Handout 17 Game Theory – Strategies and Equilibria This handout contains exercises of - Game Theory Things you should know - What are games and components of games? - What are the dominant strategy, Nash strategy, collusion strategy and Maxmin strategy? - How to find the equilibrium/solution of the game under each solution concept? Exercises Question 1 (Finding Games Solutions) In each of the following games, find the game equilibrium (that is, you must list what strategy does each player choose, and what outcome each player gets) using each of the following solution concepts. Write down your explanation as you find the solutions. - Dominant strategies - Nash strategies - Collusion strategies (if asked) - Maxmin strategies (if asked) 1. Prisoners’ Dilemma Country A and Country B can decide to build “Nuclear” or “Non-nuclear” weapons. If both countries decide to build nuclear weapons, they are wastefully utilizing the resources. If both decide not to build nuclear weapons, they employ their resources somewhere efficient. If one country has the nuclear weapon but the other has not, the country that has the nuclear weapon is more powerful, while the country that has no nuclear weapon is prone to being attacked. Country B Nuclear Non-nuclear 0,0 4 , -4 Nuclear Country A -4 , 4 2,2 Non-nuclear Find the dominant strategy equilibrium, Nash equilibrium, collusion equilibrium and Maxmin strategy equilibrium, if any exists. 2. Chicken Game Two drivers are driving from the opposite ends on the center of an unmarked road. As they come into visual, they can choose to swerve from each other or drive straight. Swerving the car takes effort but the crash is avoided with certainty, and accident can also be avoided if one driver swerves but the other drive straight. However, if both drive straight, they will crash. Driver B Swerve Drive Straight -1 , -1 -2 , 0 Swerve Driver A 0 , -2 -10 , -10 Drive Straight Find the dominant strategy equilibrium, Nash equilibrium and Maxmin strategy equilibrium, if any exists. 3. Battle of the Sexes The couple is considering what to do on their date night but they don’t know what they have agreed upon. The girl prefers to Netflix but the guy prefers to Chill. They have to choose the same activity to enjoy their time, even if one prefers the other activity. If they don’t choose the same activity, they both lose. 1 The Girl Netflix Chill 1,2 0,0 Netflix The Guy 0 , 0 2,1 Chill Find the dominant strategy equilibrium, Nash equilibrium, Collusion equilibrium and Maxmin strategy equilibrium, if any exists. 4. Rock-Paper-Scissors A rock-paper-scissors game can be written in the game matrix as the following Player B Rock Paper Scissors 1 , -1 0,0 -1 , 1 Rock -1 , 1 Player A 1 , -1 0,0 Paper 0,0 -1 , 1 1 , -1 Scissors Find the dominant strategy equilibrium and Nash equilibrium, if any. 5. Find the dominant strategy equilibrium, Nash equilibrium, collusion equilibrium and Maxmin strategy equilibrium, if any exists. Player B Player B Left Right Left Right 2,1 0,0 Up 1,0 1,1 Up Player A Player A -1 , -1 1,2 Down -5 , 0 5,1 Down Problem 2 Consider the following game table, then answer the questions Player 2 Left Right 1 , -1 -1 , 1 Up Player 1 1 , -1 Down X,Y 1. 2. 3. 4. 5. 6. Find Nash equilibrium if X = –1 and Y = 1. For what value of X is “Down” the dominant strategy for Player 1? For what values of Y is “Left” the dominant strategy for Player 2? For what values of X guarantees that “Up” is the Maxmin strategy for Player 1? For what values of Y guarantees that “Right” is the Maxmin strategy for Player 2? For what values of X guarantees that “Down” is the Maxmin strategy for Player 1? 2