# Tutorial 1 in the A4-size paper note: all tutorial questions from tutorial 1 – tutorial 4

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## Tutorial 3

(Problem Set: Chapter 7

Business School, Shanghai Dianji University, China

Students are demanded to complete tutorial 1 in the A4-size paper (note: all tutorial questions from tutorial 1 – tutorial 4 must be binding together and submitted to lecturer for recording purpose on Duifene platform by 13th Dec 2021). Please kindly notify the answer to specified tutorial questions takes 30% of the final score (any single tutorial is not submitted will cost you 7.5% of the final score).

## Question 1: Game Tree, Complete Plan of Action and Rollback Analysis

In the following games, how many pure strategies (complete plans of action) are available to each player? List out all the pure strategies for each player. Identify the rollback equilibrium outcome and the complete equilibrium strategy for each player.

## Question 2: Game Tree and Rollback Analysis

Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. On each turn, each player may remove either one, two, three, or four matchsticks. The player to remove the last matchstick wins the game. Suppose the players have reached a point where it is Fred’s move and there are just five matchsticks left.

1. Draw the game tree for the game starting with five matchsticks.

2. Find the rollback equilibria for this game starting with five matchsticks.

3. Would you say this five-matchstick game has a first-mover advantage or a second-mover advantage?

4. Explain why you found more than one rollback equilibrium.

## Question 3: Change the Rule of the Games

Consider the game represented by the game tree below. The first mover, Player 1, may move either Up or Down, after which Player 2 may move either Left or Right. Payoffs for the possible outcomes appear below. Re-express this game in strategic (table) form (i.e., the game rule is changed). Determine all the pure-strategy Nash equilibria in the game. If there are multiple equilibria, indicate which one is subgame-perfect.

## Tutorial 4

(Problem Set: Chapter 9 –

Business School, Shanghai Dianji University, China

Students are demanded to complete tutorial 1 in the A4-size paper (note: all tutorial questions from tutorial 1 – tutorial 4 must be binding together and submitted to lecturer for recording purpose on Duifene platform by 13th Dec 2021). Please kindly notify the answer to specified tutorial questions takes 30% of the final score (any single tutorial is not submitted will cost you 7.5% of the final score).

## Question 1: Mixed-Strategies in Simultaneous-move Game and Best Response Panel

In football the offense can either run the ball or pass the ball, whereas the defense can either anticipate (and prepare for) a run or anticipate (and prepare for) a pass. Assume that the expected payoffs (in yards) for the two teams on any given down are as follows:

1. Determine the pure strategy Nash equilibria if there are any.

2. Determine the unique mixed-strategy Nash equilibrium to this game.

3. How many yards is the offense expected to gain in the equilibrium.