DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Noncooperative games are examined from the point of view of an outside observer who believes that... more Noncooperative games are examined from the point of view of an outside observer who believes that the players are rational and that they know at least as much as the observer. The observer is assumed to be able to observe many instances of the play of the game; these instances are identical in the sense that the observer cannot distinguish between the settings in which different plays occur. If the observer does not believe that he will be able to offer beneficial advice then he must believe that the players are playing a correlated equilibrium, though he may not initially know which correlated equilibrium. If the observer also believes that, in a certain sense, there is nothing connecting the players in a particular instance of the game then he must believe that the correlated equilibrium they are playing is, in fact, a Nash equilibrium.
The game theorists who defined the equilibrium concepts that we now think of as various forms of ... more The game theorists who defined the equilibrium concepts that we now think of as various forms of backwards induction, namely subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), explicitly restricted their analysis to games with perfect recall. In spite of this the concepts are well defined, exactly as they defined them, even in games without perfect recall. There is now a small literature examining the behaviour of these concepts in games without perfect recall. Jeff Kline (2005) and Joe Halpern and Rafael Pass (2008) look at what happens in games without perfect recall to solutions defined in exactly the same way as they were defined in games with perfect recall. The Nature of Our Project What we want to do is to give definitions of the "backward induction" solutions-in this paper sequential equilibrium and quasi-perfect equilibrium-that 1. "coincides" with the original definitions in games with perfect recall, and
There are four types of dominance depending on whether domination is strict or weak and whether t... more There are four types of dominance depending on whether domination is strict or weak and whether the dominating strategy is pure or mixed. Letting d vary over these four types of dominance, we say that a player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows. For weak dominance by a mixed strategy, Stalnaker (1994) introduced a process of iterative maximal elimination of certain profiles that we call here flaws. We define here, analogously, d-flaws for each type of dominance d, and show that for each d, iterative elimination of d-flaws is order independent. We then show that the characterization of common knowledge of d-dominance rationality is the same for each d. A strategy profile can be played when d-dominance rationality is commonly known if and only if it survives an iterative elimination of d-flaws.
We use a spokes model to analyze …rms' customization incentives when facing the choices of standa... more We use a spokes model to analyze …rms' customization incentives when facing the choices of standard and niche products. Products at or near the end of the spokes are customized products, while products near the origin are more standardized products that cater to the taste of many consumers. Our results indicate that although monopolist always o¤ers the standard product, if a …rm anticipates entry, it may choose to stake claim to a customized product. For low transportation costs, the early entrant chooses the standard product. But this equilibrium is characterized by aggressive pricing behavior. JEL Classi…cation: L11, L13.
In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in un... more In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of "indistinguishable" and "more informative." We also show that any payoff associated with a (correlated) distribution on strategy vectors in the stage game such that neither player can profitably deviate from one of his strategies to another that is indistinguishable and more informative is the payoff of a correlated equilibrium of the supergame.
We consider a privately informed seller who sells an indivisible product to a number of competing... more We consider a privately informed seller who sells an indivisible product to a number of competing buyers. For a cost a buyer can obtain an imperfect estimate of the quality of a seller's product. This estimate is not verifiable and so cannot be the basis of contractual commitments. We examine the intuition that the market could act to separate the types of the seller, with sellers of a high quality product responding to purchase offers that offer a high price but require a high value of the estimate of the quality before a trade takes place and sellers of a low value quality product responding to purchase offers that offer a lower price but do not test the product.
In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in un... more In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of ``indistinguishable'' and ``more informative.'' We also show that any payoff associated with a (correlated) distribution
This paper examines the question of the extent to which it is true that any equilibrium that is q... more This paper examines the question of the extent to which it is true that any equilibrium that is quasi-perfect in any extensive form game having a given normal form is necessarily proper. If one fixes not only the equilibrium in question but also a a sequence of completely mixed strategies converging to that equilibrium then indeed the notions are equivalent.
Hillas (1990) introduced a definition of strategic stability based on perturbations of the best r... more Hillas (1990) introduced a definition of strategic stability based on perturbations of the best reply correspondence that satisfies all of the requirements given by Kohlberg and Mertens (1986). Hillas et al. (2001) point out though that the proofs of the iterated dominance and forward induction properties were not correct. They also provide a proof of the IIS property, a stronger version of both iterated dominance and forward induction, using the results of that paper. In this note we provide a direct proof of the IIS property.
... The primary purpose of this paper is to provide a definition of stability that does satisfy a... more ... The primary purpose of this paper is to provide a definition of stability that does satisfy all of ... those of Kohlberg and Mertens in the set of perturbations we consider and on the topology we place on ... In the definition we develop in Sections 4 and 5 we use a set of perturbations very ...
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Noncooperative games are examined from the point of view of an outside observer who believes that... more Noncooperative games are examined from the point of view of an outside observer who believes that the players are rational and that they know at least as much as the observer. The observer is assumed to be able to observe many instances of the play of the game; these instances are identical in the sense that the observer cannot distinguish between the settings in which different plays occur. If the observer does not believe that he will be able to offer beneficial advice then he must believe that the players are playing a correlated equilibrium, though he may not initially know which correlated equilibrium. If the observer also believes that, in a certain sense, there is nothing connecting the players in a particular instance of the game then he must believe that the correlated equilibrium they are playing is, in fact, a Nash equilibrium.
The game theorists who defined the equilibrium concepts that we now think of as various forms of ... more The game theorists who defined the equilibrium concepts that we now think of as various forms of backwards induction, namely subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), explicitly restricted their analysis to games with perfect recall. In spite of this the concepts are well defined, exactly as they defined them, even in games without perfect recall. There is now a small literature examining the behaviour of these concepts in games without perfect recall. Jeff Kline (2005) and Joe Halpern and Rafael Pass (2008) look at what happens in games without perfect recall to solutions defined in exactly the same way as they were defined in games with perfect recall. The Nature of Our Project What we want to do is to give definitions of the "backward induction" solutions-in this paper sequential equilibrium and quasi-perfect equilibrium-that 1. "coincides" with the original definitions in games with perfect recall, and
There are four types of dominance depending on whether domination is strict or weak and whether t... more There are four types of dominance depending on whether domination is strict or weak and whether the dominating strategy is pure or mixed. Letting d vary over these four types of dominance, we say that a player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows. For weak dominance by a mixed strategy, Stalnaker (1994) introduced a process of iterative maximal elimination of certain profiles that we call here flaws. We define here, analogously, d-flaws for each type of dominance d, and show that for each d, iterative elimination of d-flaws is order independent. We then show that the characterization of common knowledge of d-dominance rationality is the same for each d. A strategy profile can be played when d-dominance rationality is commonly known if and only if it survives an iterative elimination of d-flaws.
We use a spokes model to analyze …rms' customization incentives when facing the choices of standa... more We use a spokes model to analyze …rms' customization incentives when facing the choices of standard and niche products. Products at or near the end of the spokes are customized products, while products near the origin are more standardized products that cater to the taste of many consumers. Our results indicate that although monopolist always o¤ers the standard product, if a …rm anticipates entry, it may choose to stake claim to a customized product. For low transportation costs, the early entrant chooses the standard product. But this equilibrium is characterized by aggressive pricing behavior. JEL Classi…cation: L11, L13.
In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in un... more In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of "indistinguishable" and "more informative." We also show that any payoff associated with a (correlated) distribution on strategy vectors in the stage game such that neither player can profitably deviate from one of his strategies to another that is indistinguishable and more informative is the payoff of a correlated equilibrium of the supergame.
We consider a privately informed seller who sells an indivisible product to a number of competing... more We consider a privately informed seller who sells an indivisible product to a number of competing buyers. For a cost a buyer can obtain an imperfect estimate of the quality of a seller's product. This estimate is not verifiable and so cannot be the basis of contractual commitments. We examine the intuition that the market could act to separate the types of the seller, with sellers of a high quality product responding to purchase offers that offer a high price but require a high value of the estimate of the quality before a trade takes place and sellers of a low value quality product responding to purchase offers that offer a lower price but do not test the product.
In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in un... more In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of ``indistinguishable'' and ``more informative.'' We also show that any payoff associated with a (correlated) distribution
This paper examines the question of the extent to which it is true that any equilibrium that is q... more This paper examines the question of the extent to which it is true that any equilibrium that is quasi-perfect in any extensive form game having a given normal form is necessarily proper. If one fixes not only the equilibrium in question but also a a sequence of completely mixed strategies converging to that equilibrium then indeed the notions are equivalent.
Hillas (1990) introduced a definition of strategic stability based on perturbations of the best r... more Hillas (1990) introduced a definition of strategic stability based on perturbations of the best reply correspondence that satisfies all of the requirements given by Kohlberg and Mertens (1986). Hillas et al. (2001) point out though that the proofs of the iterated dominance and forward induction properties were not correct. They also provide a proof of the IIS property, a stronger version of both iterated dominance and forward induction, using the results of that paper. In this note we provide a direct proof of the IIS property.
... The primary purpose of this paper is to provide a definition of stability that does satisfy a... more ... The primary purpose of this paper is to provide a definition of stability that does satisfy all of ... those of Kohlberg and Mertens in the set of perturbations we consider and on the topology we place on ... In the definition we develop in Sections 4 and 5 we use a set of perturbations very ...
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