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ECON7073, Semester 1, 2023

Tutorial problems for Lecture 1

1. Two people enter a bus. Two adjacent cramped seats are free. Each person must decide whether to sit or stand. Sitting alone is more comfortable than sitting next to the other person, which is more comfortable than standing.

(i) Suppose that each person cares only about her own comfort. Model the situation as a simultaneous- move game. Do the players have a strictly dominant strategy? Find all Nash equilibria of the game. Is the game a Prisoners’ Dilemma?

(ii) Now suppose instead that each person is altruistic, ranking the outcomes according the other

person’s comfort, but, out of politeness, prefers to stand than to sit if the other person stands. Model the situation as a simultaneous-move game. Do the players have a strictly dominant strategy? Find all Nash equilibria of the game. Is the game a Prisoners’ Dilemma?

(iii) Compare the people’s comfort in the equilibria of the two games from parts (i) and (ii).

2. Each of two players simultaneously announces either Rock, or Paper, or Scissors. Paper beats (wraps) Rock, Rock beats (blunts) Scissors, and Scissors beats (cuts) Paper. If the two players name the same object, there is a tie. Otherwise, the player who names the object that “beats” the other player’s named object wins, and the other player loses. Hence, there are only three possible outcomes, player 1 wins (and 2 loses), player 2 wins (and 1 loses), and a tie. Denote these three outcomes by 1W, 2W, and T, respectively.

(i) Draw a table where the rows are indexed by player 1’s action choices, the columns by player 2’s action choices, and each cell contains the respective outcome, 1W, 2W, or T, from the game.

(ii) Now assume that the losing player in any of the three outcomes must pay the winning player

1 dollar, and that no money changes hand in the case of a tie. If a player prefers winning to losing and to a tie, and a tie to losing, explain why the net monetary amounts changing hands can be used to represent such preferences in the associated game, and sketch the resulting payoff matrix.

(iii) Determine which of the three outcomes are Pareto efficient, and explain your answers.

(iv) Determine whether any of a player’s three actions strictly dominates any of the other actions,

and explain your answers.

(v) Show that the game possesses no Nash equilibrium (as defined in the lecture). Discuss how you think any players would play this game.

3. A three-player game where the players simultaneously choose two possible actions can be represented by two separate payoff matrices, where player 1 chooses a row, player 2 a column, a player 3 a matrix. For example, consider the three-player simultaneous-move game depicted by the following two matrices:

Player 2

Left Right

7, 2, 2	0, 0, 2
9, 4, 8	0, 0, 0

Player 3 – Left Matrix Player 3 – Right Matrix

In each cell, the first number is the payoff of player 1 (the row player), the second the payoff of player 2 (the column player), and the third the payoff of player 3 (the matrix player).

(i) Find all Nash equilibria of this game.

(ii) For each Nash equilibrium you found, determine whether it is Pareto efficient or not.

4. We say that a strategy A of a player is weakly dominated by another strategy B (or equivalently, that B weakly dominates A) if the payoff from playing B is greater than or equal to that from playing A for every strategy profile of the other players, and is strictly greater for at least one strategy profile of the other players. A strategy of a player is weakly dominant if it weakly dominates every one of his other strategies.

Now consider the simultaneous-move game depicted below:

Evelyn

Left Right