On Incentive Compatibility in Dynamic Mechanism Design With Exit Option in a Markovian Environment

This paper studies dynamic mechanism design in a quasilinear Markovian environment and analyzes a direct mechanism model of a principal-agent framework in which the agent is allowed to exit at any period. We consider that the agent's private information, referred to as state, evolves over time. The agent makes decisions of whether to stop or continue and what to report at each period. The principal, on the other hand, chooses decision rules consisting of an allocation rule and a set of payment rules to maximize her ex-ante expected payoff. In order to influence the agent's stopping decision, one of the terminal payment rules is posted-price, i.e., it depends only on the realized stopping time of the agent. We define the incentive compatibility in this dynamic environment in terms of Bellman equations, which is then simplified by establishing a one-shot deviation principle. Given the optimality of the stopping rule, a sufficient condition for incentive compatibility is obtained by constructing the state-dependent payment rules in terms of a set of functions parameterized by the allocation rule. A necessary condition is derived from envelope theorem, which explicitly formulates the state-dependent payment rules in terms of allocation rules. A class of monotone environment is considered to characterize the optimal stopping by a threshold rule. The posted-price payment rules are then pinned down in terms of the allocation rule and the threshold function up to a constant. The incentive compatibility constraints restrict the design of the posted-price payment rule by a regular condition.

[1]  Roger B. Myerson,et al.  Optimal Auction Design , 1981, Math. Oper. Res..

[2]  Lones Smith,et al.  Dynamic Matching and Evolving Reputations , 2009 .

[3]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal Stopping and Free-Boundary Problems , 2006 .

[4]  Mallesh M. Pai,et al.  Optimal Dynamic Auctions , 2008 .

[5]  Philipp Strack,et al.  An Inverse Optimal Stopping Problem for Diffusion Processes , 2014, Math. Oper. Res..

[6]  Sham M. Kakade,et al.  Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism , 2013, Oper. Res..

[7]  Rakesh V. Vohra,et al.  Optimal Dynamic Auctions and Simple Index Rules , 2013, Math. Oper. Res..

[8]  D. Bergemann,et al.  The Dynamic Pivot Mechanism , 2008 .

[9]  L. Dubins,et al.  Countably additive gambling and optimal stopping , 1977 .

[10]  M. Said Auctions with Dynamic Populations: Efficiency and Revenue Maximization , 2012 .

[11]  Mustafa Akan,et al.  Revenue management by sequential screening , 2015, J. Econ. Theory.

[12]  J. Rochet A necessary and sufficient condition for rationalizability in a quasi-linear context , 1987 .

[13]  X. Zhou,et al.  Optimal Exit Time from Casino Gambling: Strategies of Pre-Committed and Naive Gamblers , 2019 .

[14]  Stéphane Villeneuve,et al.  On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems , 2007, Journal of Applied Probability.

[15]  Hung-Yu Wei,et al.  Dynamic Auction Mechanism for Cloud Resource Allocation , 2010, 2010 10th IEEE/ACM International Conference on Cluster, Cloud and Grid Computing.

[16]  Noah Williams,et al.  Persistent Private Information , 2008 .

[17]  William Vickrey,et al.  Counterspeculation, Auctions, And Competitive Sealed Tenders , 1961 .

[18]  V. Veeravalli,et al.  Asymptotically Optimal Quickest Change Detection in Distributed Sensor Systems , 2008 .

[19]  Xiaodong Wang,et al.  Quickest Detection of False Data Injection Attack in Wide-Area Smart Grids , 2015, IEEE Transactions on Smart Grid.

[20]  Andrzej Skrzypacz,et al.  Revenue Management with Forward-Looking Buyers , 2016, Journal of Political Economy.

[21]  Garrett J. van Ryzin,et al.  Optimal Dynamic Auctions for Revenue Management , 2002, Manuf. Serv. Oper. Manag..

[22]  I. Segal,et al.  Dynamic Mechanism Design: Incentive Compatibility, Profit Maximization and Information Disclosure , 2009 .

[23]  Damien Lamberton,et al.  Optimal stopping and American options , 2009 .

[24]  D. Silvestrov,et al.  Optimal Stopping and Reselling of European Options , 2010 .

[25]  Péter Eso,et al.  Optimal Information Disclosure in Auctions and the Handicap Auction , 2007 .

[26]  R. Deb Optimal Contracting of New Experience Goods , 2008 .

[27]  S. Kakade,et al.  Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism , 2013 .

[28]  Mohammad Akbarpour,et al.  Dynamic matching market design , 2014, EC.

[29]  I. Segal,et al.  Dynamic Mechanism Design: A Myersonian Approach , 2014 .

[30]  Amin Saberi,et al.  Dynamic Pay-Per-Action Mechanisms and Applications to Online Advertising , 2013, Oper. Res..

[31]  David C. Parkes,et al.  An MDP-Based Approach to Online Mechanism Design , 2003, NIPS.

[32]  P. Courty,et al.  Sequential Screening , 1998 .

[33]  D. Blackwell Discounted Dynamic Programming , 1965 .

[34]  D. P. Baron,et al.  Regulation and information in a continuing relationship , 1984 .

[35]  Alex Gershkov,et al.  Dynamic Revenue Maximization with Heterogeneous Objects: A Mechanism Design Approach , 2009 .

[36]  R. Deb,et al.  Dynamic Screening with Limited Commitment , 2015 .

[37]  Thomas A. Weber,et al.  Efficient Dynamic Allocation with Uncertain Valuations , 2005 .

[38]  Philipp Strack,et al.  Optimal Stopping with Private Information , 2014, J. Econ. Theory.

[39]  Yuzhe Zhang,et al.  Dynamic Contracting with Persistent Shocks , 2008, J. Econ. Theory.

[40]  Katsunori Ano Optimal Stopping Problem with Uncertain Stopping and its Application to Discrete Options , 2009 .

[41]  M. Jackson Mechanism Theory , 2014 .