Monotone Comparative Statics Research Articles

Precautionary risk-reduction and saving decisions: Two sides of the same coin?

We provide new results about the comparative static effects of income risk and interest rate risk on optimal risk-reduction and saving decisions. We combine arguments from the risk apportionment literature with monotone comparative statics. Risk reduction and saving are Edgeworth-Pareto substitutes for (mixed) risk averters and Edgeworth-Pareto complements for (mixed) risk lovers. For changes in income risk, risk reduction and saving are Nth-degree risk complements for risk lovers. For changes in interest rate risk, risk reduction and saving are Nth-degree risk substitutes for risk averters. The individual's risk attitude and the source of risk thus co-determine the effects of risk changes on optimal. We also discuss several extensions including multiple loss states, higher-order risk reduction, stochastic dominance, non-separable utility, and inflation risk.

Submodularity and supermodularity in contest games

AbstractThis paper presents various examples of two‐player submodular or supermodular contest games. Emphasizing the three main elements of a contest model, our examples revolve around situations where (i) contest success function allows for a draw, (ii) winning prize is not exogenously given but rather jointly produced, or (iii) individual effort cost also depends on the rival's effort. We then illustrate how submodularity and supermodularity can be used to study the existence of equilibrium, the order structure of the equilibrium set, and monotone comparative statics in such contest game examples.

Interval-valued quasisupermodular function and monotone comparative statics

In this study, we consider monotone comparative statics under interval uncertainty. We introduce interval-valued supermodular functions and interval-valued quasisupermodular functions with respect to a partial order relation on intervals. In addition, we obtain some sufficient conditions for monotone comparative statics under interval uncertainty. We also apply our results to discuss the monotone comparative statics of interval game with strategic complements.

Improvement of Reinforcement Learning With Supermodularity.

Reinforcement learning (RL) is a promising approach to tackling learning and decision-making problems in a dynamic environment. Most studies on RL focus on the improvement of state evaluation or action evaluation. In this article, we investigate how to reduce action space by using supermodularity. We consider the decision tasks in the multistage decision process as a collection of parameterized optimization problems, where state parameters dynamically vary along with the time or stage. The optimal solutions of these parameterized optimization problems correspond to the optimal actions in RL. For a given Markov decision process (MDP) with supermodularity, the monotonicity of the optimal action set and the optimal selection with respect to state parameters can be obtained by using the monotone comparative statics. Accordingly, we propose a monotonicity cut to remove unpromising actions from the action space. Taking bin packing problem (BPP) as an example, we show how the supermodularity and monotonicity cut work in RL. Finally, we evaluate the monotonicity cut on the benchmark datasets reported in the literature and compare the proposed RL with some popular baseline algorithms. The results show that the monotonicity cut can effectively improve the performance of RL.

Strategic mistakes

To study the equilibrium implications of decision frictions, we introduce a new class of control costs in continuum-player, continuum-action games in which agents interact via an aggregate of the actions of others. The costs that we study accommodate a rich class of decision frictions, including ex post misoptimization, imperfect ex ante planning, cognitive constraints that depend endogenously on the behavior of others, and consideration sets. We provide primitive conditions such that equilibria exist, are unique, are efficient, and feature monotone comparative statics for action distributions, aggregates, and the size of agents' mistakes. We apply the model to make robust equilibrium predictions in a monetary business-cycle model of price-setting with planning frictions and a model of consumption and savings during a liquidity trap when endogenous stress worsens decisions.

Monotone Comparative Statics and Bounds on Strategic Influence

Monotone Comparative Statics and Bounds on Strategic Influence

Directional Monotone Comparative Statics in Function Spaces

Directional Monotone Comparative Statics in Function Spaces

Streaming Stimulates the Live Concert Industry: Evidence from YouTube

The removal of Warner Music content from YouTube in the first three quarters of 2009 constitutes a plausible natural experiment to investigate the impact of streaming on live concert sales. This Warner-YouTube blackout had statistically and economically negative effects on Warner artists relative to non-Warner artists. Specifically, relative revenues and prices were lower and relative attendance was not higher. These effects were stronger among artists who recently had a song in the Billboard Hot 100 and among those who were more frequently searched on YouTube just prior to the blackout. These findings suggest that the diffusion of streaming has stimulated the demand for live concerts. The evidence is also consistent with a differentiated Bertrand model of ticket pricing in which prices are strategic complements and prices and streaming penetration gives rise to increasing differences in the artist profit function. This suggests that concerts and streaming are complements in demand for a given artist, and that concerts by different artists are substitutes. More broadly, the paper is an example of how the results from the monotone comparative statics literature can be adapted for use with difference-in-differences estimation.

Monotone comparative statics in the Calvert–Wittman model

In this paper, we show that when policy-motivated parties can commit to a particular platform during a uni-dimensional electoral contest where valence issues do not arise there must be a positive association between the policies preferred by candidates and the policies adopted in expectation in the lowest and the highest equilibria of the electoral contest. We also show that this need not be so if the parties cannot commit to a particular policy. The implication is that evidence of a negative relationship between enacted and preferred policies is suggestive of parties that hold positions from which they would like to move from yet are unable to do so.

Comparative Statics Analysis of An Inventory Management Model with Dynamic Pricing, Market Environment Fluctuation, and Delayed Differentiation

We consider a general joint pricing and inventory management model with delayed differentiation, in which a firm serves a market with multiple products made from a generic one. The firm holds inventory for the generic product which is produced using multiple resources. Moreover, the market size, the attractiveness of each product, the firm's productivity, and the procurement cost of each resource all evolve over the planning horizon driven by an exogenous Markov process. Comparative statics analysis is essential for studying this model, offering insights on the impact of market environment fluctuation upon the firm's optimal pricing and inventory policy. Hence, we propose a new approach that combines the advantages of implicit function theorem (IFT) and monotone comparative statics (MCS) approaches to perform comparative statics analysis in our joint pricing and inventory management model under market environment fluctuation. The new approach applies to our model where the standard IFT and MCS methods are not easily amenable. Using our new comparative statics approach, we characterize the optimal pricing and production policy of the firm, and offer insights on how the firm should adaptively respond to market environment fluctuations.

Iterative Monotone Comparative Statics

Iterative Monotone Comparative Statics

New Stochastic Orders and Monotone Comparative Statics of Changes in Risk under Risk Aversion

New Stochastic Orders and Monotone Comparative Statics of Changes in Risk under Risk Aversion

Monotone persuasion

We explore Bayesian persuasion environments in which the state and action spaces are ordered, allowing for complementarity between actions and types. Building on the literature on monotone comparative statics, we identify conditions that guarantee that these are optimal among all (possibly non-monotone) signal structures. When the action space is binary, supermodularity of the sender's and receiver's preferences suffices for the optimal signal to have a monotone structure. With a continuum of actions, the conditions are more intriguing. We identify a novel single-crossing condition using a virtual utility representation of the sender's payoff. We also provide a technique to compute the optimal monotone signal structure, even when this monotonicity is due to an exogenous constraint. Applications are given to quadratic loss functions with biases, capacity constraints, and credit ratings.

Green human resource management – A personnel economics perspective

The Stern-Stiglitz Report on carbon pricing emphasizes that environmental policy must not only cope with market failures, but also with government and organizational failures. Focusing on the latter, this paper investigates how some practices central to human resource management – employee selection and training, performance appraisal and rewards, employee discretion and empowerment – can be jointly deployed to direct managerial attention adequately. Using a static multi-task principal-agent model and standard monotone comparative statics tools, I derive predictions/prescriptions on how these practices should adjust as the firm's stakes in sustainability increase. These prescriptions notably include some qualifications concerning the outsourcing of environmental services and employee training, and some implications for public policy.

M♮-Convexity and Its Applications in Operations

A new approach for structural analysis of operations models with substitutability structures. In many operations models with substitutability structures, one often ends up with parametric optimization models that maximize submodular objective functions, and it is desirable to derive structural properties including monotone comparative statics of the optimal solutions or preservation of submodularity under the optimization operations. Yet, this task is challenging because the classical and commonly used results in lattice programming, applicable to optimization models with supermodular objective function maximization, do not apply. Using a key concept in discrete convex analysis, M♮-convexity, Chen and Li establish conditions under which the optimal solutions are nonincreasing in the parameters and the preservation property holds for parametric maximization models with submodular objectives, together with the development of several new fundamental properties of M♮-convexity. Their approach is powerful as demonstrated by applications in a classical multiproduct stochastic inventory model and a portfolio contract model.

Protection in numbers? Self-protection as a local public good

In many contexts with endogenous physical risks – e.g., households, neighbourhood traffic calming, production quality control – risk reduction is a local public good. Risk-reduction incentives then depend on the protected population’s size. Focusing on a household’s physical risks modelled as an i.i.d. Bernoulli trials sequence with endogenous “success” probability, I give sufficient conditions for safety to increase with the number protected via both monotone comparative statics methodology and a “first-order” approach. I utilise a recursive decomposition of a covariance involving a monotonic function of a binomial variable and first-degree stochastic dominance (FSD). Because “protection” problems are generally non-concave, I give a detailed treatment of the second-order condition, again via FSD.

The evolutionary stability of optimism, pessimism, and complete ignorance

We seek an evolutionary explanation for why in some situations humans maintain either optimistic or pessimistic attitudes toward uncertainty and are ignorant to relevant aspects of their environment. Players in strategic games face Knightian uncertainty about opponents’ actions and maximize individually their Choquet expected utility with respect to neo-additive capacities (Chateauneuf et al. 2007) allowing for both an optimistic or pessimistic attitude toward uncertainty as well as ignorance to strategic dependencies. An optimist (resp. pessimist) overweighs good (resp. bad) outcomes. A complete ignorant never reacts to opponents’ changes of actions. We focus on sub- and supermodular aggregative games and provide monotone comparative statics w.r.t. optimism/pessimism. With qualifications, we show that in finite populations, optimistic (resp. pessimistic) complete ignorance is evolutionary stable and yields a strategic advantage in submodular (resp. supermodular) games with aggregate externalities. Moreover, this evolutionary stable preference leads to Walrasian behavior in these classes of games.

Informational Herding, Optimal Experimentation, and Contrarianism

Abstract In the standard herding model, privately informed individuals sequentially see prior actions and then act. An identical action herd eventually starts and public beliefs tend to “cascade sets” where social learning stops. What behaviour is socially efficient when actions ignore informational externalities? We characterize the outcome that maximizes the discounted sum of utilities. Our four key findings are: (1) cascade sets shrink but do not vanish, and herding should occur but less readily as greater weight is attached to posterity. (2) An optimal mechanism rewards individuals mimicked by their successor. (3) Cascades cannot start after period one under a signal log-concavity condition. (4) Given this condition, efficient behaviour is contrarian, leaning against the myopically more popular actions in every period. We make two technical contributions: as value functions with learning are not smooth, we use monotone comparative statics under uncertainty to deduce optimal dynamic behaviour. We also adapt dynamic pivot mechanisms to Bayesian learning.

Climate Change and Human Resource Management

The Stern-Stiglitz Report on carbon pricing emphasizes that environmental policy must not only cope with market failures, but also with government and organizational failures. Focusing on the latter, this paper investigates how some practices central to human resource management - employee selection and training, performance appraisal and rewards, employee discretion and empowerment - can be jointly deployed to direct managerial attention adequately. Using a static multi-task principal-agent model and standard monotone comparative statics tools, I derive predictions/prescriptions on how these practices should adjust as the firm's stakes in sustainability increase. These prescriptions notably include some qualifications concerning the outsourcing of environmental services and employee training, and some implications for public policy.

The Role of Demand Information for Price-Setting Newsvendors: A Monotone Comparative Statics Approach

In this paper, we examine the role of demand forecast information for price-setting newsvendors. In our model, the newsvendor faces stochastic demand that is price-sensitive, has the discretion of setting the retail price, and receives some informative signal that helps the quantity decision. Focusing on a specific family of signal structure which satisfies the scale-location property, we find that when the newsvendor's demand forecast gets improved, the optimal retail price can either increase, decrease, or stay the same, and we provide concrete sufficient conditions on primitives under which each sort of monotonicity prevails. The analysis utilizes the monotone comparative statics approach, which does not require the solution uniqueness and largely weakens the regularity conditions commonly adopted in the literature. We offer a comprehensive investigation of a full spectrum of information structures, whereas the extant literature focuses exclusively on the comparison between the full-information and null-information cases. We also provide several commonly adopted distribution functions as applications of this approach.