Abstract
This chapter analyzes how to find equilibrium behavior when players are allowed to randomize, helping us to identify mixed strategy Nash equilibria (msNE). Finding this type of equilibrium completes our analysis in Chap. 2 where we focused on Nash equilibria involving pure strategies (not allowing for randomizations).
The original version of the chapter was revised: The erratum to the chapter is available at: 10.1007/978-3-319-32963-5_11
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Notes
- 1.
Note that we compare player \( i \)’s expected payoff from mixed strategy \( \sigma_{i} \), \( u_{i} \left( {\sigma_{i} ,\sigma_{ - i} } \right) \), against his expected payoff from selecting pure strategy \( s_{i}^{\prime } \), \( u_{i} \left( {s_{i}^{\prime } ,\sigma_{ - i} } \right) \). We could, instead, compare \( u_{i} \left( {\sigma_{i} ,\sigma_{ - i} } \right) \) against \( u_{i} \left( {\sigma_{i}^{\prime } ,\sigma_{ - i} } \right) \), where \( \sigma_{i}^{\prime } \ne \sigma_{i} \). However, for player \( i \) to play mixed strategy \( \sigma_{i}^{\prime } \), he must be indifferent between at least two pure strategies, e.g., \( s_{i}^{\prime } \) and \( s_{i}^{\prime \prime } \). Otherwise, player \( i \) would not be mixing, but choosing a pure strategy. Hence, his indifference between pure strategies \( s_{i}^{\prime } \) and \( s_{i}^{\prime \prime } \) entails that \( u_{i} \left( {s_{i}^{\prime } ,\sigma_{ - i} } \right) = u_{i} \left( {s_{i}^{\prime \prime } ,\sigma_{ - i} } \right) \), implying that it suffices to check if mixed strategy \( \sigma_{i} \) yields a higher expected payoff than all pure strategies that player \( i \) could use in his randomization. (This is a convenient result, as we will not need to compare the expected payoff of mixed strategy \( \sigma_{i} \) against all possible mixed strategies \( \sigma_{i}^{\prime } \ne \sigma_{i} \); which would entail studying the expected payoff from all feasible randomizations.)
- 2.
Nonetheless, in the specific case in which x = 30, the payoff matrix in Fig. 3.11 coincides with that in Exercise 3.2, and the probability q that makes Firm 1 indifferent reduces to \( q = \frac{25 - 30}{20 - 30} = \frac{ - 5}{ - 10} = \frac{1}{2} \), as that in Exercise 3.2.)
- 3.
The denominator is given by the probability of outcomes (D, L) and (D, R), i.e., \( r + \left( {1 - p - q - r} \right) = 1 - p - q ; \) while the numerator reflects, respectively, the probability of outcome (D, L), r, or outcome (D, R), \( 1 - p - q - r \); as depicted in the matrix of Fig. 3.38.
- 4.
Similarly as for Player 1, the denominator represents the probability of outcomes (U, R) and (D, R), i.e., \( q + \left( {1 - p - q - r} \right) = 1 - p - r; \) while the numerator indicates, respectively, the probability of outcome (U, R), q, or (D, R), \( 1 - q - p - r; \) as illustrated in the matrix of Fig. 3.38.
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© 2016 Springer International Publishing Switzerland
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Munoz-Garcia, F., Toro-Gonzalez, D. (2016). Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria. In: Strategy and Game Theory. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32963-5_3
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DOI: https://doi.org/10.1007/978-3-319-32963-5_3
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