### Citations

786 | Games with Incomplete Information Played by Bayesian Players
- Harsanyi
- 1968
(Show Context)
Citation Context ... signals, then their posteriors can differ, unless the content of the common signals is common knowledge. This is true even in the absence of ambiguity: as we show, a seemingly weak condition that ensures that players have identical beliefs when there is no ambiguity is in fact equivalent to the strong condition that the information is commonly known. This suggests that the assumption that signals are unambiguous, that is, admit only a single interpretation, is far from innocuous. One implication of our results is that one of the standard justifications for the common-prior assumption, due to Harsanyi (1968) and Aumann (1987), is not valid. Given the centrality of the common-prior assumption in economics, this reinforces the importance of developing alternative models that do not rely on the common-prior assumption. A number of authors have considered the role of ambiguity in explaining economic phenomena. Harris and Raviv (1993), for instance, show that speculative trade is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1... |

312 |
Differences in Opinion Make a Horse Race
- HARRIS, RAVIV
- 1993
(Show Context)
Citation Context ...he information is commonly known. This suggests that the assumption that signals are unambiguous, that is, admit only a single interpretation, is far from innocuous. One implication of our results is that one of the standard justifications for the common-prior assumption, due to Harsanyi (1968) and Aumann (1987), is not valid. Given the centrality of the common-prior assumption in economics, this reinforces the importance of developing alternative models that do not rely on the common-prior assumption. A number of authors have considered the role of ambiguity in explaining economic phenomena. Harris and Raviv (1993), for instance, show that speculative trade is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1998) study how players can benefit from leaving some obligations ambiguous in contracts. Blume and Board (2014) demonstrate that the strategic use of ambiguous messages can mitigate conflict and may thus be welfare enhancing. Neither of these papers models ambiguity syntactically, as we do, which makes it hard to apply the mod... |

151 | Central bank communication and monetary policy: A survey of theory and evidence. - Blinder, Ehrmann, et al. - 2008 |

144 |
The common prior assumption in economic theory
- Morris
- 1995
(Show Context)
Citation Context ...nsight into differences in beliefs. Our results imply that the common-prior assumption is hard to justify even if players have the same information and start out with identical priors. They are thus related to, but distinct from, those of Brandenburger et al. (1992), who showed that there is an equivalence between models with heterogeneous beliefs and models in which players make information processing errors. Our results imply that one does not even need to assume information processing errors to obtain heterogeneous beliefs. They can thus be viewed as formalizing a (qualitative) argument by Morris (1995, p. 234), who argues that it can be the case that players have the same information 2 but different beliefs even if players are rational and follow the rules of Bayesian updating. Our paper appears to be first to focus on the crucial role of language when it comes to ambiguity, and to drop the assumption that ambiguity is commonly understood. In doing so, our paper is one of the very few papers that provides a formal argument that the common-prior assumption can be justified only under very restrictive assumptions.2 2 Example We start with an example. There are two players, labeled 1 and 2. P... |

88 |
Incomplete Contracts and Strategic Ambiguity,American Economic Review 88
- Whinston
- 1998
(Show Context)
Citation Context ..., due to Harsanyi (1968) and Aumann (1987), is not valid. Given the centrality of the common-prior assumption in economics, this reinforces the importance of developing alternative models that do not rely on the common-prior assumption. A number of authors have considered the role of ambiguity in explaining economic phenomena. Harris and Raviv (1993), for instance, show that speculative trade is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1998) study how players can benefit from leaving some obligations ambiguous in contracts. Blume and Board (2014) demonstrate that the strategic use of ambiguous messages can mitigate conflict and may thus be welfare enhancing. Neither of these papers models ambiguity syntactically, as we do, which makes it hard to apply the models to related but different situations. Grant et al. (2009) model contracting in the face of ambiguity. They model ambiguity by assuming that unambiguous terms do not fully specify the state of the world. Board and Chung (2012) use a simplified model of the one presented her... |

73 | Updating probabilities. - Grunwald, Halpern - 2003 |

35 |
Central bank communication and policy effectiveness. In The Greenspan Era: Lessons for the Future,
- Woodford
- 2005
(Show Context)
Citation Context ... certain that that is the case: both players assign probability 1 2 in ω2 to the event that the other player has observed a different color than he has. In Section 4, we show that this is not the case: we demonstrate that in the presence of ambiguity, players can have different beliefs about unambiguous propositions, even if they have a common prior, have the same information, and know that they have the same information, unless there is common knowledge of the information that they receive. 3It has been argued that such divergence in expectations reduces the effectiveness of monetary policy (Woodford, 2005). In recent years, the Federal Reserve has indeed taken measures to increase transparency not only of their decision-making process, but also of their interpretation of macroeconomic indicators (Bernanke, 2013). As suggested by the example, some ambiguity will most likely remain. 4 3 Framework To capture ambiguity, we explicitly model players’ language, that is, the syntax. The symbols in this language are given meaning in a semantic model. We discuss the syntax in Section 3.1, and the semantics in Section 3.2. We go on to define ambiguity and information structures in Sections 3.3 and 3.4. Mu... |

23 | Intransitivity and vagueness.
- Halpern
- 2009
(Show Context)
Citation Context ...to be an abbreviation for EB(EBm−1ϕ); EBmϕ says that ϕ is mth-order mutual belief. 3.2 Semantics The intended reading of formulas like ϕ ∨ ψ and CBϕ that we gave above is supposed to correspond to intuitions that we have regarding words like “or” and “common belief.” These intuitions are captured by providing a semantic model for the formulas in the language, that is, a method for deciding whether a given formula is true or false. To model that statements can be ambiguous, we want to allow for the possibility that players interpret statements differently. We build on an approach used earlier (Halpern, 2009; Grove and Halpern, 1993), where formulas are interpreted relative to a player. This means that players can disagree on the meaning of a statement. More specifically, the semantic model we adopt is an epistemic probability structure. An (epistemic probability) structure (over a set of primitive propositions Φ) has the form M = (Ω, µ, (Πj)j∈N , (πj)j∈N), where Ω is the state space, assumed to be countable, and µ is the common prior on Ω (defined on the power set of Ω). For each player i ∈ N , Πi is a partition of Ω, and πi is an interpretation that associates with each state a truth assignment... |

16 |
Correlated equilibria as an expression of Bayesian rationality.
- Aumann
- 1987
(Show Context)
Citation Context ... posteriors can differ, unless the content of the common signals is common knowledge. This is true even in the absence of ambiguity: as we show, a seemingly weak condition that ensures that players have identical beliefs when there is no ambiguity is in fact equivalent to the strong condition that the information is commonly known. This suggests that the assumption that signals are unambiguous, that is, admit only a single interpretation, is far from innocuous. One implication of our results is that one of the standard justifications for the common-prior assumption, due to Harsanyi (1968) and Aumann (1987), is not valid. Given the centrality of the common-prior assumption in economics, this reinforces the importance of developing alternative models that do not rely on the common-prior assumption. A number of authors have considered the role of ambiguity in explaining economic phenomena. Harris and Raviv (1993), for instance, show that speculative trade is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1998) study how pla... |

13 | Naming and identity in epistemic logics, Part I: the propositional case.
- Grove, Halpern
- 1993
(Show Context)
Citation Context ...iation for EB(EBm−1ϕ); EBmϕ says that ϕ is mth-order mutual belief. 3.2 Semantics The intended reading of formulas like ϕ ∨ ψ and CBϕ that we gave above is supposed to correspond to intuitions that we have regarding words like “or” and “common belief.” These intuitions are captured by providing a semantic model for the formulas in the language, that is, a method for deciding whether a given formula is true or false. To model that statements can be ambiguous, we want to allow for the possibility that players interpret statements differently. We build on an approach used earlier (Halpern, 2009; Grove and Halpern, 1993), where formulas are interpreted relative to a player. This means that players can disagree on the meaning of a statement. More specifically, the semantic model we adopt is an epistemic probability structure. An (epistemic probability) structure (over a set of primitive propositions Φ) has the form M = (Ω, µ, (Πj)j∈N , (πj)j∈N), where Ω is the state space, assumed to be countable, and µ is the common prior on Ω (defined on the power set of Ω). For each player i ∈ N , Πi is a partition of Ω, and πi is an interpretation that associates with each state a truth assignment to the primitive proposit... |

8 | A century of U.S. central banking: Goals, frameworks, accountability.
- Bernanke
- 2013
(Show Context)
Citation Context ...strate that in the presence of ambiguity, players can have different beliefs about unambiguous propositions, even if they have a common prior, have the same information, and know that they have the same information, unless there is common knowledge of the information that they receive. 3It has been argued that such divergence in expectations reduces the effectiveness of monetary policy (Woodford, 2005). In recent years, the Federal Reserve has indeed taken measures to increase transparency not only of their decision-making process, but also of their interpretation of macroeconomic indicators (Bernanke, 2013). As suggested by the example, some ambiguity will most likely remain. 4 3 Framework To capture ambiguity, we explicitly model players’ language, that is, the syntax. The symbols in this language are given meaning in a semantic model. We discuss the syntax in Section 3.1, and the semantics in Section 3.2. We go on to define ambiguity and information structures in Sections 3.3 and 3.4. Much of this material in this section is taken from our companion paper (Halpern and Kets, 2014), where we study the logic of ambiguity. We have simplified some of the definitions here so as to allow us to better... |

4 | A matter of interpretation: Bargaining over ambiguous contracts. Working paper,
- Grant, Kline, et al.
- 2009
(Show Context)
Citation Context ...de is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1998) study how players can benefit from leaving some obligations ambiguous in contracts. Blume and Board (2014) demonstrate that the strategic use of ambiguous messages can mitigate conflict and may thus be welfare enhancing. Neither of these papers models ambiguity syntactically, as we do, which makes it hard to apply the models to related but different situations. Grant et al. (2009) model contracting in the face of ambiguity. They model ambiguity by assuming that unambiguous terms do not fully specify the state of the world. Board and Chung (2012) use a simplified model of the one presented here to study ambiguous contracts. Rather than focusing on specific applications, we characterize the implications of ambiguity generally, and illustrate how ambiguity can give more insight into differences in beliefs. Our results imply that the common-prior assumption is hard to justify even if players have the same information and start out with identical priors. They are thus relat... |

2 | Intentional vagueness.
- Blume, Board
- 2014
(Show Context)
Citation Context ...economics, this reinforces the importance of developing alternative models that do not rely on the common-prior assumption. A number of authors have considered the role of ambiguity in explaining economic phenomena. Harris and Raviv (1993), for instance, show that speculative trade is possible when traders have a common prior and observe public signals if they use different statistical models to update their beliefs, which they interpret as traders interpreting signals differently. Bernheim and Whinston (1998) study how players can benefit from leaving some obligations ambiguous in contracts. Blume and Board (2014) demonstrate that the strategic use of ambiguous messages can mitigate conflict and may thus be welfare enhancing. Neither of these papers models ambiguity syntactically, as we do, which makes it hard to apply the models to related but different situations. Grant et al. (2009) model contracting in the face of ambiguity. They model ambiguity by assuming that unambiguous terms do not fully specify the state of the world. Board and Chung (2012) use a simplified model of the one presented here to study ambiguous contracts. Rather than focusing on specific applications, we characterize the implicat... |

2 | How common are common priors?
- Hellman, Samet
- 2012
(Show Context)
Citation Context ...it). For concreteness, assume that there are three states, ω1, ω2, and ω3. In state ω1, the wavelength is 400 nanometers, so that player 1 would say that the car is burgundy, while player 2 would say that the car is rosewood. In state ω2, the wavelength is 403 nanometers, so both players would say that the car is burgundy. In ω3, the wavelength is 405 nanometers, so that player 1 would say that the color is scarlet, and player 2 would say that the color is burgundy. 2See Morris (1995) and references therein for various philosophical objections to the common-prior assumption. Nyarko (2010) and Hellman and Samet (2012) show that the set of states in which the common-prior assumption holds is small in a measure-theoretic and a topological sense, respectively. Rather than showing that the set of states that satisfies the common-prior assumption is small in some sense, we ask whether players’ posteriors can remain the same under realistic scenarios. 3 There is also an unambiguous proposition q: each player thinks q is true in ω1 and ω2, and false in ω3. For example, q could be “the wavelength of the car is between 399 and 404 nanometers.” Players’ information is given by the color of the car. This means when a... |

2 |
Most games violate the common priors doctrine.
- Nyarko
- 2010
(Show Context)
Citation Context ...y were to measure it). For concreteness, assume that there are three states, ω1, ω2, and ω3. In state ω1, the wavelength is 400 nanometers, so that player 1 would say that the car is burgundy, while player 2 would say that the car is rosewood. In state ω2, the wavelength is 403 nanometers, so both players would say that the car is burgundy. In ω3, the wavelength is 405 nanometers, so that player 1 would say that the color is scarlet, and player 2 would say that the color is burgundy. 2See Morris (1995) and references therein for various philosophical objections to the common-prior assumption. Nyarko (2010) and Hellman and Samet (2012) show that the set of states in which the common-prior assumption holds is small in a measure-theoretic and a topological sense, respectively. Rather than showing that the set of states that satisfies the common-prior assumption is small in some sense, we ask whether players’ posteriors can remain the same under realistic scenarios. 3 There is also an unambiguous proposition q: each player thinks q is true in ω1 and ω2, and false in ω3. For example, q could be “the wavelength of the car is between 399 and 404 nanometers.” Players’ information is given by the color ... |

1 |
Contract interpretation in insurance law. Working paper,
- Board, Chung
- 2012
(Show Context)
Citation Context ...traders interpreting signals differently. Bernheim and Whinston (1998) study how players can benefit from leaving some obligations ambiguous in contracts. Blume and Board (2014) demonstrate that the strategic use of ambiguous messages can mitigate conflict and may thus be welfare enhancing. Neither of these papers models ambiguity syntactically, as we do, which makes it hard to apply the models to related but different situations. Grant et al. (2009) model contracting in the face of ambiguity. They model ambiguity by assuming that unambiguous terms do not fully specify the state of the world. Board and Chung (2012) use a simplified model of the one presented here to study ambiguous contracts. Rather than focusing on specific applications, we characterize the implications of ambiguity generally, and illustrate how ambiguity can give more insight into differences in beliefs. Our results imply that the common-prior assumption is hard to justify even if players have the same information and start out with identical priors. They are thus related to, but distinct from, those of Brandenburger et al. (1992), who showed that there is an equivalence between models with heterogeneous beliefs and models in which pl... |

1 |
Correlated equilibria with generalized information structures.
- Brandenburger, Dekel, et al.
- 1992
(Show Context)
Citation Context ...mbiguity. They model ambiguity by assuming that unambiguous terms do not fully specify the state of the world. Board and Chung (2012) use a simplified model of the one presented here to study ambiguous contracts. Rather than focusing on specific applications, we characterize the implications of ambiguity generally, and illustrate how ambiguity can give more insight into differences in beliefs. Our results imply that the common-prior assumption is hard to justify even if players have the same information and start out with identical priors. They are thus related to, but distinct from, those of Brandenburger et al. (1992), who showed that there is an equivalence between models with heterogeneous beliefs and models in which players make information processing errors. Our results imply that one does not even need to assume information processing errors to obtain heterogeneous beliefs. They can thus be viewed as formalizing a (qualitative) argument by Morris (1995, p. 234), who argues that it can be the case that players have the same information 2 but different beliefs even if players are rational and follow the rules of Bayesian updating. Our paper appears to be first to focus on the crucial role of language wh... |

1 | A logic for reasoning about ambiguity.
- Halpern, Kets
- 2014
(Show Context)
Citation Context ...rease transparency not only of their decision-making process, but also of their interpretation of macroeconomic indicators (Bernanke, 2013). As suggested by the example, some ambiguity will most likely remain. 4 3 Framework To capture ambiguity, we explicitly model players’ language, that is, the syntax. The symbols in this language are given meaning in a semantic model. We discuss the syntax in Section 3.1, and the semantics in Section 3.2. We go on to define ambiguity and information structures in Sections 3.3 and 3.4. Much of this material in this section is taken from our companion paper (Halpern and Kets, 2014), where we study the logic of ambiguity. We have simplified some of the definitions here so as to allow us to better focus on the game-theoretic applications. 3.1 Syntax We start by defining the syntax of a formal language that describes all relevant situations. We want a logic where players use a fixed common language, but each player may interpret formulas in the language differently. We also want to allow the players to be able to reason about (probabilistic) beliefs, so as to be able to study the possibility of heterogeneous beliefs. There is a finite, nonempty set N = {1, . . . , n} of pl... |