Equilibrium concepts for time‐inconsistent stopping problems in continuous time

A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.     

Optimal equilibria for time‐inconsistent stopping problems in continuous time

For an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.     

OVERLAPPING SETS OF PRIORS AND THE EXISTENCE OF EFFICIENT ALLOCATIONS AND EQUILIBRIA FOR RISK MEASURES

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short‐selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no‐arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow–Debreu equilibria.     

Learning equilibrium mean-variance strategy

We study a dynamic mean-variance portfolio optimization problem under the reinforcement learning framework, where an entropy regularizer is introduced to induce exploration. Due to the time–inconsistency involved in a mean-variance criterion, we aim to learn an equilibrium policy. Under an incomplete market setting, we obtain a semi-analytical, exploratory, equilibrium mean-variance policy that turns out to follow a Gaussian distribution. We then focus on a Gaussian mean return model and propose a reinforcement learning algorithm to find the equilibrium policy. Thanks to a thoroughly designed policy iteration procedure in our algorithm, we prove the convergence of our algorithm under mild conditions, despite that dynamic programming principle and the usual policy improvement theorem failing to hold for an equilibrium policy. Numerical experiments are given to demonstrate our algorithm. The design and implementation of our reinforcement learning algorithm apply to a general market setup.     

Arrow–Debreu equilibria for rank‐dependent utilities with heterogeneous probability weighting

We study Arrow–Debreu equilibria for a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents having heterogeneous probability weighting and outcome utility functions. In particular, we allow the economy to have a mix of expected utility agents and rank‐dependent utility ones, with nonconvex probability weighting functions. The standard approach for convex economy equilibria fails due to the incompatibility with second‐order stochastic dominance. The representative agent approach devised in Xia and Zhou (2016) does not work either due to the heterogeneity of the weighting functions. We overcome these difficulties by considering the comonotone allocations, on which the rank‐dependent utilities become concave. Accordingly, we introduce the notion of comonotone Pareto optima, and derive their characterizing conditions. With the aid of the auxiliary problem of price equilibria with transfers, we provide a sufficient condition in terms of the model primitives under which an Arrow–Debreu equilibrium exists, along with the explicit expression of the state‐price density in equilibrium. This new, general sufficient condition distinguishes the paper from previous related studies with homogeneous and/or convex probability weightings.     

Mean‐field games with differing beliefs for algorithmic trading

Even when confronted with the same data, agents often disagree on a model of the real world. Here, we address the question of how interacting heterogeneous agents, who disagree on what model the real world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for the permanent impact they have on prices. This leads to a large stochastic game, where each agents performance criteria are computed under a different probability measure. We analyze the mean‐field game (MFG) limit of the stochastic game and show that the Nash equilibrium is given by the solution to a nonstandard vector‐valued forward–backward stochastic differential equation. Under some mild assumptions, we construct the solution in terms of expectations of the filtered states. Furthermore, we prove that the MFG strategy forms an ε‐Nash equilibrium for the finite player game. Finally, we present a least square Monte Carlo based algorithm for computing the equilibria and show through simulations that increasing disagreement may increase price volatility and trading activity.     

On the Existence of Linear Equilibria in Models of Market Making

We derive necessary and sufficient conditions for a linear equilibrium in three types of competitive market making models: Kyle type models (when market makers only observe aggregate net order flow), Glosten–Milgrom and Easley–O'Hara type models (when market makers observe and trade one order at a time), and call markets models (individual order models when market makers observe a number of orders before pricing and executing any of them). We study two cases: when privately informed (strategic) traders are symmetrically informed and when they have differential information. We derive necessary and sufficient conditions on the distributions of the random variables for a linear equilibrium. We also explore those features of the equilibrium that depend on linearity as opposed to the particular distributional assumptions and we provide a large number of examples of linear equilibria for each of the models.     

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.     

A STATE‐CONSTRAINED DIFFERENTIAL GAME ARISING IN OPTIMAL PORTFOLIO LIQUIDATION

We consider n risk‐averse agents who compete for liquidity in an Almgren–Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed‐form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.     

The Impact of Ethical Leadership, the Internal Audit Function, and Moral Intensity on a Financial Reporting Decision

Two elements of corporate governance??the strength of ethical executive leadership and the internal audit function (IAF hereafter)??provide guidance to accounting managers making decisions involving uncertainty. We examine the joint effect of these two factors, manipulated at two levels (strong, weak), in an experiment in which accounting professionals decide whether to book a questionable journal entry (i.e., a journal entry for which a reasonable business case can be made but there is no supporting documentation). We find that ethical leadership and the IAF interact to determine the likelihood that accountants book the entry. Specifically, accountants are less likely to book a questionable journal entry when there is a weak ethical leader and a strong IAF compared to all other conditions. In addition, we find that accountants question the appropriateness and ethicalness of the request to book an undocumented journal entry more in the weak ethical leader and strong IAF condition than in the other conditions. These results suggest that the IAF has a different impact on financial reporting decisions depending on the ethicalness of executive leadership and that a strong IAF may cause accountants to question the appropriateness and ethicalness of an undocumented journal entry when combined with weak ethical leadership. We also find that the interactive effect of ethical leadership and the IAF on an accountant??s decision is fully mediated by his/her perception of the moral intensity of the issue. Thus, accountants, who perceive greater moral intensity associated with booking the entry, are less willing to do so.     

Duopolistic Price Competition with Captives

We extend the Bertrand duopolistic competition to include captives. These are consumers that have no choice between the suppliers. Usual population of shoppers are modeled performing a sequential search in order to decide where to buy a homogenous good. These two simple departures from the original setup have sharp consequences. First, we find that duopolistic price competition is not robust to inclusion of captives. The equilibrium results starkly differ and the only possible equilibrium now includes duopolists charging monopolistic prices. Second, addition of sequential search introduces multiplicity of pure strategy Nash equilibria. In this setup, we observe perverse optimal response to competitor's price changes. Notably, we find that the firm might want to reduce the price in response to the competitor's price increase, which is at odds with the usual undercutting principle. Third, we investigate the behavior of equilibrium prices depending on the heterogeneity in consumer risk attitudes. We find that the higher consumer heterogeneity with respect to acceptance of risky gambles leads to higher prices in equilibrium.     

Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\widehat{u},C)$ of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.     

Optimal Timing of Advertising with Demand Spillovers

We construct a model of a horizontally differentiated duopoly with demand spillovers in which advertising influences the willingness-to-pay of consumers for products and thereby affects not only market share, but also the level of market demand. Furthermore, firms decide the timing as well as the level of advertising. We first derive a subgame perfect Nash equilibrium and Stackelberg equilibria in the advertising competition. Then, using the framework of an endogenous timing decision game with an observable delay (i.e., Hamilton and Slutsky, Games Econ Behavior 2: 29–46, 1990), we consider the optimal timing of advertising. We demonstrate that the optimal timing depends on the degree of demand spillovers and the product substitutability. In particular, if there are sufficient asymmetric demand spillovers between firms, there is a unique Stackelberg equilibrium in the advertising competition, in which the firm providing the product with small (large) demand spillovers chooses to invest in advertising early (late), regardless of the mode of competition.     

Predictable forward performance processes: Infrequent evaluation and applications to human-machine interactions

We study discrete-time predictable forward processes when trading times do not coincide with performance evaluation times in a binomial tree model for the financial market. The key step in the construction of these processes is to solve a linear functional equation of higher order associated with the inverse problem driving the evolution of the predictable forward process. We provide sufficient conditions for the existence and uniqueness and an explicit construction of the predictable forward process under these conditions. Furthermore, we find that these processes are inherently myopic in the sense that optimal strategies do not make use of future model parameters even if these are known. Finally, we argue that predictable forward preferences are a viable framework to model human-machine interactions occurring in automated trading or robo-advising. For both applications, we determine an optimal interaction schedule of a human agent interacting infrequently with a machine that is in charge of trading.     

Managing Channel Profits: The Role of Managerial Incentives

The existing literature has examined how manufacturers can enhance profits by employing specific channel structures and channel coordination mechanisms. In this paper, we examine the implications of strategically designed managerial incentives for channel performance in a duopoly. We first analyze how equilibrium outcomes (especially manufacturer profits) are altered when the manufacturers provide their channel managers with strategically designed incentives. Following that, we examine how optimal channel structure decisions are altered when manufacturers provide their managers with strategic incentives, i.e., we examine how strategic incentives moderate optimal channel structure decisions. In contrast with the existing literature, we find that an asymmetric channel structure with one manufacturer employing a profit-maximizing retailer and the other integrated manufacturer providing strategic incentives for the channel manager in charge of pricing, is an equilibrium outcome under certain conditions. We then compare how the implications of strategic incentives differ from those of channel structure decisions and channel coordination initiatives, and discuss when and why strategic incentives yield superior outcomes from the manufacturer’s perspective. Our results shed light on the sparsely researched role of managerial incentives in the channel context.     

COMONOTONIC MEASURES OF MULTIVARIATE RISKS

We propose a multivariate extension of a well‐known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invariance, subadditivity, and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we reformulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.     

Coalition Formation and Stability

This paper aims to develop, for any cooperative game, a solution notion that enjoys stability and consists of a coalition structure and an associated payoff vector derived from the Shapley value. To this end, two concepts are combined: those of strong Nash equilibrium and Aumann–Drèze coalitional value. In particular, we are interested in conditions ensuring that the grand coalition is the best preference for all players. Monotonicity, convexity, cohesiveness and other conditions are used to provide several theoretical results that we apply to numerical examples including real-world economic situations.     

Speculative attacks: unique equilibrium and transparency

Models with multiple equilibria are a popular way to explain currency attacks. Morris and Shin (1998) have shown that, in the context of those models, unique equilibria may prevail once noisy private information is introduced. In this paper, we apply the method of Morris and Shin to a broader class of probability distributions and show—using the technique of iterated elimination of dominated strategies—that their results continue to hold, even if we allow for sunspots and individual uncertainty about strategic behavior of other agents. We provide a clear exposition of the logic of this model and we analyze the impact of transparency on the probability of a speculative attack. For the case of uniform distribution of noisy signals, we show that increased transparency of government policy reduces the likelihood of attacks.     

Strategic Compatibility Choice,Network Alliance,and Welfare

Based on a simple model of compatibility choice under differentiated Cournot duopoly with network externalities, we consider how the levels of a network externality and product substitutability affect the choice of compatibility. In particular, if the level of network externality is larger than that of product substitutability, there are multiple equilibria involving imperfect and perfect compatibility. Furthermore, we demonstrate the conditions for constructing such a network alliance so that firms provide perfectly compatible products. The network alliance is stable and socially optimal.