We study the subgame perfect equilibria of two player stochastic games with perfect monitoring and geometric discounting. A novel algorithm is developed for calculating the discounted payoffs that can be attained in equilibrium. This algorithm generates a sequence of tuples of payoffs vectors, one payoff for each state, that move around the equilibrium payoff sets in a clockwise manner. The trajectory of these "pivot" payoffs asymptotically traces the boundary of the equilibrium payoff correspondence. We also provide an implementation of our algorithm, and preliminary simulations indicate that it is more efficient than existing methods. The theoretical results that underlie the algorithm also yield a bound on the number of extremal equilibrium payoffs.
{"title":"A 'Pencil Sharpening' Algorithm for Two Player Stochastic Games with Perfect Monitoring","authors":"Dilip Abreu, Benjamin Brooks, Yuliy Sannikov","doi":"10.2139/ssrn.2736574","DOIUrl":"https://doi.org/10.2139/ssrn.2736574","url":null,"abstract":"We study the subgame perfect equilibria of two player stochastic games with perfect monitoring and geometric discounting. A novel algorithm is developed for calculating the discounted payoffs that can be attained in equilibrium. This algorithm generates a sequence of tuples of payoffs vectors, one payoff for each state, that move around the equilibrium payoff sets in a clockwise manner. The trajectory of these \"pivot\" payoffs asymptotically traces the boundary of the equilibrium payoff correspondence. We also provide an implementation of our algorithm, and preliminary simulations indicate that it is more efficient than existing methods. The theoretical results that underlie the algorithm also yield a bound on the number of extremal equilibrium payoffs.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122674210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vianney Perchet, P. Rigollet, Sylvain Chassang, E. Snowberg
Motivated by practical applications, chiefly clinical trials, we study the regret achievable for stochastic bandits under the constraint that the employed policy must split trials into a small number of batches. We propose a simple policy, and show that a very small number of batches gives close to minimax optimal regret bounds. As a byproduct, we derive optimal policies with low switching cost for stochastic bandits.
{"title":"Batched Bandit Problems","authors":"Vianney Perchet, P. Rigollet, Sylvain Chassang, E. Snowberg","doi":"10.1214/15-AOS1381","DOIUrl":"https://doi.org/10.1214/15-AOS1381","url":null,"abstract":"Motivated by practical applications, chiefly clinical trials, we study the regret achievable for stochastic bandits under the constraint that the employed policy must split trials into a small number of batches. We propose a simple policy, and show that a very small number of batches gives close to minimax optimal regret bounds. As a byproduct, we derive optimal policies with low switching cost for stochastic bandits.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"2083 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129844600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study how the outcomes of a private-value first price auction can vary with bidders' information, for a fixed distribution of private values. In a two bidder, two value, setting, we characterize all combinations of bidder surplus and revenue that can arise, and identify the information structure that minimizes revenue. The extremal information structure that minimizes revenue entails each bidder observing a noisy and correlated signal about the other bidder's value. In the general environment with many bidders and many values, we characterize the minimum bidder surplus of each bidder and maximum revenue across all information structures. The extremal information structure that simultaneously attains these bounds entails an efficient allocation, bidders knowing whether they will win or lose, losers bidding their true value and winners being induced to bid high by partial information about the highest losing bid. Our analysis uses a linear algebraic characterization of equilibria across all information structures, and we report simulations of properties of the set of all equilibria.
{"title":"Extremal Information Structures in the First Price Auction","authors":"D. Bergemann, Benjamin Brooks, S. Morris","doi":"10.2139/ssrn.2354494","DOIUrl":"https://doi.org/10.2139/ssrn.2354494","url":null,"abstract":"We study how the outcomes of a private-value first price auction can vary with bidders' information, for a fixed distribution of private values. In a two bidder, two value, setting, we characterize all combinations of bidder surplus and revenue that can arise, and identify the information structure that minimizes revenue. The extremal information structure that minimizes revenue entails each bidder observing a noisy and correlated signal about the other bidder's value. In the general environment with many bidders and many values, we characterize the minimum bidder surplus of each bidder and maximum revenue across all information structures. The extremal information structure that simultaneously attains these bounds entails an efficient allocation, bidders knowing whether they will win or lose, losers bidding their true value and winners being induced to bid high by partial information about the highest losing bid. Our analysis uses a linear algebraic characterization of equilibria across all information structures, and we report simulations of properties of the set of all equilibria.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124675303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper proves that the additive representation of Dekel-Lipman-Rustichini (2001) is consistent with any preference relation among the deterministic alternatives in their model. The result yields an additive representation which relaxes both the monotonicity and ordinal sub-modularity axioms in Kreps (1979) flexibility representation theorem.
{"title":"Additive Representation for Preferences Over Menus in Finite Choice Settings","authors":"Leandro Gorno","doi":"10.2139/ssrn.1755250","DOIUrl":"https://doi.org/10.2139/ssrn.1755250","url":null,"abstract":"This paper proves that the additive representation of Dekel-Lipman-Rustichini (2001) is consistent with any preference relation among the deterministic alternatives in their model. The result yields an additive representation which relaxes both the monotonicity and ordinal sub-modularity axioms in Kreps (1979) flexibility representation theorem.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132685760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An early death is, undoubtedly, a serious disadvantage. However, the compensation of short-lived individuals has remained so far largely unexplored, probably because it appears infeasible. Indeed, short-lived agents can hardly be identified ex ante, and cannot be compensated ex post. We argue that, despite those difficulties, a compensation can be carried out by encouraging early consumption in the life cycle. In a model with heterogeneous preferences and longevities, we show how a specific social criterion can be derived from intuitive principles, and we study the corresponding optimal policy under various informational assumptions.
{"title":"Compensating the Dead","authors":"M. Fleurbaey, Marie‐Louise Leroux, G. Ponthiere","doi":"10.2139/ssrn.1969441","DOIUrl":"https://doi.org/10.2139/ssrn.1969441","url":null,"abstract":"An early death is, undoubtedly, a serious disadvantage. However, the compensation of short-lived individuals has remained so far largely unexplored, probably because it appears infeasible. Indeed, short-lived agents can hardly be identified ex ante, and cannot be compensated ex post. We argue that, despite those difficulties, a compensation can be carried out by encouraging early consumption in the life cycle. In a model with heterogeneous preferences and longevities, we show how a specific social criterion can be derived from intuitive principles, and we study the corresponding optimal policy under various informational assumptions.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122647776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Supplementary Material: Proof of Theorem 1 Without Cheap Talk","authors":"Takuo Sugaya","doi":"10.2139/SSRN.1789793","DOIUrl":"https://doi.org/10.2139/SSRN.1789793","url":null,"abstract":"Supplementary material to \"Folk Theorem in Repeated Games with Private Monitoring\" available at http://ssrn.com/abstract=1789775.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123420161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In conventional epistemic analysis of solution concepts in complete information games, complete information is implicitly interpreted to mean common certainty of (i) a mapping from action profiles to outcomes; (ii) players' (unconditional) preferences over outcomes; and (iii) players' preferences over outcomes conditional on others' actions. We characterize a new solution concept preference-correlated rationalizability that captures common certainty of (i) and (ii) but not (iii). We show that it is badly behaved, with failures of upper hemicontinuity giving rise to counter-intuitive results. We discuss restrictions that restore well-behaved results.
{"title":"Common Certainty of Rationality Revisited","authors":"S. Morris, S. Takahashi","doi":"10.2139/ssrn.1789773","DOIUrl":"https://doi.org/10.2139/ssrn.1789773","url":null,"abstract":"In conventional epistemic analysis of solution concepts in complete information games, complete information is implicitly interpreted to mean common certainty of (i) a mapping from action profiles to outcomes; (ii) players' (unconditional) preferences over outcomes; and (iii) players' preferences over outcomes conditional on others' actions. We characterize a new solution concept preference-correlated rationalizability that captures common certainty of (i) and (ii) but not (iii). We show that it is badly behaved, with failures of upper hemicontinuity giving rise to counter-intuitive results. We discuss restrictions that restore well-behaved results.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117015551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We illustrate the corrosive effect of even small amounts of adverse selection in an asset market and show how it can lead to the total breakdown of trade. The problem is the failure of "market confidence," defined as approximate common knowledge of an upper bound on expected losses. Small probability events can unravel market confidence. We discuss the role of contagious adverse selection and the problem of "toxic assets" in the recent financial crisis. (JEL D82, G01, G12, G14)
{"title":"Contagious Adverse Selection","authors":"S. Morris, H. Shin","doi":"10.2139/ssrn.1729236","DOIUrl":"https://doi.org/10.2139/ssrn.1729236","url":null,"abstract":"We illustrate the corrosive effect of even small amounts of adverse selection in an asset market and show how it can lead to the total breakdown of trade. The problem is the failure of \"market confidence,\" defined as approximate common knowledge of an upper bound on expected losses. Small probability events can unravel market confidence. We discuss the role of contagious adverse selection and the problem of \"toxic assets\" in the recent financial crisis. (JEL D82, G01, G12, G14)","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"514 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115345757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Johannes Horner, Takuo Sugaya, S. Takahashi, N. Vieille
We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players') equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period's state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.
{"title":"Recursive Methods in Discounted Stochastic Games: An Algorithm for δ → 1 and a Folk Theorem","authors":"Johannes Horner, Takuo Sugaya, S. Takahashi, N. Vieille","doi":"10.2139/ssrn.1729299","DOIUrl":"https://doi.org/10.2139/ssrn.1729299","url":null,"abstract":"We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players') equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period's state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134363393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores the pricing of debt in a financial system where the assets that borrowers hold to meet their obligations include claims against other borrowers. Assessing financial claims in a system context captures features that are missing in a partial equilibrium setting. It is possible for spreads to fall as debts rise, as debt-fuelled increases in asset prices and stronger balance sheets reinforce each other. Conversely, it is possible that de-leveraging leads to increases in spreads, as is often observed during crises.
{"title":"Risk and Liquidity in a System Context","authors":"H. Shin","doi":"10.2139/ssrn.898411","DOIUrl":"https://doi.org/10.2139/ssrn.898411","url":null,"abstract":"This paper explores the pricing of debt in a financial system where the assets that borrowers hold to meet their obligations include claims against other borrowers. Assessing financial claims in a system context captures features that are missing in a partial equilibrium setting. It is possible for spreads to fall as debts rise, as debt-fuelled increases in asset prices and stronger balance sheets reinforce each other. Conversely, it is possible that de-leveraging leads to increases in spreads, as is often observed during crises.","PeriodicalId":281179,"journal":{"name":"Princeton University William S. Dietrich II Economic Theory Center Research Paper Series","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115818760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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