Abstract
In this chapter, we explain Harsanyi’s Bayesian framework for games with incomplete information. For normal-form games with incomplete information, Bayesian games and Bayesian Nash equilibrium are defined.
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Notes
- 1.
To be precise, mathematical equivalence between the infinite hierarchy of beliefs model and Harsanyi’s ‘type’ model needs to be proved. See Mertens and Zamir [12].
- 2.
- 3.
Bayes’ rule generates the conditional probability of an event A given that the event B has occurred from the prior probability distribution Pr in such a way that
$$ Pr(A\mid B) = \frac{Pr(A \cap B)}{Pr(B)}.$$ - 4.
It is possible to extend the Bayesian framework and the Bayesian Nash equilibrium to infinite type spaces with appropriate probability measures.
- 5.
The combination \((NN', N)\) is a Bayesian Nash equilibrium for any P.
- 6.
Also called the open or oral auction.
- 7.
Generic games is a set of games (interpreted as \(n \times |S|\)-dimensional payoff vectors) which excludes measure 0 sets in the \(n \times |S|\)-dimensional Eucledian space. They are not all games, and therefore counter examples exist. For more details, see Harsanyi [8], Chap. 6 of Fudenberg and Tirole [3], and Govindan et al. [4].
- 8.
To be precise, we need to restrict the perturbation structure as well as the target Nash equilibrium.
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Fujiwara-Greve, T. (2015). Bayesian Nash Equilibrium. In: Non-Cooperative Game Theory. Monographs in Mathematical Economics, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55645-9_6
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DOI: https://doi.org/10.1007/978-4-431-55645-9_6
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