Schedule for: 24w5277 - New Trends and Challenges in Stochastic Differential Games

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Static potential games, pioneered by Monderer and Shapley (1996) are non-cooperative games in which there exists an auxiliary function called static potential function, so that any player’s change in utility function upon unilaterally deviating from her policy can be evaluated through the change in the value of this potential function. The introduction of the potential function is powerful as it simplifies the otherwise challenging task of searching for Nash equilibria in multi-agent non-cooperative games: maximizers of potential functions lead to the game’s Nash equilibria. In this talk, we propose an analogous and new framework called α-potential game for dynamic N-player games, with the potential function in the static setting replaced by an α-potential function. We will present the analytical criteria for any game to be an α-potential game, and identify several important classes of Markov α-potential games. We will provide detailed analysis for games with mean-field interactions, distributed games, and the crowd aversion games, in which α is shown to depend on the number of players, the admissible policies, and the cost structure. We will also show the changes of α from open-loop to closed-loop settings.

The recent work by Cvitani´c, Possama¨ı, and Touzi (2018) [1] presents a general approach for continuous-time principal-agent problems, through dynamic programming and so-called second order backward stochastic differential equations (2BSDEs). In this paper, we provide an alternative formulation of the principal-agent problem, which can be solved using more straightforward techniques, simply relying on the theory of BSDEs. This reformulation is strongly inspired by an important remark in [1], namely that if the principal observes the output process X in continuous-time, she can compute its quadratic variation pathwise. While in [1] this information is used in the contract, our reformulation consist in assuming that the principal could directly control this process, in a ‘first-best’ fashion. The resolution approach for this alternative problem actually follows the line of the so-called ‘Sannikov’s trick’ in the literature on continuous-time principal-agent problems, as originally introduced by Sannikov (2008) [2]. We then show that the solution to this ‘first-best’ formulation is identical to the solution of the original problem. More precisely, using the contract’s form introduced in [1] as penalisation contracts, we highlight that this ‘first-best’ scenario can be achieved even if the principal cannot directly control the quadratic variation. Nevertheless, we do not have to rely on the theory of 2BSDEs to prove that such contracts are optimal, as their optimality is ensured by showing that the ‘first-best’ scenario is achieved. We believe that this more straightforward approach to solve general continuous-time principal-agent problems with volatility control will facilitate the dissemination of these problems across many fields, and its extension to more intricate principal-agent problems, for example with many (or even a continuum of) agents, or to more general, non-necessarily continuous, output processes. References [1] J. Cvitani´c, D. Possama¨ı, and N. Touzi. Dynamic programming approach to principal–agent problems. Finance and Stochastics, 22(1):1–37, 2018. [2] Y. Sannikov. A continuous–time version of the principal–agent problem. The Review of Economic Studies, 75 (3):957–984, 2008.

Optimal stopping is the problem of finding the right time to take a particular action in a stochastic system, in order to maximize an expected reward. It has applications in areas such as finance, healthcare, and statistics. In this talk, we investigate how to learn to make optimal stopping decisions in an unknown stochastic system via a novel reinforcement learning framework. More specifically, in order to encourage exploration in the stochastic system, we randomize the stopping time through cumulative residual entropy, resulting in a singular control problem with special structures. We will discuss the regularity of the solution, the convergence of our proposed learning algorithm, and the algorithm performance on a real option example. This is based on joint work with Jodi Dianetti and Giorgio Ferrari (Bielefeld University).

We consider mean field Stackelberg games with a leader and a large number of followers. We adopt a dynamic programming approach to derive a pair of HJB equations (also called master equations). This approach has the advantage of obtaining time consistent strategies while existing approaches via the stochastic maximum principle face time inconsistency. For the linear-quadratic (LQ) case, our solution reduces to a pair of coupled Riccati differential equations. To analyze the performance of the infinite population-based decentralized strategies applied to finite populations, we adopt a notion of t-selves of each player and view the game as being played by a stream of players. Subsequently, we establish asymptotic equilibrium properties of the decentralize strategies. The talk is based on joint work with Xuwei Yang.

I will present two independent contributions made recently on some modeling aspects of MFG. The first one, in collaboration with M. Rakotomalala, is new way to model MFG in which an underlying graph describes the interaction between the players. By working with the notion of geometric graph, we are able to model situation in which this interaction structure can evolve in time, and even be the consequence of strategic decision of the players. The second one, in collaboration with C. Meynard, is systematic study of common noise in MFG with the common randomness affects players through an exterior factors (prices, environmental parameters,...). We analyzed the well-posedness of the master equation, with results in the case in which the dynamics of the common noise depend on the decisions of the players.

We consider time-inconsistent stopping problems for a continuous-time Markov chain under finite time horizon with non-exponential discounting. We provide an example indicating that strong equilibria may not exist in general. As a result, we propose a notion of equilibrium called almost strong equilibrium (ASE), which is a weak equilibrium and satisfies the condition of strong equilibria except at the boundary points of the associated stopping region. We provide an iteration procedure and show that this procedure leads to an ASE. Moreover, we prove that this ASE is the unique ASE among all regular stopping policies under finite horizon T < ∞. In contrast, we show that strong equilibria (and thus ASE) exist and may not be unique for the infinite horizon case T = ∞. Furthermore, we show that the limit of the finite-horizon ASE as T → ∞ is a weak equilibrium for the infinite-horizon problem, and may not be a strong equilibrium or ASE.

We introduce a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle; the reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus; the competition game between the two reinsurers is settled by a non-cooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semi-closed form. Additionally, we obtain interesting sensitivity results for the equilibrium strategies through both an analytical and numerical study.

Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.

The solutions to Stefan problem with Gibbs-Thomson law (i.e., with surface tension effect) are well known to exhibit singularities which, in particular, lead to jumps of the associated free boundary along the time variable. The correct times, directions and sizes of such jumps are only well understood under the assumption of radial symmetry, under which the free boundary is a sphere with varying radius. The characterization of such jumps in a general multidimensional setting has remained an open question until recently. In our ongoing work with M. Shkolnikov and Y. Guo, we have derived a separate (hyperbolic) partial differential equation — referred to as the cascade equation — whose solutions describe the jumps of the solutions to the Stefan problem without any symmetry assumptions. It turns out that a solution of the cascade equation corresponds to a maximal element of the set of all equilibria in a family of (first-order local) mean field games. In this talk, I will present and justify the cascade equation, will show its connection to the mean field games, and will prove the existence of a solution to the cascade equation. If time permits, I will also show how these results can be used to construct a solution to the Stefan problem itself.

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In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games. This is based on a joint work with Jianfeng Zhang.

Applying the theory of quantum filtering, the author initiated in [1] the theory of quantum dynamic games with strategies chosen by players in real time. As it turns out, quantum filtering allows one to reduce the study of quantum dynamic games to stochastic differential games with diffusive or jump-type noises on complex Riemannian manifolds (generally speaking infinite-dimensional). A study of the corresponding games with a large number of players (initiated in [2] and [3]) leads to quantum mean-field games that can be eventually reformulated as a specific class of controlled infinite-dimensional McKean-Vlasov diffusions on complex projective spaces. A variety of new problems and challenges arises from this development, including a study of the new class of nonlinear SDEs in Banach spaces with singular coefficients (law of large number limits of stochastic quantum master equations for mixed states). In the talk we shall demonstrate some basic features of this new theory providing also some solvable examples. [1] Vassili N. Kolokoltsov. Dynamic Quantum Games. Dynamic Games and Applications. v. 12 (2022), 552-573. [2] Vassili N. Kolokoltsov. Quantum mean field games. Annals Applied Probability 32:3 (2022), 2254 - 2288. [3] Vassili N. Kolokoltsov. Quantum Mean-Field Games with the Observations of Counting Type. Games (2021), 12, 7. https://doi.org/10.3390/g12010007

We consider both N-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the N-player game. The talk is based on joint work with Gunaxing Fu and Paul Hager.

In this talk, I will first present a new approach to solve mean-field games, based on the reformulation of the stochastic control problems in terms of linear programming problems. I will review the linear programming approach for MFG of control and optimal stopping, discuss the numerical algorithms and some applications to energy transition. In particular, I will focus on a specific model related to the dynamics of the electric market structure in the long term under possible uncertainty on the scenario, determined by the progressive replacement of conventional power generation with renewable energy sources.

We study the long-time regime of the prediction with expert advice problems in both full information and adversarial bandit feedback setting. In the adversarial bandit feedback setting, we show that the problem leads to second order parabolic PDEs in the Wasserstein space. We establish the wellposedness of weak solutions for these PDEs and establish the connection between these PDEs, mean-field control and filtering problems. Based on joint works with Ibrahim Ekren and Xin Zhang.

This work aims to study an extension of the celebrated Sannikov’s Principal-Agent problem to the multi-Agents case. In this framework, the contracts proposed by the Principal consist in a running payment, a retirement time and a final payment at retirement. After discussing how the Principal may derive optimal contracts in the N-Agent case, we explore the corresponding mean field model, with a continuous infinity of Agents. We then prove that the Principal’s problem can be reduced to a mixed control-and-stopping mean field problem, and we derive a semi-explicit solution of the first best contracting problem. This is a joint work with Thibaut Mastrolia and Nizar Touzi.

We use mean field games (MFGs) to investigate approximations of $N$-player games with heterogeneous symmetrically continuous closed-loop actions. Centered around the notion of regret, we conduct non-asymptotic analysis on the approximation capability of MFGs from the perspective of state-action distributions without requiring the uniqueness of equilibria. Under suitable assumptions, we first show that scenarios in the $N$-player games with large $N$ and small average regrets can be well approximated by approximate solutions of MFGs with relatively small regrets. We then show that $\delta$-mean field equilibria can be used to construct $\varepsilon$-equilibria in $N$-player games. If time permits, we will discuss the incorporation of risk aversion into MFGs. This is based on a joint work with Sebastian Jaimungal (UToronto).

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

We study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J., 1989] and the seminal work by Buckdahn and Li [SIAM J. Control Optim., 2008], the involved coefficients may be random, going beyond the Markovian framework and leading to the random upper and lower value functions. We first prove the dynamic programming principle for the game, and then under the standard Lipschitz continuity assumptions on the coefficients, the upper and lower value functions are shown to be the viscosity solutions of the upper and the lower fully nonlinear stochastic Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. A stability property of viscosity solutions is also proved. Under certain additional regularity assumptions on the diffusion coefficient, the uniqueness of the viscosity solution is addressed as well. This is a joint work with Jinniao Qiu (University of Calgary, Canada).

Our work proposes a new framework to study multi-agent interaction in Markov games: Markov α-potential games. Markov potential games are special cases of Markov α-potential games, so are two important and practically significant classes of games: Markov congestion games and perturbed Markov team games. In this paper, α-potential functions for both games are provided and the gap α is characterized with respect to game parameters. Two algorithms–the projected gradient-ascent algorithm and the sequential maximum improvement smoothed best response dynamics–are introduced for approximating the stationary Nash equilibrium in Markov α-potential games. The Nash-regret for each algorithm is shown to scale sub-linearly in time horizon. Our analysis and numerical experiments demonstrate that simple algorithms are capable of finding approximate equilibrium in Markov α-potential games.

In this talk, I will briefly discuss about a comprehensive viscosity solution theory on a class of second order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise.

In this talk, we study the Policy Improvement Algorithm for reinforcement learning for continuous-time entropy-regularized stochastic control problems. We prove the uniform convergence both for the iterative value functions and for their derivatives. More importantly, in the finite horizon case and in the infinite horizon case with a large discount factor, we obtain the exponential rate of convergence, which is new in the literature. Our arguments rely on a simple representation formula for the derivatives of a linear PDE and are much easier than those in the existing works for convergence. The talk is based on a joint work with Jin Ma and Jianfeng Zhang.

This talk is about the price-storage dynamics in natural gas markets. A novel stochastic path-dependent volatility model is proposed with path-dependence introduced in both price volatility and storage increments. Model calibrations are conducted for both the price and storage dynamics. Further, we discuss a pricing problem of a specific discrete swing option using the dynamic programming principle, and a deep learning-based method is proposed for numerical approximations. A numerical algorithm is provided, followed by a convergence analysis result for the deep-learning approach.

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Potential game is an emerging notion and framework for studying multi-agent games, especially with heterogeneous agents. Up to date, potential games have been extensively studied mostly from the algorithmic aspect in approximating and computing the Nash equilibrium without verifying if the game is a potential game, due to the lack of analytical structure. In this paper, we aim to build an analytical framework for dynamic potential games. We prove that a game is a potential game if and only if each agent's value function can be decomposed as a potential function and a residual term that solely dependent on other agents' actions. This decomposition enables us to identify and analyze a new and important class of potential games called the distributed game. Moreover, by an appropriate notion of functional derivatives, we prove that a game is a potential game if the value function has a symmetric Jacobian. Consequently, for a general class of continuous-time stochastic games, their potential functions can be further characterized from both the probabilistic and the PDE approaches. The consistency of these two characterisations are shown in a class of linear-quadratic games. The talk is based on joint work with Xin Guo: https://arxiv.org/abs/2310.02259

In this talk, we introduce a notion of viscosity solutions for fully nonlinear second order path-dependent partial differential equations in the spirit of [Zhou, Ann. Appl. Probab., 33 (2023), 5564-5612]. We prove the existence, comparison principle, consistency and stability for the viscosity solutions. Application to path-dependent stochastic differential games is given. This is based on a joint work with Shanjian Tang.

In this paper, we provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt "closed-loop strategies", i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the-second-order-backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with stochastic target constraints. Thereafter, adapting the methodology developed by Soner and Touzi or Bouchard, Élie, and Imbert, the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton–Jacobi–Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature. This is a joint work with Camilo Hernández, Nicolás Hernández Santibáñez, and Emma Hubert.

In this talk we will present a collection of recent results on mean field games and the corresponding master equations governed by non-separable Hamiltonians and potentially degenerate idiosyncratic noise. We will show how the notion of displacement monotonicity ensures global in time well-posedness theories and give particular stabilization effects when one considers long time behavior of the solutions. The talk will be based on several works in collaborations with M. Bansil, M. Cirant, W. Gangbo, C. Mou and J. Zhang.

We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space S. We show that the supremums over maps and plans coincide, provided that the law ν of the input random variable Y is atomless. An optimal map X exists if ν does not charge any c−c surface (the graph of a difference of convex functions) with strictly positive normal vectors in the sense of the S-space. The optimal map X is unique if ν does not charge c−c surfaces with nonnegative normal vectors in the S-space. As an application, we derive sharp conditions for the existence and uniqueness of equilibrium in a multi-asset version of the model with insider from Rochet and Vila [10]. In the linear-Gaussian case, we characterize Kyle's lambda, the sensitivity of price to trading volume, as the unique positive solution of a non-symmetric algebraic Riccati equation. Based on joint work with Dmitry Kramkov.

Adapted optimal transport has emerged to be a useful tool for quantifying distributional uncertainty and the sensitivity of stochastic optimization problems in contexts where the flow of information in time plays a crucial role. While a rich theory has been developed, the collection of explicit examples is still limited. In this talk we study the adapted (bicausal) optimal transport between real-valued Gaussian processes in discrete time. We characterize the optimal coupling, compare it with the classic (Bures-Wasserstein) case, and discuss some geometric implications. Joint work with Madhu Gunasingam (U of T).

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We provide an It\^o's formula for $C^1$-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the $C^1$-It\^o's formula in Gozzi and Russo (2006) to this context. As the first application, we study a class of McKean-Vlasov optimal control problems, and establish a verification theorem which only requires $C^1$-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result. We provide an example in which the required regularity is checked by using only probabilistic tools.

We prove convergence results of the simulation of the density solution to the McKean-Vlasov equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

We propose two methods to solve the master equation for finite-state mean field games (MFGs). Solving MFGs provides approximate Nash equilibria for stochastic, differential games with finite but large populations of agents. The master equation is a partial differential equation (PDE) whose solution characterizes MFG equilibria for any possible initial distribution. The first method we propose relies on backward induction while the second one directly tackles the PDE without discretizing time. For both approaches, we prove two types of results: there exist neural networks that make the loss functions of the algorithms arbitrarily small and, conversely, if the losses are small, then the neural networks are good approximations of the master equation solution. We conclude with numerical experiments on benchmark problems from the literature in dimension up to 15, and a comparison with solutions computed by a classical method for fixed initial distributions. This is joint work with Asaf Cohen and Ethan Zell (University of Michigan, Ann Arbor).

Motivated by the mean-field optimization model of the training of two-layer neural networks, we propose a novel method to approximate the invariant measures of a class of McKean-Vlasov diffusions. We introduce a proxy process that substitutes the mean-field interaction with self-interaction through a weighted occupation measure of the particle's past. If the McKean-Vlasov diffusion is the gradient flow of a convex mean-field potential functional, we show that the self-interacting process exponentially converges towards its unique invariant measure close to that of the McKean-Vlasov diffusion. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron.

This talk explores a novel consensus-based optimization approach for nonconvex-nonconcave min-max problems, utilizing a multi-particle, metaheuristic, derivative-free method capable of provably finding global solutions. Inspired by swarm intelligence, the approach features two groups of interacting particles. One group is tasked with minimization over one variable, while the other focuses on maximization over another. This paradigm permits for a passage to the mean-field limit, which makes the method amenable to theoretical analysis and it allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function.

Building upon the dynamic programming principle for set valued functions arising from many applications, in this paper we propose a new notion of set valued PDEs. The key component in the theory is a set valued Itô formula, characterizing the flows on the surface of the dynamic sets. In the contexts of multivariate control problems, we establish the wellposedness of the set valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, a moving scalarization for certain time inconsistent problems is constructed by using the classical solution of the set valued HJB equation. This is a joint work with Jianfeng Zhang.

In this talk we discuss mean field games in discrete time on general probability spaces. Using dynamic programming and a forward-backward algorithm, we will construct mean field equilibria of multi period models as concatenation of equilibria of one-step games. We will also present results on convergence of discrete time games to continuous time counterparts. The talk is based on a joint work with J. Dianetti, M. Nendel and S. Wang

We study the daily operation of hybrid energy resources that couple a renewable generator with a battery energy storage system (BESS). We propose a stochastic control formulation for optimal dispatch of BESS to maximize the reliability of the hybrid asset relative to a given day-ahead dispatch target or forecast. We develop a machine-learning algorithm based on Gaussian Process regression to efficiently find the dynamic feedback control map. Several numerical case studies based on realistic asset simulations will be discussed. Time permitting, I will address extensions to alternative objectives, such as peak shaving; accounting for battery aging; and strategic interactions among many agents.

This paper examines a continuous-time principal-agent model in which agent’s preference exhibits habit formation over consumption. As agent’s concern over the standard of living strengthens, his continuation utility is less sensitive to current wealth but more sensitive to the standard of living, leading to lower demand for risk-sharing compensation. The optimal contract has lower pay-for-performance but incentivizes agent’s higher effort. In the Leland (1994) capital structure model, agent’s habit formation preference combined with the optimal contract lowers firm’s leverage and mitigates the debt-overhang problem.

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)