On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria

Berthelsen, Marie Louisa Tølbøll; Hansen, Kristoffer Arnsfelt

doi:10.1007/s00224-022-10080-1

Cited by 7 publications

(6 citation statements)

References 23 publications

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“…Yet, it is also relevant for studying polytopes [48,26]. There is a series of papers related to Nash-Equilibria [9,52,30,12,11 Covering polygons with convex polygons [1], and training neural networks [3,10,62]. Practical Implications.…”

Section: Scopementioning

confidence: 99%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

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A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

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Section: Scopementioning

confidence: 99%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

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“…A (1) ) be an m-person symmetric game and A defined as (4). We construct a tensor complementary problem as follows, y 0, Ay m−1 − 1 n 0, y (Ay…”

Section: Let G = ([M]; [N];mentioning

confidence: 99%

“…Theorem 3. Suppose that G = ([m]; [n]; A (1) ) be an m-person symmetric game and A defined as (4). 1) ), then y * ∈ SOL (−1 n , A), where…”

Section: Let G = ([M]; [N];mentioning

confidence: 99%

“…For example, Nash [1,2] proposed a very important concept of equilibrium, called Nash equilibrium, which is a strategy profile in which each player's strategy is an optimal response to the strategies of the other players. This concept has a nice property that there is at least one Nash equilibrium for every finite game, and every finite symmetric game leads to a symmetric Nash equilibrium (an equilibrium in which all players use the same strategy) [1][2][3][4]. Some researchers were devoted to symmetric games; for additional statements, one can refer to [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

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Computing Nash Equilibria for Multiplayer Symmetric Games Based on Tensor Form

Liu¹,

Liao²

2023

Preprint

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In an $m$-person symmetric game, all players are identical and indistinguishable. In this paper, we find that the payoff tensor of the player $k$ in an $m$-person symmetric game is $k$-mode symmetric, and the payoff tensors of two different individuals are the transpose of each other. Furthermore, we reformulate the $m$-person symmetric game as a tensor complementary problem and demonstrate that locating a symmetric Nash equilibrium is equivalent to finding a solution to the resulting tensor complementary problem. Finally, we use the hyperplane projection algorithm to solve the resulting tensor complementary problem, and we present some numerical results to find the symmetric Nash equilibrium.

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“…For example, Nash [1,2] proposed a very important concept of equilibrium, called the Nash equilibrium, which is a strategy profile in which each player's strategy is an optimal response to the strategies of the other players. This concept has the nice property that there is at least one Nash equilibrium for every finite game, and every finite symmetric game leads to a symmetric Nash equilibrium (an equilibrium in which all players use the same strategy) [1][2][3][4]. Some researchers were devoted to symmetric games; for additional statements, one can refer to [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Computing Nash Equilibria for Multiplayer Symmetric Games Based on Tensor Form

Liu

Liao

2023

Mathematics

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In an m-person symmetric game, all players are identical and indistinguishable. In this paper, we find that the payoff tensor of the player k in an m-person symmetric game is k-mode symmetric, and the payoff tensors of two different individuals are the transpose of each other. Furthermore, we reformulate the m-person symmetric game as a tensor complementary problem and demonstrate that locating a symmetric Nash equilibrium is equivalent to finding a solution to the resulting tensor complementary problem. Finally, we use the hyperplane projection algorithm to solve the resulting tensor complementary problem, and we present some numerical results to find the symmetric Nash equilibrium.

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On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria

Cited by 7 publications

References 23 publications

On Classifying Continuous Constraint Satisfaction Problems

On Classifying Continuous Constraint Satisfaction Problems

Computing Nash Equilibria for Multiplayer Symmetric Games Based on Tensor Form

Computing Nash Equilibria for Multiplayer Symmetric Games Based on Tensor Form

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