# When Do Types Induce the Same Belief Hierarchy?

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Belief Hierarchies and Types

#### 2.1. Type Structures

**Definition**

**1.**

#### 2.2. From Type Structures to Belief Hierarchies

#### 2.3. Example

## 3. Hierarchy and Type Morphisms

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**1.**

## 4. Generalized Type Morphisms

**Definition**

**4.**

**Lemma**

**1.**

**Definition**

**5.**

**Theorem**

**1.**

**Procedure**

**1.**

**Proposition**

**2.**

## 5. Characterizing Types with the Same Belief Hierarchy

**Lemma**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

## 6. Finite Type Structures

**Procedure**

**2**

**Example**

**1.**

**Proposition**

**5**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.1.1. Every Hierarchy Morphism Is a Generalized Type Morphism

#### Appendix A.1.2. Every Generalized Type Morphism Is a Hierarchy Morphism

#### Appendix A.2. Proof of Proposition 2

#### Appendix A.3. Proof of Lemma 2

**Lemma**

**3.**

**Proof.**

#### Appendix A.4. Proof of Proposition 3

## References

- Carlsson, H.; van Damme, E. Global Games and Equilibrium Selection. Econometrica
**1993**, 61, 989–1018. [Google Scholar] [CrossRef] - Feinberg, Y.; Skrzypacz, A. Uncertainty about uncertainty and delay in bargaining. Econometrica
**2005**, 73, 69–91. [Google Scholar] [CrossRef] - Friedenberg, A. Bargaining under Strategic Uncertainty; Working Paper; Arizona State University: Tempe, AZ, USA, 2014. [Google Scholar]
- Geanakoplos, J.D.; Polemarchakis, H.M. We can’t disagree forever. J. Econ. Theory
**1982**, 28, 192–200. [Google Scholar] [CrossRef] - Neeman, Z. The relevance of private information in mechanism design. J. Econ. Theory
**2004**, 117, 55–77. [Google Scholar] [CrossRef] - Brandenburger, A.; Dekel, E. Rationalizability and Correlated Equilibria. Econometrica
**1987**, 55, 1391–1402. [Google Scholar] [CrossRef] - Tan, T.; Werlang, S. The Bayesian foundations of solution concepts in games. J. Econ. Theory
**1988**, 45, 370–391. [Google Scholar] [CrossRef] - Aumann, R.; Brandenburger, A. Epistemic conditions for Nash equilibrium. Econometrica
**1995**, 63, 1161–1180. [Google Scholar] [CrossRef] - Perea, A. A one-person doxastic characterization of Nash strategies. Synthese
**2007**, 158, 251–271. [Google Scholar] [CrossRef] - Battigalli, P.; Siniscalchi, M. Strong belief and forward-induction reasoning. J. Econ. Theory
**2002**, 106, 356–391. [Google Scholar] [CrossRef] - Harsanyi, J.C. Games on incomplete information played by Bayesian players. Part I. Manag. Sci.
**1967**, 14, 159–182. [Google Scholar] [CrossRef] - Friedenberg, A.; Meier, M. On the relationship between hierarchy and type morphisms. Econ. Theory
**2011**, 46, 377–399. [Google Scholar] [CrossRef][Green Version] - Mertens, J.F.; Zamir, S. Formulation of Bayesian analysis for games with incomplete information. Int. J. Game Theory
**1985**, 14, 1–29. [Google Scholar] [CrossRef] - Heifetz, A.; Samet, D. Topology-Free Typology of Beliefs. J. Econ. Theory
**1998**, 82, 324–341. [Google Scholar] [CrossRef] - Dekel, E.; Fudenberg, D.; Morris, S. Interim Correlated Rationalizability. Theor. Econ.
**2007**, 2, 15–40. [Google Scholar] - Ely, J.; Peski, M. Hierarchies of beliefs and interim rationalizability. Theor. Econ.
**2006**, 1, 19–65. [Google Scholar] [CrossRef] - Liu, Q. On redundant types and Bayesian formulation of incomplete information. J. Econ. Theory
**2009**, 144, 2115–2145. [Google Scholar] [CrossRef] - Friedenberg, A.; Meier, M. The context of the game. Econ. Theory
**2016**, in press. [Google Scholar] - Yildiz, M. Invariance to Representation of Information. Games Econ. Behav.
**2015**, 94, 142–156. [Google Scholar] [CrossRef] - Brandenburger, A.; Keisler, H.J. An Impossibility Theorem on Beliefs in Games. Stud. Log.
**2006**, 84, 211–240. [Google Scholar] [CrossRef] - Friedenberg, A. When Do Type Structures Contain All Hierarchies of Beliefs? Games Econ. Behav.
**2010**, 68, 108–129. [Google Scholar] [CrossRef] - Kets, W. Bounded Reasoning and Higher-Order Uncertainty; Working Paper; Northwestern University: Evanston, IL, USA, 2011. [Google Scholar]
- Friedenberg, A.; Keisler, H.J. Iterated Dominance Revisited; Working Paper; Arizona State University: Tempe, AZ, USA; University of Wisconsin-Madison: Madison, WI, USA, 2011. [Google Scholar]
- Heifetz, A.; Samet, D. Knowledge Spaces with Arbitrarily High Rank. Games Econ. Behav.
**1998**, 22, 260–273. [Google Scholar] [CrossRef] - Battigalli, P.; Siniscalchi, M. Hierarchies of conditional beliefs and interactive epistemology in dynamic games. J. Econ. Theory
**1999**, 88, 188–230. [Google Scholar] [CrossRef] - Blume, L.; Brandenburger, A.; Dekel, E. Lexicographic Probabilities and Choice Under Uncertainty. Econometrica
**1991**, 59, 61–79. [Google Scholar] [CrossRef] - Heifetz, A.; Kets, W. Robust Multiplicity with a Grain of Naiveté; Working Paper; Northwestern University: Evanston, IL, USA, 2013. [Google Scholar]
- Aliprantis, C.D.; Border, K.C. Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed.; Springer: Berlin, Germany, 2005. [Google Scholar]
- Parthasarathy, K. Probability Measures on Metric Spaces; AMS Chelsea Publishing: Providence, RI, USA, 2005. [Google Scholar]

^{1}We follow the terminology of Friedenberg and Meier [12] here. A stronger condition for $\mathcal{T}$ to be contained in ${\mathcal{T}}^{\prime}$ is that $\mathcal{T}$ can be embedded (using a type morphism) into ${\mathcal{T}}^{\prime}$ as a belief-closed subset [13]. Our results can be used to characterize conditions under which $\mathcal{T}$ is contained in ${\mathcal{T}}^{\prime}$ in this stronger sense in a straightforward way.^{2}Clearly, ${t}_{i}$ and ${t}_{i}^{\prime}$ generate the same n-th-order belief if and only if ${h}_{i}^{T,n}\left({t}_{i}\right)={h}_{i}^{T,n}\left({t}_{i}^{\prime}\right)$.^{3}That is, for each $E\in {\Sigma}_{i}^{{T}^{\prime}}$, we have $\{{t}_{i}\in {T}_{i}\mid {\phi}_{i}\left({t}_{i}\right)\in E\}\in {\Sigma}_{i}^{T}$.^{4}Since ${\Sigma}^{T}$ is closed under $\mathcal{T}$ (by measurability of the belief maps ${b}_{i}$), the intersection is nonempty. It is easy to verify that the intersection is a σ-algebra.^{5}That is, for each $E\in {\mathcal{F}}_{i}^{{T}^{\prime}}$, we have $\{{t}_{i}\in {T}_{i}\mid {\phi}_{i}\left({t}_{i}\right)\in E\}\in {\Sigma}_{i}^{T}$.^{6}Clearly, for any $y\in Y$, if there is an atom a that contains y (i.e., $y\in a$), then this atom is unique.^{7}This is with some abuse of notation, since ${b}_{i}^{*}$ is defined on ${X}_{i}\times {T}_{-i}^{*}$, while ${b}_{i}^{1}$ and ${b}_{i}^{2}$ are defined on ${X}_{i}\times {T}_{-i}^{1}$ and ${X}_{i}\times {T}_{-i}^{2}$, respectively. By defining the σ-algebra ${\Sigma}_{j}^{{T}^{*}}$ on ${T}_{j}^{*}$ as above, the extension of ${b}_{i}^{1}$ and ${b}_{i}^{2}$ to the larger domain is well defined.

Type Structure $\mathcal{T}$ |

${T}_{1}=\{{t}_{1},{t}_{1}^{\prime},{t}_{1}^{\u2033}\},\phantom{\rule{1.em}{0ex}}{T}_{2}=\{{t}_{2},{t}_{2}^{\prime},{t}_{2}^{\u2033}\}$ |

${b}_{1}\left({t}_{1}\right)=\frac{1}{2}(c,{t}_{2})+\frac{1}{2}(d,{t}_{2}^{\prime})$ |

${b}_{1}\left({t}_{1}^{\prime}\right)=\frac{1}{6}(c,{t}_{2})+\frac{1}{3}(c,{t}_{2}^{\u2033})+\frac{1}{2}(d,{t}_{2}^{\prime})$ |

${b}_{1}\left({t}_{1}^{\u2033}\right)=\frac{1}{2}(c,{t}_{2}^{\prime})+\frac{1}{2}(d,{t}_{2}^{\u2033})$ |

${b}_{2}\left({t}_{2}\right)=\frac{1}{4}(e,{t}_{1})+\frac{1}{2}(e,{t}_{1}^{\prime})+\frac{1}{4}(f,{t}_{1}^{\u2033})$ |

${b}_{2}\left({t}_{2}^{\prime}\right)=\frac{1}{8}(e,{t}_{1})+\frac{1}{8}(e,{t}_{1}^{\prime})+\frac{3}{4}(f,{t}_{1}^{\u2033})$ |

${b}_{2}\left({t}_{2}^{\u2033}\right)=\frac{3}{8}(e,{t}_{1})+\frac{3}{8}(e,{t}_{1}^{\prime})+\frac{1}{4}(f,{t}_{1}^{\u2033})$ |

Type Structure ${\mathcal{T}}^{\prime}$ |

${R}_{1}=\{{r}_{1},{r}_{1}^{\prime},{r}_{1}^{\u2033}\},$ ${R}_{2}=\{{r}_{2},{r}_{2}^{\prime},{r}_{2}^{\u2033}\}$ |

${\beta}_{1}\left({r}_{1}\right)=\frac{1}{4}(c,{r}_{2})+\frac{1}{4}(c,{r}_{2}^{\u2033})+\frac{1}{2}(d,{r}_{2}^{\prime})$ |

${\beta}_{1}\left({r}_{1}^{\prime}\right)=\frac{1}{2}(c,{r}_{2}^{\prime})+\frac{1}{8}(d,{r}_{2})+\frac{3}{8}(d,{r}_{2}^{\u2033})$ |

${\beta}_{1}\left({r}_{1}^{\u2033}\right)=\frac{1}{2}(c,{r}_{2}^{\prime})+\frac{3}{8}(d,{r}_{2})+\frac{1}{8}(d,{r}_{2}^{\u2033})$ |

${\beta}_{2}\left({r}_{2}\right)=\frac{1}{4}(e,{r}_{1}^{\prime})+\frac{3}{4}(f,{r}_{1})$ |

${\beta}_{2}\left({r}_{2}^{\prime}\right)=\frac{3}{4}(e,{r}_{1}^{\prime})+\frac{1}{4}(f,{r}_{1})$ |

${\beta}_{2}\left({r}_{2}^{\u2033}\right)=\frac{1}{8}(e,{r}_{1}^{\prime})+\frac{1}{8}(e,{r}_{1}^{\u2033})+\frac{3}{4}(f,{r}_{1})$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Perea, A.; Kets, W. When Do Types Induce the Same Belief Hierarchy? *Games* **2016**, *7*, 28.
https://doi.org/10.3390/g7040028

**AMA Style**

Perea A, Kets W. When Do Types Induce the Same Belief Hierarchy? *Games*. 2016; 7(4):28.
https://doi.org/10.3390/g7040028

**Chicago/Turabian Style**

Perea, Andrés, and Willemien Kets. 2016. "When Do Types Induce the Same Belief Hierarchy?" *Games* 7, no. 4: 28.
https://doi.org/10.3390/g7040028