# Philipp Strack Publications

## Abstract

A key part of decentralized consensus protocols is a procedure for random selection, which is the source of the majority of miners cost and wasteful energy consumption in Bitcoin. We provide a simple economic model for random selection mechanism and show that any PoW protocol with natural desirable properties is outcome equivalent to the random selection mechanism used in Bitcoin.

## Abstract

Bitcoin’s main innovation lies in allowing a decentralized system that relies on anonymous, proﬁt driven miners who can freely join the system. We formalize these properties in three axioms: anonymity of miners, no incentives for miners to consolidate, and no incentive to assuming multiple fake identities. This novel axiomatic formalization allows us to characterize which other protocols are feasible: Every protocol with these properties must have the same reward scheme as Bitcoin. This implies an impossibility result for risk-averse miners: no protocol satisﬁes the aforementioned constraints simultaneously without giving miners a strict incentive to merge. Furthermore, any protocol either gives up on some degree of decentralization or its reward scheme is equivalent to Bitcoin’s.

## Abstract

A single seller faces a sequence of buyers with unit demand. The buyers are forwardlooking and long-lived but vanish (and are replaced) at a constant rate. The arrival time and the valuation is private information of each buyer and unobservable to the seller. Any incentive compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.

We derive the optimal stationary mechanism in closed form and characterize its qualitative structure. As the arrival time is private information, the buyer can choose the time at which he reports his arrival. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem. The stopping time determines the time at which the buyer decides to participate in the mechanism. The resulting value function of each buyer cannot be too convex and must be continuously differentiable everywhere, reflecting the option value of delaying participation. The optimal mechanism thus induces progressive participation by each buyer: he participates either immediately or at a future random time.

## Abstract

A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived. The arrival time and the valuation is private information of each buyer. Any incentive compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.

We derive the optimal stationary mechanism in closed form and characterize its qualitative structure. As the arrival time is private information, the buyer can choose the time at which he reports his arrival. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem. The stopping time determines the time at which the buyer decides to participate in the mechanism. The resulting value function of each buyer cannot be too convex and must be continuously differentiable everywhere, reflecting the option value of delaying participation. The optimal mechanism thus induces progressive participation by each buyer: he participates either immediately or at a future random time.

## Abstract

A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived but vanish (and are replaced) at a constant rate. The arrival time and the valuation is private information of each buyer and unobservable to the seller. Any incentive compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.

We derive the optimal stationary mechanism, characterize its qualitative structure, and derive a closed-form solution. As the arrival time is private information, the buyer can choose the time at which he reports his arrival. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem. The stopping time determines the time at which the buyer decides to participate in the mechanism. The resulting value function of each buyer cannot be too convex and must be continuously differentiable everywhere, reflecting the option value of delaying participation. The optimal mechanism thus induces progressive participation by each buyer: he participates either immediately or at a future random time.

## Abstract

A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived but vanish (and are replaced) at a constant rate. The arrival time and the valuation is private information of each buyer and unobservable to the seller. Any incentive-compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.

We derive the optimal stationary mechanism, characterize its qualitative structure and derive a closed-form solution. As the arrival time is private information, the agent can choose the time at which he reports his arrival. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem. The stopping time determines the time at which the agent decides to participate in the mechanism. The resulting value function of each agent can not be too convex and has to be continuously differentiable everywhere, reflecting the option value of delaying participation. The optimal mechanism thus induces progressive participation by each agent: he participates either immediately or at a future random time.

## Abstract

A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived. Each buyer has private information about his arrival time and valuation where the latter evolves according to a geometric Brownian motion. Any incentive-compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.

We establish that the optimal stationary allocation policy can be implemented by a simple posted price. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem which determines the first time at which the buyer participates in the mechanism. The optimal mechanism thus induces progressive participation by each buyer: he either participates immediately or at a future random time.

## Abstract

We characterize the proﬁt-maximizing mechanism for repeatedly selling a non-durable good in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. In the proﬁt-maximizing mechanism the allocation is distorted in favor of agents with high initial types.

We derive the optimal mechanism in closed form, which enables us to compare the distortion in various examples. The case where the valuation of the agents follows an arithmetic/geometric Brownian motion, Ornstein-Uhlenbeck process, or is derived from a Bayesian learning model are discussed. We show that depending on the nature of the private information and the valuation process the distortion might increase or decrease over time.

## Abstract

We characterize the proﬁt-maximizing mechanism for repeatedly selling a non-durable good in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. In the proﬁt-maximizing mechanism the allocation is distorted in favor of agents with high initial types.

We derive the optimal mechanism in closed form, which enables us to compare the distortion in various examples. The case where the valuation of the agents follows an arithmetic/geometric Brownian motion, Ornstein-Uhlenbeck process, or is derived from a Bayesian learning model are discussed. We show that depending on the nature of the private information and the valuation process the distortion might increase or decrease over time.

## Abstract

We characterize the revenue-maximizing mechanism for time separable allocation problems in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. The leading example is the repeated sales of a good or a service.

We derive the optimal dynamic mechanism, analyze its qualitative structure and frequently derive its closed form solution. This enables us to compare the distortion in various settings. In particular, we discuss the cases where the type of each agent follows an arithmetic or geometric Brownian motion or a mean reverting process. We show that depending on the nature of the private information the distortion might increase or decrease over time.

## Abstract

We characterize the revenue-maximizing mechanism for time separable allocation problems in continuous time. The willingness-to-pay of each agent is private information and changes over time.

We derive the dynamic revenue-maximizing mechanism, analyze its qualitative structure and frequently derive its closed form solution. In the leading example of repeat sales of a good or service, we establish that commonly observed contract features such as at rates, free consumption units and two-part tariﬀs emerge as part of the optimal contract. We investigate in detail the environments in which the type of each agent follows an arithmetic or geometric Brownian motion or a mean-reverting process. We analyze the allocative distortions and show that depending on the nature of the private information the distortion might increase or decrease over time.