Relative controllability of linear difference equations

In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time $T$ in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces $L^p$ and $\mathcal C^k$. We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are "less rationally dependent" than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay

On the Notion of Social Institutions

We argue that it is natural to study social institutions within the framework of standard game theory (i.e., only by resorting to concepts like players, actions, strategies, information sets, payoff functions, and stochastic processes describing the moves of nature, which constitute a stochastic game when combined) concepts like social norms, and mechanisms can be easily accommodated, as well as philosophical/ sociological definitions of social institutions Focusing on strategies rather than on mechanisms have two advantages: First, focusing on strategies allows us to distinguish between those aspects that are behavioral in nature and are subject to alternative design, and those that are part of the environment. Second, considering strategies allows for a more detailed look into the way an outcome function is genuinely implemented (Hurwicz (1996, p.123)).

Nash Equilibria of Games with a Continuum of Players

We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of $(\epsilon,\epsilon)$ - equilibrium --- in which the fraction of players not $\epsilon$ - optimizing is less than $\epsilon$ --- we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an $(\epsilon_n,\epsilon_n)$ - equilibria, with $\epsilon_n$ converging to zero. The same holds for $\epsilon$ - equilibrium --- in which almost all players are $\epsilon$ - optimizing --- provided that either players' payoff functions are equicontinuous or players' action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance.Nash equilibrium, Games with a continuum of players, Games with finitely many players, approximate equilibria.

α′\alpha'-Cosmology: solutions and stability analysis

We review O$(d,d)$ Covariant String Cosmology to all orders in $\alpha'$ in the presence of matter and study its solutions. We show that the perturbative analysis for a constant dilaton in the absence of a dilatonic charge does not lead to a time-independet equation of state. Meanwhile, the non-perturbative equations of motion allow de Sitter solutions in the String frame parametrized by the equation of state and the dilatonic charge. Among this set of solutions, we show that a cosmological constant equation of state implies a de Sitter solution both in String and Einstein frames while a winding equation of state implies a de Sitter solution in the former and a static phase in the latter. We also consider the stability of these solutions under homogeneous linear perturbations and show that they are not unstable, therefore defining viable cosmological scenarios.Comment: 13 pages, 1 figure, references updated, typos fixe

A Simple Proof of a Theorem by Harris

We present a simple proof of existence of subgame perfect equilibria in games with perfect information.

On a Theorem by Mas-Colell

We consider anonymous games with a Lebesgue space of players in which either the action space or players' characteristics are denumer- able. Our main result shows that the set of equilibrium distributions over actions coincides with the set of distributions induced by equilib- rium strategies. This result, together with Mas-Colell (1984)'s theorem, implies that any continuous, denumerable game has an equilibrium strategy. In particular, the theorems of Khan and Sun (1995) and Khan, Rath, and Sun (1997) can be obtained as corollaries of Mas-Colell's.

On the Existence of Equilibria in Discontinuous Games: Three Counterexamples

We study whether we can weaken the conditions given in Reny (1999) and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy $\varepsilon-$equilibria for all epsilon>0. We show by examples that there are: (1) quasiconcave, payoff secure games without pure strategy epsilon-equilibria for small enough epsilon>0 (and hence, without pure strategy Nash equilibria), (2) quasiconcave, reciprocally upper semicontinuous games without pure strategy epsilon-equilibria for small enough epsilon>0, and (3) payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe (1957), also shows that non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.Discontinuous Games, Nash Equilibrium

Social Norms and Monetary Trading

Random matching models have been used in Monetary Economics to argue that money can increase the well being of all agents in the economy. If the model features a finite number of agents it will be shown that there is an equilibrium, analogous to the contagious equilibria described in Kandori (1992), that Pareto dominates the monetary one. However it will be shown also that monetaty equilibria have two important advantages: firstly, they are more plausible in large economies in the sense that the lowest discount factor compatible with monetary equilibria doesn't depend on the population size, which is not the case with contagious equilibria; secondly, it is more stable to finite deviations in the following sense: no matter what the past has been, future play of the equilibrium strategies will give players the same payoff as if the equilibrium strategies were always followed.Monetary trading, social norms

Symmetric Approximate Equilibrium Distributions with Finite Support

We show that a distribution of a game with a continuum of players is an equilibrium distribution if and only if there exists a sequence of symmetric approximate equilibrium distributions of games with finite support that converges to it. Thus, although not all games have symmetric equilibrium distributions, this result shows that all equilibrium distributions can be characterized by symmetric distributions of simpler games (i.e., games with a finite number of characteristics).Equilibrium distributions, games with a continuum of players, symmetric distributions

On the Existence of Equilibrium Bank Runs in a Diamond-Dybvig Environment

In a version of the Diamond and Dybvig (1983) model with aggregate uncertainty, we show that there exists an equilibrium with the following properties: all consumers deposit at the bank, all patient consumers wait for the last period to withdraw, and the bank fails with strictly positive probability. Furthermore, we show that the probability of a bank failure remains bounded away from zero as the number of consumers increases. We interpret such an equilibrium as reflecting a bank run, defined as an episode in which a large number of people withdraw their deposits from a bank, forcing it to fail. Our results show that we can have equilibrium bank runs with consumers poorly informed about the true state of nature, a sequential service constraint, an infinite marginal utility of consumption at zero, and without consumers' panic and sunspots. We therefore think that aggregate risk in Diamond-Dybvig-like environments can be an important element to explain bank runs.Bank runs, aggregate uncertainty