Stochastic games and monotone comparative statics

In Chapter 1, I study decision problems under uncertainty involving the choice of a rule mapping states into actions. I show that for any rule, there exists an increasing rule inducing larger expected payoffs for all payoff functions that are supermodular in action and state. I present five applications. The main ones are 1. A planner implementing a subsidy based on household data, subject to a budget constraint, may improve any schedule failing to transfer more to households with data indicating larger returns, even without knowing their specific utility functions. 2. A monopolist price-discriminating in a market where wealthier buyers are less pricesensitive should charge them more, even if there is a positive demand externality. Shifting high prices onto wealthier buyers while preserving the price distribution is profitable. 3. If an insurer selling a single product to several buyers has access to a signal of the shocks affecting them, she should offer an insurance that pays more when signals indicating larger shocks are observed, even if the shocks and the buyers’ degrees of risk-aversion are heterogeneous. In Chapters 2 and 3, I analyse a new class of dynamic public-good games arising among innovators of a common technology who share improvements with one other. Innovators could be firms part of an R&D; alliance or an informal disclosure agreement. The game is novel as firms induce randomly-sized increments in the stock (innovations), and their payoffs are a general function of private effort (R&D;) and the current stock (the technology level). In Chapter 2, I characterise the social-welfare benchmark, and compute the unique symmetric Markov equilibrium of the game. I show that, in the welfare benchmark, agents decrease their effort and are better-off as the stock grows. In contrast, innovations may be detrimental in the symmetric equilibrium. Namely, if the opportunity cost of effort does not depend on the current stock, then any innovation is beneficial. However, if the cost of effort is sufficiently increasing in the stock, and there are sufficiently many agents, moderate innovations are detrimental early on. In Chapter 3, I characterise the payoff-maximising equilibrium of the two-player game with binary effort, when strategies may depend on time and the past trajectory of the stock. I show that the equilibrium is asymmetric and non-Markov. In particular, only one agent exerts effort after the stock exceeds a certain cutoff and, before this occurs, agents face an equal chance of being the ones to do so. Moreover, I extend the game by allowing agents to conceal the innovations they obtain for an arbitrary period of time, foregoing their benefits. I derive necessary and sufficient conditions for the symmetric equilibrium of the baseline model with full disclosure to carry over to the game with concealment. I conjecture that, whether or not this is the case, there exists an equilibrium in which innovations are disclosed if and only if they exceed a common cutoff. Moreover, this equilibrium is more efficient than the old one. In Chapter 4, my coauthor and me show that the set of all preferences over a totallyranked set of alternatives is a complete lattice when ordered according to single-crossing dominance, and characterise joins and meets of arbitrary sets of preferences. We also give necessary and sufficient conditions on the underlying pre-order of the alternatives for the existence of joins and meets, as well as for their uniqueness. We present five applications. In particular 1. We derive monotone comparative statics results showing that (a) when the set of preferences of a group agents increases, so does the set of alternatives preferred by all agents, and (b) when a set known to contain the true preferences of an agent increases, so does the smallest set guaranteed to contain her preferred alternative(s). 2. We characterise a general class of ‘maxmin’ preferences as precisely minimum upper bounds with respect to ‘more ambiguity-averse than’. 3. We characterise when aggregation of individual preferences (respecting a suitable Pareto criterion) is possible if, for some given pairs of alternatives, society may rank one above the other only with the consent of all individuals.