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.

Two-Player Non-Zero-Sum Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Nash Equilibrium ◽  
Security Policy ◽  
The Other ◽  
Bimatrix Games ◽  
Zero Sum Games ◽  
Best Response ◽  
Key Concepts ◽  
Player Game ◽  
Zero Sum ◽  
Action Spaces

This chapter defines a number of key concepts for non-zero-sum games involving two players. It begins by considering a two-player game G in which two players P₁ and P₂ are allowed to select policies within action spaces Γ‎₁ and Γ‎₂, respectively. Each player wants to minimize their own outcome, and does not care about the outcome of the other player. The chapter proceeds by discussing the security policy and Nash equilibrium for two-player non-zero-sum games, bimatrix games, admissible Nash equilibrium, and mixed policy. It also explores the order interchangeability property for Nash equilibria in best-response equivalent games before concluding with practice exercises and their corresponding solutions, along with additional exercises.

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.

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.

John von Neumann's Contribution to Modern Game Theory

2004 ◽  
Vol 54 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Ferenc Forgó
Keyword(s):  
Game Theory ◽  
Cooperative Games ◽  
Minimax Theorem ◽  
Stable Set ◽  
Zero Sum Games ◽  
Side Payments ◽  
Basic Concepts ◽  
Definition Of ◽  
Zero Sum ◽  
Abstract Games

The paper gives a brief account of von Neumann's contribution to the foundation of game theory: definition of abstract games, the minimax theorem for two-person zero-sum games and the stable set solution for cooperative games with side payments. The presentation is self-contained, uses very little mathematical formalism and caters to the nonspecialist. Basic concepts and their implications are in focus. It is also indicated how von Neumann's groundbreaking work initiated further research, and a few unsolved problems are also mentioned.

VON NEUMANN, VILLE, AND THE MINIMAX THEOREM

2010 ◽  
Vol 12 (02) ◽  
pp. 115-137 ◽  
Author(s):  
HICHEM BEN-EL-MECHAIEKH ◽  
ROBERT W. DIMAND
Keyword(s):  
Fixed Point ◽  
Minimax Theorem ◽  
Von Neumann ◽  
Zero Sum Games ◽  
Coincidence Theory ◽  
Point Solution ◽  
Zero Sum ◽  
Academy Of Sciences ◽  
First Time

Von Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero sum games) in 1928. While his second article on the minimax theorem, stating the proof, has long been translated from German, his first announcement of his result (communicated in French to the Academy of Sciences in Paris by Borel, who had posed the problem settled by Von Neumann's proof) is translated here for the first time. The proof presented by Von Neumann and Morgenstern (1944) is not Von Neumann's rather involved proof of 1928, but is based on what they called "The Theorem of the Alternative for Matrices" which is in essence a reformulation of an elegant and elementary result by Borel's student Jean Ville in 1938. Ville's argument was the first to bring to light the simplifying role of convexity and to highlight the connection between the existence of minimax and the solvability of systems of linear inequalities. It by-passes nontrivial topological fixed point arguments and allows the treatment of minimax by simpler geometric methods. This approach has inspired a number of seminal contributions in convex analysis including fixed point and coincidence theory for set-valued mappings. Ville's contributions are discussed briefly and von Neuman's original communication, Ville's note, and Borel's commentary on it are translated here for the first time.

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