Mixed strategy - Revision history

Toscool at 13:07, 23 January 2021

2021-01-23T13:07:26Z

Toscool: /* Mixed strategy in Rock, paper, scissors */

2021-01-08T13:23:16Z

‎Mixed strategy in Rock, paper, scissors

Toscool: /* Mixed strategy in Rock, paper, scissors */

2021-01-08T13:22:54Z

‎Mixed strategy in Rock, paper, scissors

Toscool: /* Definition */

2021-01-08T13:19:50Z

‎Definition

Toscool: /* Definition */

2021-01-08T13:14:53Z

‎Definition

Toscool: /* History */

2021-01-08T13:12:09Z

‎History

Toscool: /* Introduction */

2021-01-08T13:09:53Z

‎Introduction

Toscool: /* Introduction */

2021-01-08T13:09:31Z

‎Introduction

Toscool: /* Introduction */

2021-01-08T13:09:00Z

‎Introduction

Toscool at 10:49, 6 January 2021

2021-01-06T10:49:29Z

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 <references/>
 https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/chicken.html
 https://en.wikipedia.org/wiki/Strategy_(game_theory)
 https://saylordotorg.github.io/text_introduction-to-economic-analysis/s17-03-mixed-strategies.html
 https://en.wikipedia.org/wiki/Theory_of_Games_and_Economic_Behavior
 https://www.jstor.org/stable/1969529?seq=1

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 We can note that in the game between Bart and Lisa there is '''no pure Nash equilibrium''', because the gains are directly opposed. if Bart received a big utility, Lisa received a small one. If one player knows what will do his opponent, he will directly win. That is why the game has no Nash equilibrium in '''pure strategy'''.
-==Mixed strategy in Rock, paper, scissors==
+===Mixed strategy in Rock, paper, scissors===
 '''The mixed strategy in this game give you the opportunity to protect yourself against the other.'''

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 We can note that the probability of a collision (Don't, Don't) is just 1/16 in the mixed strategy equilibrium. In the chicken game the mixed strategy equilibrium is more likely than in the [[#Mixed strategy in battle of the sexes|Battle of the sexes]]. The whole point in this game is to find out who will yield, which means that it isn't known in advance. This means that the mixed strategy equilibrium is the more reasonable equilibrium.
-==Mixed strategy in Rock, paper, scissors==
+==Rock, paper, scissors==
 === Definition ===
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 We can note that in the game between Bart and Lisa there is '''no pure Nash equilibrium''', because the gains are directly opposed. if Bart received a big utility, Lisa received a small one. If one player knows what will do his opponent, he will directly win. That is why the game has no Nash equilibrium in '''pure strategy'''.
 '''The mixed strategy in this game give you the opportunity to protect yourself against the other.'''

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 [[File:Chicken1.jpeg|500px|center]]
-The chicken game in which two players drive two very fast cars towards each other from opposite ends of a long straight road. If one of them swerves before the other, he is called a chicken and will have a payoff of -1. Of course, if neither swerves, they will crash and will get both -4. This is the worst possible payoff. The best payoff is to have your opponent be the chicken, so we assign this a value 1. The last possibility is that both drivers swerve. Then, neither has less honor than the other, so this is a better option than being the chicken. We could assign a payoff matrix to this:
+The chicken game is a game in which two players drive two very fast cars towards each other from opposite ends of a long straight road. If one of them swerves before the other, he is called a chicken and will have a payoff of -1. Of course, if neither swerves, they will crash and will get both -4. This is the worst possible payoff. The best payoff is to have your opponent be the chicken, so we assign this a value 1. The last possibility is that both drivers swerve. Then, neither has less honor than the other, so this is a better option than being the chicken. We could assign a payoff matrix to this:
 [[File:Chicken2.jpg|400px|center]]
 There is again two pure Nash equilibrium here: (Swerve, Don't) and (Don't, Swerve), you can find them by using the [[Nash_equilibrium|Nash equilibrium]] course.
 ===Mixed strategy in Chicken game===

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 === Definition ===
 Mixed strategy:
-if the player <math>i</math> has <math>K</math> strategies <math>si1, si2, ...siK</math> available, a mixed strategy is a distribution of probabilities <math>pi=(pi1, pi2, ...piK)</math> where <math>pi1</math> is the probability for <math>i</math> to choose the strategy <math>si1</math>
+if the player <math>i</math> has <math>K</math> strategies <math>si1, si2, ...siK</math> available, a mixed strategy is a distribution of probabilities <math>pi=(pi1, pi2, ...piK)</math> where <math>pi1</math> is the probability for <math>i</math> to choose the strategy <math>si1</math>. Also because the mixed strategy is a distribution of probability there is:
 <math>\sum_{j=1}^K pij = 1</math>

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 === History ===
-The concept of a mixed-strategy was introduced by John von Neumann and Oskar Morgenstern in their 1944 book ''The Theory of Games and Economic Behavior''<ref name="john" />, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions.
+The concept of a mixed-strategy was introduced by John von Neumann and Oskar Morgenstern in their 1944 book ''The Theory of Games and Economic Behavior''<ref name="john" />, but their analysis was restricted to the zero-sum games. They showed that a mixed-strategy Nash equilibrium will always exist for any zero-sum game with a finite set of actions.
-In his famous paper in 1950, John Forbes Nash go further and proved that there is an equilibrium for '''every finite game'''. It can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy.
+In his famous paper in 1950, John Forbes Nash go further and proved that there is an equilibrium for '''every finite game''' and not only the zero-sum game. Now we can divide Nash equilibria into two types.
 === Definition ===

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 |}
-=== History ===
 * A '''pure strategy''' is an unconditional, defined choice that a person makes in a situation or game.
 * A '''mixed strategy''' is an assignment of probability to all choices in the strategy set.
 The concept of a mixed-strategy was introduced by John von Neumann and Oskar Morgenstern in their 1944 book ''The Theory of Games and Economic Behavior''<ref name="john" />, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions.

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 ==Introduction==
 {| class="wikitable" align=right width="220"
 ! scope="col" width="50px" |John Forbes Nash Jr.
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 * May 13, 2015 (aged 86)
-=== History ===
 John Forbes Nash Jr. was an American mathematician who made fundamental contributions mainly to [[Game_theory|Game theory]]. Nash's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life
 |}
 * A '''pure strategy''' is an unconditional, defined choice that a person makes in a situation or game.
 * A '''mixed strategy''' is an assignment of probability to all choices in the strategy set.