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Sample Questions and Answers for Module Seven
Problem One: Use the following payoff matrix for a simultaneous-move, one-shot game to answer the accompanying questions.
Player 2
Strategy C D E F
Player 1
A 20, 18 15, 19 16, 12 10, 10
B 18, 20 14, 16 12, 14 8, 12
a. What is Player 1’s optimal strategy?
Player 1’s optimal strategy is strategy A. Notice that Player 1’s payoff is higher for strategy A than for strategy B no matter which strategy Player 2 selects.
b. What is Player 2’s optimal strategy?
Player 2’s optimal strategy is strategy D. Player 2 should recognize that Player 1 will select strategy A.
Knowing that Player 1 will select strategy A, the best that Player 2 can do is to select strategy D.
c. Does either player have a dominant strategy?
Yes, Player 1 has a dominant strategy. Its payoff is always better selecting strategy A than strategy B, no matter which strategy Player 2 selects.
Player 2 does not have a dominant strategy. The strategy that gives it the highest payoff depends is different depending on which strategy Player 1 selects. For example, when Player 1 selects strategy A, Player 2’s best outcome is when it selects strategy D. When Player 1 selects strategy B, Player 2’s best outcome is when it selects strategy C.
d. What is Player 1’s equilibrium payoff?
Player 1’s equilibrium payoff is 15.
e. What is Player 2’s equilibrium payoff?
Player 2’s equilibrium payoff is 19.
Without knowing what the other player is likely to do, assuming players tend to be risk averse, the most likely outcome is (15,15).
b. Suppose Player 1 is permitted to “communicate” by uttering one syllable before the players simultaneously and independently make their decisions. What should Player 1 utter? What outcome do you think would occur as a result?
If communications or coordination or collusion exists, the most likely outcome is (20, 20). If Player 1 is allowed to communicate to Player 2, Player 1 should communicate or indicate that it is pursuing strategy A. It will then be in Player 2’s best interest to pursue strategy X.
c. Suppose Player 2 can choose its strategy before Player 1, that Player 1 observes Player 2’s choice before making her decision, and that this move structure is known by both players. What outcome would you expect?
If this game is sequential, the first player, such as Player 2, would select strategy Y. Using backward induction and assuming that Player 1 will act rationally, Player 2 will select Y knowing that once Player 1 knows that is the strategy selected by Player 2, it will select strategy A and both will be better off.
Problem Three: Suppose AutoOne and AutoTwo must decide whether to or not to make a new safety device airbags standard equipment on their new cars next year. The new device raises the price of each car by $500. If both firms make the device part of standard equipment, each company will earn profits of $1 billion. If neither company adopts the device, each company will earn $0.5 billion. If one company adopts the device as standard equipment and the other does not, the adopting company will earn a profit of $1.5 billion and the other company will earn $0.1 billion.
Construct a payoff matrix for a one-shot game using the above information:
The normal form game looks like this:
AutoOne
AutoTwo
Strategy Device Standard
Device Not Standard
Device Standard
Device Not Standard
If you were a decision maker at AutoOne, would you make side-impact airbags standard equipment?
Yes. In this case, AutoOne has a dominant strategy to make the device standard. Its payoff is higher including the device in its list of standard equipment no matter which strategy AutoTwo selects. The same is true for AutoTwo. It also has a dominant strategy to include the device as standard equipment.
If AutoOne and AutoTwo are able to cooperate, would you expect this same outcome?
Problem Four: Boeing and Airbus have been asked to submit bids for supplying 10
airplanes to Airline A. Assume for simplicity that each company can submit one of two
bids, a high bid or a low bid, for supplying the aircraft. If both submit a low bid, each firm
supplies 5 airplanes and earns a profit of $1 million. If both firms submit high bids, each
supplies 5 airplanes and earns a profit of $5 million. If one submits a high bid and the
other submits a low bid, the one submitting the low bid supplies all 10 airplanes and earns a profit of $2 million and the other does not supply any aircraft and earns a profit
of $0.
Construct a payoff matrix for a one-shot game using the above information:
The one-shot form of this game looks as follows:
Airbus
Boeing
Strategy: Price Low High
Low $1, $1 $2, $
High $0, $2 $1.5, $1.
What is the Nash equilibrium for a one-shot game?
The Nash equilibrium is both Airbus and Boeing submitting low bids. Both have dominant strategies to bid low.
Would your answer differ if Boeing and Airbus resubmit price quotes year after year if they know there is a 40% possibility that in the future, the airline will not request bids?
Yes, the possibility of implicit coordination exists if both players recognize they would be better off if each submits high bids. This depends on the net present value of cooperating today versus cheating today. The higher the probability the airline will not continue purchasing aircraft, the greater the incentive to cheat today.
If there is a 40% chance the airline will no longer purchase new airplanes, the profits of a firm that conforms to the collusive strategy (high price) under the usual trigger strategies (firms agree to charge the high price so long as no player deviated in the past, otherwise charge a low price) are:
A firm that cheats earns $2 million today and $1 million forever after:In ,
In this case, since the profits associated with cooperating exceed the profits from collusive behavior, collusion can be sustained as a Nash equilibrium.
