Abstract
This chapter analyzes behavior in relatively simple strategic settings: simultaneous-move games of complete information . Let us define the two building blocks of this chapter: best responses and Nash equilibrium.
The original version of this chapter was revised: Post-publication author corrections have been incorporated. The erratum to this chapter is available at 10.1007/978-3-319-32963-5_12
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25 May 2018
This chapter analyzes behavior in relatively simple strategic settings: simultaneous-move games of complete information . Let us define the two building blocks of this chapter: best responses and Nash equilibrium.
Notes
- 1.
While the Nash equilibrium solution concept allows for many applications in the area of industrial organization, we only explore some basic examples in this chapter, relegating many others to Chap. 5 (Applications to Industrial Organization).
- 2.
A common trick many students use in order to be able to focus on the fact that we are examining the case in which player 2 confesses (in the left-hand column) is to cover with their hand (or a piece of paper) the columns in which player 2 selects strategies different from Confess (in this case, that means covering Not confess, but in larger matrices it would imply covering all columns except for the one we are analyzing at that point.) Once we focus on the column corresponding to Confess, player 1’s best response becomes a straightforward comparison of his payoff from Confess, −5, and that from Not confess, −15, which helps us underline the largest of the two payoffs, i.e., −5.
- 3.
In this case, you can also focus on the column corresponding to Not confess by covering the column of Confess with your hand. This would allow you to easily compare player 1’s payoff from Confess, 0, and Not confess, −1, underlining the largest of the two, i.e., 0.
- 4.
Similarly as for player 2, you can now focus on the row selected by player 1 by covering with your hand the row he did not select. For instance, when player 1 chooses Confess, you can cover the row corresponding to Not confess, which allows for an immediate comparison of the payoff when player 2 responds with Confess, −5, and when he does not, −15, and underline the largest of the two, i.e., −5. An analogous argument applies to the case in which player 1 selects Not confess, where you can cover the row corresponding to Confess with your hand.
- 5.
In both of these Nash equilibria, firms are playing mutual best responses, and thus no firm has incentives to unilaterally deviate.
- 6.
However, no firm has incentives to unilaterally move from technology B to A when its competitor is selecting technology B.
- 7.
In order to obtain the output level of firm j that forces firm i to be inactive, set qi = 0 on firm i’s best response function, and solve for qj. The output you obtain should coincide with the horizontal intercept of firm i’s best response function in Fig. 2.9.
- 8.
For instance, if \( x_{j} = 5 \) and \( k = 2 \), then player \( i \) has incentives to write an estimate of \( x_{i} = 5 - 2 = 3 \), but not lower than 3 since his payoff, \( x_{i} + k \), is increasing in his own estimate \( x_{i} \).
- 9.
Visually, this implies fixing your attention on the first row of the left-hand matrix, and horizontally search for which strategy of player 2 (column) provides this player with the highest payoff.
- 10.
For instance, in finding \( BR_{1} \left( {x,\,A} \right) \), we fix the matrix in which player 3 selects A (left matrix), and the column that player 2 selects x (left-hand column), and compare the payoffs that player 1 would obtain from responding with the first row (a), $2, the second row (b), $3, or with the third row (c), $1. Hence, \( BR_{1} \left( {x,\,A} \right) = b \). A similar argument applies to other best responses of player 1.
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Munoz-Garcia, F., Toro-Gonzalez, D. (2016). Pure Strategy Nash Equilibrium and Simultaneous-Move Games with Complete Information. In: Strategy and Game Theory. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32963-5_2
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DOI: https://doi.org/10.1007/978-3-319-32963-5_2
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