ECON 1001 Tutorial 10 Q 1A dominant strategy

ECON 1001 Tutorial 10

Q 1)A dominant strategy occurs when A) One player has a strategy that yields the highest payoff independent of the other player’s choice. B) Both players have a strategy that yields the highest payoff independent of the other’s choice. C) Both players make the same choice. D) The payoff to a strategy depends on the choice made by the other player. E) Each player has a single strategy. Ans: A

• Let’s illustrate this by an example: • Player 1’s dominant strategy is {Top}, because it gives him a higher payoff than {Bottom}, no matter what Player 2 chooses. • Player 2’s dominant strategy is {Right}. 2 Left Right Top (100, 30) (80, 90) Bottom (60, 60) (70, 100) 1

• Therefore, a dominant strategy is a strategy that yields the highest payoff compared to other available strategies, no matter what the other player’s choice is. • A rational player will always choose to play his dominant strategy (if there is any in the game), because this maximises his payoff. • The other strategy available to the player that yield a payoff strictly smaller than that of the dominant strategy is called a ‘dominated strategy’ (e. g. Player 2’s [Left}) • Dominant strategies may not exist in all games. It all depends on the payoff matrix.

Q 2) The prisoner’s dilemma refers to games where A) Neither player has a dominant strategy. B) One player has a dominant strategy and the other does not. C) Both players have a dominant strategy. D) Both players have a dominant strategy which results in the largest possible payoff. E) Both players have a dominant strategy which results in a lower payoff than their dominated strategies. Ans: E

• The prisoner’s dilemma is a coordination game. • Both players have a dominant strategy, but the result of which is a lower payoff than the dominated strategies. 2 1 Confess Deny (-3, -3)* (0, -6) (-6, (-1, -1) 0)

Q 3) MC for both Firms M and N is 0. If Firms M and N decide to collude and work as a pure monopolist, what will M’s econ profit be? P Demand A) B) C) D) E) $0 $50 $100 $150 $200 Q Ans: C

• The monopolist maximises profit by producing a quantity where MC = MR, and set the price according to the willingness to pay (Demand) P Demand • • The profit-max output level is 100, and the profit will be $200. Since each firm is halving the quantity, they each earns an econ profit of $100. $2 100 Q

Q 4) If Firm M cheats on N and reduces its price to $1. How many units will Firm N sell? A) B) C) D) E) 200 150 100 50 0 Ans: E P Demand $2 100 Q

• • If Firm M cheats and charges $1/unit, the quantity demanded by the market would be 150. P At this point, M is charging $1 and N is charging $2 for the same product. Demand $2 • • All customers will buy from Firm M, and hence, Firm N will have no sales at all. Firm M is going to make a profit of $150. $1 100 150 Q

• • If Firm N is allowed to respond to Firm M’s cheating, it may lower is price to $0. 5/unit, the quantity demanded by the market would be 175. At this point, if M is charging $1, all customers will buy from Firm N, and hence, Firm M will have no sales at all. • Firm N is going to make a profit of $75. • … The story continues P Demand $2 $1 100 150 Q

Q 5) The game has ? Nash Equilibrium. A) B) C) D) E) 0 1 2 3 4 Ans: C

• Let’s look at the payoff matrix to find out the N. E. • {C, C} and {D, C} are the Nash Equilibria. • Hence, there are 2 N. E. in this game. • The N. E. is also known as pure strategy N. E. , the adjective “pure strategy” is to distinguish it from the alternative of “mixed strategy” N. E. A mixed strategy N. E. is a N. E. in which players will randomly choose between two or more strategies with some probability. Jordan Comedy Documentary Comedy (3, 5) (1, 1) Documentary (2, 2) (5, 3) Lee

Q 6)By allowing for a timing element in this game, i. e. , letting either Jordan or Lee buy a ticket first and then letting the other choose second, assuming rational players, the equilibrium is ? , based on ? . A) B) C) D) E) Still uncertain; who buys the 2 nd ticket. Now determinant; who buys the 1 st ticket. Now determinant; who buys the 2 nd ticket. Still uncertain; who buys the 1 st ticket. Now determinant; who is more cooperative. Ans: B

• By allowing a timing element, the game is now a sequential game. • That means, one player moves first, and buys the first ticket. • The other player observes any action taken (i. e. knows what ticket has been bought), and then makes his / her decision. • Actions are not taken simultaneously anymore.

• Whoever chooses an action can now predict how the other player is going to react. • E. g. If Lee chooses {Comedy}, he can be sure that Jordan will choose {Comedy} as well, because this gives Jordan a higher payoff than picking {Documentary}. • Therefore, the first mover has the advantage (called First Mover Advantage) to take actions first, hence securing his or her own payoff by predicting the response from the other player.

• A rational (self-interested) player will always pick the action that maximises his or her own payoff (irregardless of others’) • Hence, if Lee is to move first, he will pick {Documentary}, because {D, D} gives him the highest possible payoff. • If Jordan is to move first, she will pick {Comedy}, because {C, C} gives her the highest possible payoff. • Therefore, the result is now determinant, as soon as we know who is buying the 1 st ticket.

Q 7)Suppose Candidate X is running against Candidate Y. If Candidate Z enters the race, A) Approximately half of the voters who were going to vote for X will now vote for Z. B) Fewer than half of the voters who were going to vote for Y will now vote for Z. C) All of the voters who were going to vote for Y will now vote for Z. D) Most of the voters who were going to vote for Y will now vote for Z. E) X will certainly win because Y and Z compete for the same voters. Ans: D

• Originally, before Z joins the election, • Assuming voters in between 2 candidates are shared equally. • Area covered in RED are voters voting for X. • Area covered in BLUE are voters voting for Y 0 25 X 50 Y 75 100

• With Z joining the election, the area in green are voters voting for Z. • All voters in the green area used to vote for Y. • Hence, (D) is the answer. 0 25 X 50 Y Z 75 100

Q 8) A commitment problem exists when A) Players cannot make credible threats or promises. B) Players cannot make threats. C) There is a Prisoner’s Dilemma. D) Players cannot make promises. E) Players are playing games in which timing does not matter. Ans: A

• In games like the prisoner’s dilemma, players have trouble arriving at the better outcomes for both players…. Because – Both players are unable to make credible commitments that they will choose a strategy that will ensue a better outcomes for both players (either in the form of credible threats or credible promises) • This is known as the commitment problem.

Q 9) Suppose Dean promises Matthew that he will always select the upper branch of either Y or Z. If Matthew believes Dean and Dean does in fact keep his promise, the outcome of the game is A) B) C) D) E) Unpredictable. Matthew and Dean both get $1, 000. Matthew gets $500; Dean gets $1, 500. Matthew gets $1. 5 m; Dean gets $1 m. Matthew gets $400; Dean gets $1. 5 m. Ans: D

• If Dean will indeed goes for the upper branch, then Matthew can either earn $1, 000 by choosing the upper branch (i. e. , arriving the node Y), or $1. 5 m by picking the lower branch (i. e. , arriving the node Z). • As Matthew is a rational individual, he will choose a lower branch (i. e. , arriving the node Z). (1000, 1000) Dean Y (500, 1500) X Matthew * Z Dean (1. 5 m, 1 m) (400, 1. 5 m)

Q 10) Suppose Dean promises Matthew that he will always select the upper branch of either Y or Z. Dean offers to sign a legally binding contract that penalises him if he fails to choose the upper branch of Y or Z. For the contract to make Dean’s promise credible, the value of the penalty must be A) B) C) D) E) Any positive number. More than $1. 5 m. Less that $100. More than $0. 5 m. More than $500. Ans: D

• If Dean will indeed goes for the upper branch, then Matthew is better off picking the lower branch (i. e. , arriving at node Z), because he can then have a payoff of $1. 5 m (compared to $1000 from the upper branch, i. e. arriving at node Y) • As Matthew picks the lower branch (i. e. , arriving at node Z), there is a tendency for Dean to the lower branch (i. e. , arriving the payoff of (400 for Matthew and 1. 5 m for Dean) -- for a higher payoff (compared with 1 m for Dean). • The penalty of breaching the promise should then be at least $0. 5 m (say $0. 6 m). The penalty will reduce the payoff to Dean (becomes 1. 5 -0. 6 = 0. 9) when Dean chooses the lower branch at node Z. Thus, Dean will choose the upper branch at node Z. (1000, 1000) Dean Y (500, 1500) X Matthew * Z Dean (1. 5 m, 1 m) (400, 0. 9 m)