Gambling in contests with regret

This paper discusses the gambling contest introduced in Seel & Strack (Gambling in contests, Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 375, Mar 2012.) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel & Strack model consists of $n$-agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximise their probability of winning the contest. We give a new derivation of the results of Seel & Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem in which in the case when an agent is penalised when their strategy is suboptimal, in the sense that they do not win the contest, but there existed an alternative strategy which would have resulted in victory

Gambling in contests with random initial law

This paper studies a variant of the contest model introduced in Seel and Strack [J. Econom. Theory 148 (2013) 2033-2048]. In the Seel-Strack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the agent who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the two-player contest, this randomised value has a uniform distribution on $[0,2x_0]$. In this paper, we consider a variant of the problem whereby the starting values of the Brownian motions are independent, nonnegative random variables that have a common law $\mu$. We consider a two-player contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $\nu$, where $\nu$ is greater than or equal to $\mu$ in convex order; $\nu$ has an atom at zero of the same size as any atom of $\mu$ at zero, and otherwise is atom free; on $(0,\infty)$ $\nu$ has a decreasing density; and the density of $\nu$ only decreases at points where the convex order constraint is binding.Comment: Published at http://dx.doi.org/10.1214/14-AAP1088 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Configurational temperature of charge-stabilized colloidal monolayers

Recent theoretical advances show that the temperature of a system in equilibriumcan be measured from static snapshots of its constituents' instantaneous configurations, withoutregard to their dynamics. We report the first measurements of the configurational temperature in an experimental system. In particular, we introduce a hierarchy of hyperconfigurational temperature definitions, which we use to analyze monolayers of charge-stabilized colloidal spheres. Equality of the hyperconfigurational and bulk thermodynamic temperatures provides previously lacking thermodynamic self-consistency checks for the measured colloidal pair potentials, and thereby casts new light on anomalous like-charge colloidal attractions induced by geometric confinement.Comment: 4 pages, 3 figure

GIT Compactifications of M_{0,n} and Flips

We use geometric invariant theory (GIT) to construct a large class of compactifications of the moduli space M_{0,n}. These compactifications include many previously known examples, as well as many new ones. As a consequence of our GIT approach, we exhibit explicit flips and divisorial contractions between these spaces.Comment: Final version to appear in Advance