Continuous-Time Games of Timing

We address the question of existence of equilibrium in general timing games of complete information. Under weak assumptions, any two-player timing game has a subgame perfect e-equilibrium, for each e > 0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.Timing games; war of attrition; preemption games; subgame perfect equilibrium

Subgame-Perfect Equilibria in Stochastic Timing Games

We introduce a notion of subgames for stochastic timing games and the related notion of subgame-perfect equilibrium in possibly mixed strategies. While a good notion of subgame-perfect equilibrium for continuous-time games is not available in general, we argue that our model is the appropriate version for timing games. We show that the notion coincides with the usual one for discrete-time games. Many timing games in continuous time have only equilibria in mixed strategies -- in particular preemption games, which often occur in the strategic real option literature. We provide a sound foundation for some workhorse equilibria of that literature, which has been lacking as we show. We obtain a general constructive existence result for subgame-perfect equilibria in preemption games and illustrate our findings by several explicit applications.Comment: 27 pages, 1 figur

Caller Number Five and related timing games

There are two varieties of timing games in economics: wars of attrition, in which having more predecessors helps, and pre-emption games, in which having more predecessors hurts. This paper introduces and explores a spanning class with rank-order payoffs that subsumes both varieties as special cases. We assume time is continuous, actions are unobserved, and information is complete, and explore how equilibria of the games, in which there is shifting between phases of slow and explosive (positive probability) stopping, capture many economic and social timing phenomena. Inspired by auction theory, we first show how each symmetric Nash equilibrium is equivalent to a different "potential function.'' By using this function, we straightforwardly obtain existence and characterization results. Descartes' Rule of Signs bounds the number of phase transitions. We describe how adjacent timing game phases interact: war of attrition phases are not played out as long as they would be in isolation, but instead are cut short by pre-emptive atoms. We bound the number of equilibria, and compute the payoff and duration of each equilibrium.Games of timing, war of attrition, preemption game

Caller Number Five and Related Timing Games

There are two varieties of timing games in economics: Having more predecessors helps in a war of attrition and hurts in a pre-emption game. This paper introduces and explores a spanning class with rank-order payoffs} that subsumes both as special cases. We assume a continuous time setting with unobserved actions and complete information, and explore how equilibria of these games capture many economic and social timing phenomena --- shifting between phases of slow and explosive (positive probability) stopping. Inspired by auction theory, we first show how the symmetric Nash equilibria are each equivalent to a different "potential function". This device straightforwardly yields existence and characterization results. The Descartes Rule of Signs, e.g., bounds the number phase transitions. We describe how adjacent timing game phases interact: War of attrition phases are not played out as long as they would be in isolation, but instead are cut short by pre-emptive atoms. We bound the number of equilibria, and compute the payoff and duration of each equilibrium.Games of Timing, War of Attrition, Preemption Game.

Equilibria in Continuous Time Preemption Games with Markovian Payoffs

This paper studies timing games in continuous time where payoffs are stochastic and strongly Markovian. The main interest is in characterizing equilibria where players preempt each other along almost every sample path. It is found that the existence of such preemption equilibria depends crucially on whether there is a coordination mechanism that allows for rent equalization or not, and whether the stochastic payoffs admit upward jumps. Through numerical examples it is argued that the possibility of such coordination improves social welfare and that the welfare loss due to preemption decreases in uncertainty.Timing Games, Real Options, Preemption

Subgame-Perfect Equilibria in Stochastic Timing Games

Riedel F, Steg J-H. Subgame-Perfect Equilibria in Stochastic Timing Games. Center for Mathematical Economics Working Papers. Vol 524. Bielefeld: Center for Mathematical Economics; 2014.We introduce a notion of subgames for stochastic timing games and the related notion of subgame-perfect equilibrium in possibly mixed strategies. While a good notion of subgame-perfect equilibrium for continuous-time games is not available in general, we argue that our model is the appropriate version for timing games. We show that the notion coincides with the usual one for discrete-time games. Many timing games in continuous time have only equilibria in mixed strategies – in particular preemption games, which often occur in the strategic real option literature. We provide a sound foundation for some workhorse equilibria of that literature, which has been lacking as we show. We obtain a general constructive existence result for subgame-perfect equilibria in preemption games and illustrate our findings by several explicit applications

Subgame-perfect equilibria in stochastic timing games

Abstract: We develop a notion of subgames and the related notion of subgame-perfect equilibrium – possibly in mixed strategies – for stochastic timing games. To capture all situations that can arise in continuous-time models, it is necessary to consider stopping times as the starting dates of subgames. We generalize Fudenberg and Tirole’s (1985) mixed-strategy extensions to make them applicable to stochastic timing games and thereby provide a sound basis for subgame-perfect equilibria of preemption games. Sufficient conditions for equilibrium existence are presented, and examples illustrate their application as well as the fact that intuitive arguments can break down in the presence of stochastic processes with jumps

Continuois Time Contests

This paper introduces a contest model in which each player decides when to stop a privately observed Brownian motion with drift and incurs costs depending on his stopping time. The player who stops his process at the highest value wins a prize. Applications of the model include procurement contests and competitions for grants. We prove existence and uniqueness of the Nash equilibrium outcome, even if players have to choose bounded stopping times. We derive the equilibrium distribution in closed form. If the noise vanishes, the equilibrium outcome converges to - and thus selects - the symmetric equilibrium outcome of an all-pay auction. For two players and constant costs, each player’s profits increase if costs for both players increase, variance increases, or drift decreases. Intuitively, patience becomes a more important factor for contest success, which reduces informational rents