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 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 Random Actions in Experimental Zero-Sum Games

2021 ◽  
Vol 13 (1(J)) ◽  
pp. 69-81
Author(s):  
Jung S. You
Keyword(s):  
Nash Equilibrium ◽  
Mixed Strategy ◽  
Theoretical Models ◽  
Zero Sum Games ◽  
Strategy Equilibrium ◽  
Mixed Strategy Nash Equilibrium ◽  
Matching Pennies ◽  
Business Politics ◽  
The Game Theory ◽  
Zero Sum

A mixed strategy, a strategy of unpredictable actions, is applicable to business, politics, and sports. Playing mixed strategies, however, poses a challenge, as the game theory involves calculating probabilities and executing random actions. I test i.i.d. hypotheses of the mixed strategy Nash equilibrium with the simplest experiments in which student participants play zero-sum games in multiple iterations and possibly figure out the optimal mixed strategy (equilibrium) through the games. My results confirm that most players behave differently from the Nash equilibrium prediction for the simplest 2x2 zero-sum game (matching-pennies) and 3x3 zero-sum game (e.g., the rock-paper-scissors game). The results indicate the need to further develop theoretical models that explain a non-Nash equilibrium behavior.

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.

scholarly journals Last minute panic in zero sum games

2019 ◽  
Vol 25 ◽  
pp. 25
Author(s):  
Stefan Ankirchner ◽  
Christophette Blanchet-Scalliet ◽  
Kai Kümmel
Keyword(s):  
Nash Equilibrium ◽  
Theoretical Model ◽  
Unique Solution ◽  
Numerical Experiments ◽  
Strong Impact ◽  
Isaacs Equation ◽  
Sports Teams ◽  
Zero Sum Games ◽  
Set Up ◽  
Zero Sum

We set up a game theoretical model to analyze the optimal attacking intensity of sports teams during a game. We suppose that two teams can dynamically choose among more or less offensive actions and that the scoring probability of each team depends on both teams’ actions. We assume a zero sum setting and characterize a Nash equilibrium in terms of the unique solution of an Isaacs equation. We present results from numerical experiments showing that a change in the score has a strong impact on strategies, but not necessarily on scoring intensities. We give examples where strategies strongly depend on the score, the scoring intensities not at all.

Solving Two-Person Zero-Sum Game by Matlab

2011 ◽  
Vol 50-51 ◽  
pp. 262-265 ◽  
Author(s):  
Yan Mei Yang ◽  
Yan Guo ◽  
Li Chao Feng ◽  
Jian Yong Di
Keyword(s):  
Game Theory ◽  
Linear Programming ◽  
Special Class ◽  
The Other ◽  
Matrix Game ◽  
Matlab Software ◽  
Zero Sum Games ◽  
Programming Techniques ◽  
Zero Sum

In this article we present an overview on two-person zero-sum games, which play a central role in the development of the theory of games. Two-person zero-sum games is a special class of game theory in which one player wins what the other player loses with only two players. It is difficult to solve 2-person matrix game with the order m×n(m≥3,n≥3). The aim of the article is to determine the method on how to solve a 2-person matrix game by linear programming function linprog() in matlab. With linear programming techniques in the Matlab software, we present effective method for solving large zero-sum games problems.

Maximum Drawdown and Risk Tolerances

2015 ◽  
Vol 18 (01) ◽  
pp. 1550003
Author(s):  
Mohammad Reza Tavakoli Baghdadabad
Keyword(s):  
Lower Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Management Styles ◽  
Portfolio Returns ◽  
Investment Opportunity Set ◽  
Conventional Models ◽  
Maximum Drawdown ◽  
Mean Variance ◽  
Special Case

Due to the numerous studies of asymmetric portfolio returns, asymmetric risk measures have widely been used in risk management with extensive uses on the methodology of n-degree lower partial moment (LPM). Unlike the initial studies, we use the risk measure of n-degree maximum drawdown, which is a special case of n-degree LPM, to investigate the reduction impacts of n-degree maximum drawdown risk on risk tolerances generated by management styles from US equity-based mutual funds. We found that skewness does not impose any significant problems on the model of n-degree maximum drawdown. Thus, the tolerance effect of maximum drawdown risk in the n-degree M-DRM models is a decrease in fund returns. The n-degree CM-DRM optimization model decreased investors' risk more than two conventional models. Thus, the M-DRM can be accommodated with risk-averse investors' approach. The efficient set of mean-variance choices from the investment opportunity set, as described by Markowitz, shows that the n-degree CM-DRM algorithms create this set with lower risk than other algorithms. It implies that the mean-variance opportunity set generated by the n-degree CM-DRM creates lower risk for a given return than covariance and CLPM.

scholarly journals Justifying optimal play via consistency

2019 ◽  
Vol 14 (4) ◽  
pp. 1185-1201
Author(s):  
Florian Brandl ◽  
Felix Brandt
Keyword(s):  
Nash Equilibrium ◽  
Minimax Theorem ◽  
Bimatrix Games ◽  
Zero Sum Games ◽  
Zero Sum ◽  
Normative Foundations ◽  
Coherent Behavior

Developing normative foundations for optimal play in two‐player zero‐sum games has turned out to be surprisingly difficult, despite the powerful strategic implications of the minimax theorem. We characterize maximin strategies by postulating coherent behavior in varying games. The first axiom, called consequentialism, states that how probability is distributed among completely indistinguishable actions is irrelevant. The second axiom, consistency, demands that strategies that are optimal in two different games should still be optimal when there is uncertainty regarding which of the two games will actually be played. Finally, we impose a very mild rationality assumption, which merely requires that strictly dominated actions will not be played. Our characterization shows that a rational and consistent consequentialist who ascribes the same properties to his opponent has to play maximin strategies. This result can be extended to characterize Nash equilibrium in bimatrix games whenever the set of equilibria is interchangeable.

On the Size of Winning Coalitions

1974 ◽  
Vol 68 (2) ◽  
pp. 505-518 ◽  
Author(s):  
Kenneth A. Shepsle
Keyword(s):  
Coalition Structure ◽  
Solution Theory ◽  
Size Principle ◽  
Zero Sum Games ◽  
Present Essay ◽  
Reasonable Solution ◽  
Person Game ◽  
Zero Sum ◽  
Special Case

A recent note by Robert Butterworth is critical of William Riker's size principle on several important grounds. There is, however, an important omission in his analysis which this present essay aims to correct. The author goes on to tie assertions about coalition structure in n-person zero-sum games to a solution theory for such games. In the appendix to this essay the general five-person game, of which Butterworth's game is a special case, is considered in some detail. The effect, with one reasonable solution theory, is a favorable appraisal of the size principle.

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