A Characterization of Nash Equilibrium for the Games with Random Payoffs

2018 ◽  
Vol 178 (3) ◽  
pp. 998-1013 ◽  
Author(s):  
Vikas Vikram Singh ◽  
Abdel Lisser
Keyword(s):  
Nash Equilibrium ◽  
Random Payoffs

scholarly journals Multiple Activities in Networks

2018 ◽  
Vol 10 (3) ◽  
pp. 34-85 ◽  
Author(s):  
Ying-Ju Chen ◽  
Yves Zenou, ◽  
Junjie Zhou
Keyword(s):  
Nash Equilibrium ◽  
Network Model ◽  
Network Structure ◽  
Comparative Statics ◽  
Network Density ◽  
Full Characterization ◽  
The One

We consider a network model where individuals exert efforts in two types of activities that are interdependent. These activities can be either substitutes or complements. We provide a full characterization of the Nash equilibrium of this game for any network structure. We show, in particular, that quadratic games with linear best-reply functions aggregate nicely to multiple activities because equilibrium efforts obey similar formulas to that of the one-activity case. We then derive some comparative-statics results showing how own productivity affects equilibrium efforts and how network density impacts equilibrium outcomes. (JEL C72, D11, D85, Z13)

scholarly journals Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1669
Author(s):  
Jun Moon ◽  
Wonhee Kim
Keyword(s):  
Nash Equilibrium ◽  
Mean Field ◽  
Jump Diffusion ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
Field Type ◽  
The Mean ◽  
Zero Sum ◽  
Objective Functional

We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

2015 ◽  
Vol 47 (02) ◽  
pp. 355-377
Author(s):  
Qian Lin
Keyword(s):  
Nash Equilibrium ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Differential Games ◽  
Cost Functionals ◽  
Reflecting Barriers

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

Potential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Nash Equilibrium ◽  
Special Class ◽  
Potential Games ◽  
Action Spaces

This chapter introduces a special class of N-player games, the so-called potential games, for which the Nash equilibrium is guaranteed to exist and is generally easy to find. It begins by considering a game with N players P₁, P₂, . . ., P(subscript N), which are allowed to select policies from the action spaces Γ‎₁, Γ‎₂, . . ., Γ‎(subscript N), respectively. The notation is given for the outcome of the game for the player Pᵢ and all players wanting to minimize their own outcomes. The chapter goes on to discuss identical interests games, minimum vs. Nash equilibrium in potential games, bimatrix potential games, characterization of potential games, and potential games with interval action spaces. It concludes with practice exercises and their corresponding solutions, along with an additional exercise.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

2015 ◽  
Vol 47 (2) ◽  
pp. 355-377 ◽  
Author(s):  
Qian Lin
Keyword(s):  
Nash Equilibrium ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differential Games ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Differential Games ◽  
Cost Functionals ◽  
Reflecting Barriers

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

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