Abstract
Hierarchical games with strategic interactions such as the Stackelberg two-stage game epitomize a standard economic application of bilevel optimization problems. In this paper, we survey certain properties of multiple leader–follower noncooperative games, which enable the basic Stackelberg duopoly game to encompass a larger number of decision makers at each level. We focus notably on the existence, uniqueness and welfare properties of these multiple leader–follower games. We also study how this particular bilevel optimization game can be extended to a multi-level decision setting.
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Notes
- 1.
The book was published in 1934 in German, but was translated into English in 1952 by Oxford University Press and 2011 by Springer. We refer to the 2011 version, as it corresponds to the original 1934 book.
- 2.
To the best of our knowledge, the first extension of the Stackelberg duopoly was introduced by Leitmann [29], who considered a model with one leader and several followers. This was further developed by Murphy et al. [34]. It is worth noting that Stackelberg [38] had already envisaged the possibility of several market participants (see Chap. 3).
- 3.
Hamilton and Slutsky [16] provide theoretical foundations for endogenous timing in duopoly games and for the Stackelberg market outcome.
- 4.
One difficulty stems from the fact the followers’ optimal decision mappings may be mutually inconsistent (Julien [22]).
- 5.
The same problem could be rewritten as follows. Let the objective of each firm be written as \(-\Pi _{F}^{j}(x_{F}^{j},X^{-j})=C_{F}^{j}(x_{F}^{j})-p(x_{F}^{j}+X^{-j})x_{F}^{j}\) , for \(j\in \mathcal {F}_{F}\). Then, the follower’s problem might be written as \(\phi ^{j}({\mathbf {x}}_{F}^{-j},{\mathbf {x}}_{L}):=\min \{-\Pi _{L}^{i}(x_{L}^{i},X^{-i}):({\mathbf {x}}_{F}^{-j},{\mathbf {x}}_{L})\in \prod \limits _{-j}\mathcal {S}_{F}^{-j}\times \prod \limits _{i}\mathcal {S} _{L}^{i}\), \(x_{F}^{j}\in \mathcal {S}_{F}^{j}\}\). A Nash equilibrium has to be sought out between followers.
- 6.
The payoff function is strictly concave and the strategy set is compact and convex.
- 7.
This is one critical difference with the standard duopoly game in which the optimal decision of the follower coincides with their best response. Julien [22] provides a consistency condition which helps determine each optimal decision as a function of the strategy profile for the leaders. Indeed, we give a sufficient nondegeneracy condition on the determinant of the Jacobian matrix associated with the set of equations that allows us to implicitly define the best response mappings. Under this condition, the set of equations which implicitly determines the best responses is a variety of the required dimension, that is, the corresponding vector mapping which defines this set of equations is a C 1 -diffeomorphism. Here this criterion is satisfied as long as Assumptions 2.2.1 and 2.2.2 both hold. These assumptions can be weakened. It is worth noting that our notion of consistency differs from the notion of price consistency in Leyffer and Munson [30] that results in a square nonlinear complementarity problem.
- 8.
It is possible to show that the best responses are not increasing, so the game displays actions which are strategic substitutes. Please note that the condition is sufficient, so strategic complementarities could exist provided they are not too strong.
- 9.
The CE supplies are given by \(((x_{L}^{i})^{\ast },(x_{F}^{j})^{\ast })=(\alpha X^{\ast },(1-\alpha )X^{\ast })\), with α ∈ (0, 1). In what follows, we consider the symmetric outcome for which \(\alpha =\frac {1}{2 }\).
- 10.
- 11.
When the slope of the aggregate best response is zero, then the SNE can coincide with the CNE (see notably Julien [21]).
- 12.
Indeed, \(\frac {\partial S_{C}(X)}{\partial X}=-X\frac {dp(X)}{dX}>0\), with \( S_{C}(X):=\int _{0}^{X}p(z)dz-p(X)X\). In addition, if we let \( S_{P}(X):=p(X)(\vartheta _{L}\sum _{i=1}^{n_{L}}s_{L}^{i}+\vartheta _{F}\sum _{j=1}^{n_{F}}s_{F}^{j})X-\sum _{i=1}^{n_{L}}C_{L}^{i}(s_{L}^{i} \vartheta _{L}X)-\sum _{j=1}^{n_{F}}C_{F}^{j}(s_{F}^{j}\vartheta _{F}X)\), then \(\frac {dS_{P}(X)}{dX}=p(X)+X\frac {dp(X)}{dX}-[\vartheta _{L}\sum _{i=1}^{n_{L}}s_{L}^{i}\frac {dC_{L}^{i}(s_{L}^{i}\vartheta _{L}X)}{dX }+\vartheta _{F}\sum _{j=1}^{n_{F}}s_{F}^{j}\frac {dC_{F}^{j}(s_{F}^{j} \vartheta _{F}X)}{dX}]<0\) (from Assumption 2.2.2b).
- 13.
The Lerner index for any decision maker is defined in an SNE as the ratio between the excess of the price over the marginal cost and the price, that is, \(L:=\frac {p(X)-\frac {dc(x)}{dx}}{p(X)}\).
- 14.
- 15.
Lafay [28] shows that when constant marginal costs differ among firms, the price contribution by an additional entrant may not be negative since the strategies of all firms are modified when a firm no longer enters the market.
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Acknowledgements
The authors acknowledge an anonymous referee for her/his helpful comments, remarks and suggestions on an earlier version. Any remaining deficiencies are ours.
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Bazin, D., Julien, L., Musy, O. (2020). On Stackelberg–Nash Equilibria in Bilevel Optimization Games. In: Dempe, S., Zemkoho, A. (eds) Bilevel Optimization. Springer Optimization and Its Applications, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-030-52119-6_2
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