Abstract
This chapter examines again contexts of incomplete information but in sequential move games. Unlike simultaneous-move settings, sequential moves allow for players’ actions to convey or conceal the information they privately observe to players acting in subsequent stages and who did not have access to such information (uninformed players). That is, we explore the possibility that players’ actions may signal certain information to other players acting latter on in the game.
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Notes
- 1.
Note that player 2 responds with U after observing that player 1 chooses Right, and that such response cannot be made conditional on player 1’s type (since player 2 does not observe his opponent’s type). In other words, player 2 responds with U both when player 1’s type is high and when it is low (for this reason, both arrows labeled with U are shaded in Fig. 9.3 in the upper and lower part of the game tree).
- 2.
Importantly, the low-type of player 1 finds R′ to be strictly dominated by L′, since R′ yields a lower payoff than L′ regardless of player 2’s response. This is an important result, since it allows us to discard all strategy profiles in which R′ is involved, thus leaving us with only two potential candidates for PBE: the separating strategy profile RL′ and the pooling strategy profile LL′. Nonetheless, and as a practice, this exercise examined each of the four possible strategy profiles in this game.
- 3.
This game presents a simplified version of Spence’s (1973) labor market signaling game, where the worker can be of only two types (either high or low productivity), and the firm manager can only respond with two possible actions (either hiring the candidates as a manager, thus offering him a high salary, or as a cashier, at a lower salary). A richer version of this model, with a continuum of education levels and a continuum of salaries, is presented in the next chapter.
- 4.
This condition allows for different combinations of off-the-equilibrium beliefs. For instance, when γ1 < 1/2 but γ2 > 1/2, the uninformed firm is essentially placing a large probability weight on the off-the-equilibrium message of low price originating from a medium-cost firm, and not from a low- or high-cost firm. If, instead, γ1 = 1/2 but γ2 = 0, the uninformed firm believes that the off-the-equilibrium message of low price is equally likely to originate from a low- or a high-cost firm.
- 5.
References
Cho, I. and Kreps, D. (1987). Signaling games and stable equilibrium. Quarterly Journal of Economics, 102, 179–222.
Spence, M. (1973). Job market signaling. Quarterly Journal of Economics, 87, 355–374.
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Munoz-Garcia, F., Toro-Gonzalez, D. (2019). Perfect Bayesian Equilibrium and Signaling Games. In: Strategy and Game Theory. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-11902-7_9
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DOI: https://doi.org/10.1007/978-3-030-11902-7_9
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