# Dominant Strategies and Nash Equilibria in a 2x2 Game

ConsiderÂ theÂ followingÂ twoÂ personÂ game,Â betweenÂ AÂ andÂ B.Â

Â Â Â Â Â Â Â hold Not hold

Â

Â driveÂ Â Â Â Â Â Â X,Â 2Â Â Â Â Â Â Â Â Â 3,Â ZÂ

Â Â stopÂ Â Â Â Â Â 10,Â YÂ Â Â Â Â Â Â Â Â 2,Â 6Â

Â

(a.) GivenÂ anÂ exampleÂ ofÂ valuesÂ forÂ X,Â YÂ andÂ ZÂ soÂ thatÂ thereÂ isÂ aÂ dominantÂ strategyÂ equilibrium.Â

(b.) IfÂ XÂ =Â 8,Â YÂ =Â 4Â andÂ ZÂ =Â 0,Â howÂ manyÂ NashÂ equilibriumÂ doesÂ theÂ gameÂ have?Â

(c.) PickÂ valuesÂ forÂ X,Â YÂ andÂ ZÂ toÂ getÂ theÂ mostÂ possibleÂ NashÂ equilibriumÂ youÂ canÂ inÂ thisÂ game.Â Â HowÂ manyÂ NEÂ doÂ youÂ haveÂ andÂ whatÂ areÂ they?Â

https://brainmass.com/economics/oligopoly/dominant-strategies-nash-equilibria-game-618202

#### Solution Preview

Question 6c

This scenario represents the Drive or Stop options for Person A, while Person B is faced with the Hold or Not Hold options.

Player B

Hold Not Hold

Player A Drive X, 2 3, Z

Stop 10, Y 2, 6

a) In this example, I would like to create the following dominant strategies: Player B will choose Hold no matter what Player A selects (Hold is Player B's dominant strategy), and Player A will choose Drive regardless of Player B's choices (Drive is Player A's dominant strategy).

Let's start with Player B's dominant strategy. In order for this to work, this player has to generate payoffs that are higher in the Hold column than the Not Hold column. Thus, 2 has to be higher than Z, and Y has to be greater than 6.

As for ...

#### Solution Summary

This solution addresses a 2x2 game theory problem and provides information on dominant strategies and Nash equilibria based on the values given. Also, an exercise is conducted when certain values in the 2x2 matrix change, leading to different Nash equilibria (optimal outcomes) for each player.