Chapter 16: Games of imperfect and incomplete information
Short Answer Questions
1. Name and briefly explain the fundamental components of a game.
Answer: Sets of players, actions, strategies and payoffs.
2. Define the term dominant strategy equilibrium?
Answer: An outcome in a game in which each player follows a dominant strategy.
3. What are the conditions that a set of strategies need to satisfy for being perfect
equilibrium?
Answer: The Nash condition and the credibility condition.
4. Explain the situation of the Prisoners’ dilemma.
Answer: The Prisoners’ dilemma is a situation in which the two players each have a
dominant strategy, but playing these strategies leads to an outcome in which both
sides are worse off than if they collectively chose other strategies
5. Briefly explain the main difference between games of imperfect and
incomplete information.
Answer: In a game of imperfect information, a player must make a move while being
unable to observe the earlier or simultaneous move of some other player (opponent).
In a game of incomplete information, the player is unsure about some of the
underlying characteristics of the game, such as another player’s payoffs.
6. Briefly compare the concepts of equilibrium in pure and mixed strategies. Use
an example to support your answer.
Answer: Pure strategy equilibrium is defined as the deterministic optimal response of
a player, given the actions of their opponents. Pure strategy equilibria may not always
exist. Mixed strategy equilibrium is defined as the optimal response, when the player
follows a strategy that is based on the randomisation among actions of some or all
decision points. Equilibrium in mixed strategies exists always.
7. What is an infinitely repeated game? How the length of the strategic
interaction may affect the outcome of a game?
Answer: It is a game that it is repeated for an infinite number of times. When this is
the case then the incentive to cheat (in the sense of breaking an agreement) may be
reduced significantly, since the other party will have every opportunity to retaliate (or
choose not to trust the player again, thus reducing her future payoffs).
For the remaining questions support your answers with the aid of appropriate
diagrams.
8. Using a decision tree, give an example of a game in which only the player has
dominant strategy equilibrium.
Answer:
Two outcomes can arise when the firms use strategies satisfying the Nash condition.
In one, Air Lion plays produce “high” and Beta plays if Air Lion produces “high”, I
will produce “low”, and if Air Lion produces “low”, I will produce “high”. In the
other, Air Lion chooses produce “low” and Beta chooses produce “high” no matter
what Air Lion does. The first pair of strategies also satisfies the credibility condition.
The second does not – Beta’s threat to produce “high” in response to “high” is not
credible. Hence, only the first pair of strategies constitutes a perfect equilibrium for
this game.
9. X and Y are twp firms interested in selling their output in the same market. If
initially the market is a monopoly, dominated by X, explain how and under
what conditions, firm X could use its incumbency to deter the entry of Y.
Answer: When the incumbent can choose plant size, Liege Pharmaceutical can use a
large plant as a form of commitment to deter entry. In equilibrium, Liege
Pharmaceutical constructs the “large plant”, General Generic chooses “stay out”, and
Liege produces “high output”.
10. Repeat question 7, but assuming now that both firms are incumbents and
choose their output simultaneously.
Answer: When Air Lion and Beta make their output choices simultaneously, the
dashed oval around Beta’s two decision nodes represents the fact that Beta cannot
distinguish between these two points at the time that it makes its decision. Since
produce “high” is the dominant strategy for each firm, the unique equilibrium in this
game is for Air Lion and Beta each to choose produce “high”.
Essay questions
Firms X and Y are two colluding duopolists, competing in output. The two firms
have a choice of producing any of the two output levels: LOW or HIGH. Their
tacit agreement implies that at equilibrium both firms should produce the cartel’s
profit-maximising outcome (LOW, LOW).
1. Discuss the cartel’s profit maximising output choice of (LOW,LOW). Is there
an incentive to cheat and what could a player do to enhance the credibility of
such an agreement? Assume that the game is finite.
Answer: If the firms could sign a binding agreement enforced by a third party, we
would expect them to choose the outcome under which each firm chose “LOW”.
However, there would then be an incentive for either party to break the agreement
(cheat). The decision to cheat will depend on the level of the “punishment” that the
cheater will have to take over the subsequent periods. If the cost of cheating exceeds
the benefit, neither party will break the agreement.
2. How would your answer to question 1 change if the game was an infinitely
repeated one?
Answer: The present value of the cost of punishment might be higher, since the
cheater would have to accept lower payoffs (than the one offered by the (LOW,
LOW) agreement) forever. If these cumulative costs are high enough, they may
prevent either party from breaking the agreement.
3. Using a real world example of your choice, describe a mix-strategies game.
Explain the equilibrium outcome.
Answer: The goalie always wants to go the same way as the kicker, but the kicker
always wants to go the opposite direction from the goalie. Hence, there is no
equilibrium in pure strategies. There is, however, equilibrium in mixed strategies:
each player randomly goes “left” half of the time and “right” the other half. Neither
player can make himself better off by switching his strategy, given the strategy being
played by his opponent.
4. Police most often will interrogate suspects in different rooms and in isolation
from each other. Using the Prisoner’s dilemma setting (discussed in p.576)
discuss why this practice may be optimal from the Police’s point of view.
Answer: When Simon and Paul make their decisions simultaneously, the dashed oval
around Paul’s two decision nodes represents the fact that Paul cannot distinguish
between these two points at the time he makes his decision. For each robber,
“confess” is a dominant strategy. The unique equilibrium outcome is for both to
confess, even though both would be better off if they both kept quiet.
5. Would your answer to question 4 change, if the game was played repeatedly?
Fully discuss your answer.
Answer: Yes. The two players would have an incentive to co-ordinate their actions
and play (DO NOT CONFESS, DO NOT CONFESS), if there is a penalising
mechanism that removes any incentives to cheat. In infinitely repeated games, such a
mechanism could be the knowledge that if any party cheats, the other party will refuse
to cooperate in the future. With Al Capone, the preventing mechanism was that
cheaters were executed gangster style – often enough of incentive to ensure that such
agreements were preserved and that the code of silence was kept.