the firms; and that having greater capacity confers basic strategic advantage on a firm by enabling it to credibly threaten future production. The model solves an open problem in the theory of exhaustible resource economics by imposing subgame perfection in a resource oligopoly with independent stocks. It also formalizes the intuition that, when indivisibilities are important, tacit coordination of plans so as to avoid destructive competition is facilitated by establishing a convention of "taking turns" - that is, a self-enforcing norm of mutual, alternate forbearance.Supported by the Bradley Foundation, the Olin Foundation and the Center for Energy Policy Research, MIT

Nonlinear and evolutionary phenomena in deterministic growing economies

We discuss the implications of nonlinearity in competitive models of optimal endogenous growth. Departing from a simple representative agent setup with convex risk premium and investment adjustment costs, we define an open economy dynamic optimization problem and show that the optimal control solution is given by an autonomous nonlinear vector field in <3 with multiple equilibria and no optimal stable solutions. We give a thorough analytical and numerical analysis of this system qualitative dynamics and show the existence of local singularities, such as fold (saddle-node), Hopf and Fold-Hopf bifurcations of equilibria. Finally, we discuss the policy implications of global nonlinear phenomena. We focus on dynamic scenarios arising in the vicinity of Fold-Hopf bifurcations and demonstrate the existence of global dynamic phenomena arising from the complex organization of the invariant manifolds of this system. We then consider this setup in a non-cooperative differential game environment, where asymmetric players choose open loop no feedback strategies and dynamics are coupled by an aggregate risk premium mechanism. When only convex risk premium is considered, we show that these games have a specific state-separability property, where players have optimal, but naive, beliefs about the evolution of the state of the game. We argue that the existence of optimal beliefs in this fashion, provides a unique framework to study the implications of the self-confirming equilibrium (SCE) hypothesis in a dynamic game setup. We propose to answer the following question. Are players able to concur on a SCE, where their expectations are self-fulfilling? To evaluate this hypothesis we consider a simple conjecture. If beliefs bound the state-space of the game asymptotically and strategies are Lipschitz continuous, then it is possible to describe SCE solutions and evaluate the qualitative properties of equilibrium. If strategies are not smooth, which is likely in environments where belief-based solutions require players to learn a SCE, then asymptotic dynamics can be evaluated numerically as a Hidden Markov Model (HMM). We discuss this topic for a class of games where players lack the relevant information to pursue their optimal strategies and have to base their decisions on subjective beliefs. We set up one of the games proposed as a multi-objective optimization problem under uncertainty and evaluate its asymptotic solution as a multi-criteria HMM.We show that under a simple linear learning regime there is convergence to a SCE and portray strong emergence phenomena as a result of persistent uncertainty