Non-linear strategies in a linear quadratic differential game

We study non-linear Markov perfect equilibria in a two agent linear quadratic differential game. In contrast to the literature owing to Tsutsui and Mino (1990), we do not associate endogenous subsets of the state space with candidate solutions. Instead, we address the problem of unbounded-below value functions over infinite horizons by use of the `catching up optimality' criterion. We present sufficiency conditions for existence based on results in Dockner, Jorgenson, Long and Sorger (2000). Applying these to our model yields the familiar linear solution as well as a condition under which a continuum of non-linear solutions exist. As this condition is relaxed when agents are more patient, and allows more efficient steady states, it resembles a Folk Theorem for differential games. The model presented here is one of atmospheric pollution; the results apply to differential games more generally.differential game, non-linear strategies, catching up optimal, Folk Theorem

The road to extinction: commons with capital markets

Competitive agents extract in continuous time from a commons. Capital market access allows them to both save and borrow against their extraction stream. When the commons asset grows more quickly than the privately stored one, multiple equilibria are found for intermediate commons endowments. One of these has the extinction date and welfare decrease in the endowment, a resource curse. When the commons asset grows less quickly than the privately stored one, there is a unique extinction date for each endowment level. In the limit, as marginal extraction costs become constant, `jump extinctions' occur. In cases with multiple equilibria: welfare is increased for low initial stock levels when agents do not have access to capital markets, but decreased otherwise; and an extraction tax reduces welfare in the `cursed' equilibrium, increases it in the other finite extinction equilibrium and expands the set of commons stocks that are never extinguishedcommons, capital markets, extinction, resource curse, storage, multiple equilibrium, rational expectations equilibrium

The commons with capital markets

We explore commons problems when agents have access to capital markets. The commons has a high intrinsic rate of return but its fruits cannot be secured by individual agents. Resources transferred to the capital market earn lower returns, but are secure. In a two period model, we assess the consequences of market access for the commons' survival and welfare; we compare strategic and competitive equilibria. Market access generally speeds extinction, with negative welfare consequences. Against this, it allows intertemporal smoothing, a positive effect. In societies in which the former effect dominates, market liberalisation may be harmful. We reproduce the multiple equilibria found in other models of competitive agents; when agents are strategic, extinction dates are unique. Strategic agents generally earn their surplus by delaying the commons' extinction; in unusual cases, strategic agents behave as competitive ones even when their numbers are small.commons, capital markets, Washington Consensus, property rights

The Road to Extinction: Commons with Capital Markets

We study extinction in a commons problem in which agents have access to capital markets. When the commons grows more quickly than the interest rate, multiple equilibria are found for intermediate commons endowments. In one of these, welfare decreases as the resource becomes more abundant, a `re- source curse'. As marginal extraction costs become constant, market access instantly depletes the commons. Without markets - the classic environment - equilibria are unique; extinction dates and welfare increase with the endow- ment. When the endowment is either very abundant or very scarce, market access improves welfare. As marginal costs of extraction from the commons become constant, market access can reduce welfare if the subjective discount rate exceeds the interest rate.commons, capital markets, perfect foresight, extinction, resource curse, storage

The Road to Extinction: Commons with Capital Markets

We study extinction in a commons problem in which agents have access to capital markets. When the commons grows more quickly than the interest rate, multiple equilibria are found for intermediate commons endowments. In one of these, extinction is hastened and welfare decreases in the endowment, a resource curse. An extraction tax reduces welfare in this 'cursed' equilibrium, increases it in the other equilibrium in which the commons is eventually depleted, and expands the set of commons stocks that are never depleted. Capital market access harms societies with low commons endowments. In the limit, as marginal extraction costs become constant, `jump extinctions' occur. Finally, when the commons grows less quickly than the interest rate, there is a unique extinction date for each endowment level.commons, capital markets, extinction, resource curse, storage, multiple equilibria, rational expectations equilibrium

Formal representation and proof for cooperative games

In this contribution we present some work we have been doing in representing and proving theorems from the area of economics, and mainly we present work we will do in a project in which we will apply mechanised theorem proving tools to a class of economic problems for which very few general tools currently exist. For mechanised theorem proving, the research introduces the field to a new application domain with a large user base; more specifically, the researchers are collaborating with developers working on state-of-the-art theorem provers. For economics, the research will provide tools for handling a hard class of problems; more generally, as the first application of mechanised theorem proving to centrally involve economic theorists, it aims to properly introduce mechanised theorem proving techniques to the discipline.\u

The ForMaRE Project - Formal Mathematical Reasoning in Economics

The ForMaRE project applies formal mathematical reasoning to economics. We seek to increase confidence in economics' theoretical results, to aid in discovering new results, and to foster interest in formal methods, i.e. computer-aided reasoning, within economics. To formal methods, we seek to contribute user experience feedback from new audiences, as well as new challenge problems. In the first project year, we continued earlier game theory studies but then focused on auctions, where we are building a toolbox of formalisations, and have started to study matching and financial risk. In parallel to conducting research that connects economics and formal methods, we organise events and provide infrastructure to connect both communities, from fostering mutual awareness to targeted matchmaking. These efforts extend beyond economics, towards generally enabling domain experts to use mechanised reasoning.Comment: Conference on Intelligent Computer Mathematics, 8--12 July, Bath, UK. Published as number 7961 in Lecture Notes in Artificial Intelligence, Springe

An Introduction to Mechanized Reasoning

Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics journals. We introduce mechanized reasoning to economists in three ways. First, we introduce mechanized reasoning in general, describing both the techniques and their successful applications. Second, we explain how mechanized reasoning has been applied to economic problems, concentrating on the two domains that have attracted the most attention: social choice theory and auction theory. Finally, we present a detailed example of mechanized reasoning in practice by means of a proof of Vickrey's familiar theorem on second-price auctions

Budget Imbalance Criteria for Auctions: A Formalized Theorem

We present an original theorem in auction theory: it specifies general conditions under which the sum of the payments of all bidders is necessarily not identically zero, and more generally not constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such a sum is not constant. In particular, this theorem applies to the important case of a second-price Vickrey auction, where it reduces to a basic result of which a novel proof is given. To enhance the confidence in this new theorem, it has been formalized in Isabelle/HOL: the main results and definitions of the formal proof are re- produced here in common mathematical language, and are accompanied by an informal discussion about the underlying ideas.Comment: 6th Podlasie Conference on Mathematics 2014, 11 page