TY - JOUR
T1 - Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums
AU - Feinstein, Zachary
AU - Rudloff, Birgit
AU - Zhang, Jianfeng
N1 - Publisher Copyright:
© 2021 INFORMS.
PY - 2022/2
Y1 - 2022/2
N2 - Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value ø). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting.We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.
AB - Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value ø). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting.We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.
KW - Closed-loop controls
KW - Dynamic programming principle
KW - Nash equilibrium
KW - Nonzero sum game
KW - Path dependent PDE
KW - Set value
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U2 - 10.1287/moor.2021.1143
DO - 10.1287/moor.2021.1143
M3 - Article
AN - SCOPUS:85125587875
SN - 0364-765X
VL - 47
SP - 616
EP - 642
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 1
ER -