Abstract
The focus of this chapter is on games of incomplete information, including games of complete information as a special case. We will present several popular equilibrium concepts.
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Notes
- 1.
There are more general definitions of incomplete information. For the types of games covered in this book, this definition is appropriate and sufficient.
- 2.
The fact that unknown information is treated as uncertainty is referred to as awareness, i.e., all players are aware of the existence of the information.
- 3.
We can actually allow an agent’s utility function to take the form \( u_{i} \left( {x,\theta } \right) \) rather than \( u_{i} \left( {x,\theta_{i} } \right). \) All the concepts on Bayesian implementation are readily extendable to this case.
- 4.
Given the density function \( \phi \left( {\uptheta} \right) \) for all players, the conditional density function \( \phi_{ - i} (\theta_{ - i} |\theta_{i} ) \) of \( \theta_{ - i} \) conditional on the knowledge of \( \theta_{i} \) is
$$ \phi_{ - i} (\theta_{ - i} |\theta_{i} ) = \frac{\phi \left( \theta \right)}{{\phi_{i} \left( {\theta_{i} } \right)}} = \frac{\phi \left( \theta \right)}{{\mathop \int \nolimits_{{\Theta _{ - i} }} \phi \left( \theta \right)d\theta_{ - i} }}. $$The expectation operator \( E_{{\theta_{ - i} |\theta_{i} }} \) uses this conditional density function.
- 5.
See Wang (2008, 2015, Theorem 4.8).
- 6.
Condition \( \sigma^{*} (m^{*} |t) > 0 \) means that those messages that are actually been sent must be optimal for the sender. This requirement follows Crawford and Sobel (1982). An alternative in Matthews (1989) is \( \int\nolimits_{{\mathbb{T}}} {\sigma^{*} (m|t)p\left( t \right)dt > 0} . \)
- 7.
We have not seen this extension in the literature.
- 8.
This definition is similar to that in Crawford and Sobel (1982, p. 1434), except that Crawford and Sobel’s definition is not completely correct on the consistency condition (c).
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Wang, S. (2018). Incomplete Information Games. In: Microeconomic Theory. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0041-7_8
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DOI: https://doi.org/10.1007/978-981-13-0041-7_8
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