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.

Mixed Policies

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Probability Distribution ◽  
Saddle Point ◽  
Security Policy ◽  
General Type ◽  
Zero Sum Games ◽  
Security Levels ◽  
Mixed Action ◽  
Zero Sum ◽  
Saddle Point Equilibrium ◽  
Action Spaces

This chapter explores the concept of mixed policies and how the notions for pure policies can be adapted to this more general type of policies. A pure policy consists of choices of particular actions (perhaps based on some observation), whereas a mixed policy involves choosing a probability distribution to select actions (perhaps as a function of observations). The idea behind mixed policies is that the players select their actions randomly according to a previously selected probability distribution. The chapter first considers the rock-paper-scissors game as an example of mixed policy before discussing mixed action spaces, mixed security policy and saddle-point equilibrium, mixed saddle-point equilibrium vs. average security levels, and general zero-sum games. It concludes with practice exercises with corresponding solutions and an additional exercise.

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.

N-Player Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Nash Equilibrium ◽  
Normal Form ◽  
Security Policy ◽  
The Other ◽  
Security Level ◽  
Action Spaces

This chapter extends several of the concepts for two-player games to games with N-players. It begins by considering general games with N players P₁, P₂, . . ., P(subscript N), which are allowed to select policies within action spaces Γ‎₁, Γ‎₂, . . ., Γ‎(subscript N). Each player wants to minimize their own outcome, and does not care about the outcome of the other players. The chapter proceeds by discussing the security level, security policy, and Nash equilibrium for N-player games, pure N-player games in normal form, mixed policy for N-player games in normal form, and computation of the completely mixed Nash equilibrium for N-player games. A mixed Nash equilibrium is computed for a different game in which some (or all) players want to maximize instead of minimize the outcome.

The 11–20 Money Request Game: A Level-k Reasoning Study

2012 ◽  
Vol 102 (7) ◽  
pp. 3561-3573 ◽  
Author(s):  
Ayala Arad ◽  
Ariel Rubinstein
Keyword(s):  
The Other ◽  
Best Response ◽  
Starting Point ◽  
Player Game ◽  
Almost All ◽  
The Cost ◽  
Reasoning Study

We study experimentally a new two-player game: each player requests an amount between 11 and 20 shekels. He receives the requested amount and if he requests exactly one shekel less than the other player, he receives an additional 20 shekels. Level-k reasoning is appealing due to the natural starting point (requesting 20) and the straightforward best-response operation. Nevertheless, almost all subjects exhibit at most three levels of reasoning. Two variants of the game demonstrate that the depth of reasoning is not increased by enhancing the attractiveness of the level-0 strategy or by reducing the cost of undercutting the other player.

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 Best response dynamics for continuous zero--sum games

2006 ◽  
Vol 6 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Josef Hofbauer ◽  
◽  
Sylvain Sorin ◽  
Keyword(s):  
Response Dynamics ◽  
Zero Sum Games ◽  
Best Response Dynamics ◽  
Best Response ◽  
Zero Sum

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.

scholarly journals Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games

2019 ◽  
Vol 67 (3) ◽  
pp. 731-743 ◽  
Author(s):  
Lisa Hellerstein ◽  
Thomas Lidbetter ◽  
Daniel Pirutinsky
Keyword(s):  
Search Games ◽  
Zero Sum Games ◽  
Best Response ◽  
Zero Sum
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