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Equilibrium selection in multi-leader-follower games with vertical information

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Abstract

We consider two-stage multi-leader-follower games, called multi-leader-follower games with vertical information, where leaders in the first stage and followers in the second stage choose simultaneously an action, but those chosen by any leader are observed by only one “exclusive” follower. This partial unobservability leads to extensive form games that have no proper subgames but may have an infinity of Nash equilibria. So it is not possible to refine using the concept of subgame perfect Nash equilibrium and, moreover, the concept of weak perfect Bayesian equilibrium could be not useful since it does not prescribe limitations on the beliefs out of the equilibrium path. This has motivated the introduction of a selection concept for Nash equilibria based on a specific class of beliefs, called passive beliefs, that each follower has about the actions chosen by the leaders rivals of his own leader. In this paper, we illustrate the effectiveness of this concept and we investigate the existence of such a selection for significant classes of problems satisfying generalized concavity properties and conditions of minimal character on possibly discontinuous data.

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Notes

  1. In general, a weak perfect Bayesian equilibrium [(also called weak sequential equilibrium in Myerson 1991] may fall to be a subgame perfect Nash equilibrium, as emphasized by Mas-Colell et al. (1995) in Example 9.C.5 for finite extensive games. However, this is not our case because of the absence of proper subgames in the multi-leader\follower  games.

  2. It is worth mentioning that, differently from what may happen in a general extensive game (González-Díaz and Meléndez-Jiménez 2014), in our specific case the system of beliefs is inferred correctly from Bayes’ rule along the path of the equilibrium in pure strategy. This is a consequence of the fact that, if \( {{\varvec{\sigma }}}=( x_1,\ldots , x_k, {\beta }_{1},\ldots , {\beta }_{k})\) is a pure strategy profile, the unique information set of, for example, follower \(F_i\) that is on the path of \({\varvec{\sigma }}\) is the one that is associated with the action \(x_i\); then the probability of being in that information set given that \( {\varvec{\sigma }}\) has been played is equal to 1.

  3. For a proof, one can follow the same reasoning of Proposition 9.C.1 in Mas-Colell et al. (1995).

  4. This is also in line with the fact that the no signaling what you do not know property is a necessary condition for sequential equilibria [for more details see (Fudenberg and Tirole 1991b, Sect.6) or (Fudenberg and Tirole 1991a, p.350)].

  5. For the sake of simplicity we improperly identify the system of beliefs in a sequential equilibrium with passive beliefs: our Definition 4 is independent on how the simultaneous moves of the players are rearranged in the tree representation of the game. So, if we consider the game tree in Fig. 1, our definition of passive beliefs does not distinguish between nodes in an information set of follower \(F_2\) that are reached by two action profiles that differ only on the actions of follower \(F_1\).

  6. That is, if \(\underline{t}_i,\overline{t}_i\in \mathbb {R}\) and \(\underline{t}_i\le \overline{t}_i\) then \(l_i(\mathbf {x},\mathbf {y},\underline{t}_i)\le l_i(\mathbf {x},\mathbf {y},\overline{t}_i)\).

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Acknowledgements

The authors would like to thank Prof. Marco Pagnozzi for the helpful discussions and two anonymous referees whose comments helped to improve and clarify the manuscript. The first author gratefully acknowledges financial support from POR Campania FSE 2014–2020. The second author gratefully acknowledges financial support from Programma STAR Napoli “Equilibrium with ambiguity”.

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Authors and Affiliations

  1. Department of Economics and Statistics, University of Naples Federico II, Via Cinthia, 80126, Naples, Italy

    Maria Carmela Ceparano

  2. Department of Economics and Statistics & Centre for Studies in Economics and Finance, University of Naples Federico II, Via Cinthia, 80126, Naples, Italy

    Jacqueline Morgan

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  1. Maria Carmela Ceparano
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  2. Jacqueline Morgan
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Correspondence to Jacqueline Morgan.

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Ceparano, M.C., Morgan, J. Equilibrium selection in multi-leader-follower games with vertical information. TOP 25, 526–543 (2017). https://doi.org/10.1007/s11750-017-0444-5

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