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  • 1
    Electronic Resource
    Electronic Resource
    On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games (1979)
    Springer
    Journal of optimization theory and applications 28 (1979), S. 49-76 
    Details
    ISSN: 1573-2878
    Keywords: Nonzero sum linear-quadratic games ; Nash strategies ; coupled Riccati equations ; Brower's fixed-point theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933600
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Sufficient conditions for Stackelberg and Nash strategies with memory (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 233-260 
    Details
    ISSN: 1573-2878
    Keywords: Stackelberg differential games ; Nash differential games ; strategies with memory ; sufficiency conditions for game strategies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Sufficiency conditions for Stackelberg strategies for a class of deterministic differential games are derived when the players have recall of the previous trajectory. Sufficient conditions for Nash strategies when the players have recall of the trajectory are also derived. The state equation is linear, and the cost functional is quadratic. The admissible strategies are restricted to be affine in the information available.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934113
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On linear-quadratic Gaussian continuous-time Nash games (1984)
    Springer
    Journal of optimization theory and applications 42 (1984), S. 525-549 
    Details
    ISSN: 1573-2878
    Keywords: Stochastic Nash games ; linear-quadratic games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two classes of linear-quadratic Gaussian continuous-time Nash games are considered. Their main characteristic is that the σ-fields with respect to which the control actions of the players have to be measurable at each instance of time are not affected by the past controls of the players. We show that, if a solution exists, then there exists a solution linear in the information, and also show how to construct all the solutions. Several conditions guaranteeing the existence of a unique solution are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934565
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On the linear-quadratic, closed-loop, no-memory Nash game (1984)
    Springer
    Journal of optimization theory and applications 42 (1984), S. 551-560 
    Details
    ISSN: 1573-2878
    Keywords: Nash games ; linear-quadratic games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear closed-loop no-memory strategies for the LQ Nash game are considered. We exhibit a class of such problems with the property that the solution exists for any finite time interval; for the infinite time case, there exist none or a unique or many solutions, depending on the choice of the parameters. In addition, the limit of the finite time solution as the time interval increases does not have to yield the infinite time case solution. A geometric formulation of the coupled algebraic Riccati equation is given. This formulation seems to be an interesting starting point for a thorough study of these equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934566
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    On the probability of existence of pure equilibria in matrix games (1995)
    Springer
    Journal of optimization theory and applications 87 (1995), S. 419-439 
    Details
    ISSN: 1573-2878
    Keywords: Matrix games ; pure equilibria ; mixed equilibria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We examine the probability that a randomly chosen matrix game admits pure equilibria and its behavior as the number of actions of the players or the number of players increases. We show that, for zero-sum games, the probability of having pure equilibria goes to zero as the number of actions goes to infinity, but it goes to a nonzero constant for a two-player game. For many-player games, if the number of players goes to infinity, the probability of existence of pure equilibria goes to zero even if the number of actions does not go to infinity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192572
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    On the uniqueness of Nash strategies for a class of analytic differential games (1979)
    Springer
    Journal of optimization theory and applications 27 (1979), S. 309-314 
    Details
    ISSN: 1573-2878
    Keywords: Nash games ; partial differential equations ; differential games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The uniqueness of Nash equilibria is shown for the case where the data of the problem are analytic functions and the admissible strategy spaces are restricted to analytic functions of the current state and time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933234
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Performance versus informativeness in linear-quadratic Gaussian noncooperative games (1988)
    Springer
    Journal of optimization theory and applications 57 (1988), S. 161-187 
    Details
    ISSN: 1573-2878
    Keywords: Nash games ; Stackelberg games ; information structure ; informational properties ; informationally stronger players ; Kalman filter
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we study the impact of informativeness on the performance of linear quadratic Gaussian Nash and Stackelberg games. We first show that, in two-person static Nash games, if one of the players acquires more information, then this extra information is beneficial to him, provided that it is orthogonal to both players' information. A special case is that when one of the players is informationally stronger than the other, then any new information is beneficial to him. We then show that a similar result holds for dynamic Nash games. In the dynamic games, the players use strategies that are linear functions of the current estimates of the state, generated by two Kalman filters. The same properties are proved to hold in static and feedback Stackelberg games as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939334
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    On the probability of existence of pure equilibria in matrix games (1996)
    Springer
    Journal of optimization theory and applications 91 (1996), S. 729-730 
    Details
    ISSN: 1573-2878
    Keywords: Matrix games ; pure equilibria ; mixed equilibria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Previous work related to Ref. 1, not known to the author, is reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02190129
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  • 9
    Electronic Resource
    Electronic Resource
    A markov chain game with dynamic information (1988)
    Springer
    Journal of optimization theory and applications 59 (1988), S. 467-486 
    Details
    ISSN: 1573-2878
    Keywords: Markov games ; zero-sum games ; incomplete information ; pursuit-evasion ; mixed strategies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two players, not knowing each other's position, move in a domain and can flash a searchlight. The game terminates when one player is caught within the area illuminated by the flash of the other. However, if this first player is not in this area, then the other player has disclosed his position to the former one, who may be able to exploit this information. The game is considered on a finite state space and in discrete time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940310
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Learning algorithms for repeated bimatrix Nash games with incomplete information (1989)
    Springer
    Journal of optimization theory and applications 62 (1989), S. 467-488 
    Details
    ISSN: 1573-2878
    Keywords: Nash games ; learning algorithms ; bimatrix games ; games with incomplete information
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to study a particular recursive scheme for updating the actions of two players involved in a Nash game, who do not know the parameters of the game, so that the resulting costs and strategies converge to (or approach a neighborhood of) those that could be calculated in the known parameter case. We study this problem in the context of a matrix Nash game, where the elements of the matrices are unknown to both players. The essence of the contribution of this paper is twofold. On the one hand, it shows that learning algorithms which are known to work for zero-sum games or team problems can also perform well for Nash games. On the other hand, it shows that, if two players act without even knowing that they are involved in a game, but merely thinking that they try to maximize their output using the learning algorithm proposed, they end up being in Nash equilibrium.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939817
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    Paper (German National Licenses)
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