Optimal and Myopic Information Acquisition

We consider the problem of optimal dynamic information acquisition from many correlated information sources. Each period, the decision-maker jointly takes an action and allocates a fixed number of observations across the available sources. His payoff depends on the actions taken and on an unknown state. In the canonical setting of jointly normal information sources, we show that the optimal dynamic information acquisition rule proceeds myopically after finitely many periods. If signals are acquired in large blocks each period, then the optimal rule turns out to be myopic from period 1. These results demonstrate the possibility of robust and "simple" optimal information acquisition, and simplify the analysis of dynamic information acquisition in a widely used informational environment

Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics

Equilibrium and Efficiency in the Tug-of-War

We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game, the "tipping states", which are determined by the contestants’ relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes. Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single stage conflict, such structures can reduce the overall resources that are dissipated among the group of players