II1 Miren and Claudia both students of game theory at UC3M are

Given the following Normal form game:

LR A 1,1 8,0 B 0,5 3,3

Find the smallest discount factor necessary to get average payoffs equal to (3,3) in a SPNE if the game is repeated an infinite number of times. Describe the strategies that allows us to sustain such an equilibrium

There are two suspects who face the problem of the prisoner’s dilemma, with the added complication that neither suspect knows if the other is a man of honor. Say it is known with certainty that Suspect 1 is not a man of honor, but it is not clear whether or not Suspect 2 is also. If Suspect 2 is not a man of honor, the payoffs have their usual form in this game:

Suspect 2

Suspect 1

Confess Not to Confess

Confess

1, 1 0, 15

Not to Confess

15, 0 10, 10

On the other hand, if Suspect 2 is a man on honor, then he prefers to spend years in jail before he would rat on his colleague. More over, even Suspect 1 would feel bad betraying someone so honorable. For these reasons, if Suspect 2 is a man of honor the payoffs are:

Suspect 2

Suspect 1

Confess Not to Confess

Confess

1, 1 0, 15

Not to Confess

5, 20 10, 30

Denote the probability that Suspect 2 is a man of honor by p. Consider 𝑝 ∈ (0,1). (a) Identify the strategies that are dominant for Suspect 2 in this game.

(b) Identify the Bayesian-Nash equilibria in this game for each value of p.

Consider a Cournot duopoly in a market with inverse demand function given by
p(q) = 10 − q, where q = q1 + q2 is the market total quantity. Firm 1’s costs are
c1
(q1
) = 8q1 with probability 1/2, and c1
(q1
) = 0 with probability 1/2. Firm 2’s costs

are c2
(q2
) = (q2)
2
. Firm 1 knows its costs, but not Firm 2, who knows Firm 1’s

possible types and probabilities. All this is common knowledge.
(a) Represent the situation as a Bayesian game. I.e., show the set of players, the set
of types, beliefs, and utilities.
(b) Calculate the Bayesian-Nash equilibrium of the game. Calculate also the
equilibrium profits.
(c) Consider Firm 1’s low-cost type (c1
(q1
) = 0), does it gain if it can inform Firm
2 of its type in a credible way (providing proof)? What about the high cost type?
¿How does this possibility affect the equilibrium?
(d) Assume that Firm 1 only has its word (no proof) to convince Firm 2 of its type.
Does Firm 1 have any incentives to lie about its type?

Two friends share an apartment. They have to decide how to divide a total of 6 hours
in cleaning the apartment, si and watching television, 6 − si (i = 1,2). Both value a
clean apartment and both like to watch television (especially when the apartment is
clean). The utility function of Player 1 the following:

u1
(s1, s2
) = (s1 + s2

) + (6 − s1

) + (6 − s1
)(s1 + s2
).

Player 2’ utility is
u2
(s1, s2
) = a(s1 + s2

) + (6 − s2

) + (6 − s2
)(s1 + s2
).

The first term represents the utility from living in a clean apartment, the second term is
the direct utility from watching television and the last term reflects that the utility from
watching television increases when the apartment is clean. The parameter a in Player
2’s utility may take the values 1 and 4 with probabilities 1/3 and 2/3, respectively.
Player 2 knows the value of a, but Player 1 does not, although she knows the probability
distribution over the possible values.
(a) Identify all the elements of the above description to define a Bayesian game.
(b) Find the Bayesian Nash equilibria of the game.

A forward and a goalkeeper face each other in a penalty shootout. The forward is a
very skilled shooter and, if he deceives the goalkeeper, he always scores a goal. The
goalkeeper can either be very good or average with equal probabilities. If he is very
good, he will save every shot unless the forward deceives him. If he is average, he will
only save half of the shots unless he is deceived. More specifically, the forward can
shoot to the left (L) or to the right (R). The goalkeeper can also dive to the left (L) or to
the right (R). Deceiving the goalkeeper means, for example, the forward shooting to the
left and the goalkeeper diving to the right, or vice versa. If the forward scores, it means
a payoff of 1 for the forward and −1 for the goalkeeper. If he does not, payoffs are
reversed.
(a) Show the Bayesian game, with all its elements defined (indicate the payoffs in
matrix form).
(b) Show that the game does not have any Bayesian Nash equilibrium in pure
strategies.
(c) Show that (
1
2
[L] +
1
2
[R]; (
1
2
[L] +
1
2
[R],
1
2
[L] +
1
2
[R])) is a Bayesian-Nash
equilibrium in mixed strategies. (The first component of the equilibrium is the
forward's strategy and the second is the goalkeeper's strategy, with one action
per type.)

There are two suspects who face the problem of the prisoner’s dilemma, with the
added complication that neither suspect knows if the other is a man of honor. Say it is
known with certainty that Suspect 1 is not a man of honor, but it is not clear whether or
not Suspect 2 is also. If Suspect 2 is not a man of honor, the payoffs have their usual
form in this game:

Suspect 2
Confess Not to Confess

Suspect 1

Confess 1, 1 15, 0
Not to Confess 0, 15 10, 10

On the other hand, if Suspect 2 is a man on honor, then he prefers to spend years in jail
before he would rat on his colleague. More over, even Suspect 1 would feel bad
betraying someone so honorable. For these reasons, if Suspect 2 is a man of honor the
payoffs are:

Suspect 2
Confess Not to Confess

Suspect 1

Confess 1, 1 5, 20
Not to Confess 0, 15 10, 30

Denote the probability that Suspect 2 is a man of honor by p. Consider p ∈ (0,1).
(a) Identify the strategies that are dominant for Suspect 2 in this game.
(b) Identify the Bayesian-Nash equilibria in this game for each value of p.