This note studies (full) implementation of social choice functions under complete information in (correlated) rationalizable strategies. The monotonicity condition shown by Maskin (1999) to be necessary for Nash implementation is also necessary under the more stringent solution concept. We show that it is also sufficient under a mild “no worst alternative” condition. In particular, no economic condition is required.
We consider the implementation of social choice functions under complete information in rationalizable strategies. A strict (and thus stronger) version of the monotonicity condition introduced by Maskin (1999) is necessary under the solution concept of rationalizability. Assuming the social choice function is responsive (i.e., it never selects the same outcome in two distinct states), we show that it is also sufficient under a mild “no worst alternative” condition. In particular, no economic condition is required. We also discuss how our results extend when the social choice function is not responsive.
We present a dynamic model of venture capital financing, described as a sequential investment problem with uncertain outcome. Each venture has a critical, but unknown threshold beyond which it cannot progress. If the threshold is reached before the completion of the project, then the project fails, otherwise it succeeds. The investors decide sequentially about the speed of the investment and the optimal path of staged investments. We derive the dynamically optimal funding policy in response to the arrival of information during the development of the venture. We develop three types of predictions from our theoretical model and test these predictions in a large sample of venture capital investments in the U.S. for the period of 1987-2002.
First, the investment flow starts low if the failure risk is high and accelerates as the projects mature. Second, the investment flow reacts positively to information that arrives while the project is developed. We find that the investment decisions are more sensitive to the information received during the development than to the information held prior to the project launch. Third, investors distribute their investments over more funding rounds if the failure risk is larger.
We analyze sequential investment decisions in an innovative project that depend on the investor’s information about the project failure risk and its potential final value. We consider the feedback effects between learning about the project parameters and the continuous adjustment of the investment strategy. Investors decide sequentially about the speed of investment and the optimal degree of involvement. We develop three types of predictions from our theoretical model and test these predictions in a large sample of venture capital investment in the U.S. for the period of 1987-2002.
First, the investment flow starts cautiously if the failure risk is high and accelerates as the projects mature. Second, the investment flow reacts positively to information that arrives while the project is developed. We find that interim information is more significant for investment decisions than the information prior to the project launch. Third, investors distribute their investments over more funding rounds if the failure risk is larger.
We present a dynamic model of venture capital financing, described as a sequential investment problem with uncertain outcome. Each venture has a critical, but unknown threshold beyond which it cannot progress. If the threshold is reached before the completion of the project, then the project fails, otherwise it succeeds. The investors decide sequentially about the speed of the investment and the optimal path of staged investments. We derive the dynamically optimal funding policy in response to the arrival of information during the development of the venture. We develop three types of predictions from our theoretical model and test these predictions in a large sample of venture capital investments in the U.S. for the period of 1987-2002.
First, the investment flow starts low if the failure risk is high and accelerates as the projects mature. Second, the investment flow reacts positively to information that arrives while the project is developed. We find that the investment decisions are more sensitive to the information received during the development than to the information held prior to the project launch. Third, investors distribute their investments over more funding rounds if the failure risk is larger.
We consider truthful implementation of the socially efficient allocation in an independent private-value environment in which agents receive private information over time. We propose a suitable generalization of the pivot mechanism, based on the marginal contribution of each agent. In the dynamic pivot mechanism, the ex-post incentive and ex-post participation constraints are satisfied for all agents after all histories. In an environment with diverse preferences it is the unique mechanism satisfying ex-post incentive, ex-post participation and efficient exit conditions.
We develop the dynamic pivot mechanism in detail for a repeated auction of a single object in which each bidder learns over time her true valuation of the object. We show that the dynamic pivot mechanism is equivalent to a modified second price auction.
We consider truthful implementation of the socially efficient allocation in an independent private-value environment in which agents receive private information over time. We propose a suitable generalization of the pivot mechanism, based on the marginal contribution of each agent. In the dynamic pivot mechanism, the ex-post incentive and ex-post participation constraints are satisfied for all agents after all histories. In an environment with diverse preferences it is the unique mechanism satisfying ex-post incentive, ex-post participation and efficient exit conditions.
We develop the dynamic pivot mechanism in detail for a repeated auction of a single object in which each bidder learns over time her true valuation of the object. The dynamic pivot mechanism here is equivalent to a modified second price auction.
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.
Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.
Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.
We consider the problem of pricing a single object when the seller has only minimal information about the true valuation of the buyer. Specifically, the seller only knows the support of the possible valuations and has no further distributional information.
The seller is solving this choice problem under uncertainty by minimizing her regret. The pricing policy hedges against uncertainty by randomizing over a range of prices. The support of the pricing policy is bounded away from zero. Buyers with low valuations cannot generate substantial regret and are priced out of the market. We generalize the pricing policy without priors to encompass many buyers and many qualities.