On the Generic Structure and Stability of Stackelberg Equilibria

Alberto Bressan; Yilun Jiang

doi:10.1007/s10957-019-01574-z

On the Generic Structure and Stability of Stackelberg Equilibria

Alberto Bressan, Yilun Jiang

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains X⊆ R^m and Y⊆ Rⁿ. Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either X= Y= [0 , 1] , or X= R, Y= Rⁿ with n≥ 1. Using techniques from differential geometry, including a multi-jet version of Thom’s transversality theorem, we prove that, for an open dense set of cost functions F∈ C² and G∈ C³, the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.

Original language	English (US)
Pages (from-to)	840-880
Number of pages	41
Journal	Journal of Optimization Theory and Applications
Volume	183
Issue number	3
DOIs	https://doi.org/10.1007/s10957-019-01574-z
State	Published - Dec 1 2019

All Science Journal Classification (ASJC) codes

Control and Optimization
Management Science and Operations Research
Applied Mathematics

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title = "On the Generic Structure and Stability of Stackelberg Equilibria",

abstract = "We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains X⊆ Rm and Y⊆ Rn. Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either X= Y= [0 , 1] , or X= R, Y= Rn with n≥ 1. Using techniques from differential geometry, including a multi-jet version of Thom{\textquoteright}s transversality theorem, we prove that, for an open dense set of cost functions F∈ C2 and G∈ C3, the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.",

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AU - Jiang, Yilun

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N2 - We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains X⊆ Rm and Y⊆ Rn. Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either X= Y= [0 , 1] , or X= R, Y= Rn with n≥ 1. Using techniques from differential geometry, including a multi-jet version of Thom’s transversality theorem, we prove that, for an open dense set of cost functions F∈ C2 and G∈ C3, the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.

AB - We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains X⊆ Rm and Y⊆ Rn. Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either X= Y= [0 , 1] , or X= R, Y= Rn with n≥ 1. Using techniques from differential geometry, including a multi-jet version of Thom’s transversality theorem, we prove that, for an open dense set of cost functions F∈ C2 and G∈ C3, the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions.

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On the Generic Structure and Stability of Stackelberg Equilibria

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