COMPACT REPRESENTATIONS OF SEARCH IN COMPLEX DOMAINS

2005 ◽  
Vol 07 (01) ◽  
pp. 73-90
Author(s):  
YOSI BEN-ASHER ◽  
EITAN FARCHI
Keyword(s):  
Nash Equilibrium ◽  
Search Strategy ◽  
Matrix Game ◽  
Set Systems ◽  
Compact Representations ◽  
Mathematical Generalization ◽  
General Search ◽  
Pure Strategies ◽  
Zero Sum ◽  
Search Domain

We introduce a new zero-sum matrix game for modeling search in structured domains. In this game, one player tries to find a "bug" while the other tries to hide it. Both players exploit the structure of the "search" domain. Intuitively, this search game is a mathematical generalization of the well known binary search. The generalization is from searching over totally ordered sets to searching over more complex search domains such as trees, partial orders and general set systems. As there must be one row for every search strategy, and there are exponentially many ways to search even in very simple search domains, the game's matrix has exponential size ("space"). In this work we present two ways to reduce the space required to compute the Nash value (in pure strategies) of this game: • First we show that a Nash equilibrium in pure strategies can be computed by using a backward induction on the matrices of each "part" or sub structure of the search domain. This can significantly reduce the space required to represent the game. • Next, we show when general search domains can be represented as DAGs (Directed Acycliqe Graphs). As a result, the Nash equilibrium can be directly computed using the DAG. Consequently the space needed to compute the desired search strategy is reduced to O(n2) where n is the size of the search domain.

COMPACT REPRESENTATIONS OF SEARCH IN COMPLEX DOMAINS

2005 ◽  
Vol 07 (02) ◽  
pp. 171-188 ◽  
Author(s):  
YOSI BEN-ASHER ◽  
EITAN FARCHI
Keyword(s):  
Nash Equilibrium ◽  
Search Strategy ◽  
Matrix Game ◽  
Set Systems ◽  
Compact Representations ◽  
Mathematical Generalization ◽  
General Search ◽  
Pure Strategies ◽  
Zero Sum ◽  
Search Domain

We introduce a new zero-sum matrix game for modeling search in structured domains. In this game, one player tries to find a "bug" while the other tries to hide it. Both players exploit the structure of the "search" domain. Intuitively, this search game is a mathematical generalization of the well known binary search. The generalization is from searching over totally ordered sets to searching over more complex search domains such as trees, partial orders and general set systems. As there must be one row for every search strategy, and there are exponentially many ways to search even in very simple search domains, the game's matrix has exponential size ("space"). In this work we present two ways to reduce the space required to compute the Nash value (in pure strategies) of this game: • First we show that a Nash equilibrium in pure strategies can be computed by using a backward induction on the matrices of each "part" or sub structure of the search domain. This can significantly reduce the space required to represent the game. • Next, we show when general search domains can be represented as DAGs (Directed Acycliqe Graphs). As a result, the Nash equilibrium can be directly computed using the DAG. Consequently the space needed to compute the desired search strategy is reduced to O(n2) where n is the size of the search domain.

scholarly journals An Algorithmic Procedure for Finding Nash Equilibrium

2020 ◽  
Vol 40 (1) ◽  
pp. 71-85
Author(s):  
HK Das ◽  
T Saha
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Payoff Matrix ◽  
Matrix Game ◽  
Perfect Equilibrium ◽  
Repeated Use ◽  
Definition Of ◽  
Zero Sum ◽  
Mathematical Construction ◽  
Algorithmic Procedure

This paper proposes a heuristic algorithm for the computation of Nash equilibrium of a bi-matrix game, which extends the idea of a single payoff matrix of two-person zero-sum game problems. As for auxiliary but making the comparison, we also introduce here the well-known definition of Nash equilibrium and a mathematical construction via a set-valued map for finding the Nash equilibrium and illustrates them. An important feature of our algorithm is that it finds a perfect equilibrium when at the start of all actions are played. Furthermore, we can find all Nash equilibria of repeated use of this algorithm. It is found from our illustrative examples and extensive experiment on the current phenomenon that some games have a single Nash equilibrium, some possess no Nash equilibrium, and others had many Nash equilibria. These suggest that our proposed algorithm is capable of solving all types of problems. Finally, we explore the economic behaviour of game theory and its social implications to draw a conclusion stating the privilege of our algorithm. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 71-85

scholarly journals A Dynkin Game on Assets with Incomplete Information on the Return

2020 ◽  
Author(s):  
Tiziano De Angelis ◽  
Fabien Gensbittel ◽  
Stephane Villeneuve
Keyword(s):  
Nash Equilibrium ◽  
Incomplete Information ◽  
Value Function ◽  
Moving Boundary ◽  
Rate Of Return ◽  
Stopping Sets ◽  
Dynkin Game ◽  
Pure Strategies ◽  
Zero Sum ◽  
The Value Function

This paper studies a two-player zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques, we first reduce the problem to a zero-sum Dynkin game on a bidimensional diffusion (X,Y). Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of (X,Y) to a set with a moving boundary. A detailed description of the stopping sets for the two players is provided along with global C1 regularity of the value function.

How Risky is it to Deviate from Nash Equilibrium?

2016 ◽  
Vol 18 (03) ◽  
pp. 1650006 ◽  
Author(s):  
Irit Nowik
Keyword(s):  
Nash Equilibrium ◽  
Risk Measures ◽  
Risk Measure ◽  
Matrix Game ◽  
Integer Programming Problem ◽  
Zero Sum Games ◽  
Potential Damage ◽  
Zero Sum ◽  
S Model ◽  
Special Case

The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for the potential risk in deviating from the NE strategy in any two person matrix game. We present two approaches regarding the nature of deviations: Strategic and Accidental. Accordingly, we define two models: S-model and T-model. The S-model defines a new game in which players deviate in the least dangerous direction. The risk defined in the T-model can serve as a refinement for the notion of “trembling hand perfect equilibrium” introduced by R. Selten. The risk measures enable testing and evaluating predictions on the behavior of players. For example: do players deviate more from a NE that is less risky? This may be relevant to the design of experiments. We present an Integer programming problem that computes the risk for any given player and NE. In the special case of zero-sum games with a unique strictly mixed NE, we prove that the risks of the players always coincide, even if the game is far from symmetry. This result holds for any norm we use for the size of deviations. We compare our risk measures to the risk measure defined by Harsanyi and Selten which is based on criteria of stability rather than on potential damage. We show that the measures may contradict.

scholarly journals A game strategy model in the digital curling system based on NFSP

2021 ◽  
Author(s):  
Yuntao Han ◽  
Qibin Zhou ◽  
Fuqing Duan
Keyword(s):  
Reinforcement Learning ◽  
Nash Equilibrium ◽  
Action Space ◽  
Learning Networks ◽  
Game Tree ◽  
Continuous Action ◽  
Extensive Game ◽  
Strategy Model ◽  
Zero Sum ◽  
Tree Searching

AbstractThe digital curling game is a two-player zero-sum extensive game in a continuous action space. There are some challenging problems that are still not solved well, such as the uncertainty of strategy, the large game tree searching, and the use of large amounts of supervised data, etc. In this work, we combine NFSP and KR-UCT for digital curling games, where NFSP uses two adversary learning networks and can automatically produce supervised data, and KR-UCT can be used for large game tree searching in continuous action space. We propose two reward mechanisms to make reinforcement learning converge quickly. Experimental results validate the proposed method, and show the strategy model can reach the Nash equilibrium.

scholarly journals Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 132
Author(s):  
Valery Y. Glizer
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Quadratic Functional ◽  
Linear Quadratic ◽  
Feedback Controls ◽  
Cheap Control ◽  
Optimal Values ◽  
Definition Of ◽  
Zero Sum ◽  
Equilibrium Sequence

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

Creative Design Methodology and the SIT Method

1997 ◽  
Author(s):  
Roni Horowitz ◽  
Oded Maimon
Keyword(s):  
Design Methodology ◽  
Search Strategy ◽  
Sufficient Conditions ◽  
Strategy Selection ◽  
Step Procedure ◽  
Creative Solutions ◽  
General Search ◽  
Problem Reformulation ◽  
Creative Problem ◽  
The Given

Abstract The paper presents SIT (Structured Inventive Thinking) — a structured method for enhancing creative problem solving in engineering design. The method is a three step procedure: problem reformulation, general search strategy selection, and an application of idea provoking techniques. The most innovative part of the method is the problem reformulation stage. The given problem is modified through the application of objectively defined and empirically tested set of sufficient conditions for creative solutions. The paper describes the sufficient conditions and the empirical study that demonstrates their appropriateness. Then the whole SIT mechanism is presented with illustrative examples.

scholarly journals Max-min solution approach for multi-objective matrix game with fuzzy goals

2016 ◽  
Vol 26 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Sandeep Kumar
Keyword(s):  
Linear Programming ◽  
Linear Programming Problem ◽  
Optimization Problem ◽  
Matrix Game ◽  
Game Model ◽  
Solution Approach ◽  
Multi Objective ◽  
Fuzzy Goals ◽  
Fuzzy Goal ◽  
Zero Sum

In this paper, we consider a multi-objective two person zero-sum matrix game with fuzzy goals, assuming that each player has a fuzzy goal for each of the payoffs. The max-min solution is formulated for this multi-objective game model, in which the optimization problem for each player is a linear programming problem. Every developed model for each player is demonstrated through a numerical example.

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