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ECON 1001. Tutorial 10. Q1) A dominant strategy occurs when One player has a strategy that yields the highest payoff independent of the other player’s choice. Both players have a strategy that yields the highest payoff independent of the other’s choice. Both players make the same choice.

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econ 1001

ECON 1001

Tutorial 10

slide2
Q1) A dominant strategy occurs when
  • One player has a strategy that yields the highest payoff independent of the other player’s choice.
  • Both players have a strategy that yields the highest payoff independent of the other’s choice.
  • Both players make the same choice.
  • The payoff to a strategy depends on the choice made by the other player.
  • Each player has a single strategy.

Ans: A

slide3
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}.
slide4
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.
slide5
Q2) The prisoner’s dilemma refers to games where
  • Neither player has a dominant strategy.
  • One player has a dominant strategy and the other does not.
  • Both players have a dominant strategy.
  • Both players have a dominant strategy which results in the largest possible payoff.
  • Both players have a dominant strategy which results in a lower payoff than their dominated strategies.

Ans: E

slide6
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.
slide7

P

Q3) 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?

  • $0
  • $50
  • $100
  • $150
  • $200

Ans: C

Demand

Q

slide8

P

  • The monopolist maximises profit by producing a quantity where MC = MR, and set the price according to the willingness to pay (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.

Demand

$2

100

Q

slide9

P

Q4) If Firm M cheats on N and reduces its price to $1. How many units will Firm N sell?

  • 200
  • 150
  • 100
  • 50
  • 0

Ans: E

Demand

$2

100

Q

slide10

P

  • If Firm M cheats and charges $1/unit, the quantity demanded by the market would be 150.
  • At this point, M is charging $1 and N is charging $2 for the same product.
  • 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.

Demand

$2

$1

100

150

Q

slide11

P

  • 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

Demand

$2

$1

100

150

Q

slide13
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.
slide14
Q6) 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 ? .
  • Still uncertain; who buys the 2nd ticket.
  • Now determinant; who buys the 1st ticket.
  • Now determinant; who buys the 2nd ticket.
  • Still uncertain; who buys the 1st ticket.
  • Now determinant; who is more cooperative.

Ans: B

slide15
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.
slide16
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.
slide17
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 1st ticket.
slide18
Q7) Suppose Candidate X is running against Candidate Y. If Candidate Z enters the race,
  • Approximately half of the voters who were going to vote for X will now vote for Z.
  • Fewer than half of the voters who were going to vote for Y will now vote for Z.
  • All of the voters who were going to vote for Y will now vote for Z.
  • Most of the voters who were going to vote for Y will now vote for Z.
  • X will certainly win because Y and Z compete for the same voters.

Ans: D

slide19
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

50

75

100

X

Y

slide20
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

50

75

100

X

Y

Z

slide21
Q8) A commitment problem exists when
  • Players cannot make credible threats or promises.
  • Players cannot make threats.
  • There is a Prisoner’s Dilemma.
  • Players cannot make promises.
  • Players are playing games in which timing does not matter.

Ans: A

slide22
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.
slide23
Q9) 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
  • Unpredictable.
  • Matthew and Dean both get $1,000.
  • Matthew gets $500; Dean gets $1,500.
  • Matthew gets $1.5m; Dean gets $1m.
  • Matthew gets $400; Dean gets $1.5m.

Ans: D

slide24
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 $1.5m 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

*

(1.5m, 1m)

Z

Dean

(400, 1.5m)

slide25
Q10) 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
  • Any positive number.
  • More than $1.5m.
  • Less that $100.
  • More than $0.5m.
  • More than $500.

Ans: D

slide26
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 $1.5m (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 1.5m for Dean) -- for a higher payoff (compared with 1m for Dean).
  • The penalty of breaching the promise should then be at least $0.5m (say $0.6m). 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

*

(1.5m, 1m)

Z

Dean

(400, 0.9m)