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. JEL subject classification: C61, C73, D21, L12 MSC2010 subject classification: 60G40, 91A25, 91A40, 91A55

We take a general perspective on capital accumulation games with open loop strategies, as they have been formalized by Back and Paulsen (Rev. Financ. Stud. 22, 4531-4552, 2009). With such strategies, the optimization problems of the individual players are of the monotone follower type. Consequently, one can adapt available methods, in particular the approach of Bank (SIAM J. Control Optim. 44, 1529-1541, 2005. We obtain consistency in equilibrium by proving that with common assumptions from the oligopoly literature on instantaneous revenue, equilibrium determination is equivalent to solving a single monotone follower problem. In the unique open loop equilibrium, only the currently smallest firms invest. This result is valid for arbitrary initial capital levels and general stochastic shock processes, which may be non-Markovian and include jumps. We explicitly solve an example, the specification of Grenadier (Rev. Financ. Stud. 15, 691-721, 2002) with a Lévy process.Keywords Irreversible investment · Oligopoly · Equilibrium · Singular stochastic control Mathematics Subject Classification (2010) 60G40 · 91A15 · 91A25 · 93E20 JEL Classification C73 · D43 · D92

We construct subgame-perfect equilibria with mixed strategies for symmetric stochastic timing games with arbitrary strategic incentives. The strategies are qualitatively different for local first-or second-mover advantages, which we analyse in turn. When there is a local second-mover advantage, the players may conduct a war of attrition with stopping rates that we characterize in terms of the Snell envelope from the general theory of optimal stopping, which is very general but provides a clear interpretation. With a local firstmover advantage, stopping typically results from preemption and is abrupt. Equilibria may differ in the degree of preemption, precisely at which points it is triggered. We provide an algorithm to characterize where preemption is inevitable and to establish the existence of corresponding payoff-maximal symmetric equilibria.

Motivated by recent path-breaking contributions in the theory of repeated games in continuous time, this paper presents a family of discrete-time games which provides a consistent discrete-time approximation of the continuous-time limit game. Using probabilistic arguments, we prove that continuous-time games can be defined as the limit of a sequence of discrete-time games. Our convergence analysis reveals various intricacies of continuous-time games. First, we demonstrate the importance of correlated strategies in continuous-time. Second, we attach a precise meaning to the statement that a sequence of discrete-time games can be used to approximate a continuous-time game.

Consider a mechanism for the binary public good provision problem that is dominant strategy incentive compatible (DSIC), ex-post individually rational (EPIR), and ex-post budget balanced (EPBB). Suppose this mechanism has the additional property that the utility from participating in the mechanism to the lowest types is zero for all agents. Such a mechanism must be of a threshold form, in which there is a fixed threshold for each agent such that the public good is not provided if there is an agent with a value below her threshold and is provided if all agents' values exceed their respective threshold. There are mechanism that are DSIC, EPIR, and EPBB that are not of the threshold form. Mechanisms that maximize welfare subject to DSIC, EPIR, and EPBB must again have the threshold form. Finally, mechanisms that are DSIC, EPIR, EPBB and that furthermore satisfy the condition that there is at least one type profile in which all agents can block the provision of the public good, also must be of the threshold form. As we allow individuals' values for the public good to be negative and positive, our results cover examples including bilateral trade, bilateral wage negotiations, a seller selling to a group of individuals (who then have joint ownership rights), and rezoning the use of land.