¡Descarga Problems Static Games - Game Theory y más Ejercicios en PDF de Teoría de Juegos solo en Docsity!
Universidad Carlos III de Madrid
GAME THEORY
Problem set on
static games
- Two car companies are planning to launch at the same time a car to the market. Each
of them is considering whether it should offer credit to the buyers in order to reach a
larger share of customers. However; offering credit would imply incurring some costs.
Both companies prefer not to offer credit but they are afraid that the other one will do
offer and will therefore attract more clients. Suppose that the expected benefits for both
companies are the following: If both offer a credit, each gets 400 million euros. If none
of them offers credit, they get 600 million each and if one offers credit and the other one
does not; the first one will earn 800 million while the other will obtain 300. Represent
the game in the normal form.
- Two villages that belong to two different independent communities compete for a
plan that will offer tax reductions that help industrial development. If both offer the
same reductions, the tax rate decreases with no guarantees of any industrial
development. In such a case, the villages would have preferred higher taxes. The idea is
to attract firms even if it implies tax reductions. Represent this situation as a game with
a numerical example. Explain what are the relevant strategic factors.
- The city hall has to decide whether to construct a high school or a nursery in a
particular city zone. It does not have budget to carry out both projects (i.e. both the high
school and the nursery). The person in charge of managing these subjects has spoken
with two indispensable companies that can make any of these two projects: one
construction company and one carpentry company. Due to the composition of
population, the building of the high school would be greater than the one of nursery (it
requires more construction), but that one will need a park of wood games (it requires
more carpentry). In addition, each one of the companies is interested more in
participating in a certain project that in the other (the one of construction in the high
school, and the one of carpentry in the nursery), but both prefer signing the same
contract to signing different contracts, since in this case the city hall would not carry out
any project. The city hall asks them to present a project. As none of the companies has
sufficient personnel available to process both projects, they must choose one of the
projects another, without knowing which project will be chosen by the other company.
(a) Define a game in normal form whose payments reflect the expected profits of
each company in each possible situation.
(b) Compute the Nash equilibria.
- Guillermo and Miguel share an apartment where each one of them has his own room.
They want to decorate their apartment. Each one has two paintings and must decide how
many to place in his own room and how many to place in the common living room.
Suppose that the decision is made privately and that once the paintings are in their
place, they cannot be removed. Let 𝑥
𝐺
and 𝑥
𝑀
be the number of paintings that
Guillermo and Miguel, respectively, decide to place in their own room (thus, 𝑥
𝑆
𝐺
𝑀
is the number of paintings in the living room). Guillermo’s utility function is
𝐺
𝐺
𝑆
𝐺
𝑆
) and Miguel’s is 𝑢
𝑀
𝑀
𝑆
𝑀
𝑆
). Then, for
instance, if Miguel places one painting is his own room and Guillermo two in his (𝑥
𝑀
𝐺
𝑆
= 1 ) they will get utilities 𝑢
𝑀
= 2. 5 and 𝑢
𝐺
(a) Which are the strategies of each one of the roommates?
(b) Represent the game. I.e., in the usual entry matrix describe the utilities of each
player for each of the possible distributions of the paintings, depending on the
chosen strategies.
(c) Find the unique Nash equilibrium of this game. Is this a good outcome for
Guillermo and Miguel?
- Pedro and Miguel are neighbors. From his terrace, Pedro cannot see his own garden
but he has a wonderful view of Miguel’s garden. Pedro considers that maintaining
Miguel’s garden is worth 2000 euros. Pedro also considers that keeping his own garden
in a good state is worth only 500 euros. The preferences of Miguel are completely
reciprocal. Given that both gardens can be seen form the public road, the mayor pays a
subsidy of 500 euros to each house in a street in which all the gardens are maintained.
Pedro and Miguel are the two only neighbors of their street. The cost to maintain each
garden is 1000 euros. Represent the game faced by Pedro and Miguel.
- Consider the following game in the normal form:
B 1 B 2 B 3
A 1 1,1 0,0 - 1,
A 2 0,0 0,6 10,- 1
A 3 2,0 10,- 1 - 1,- 1
(a) Which strategies survive to the iterated elimination of dominated strategies?
(b) Which are the Nash equilibria?
- Consider the following game in the normal form:
L R
T 4,4 0,
M 2,1 1,
B a,b c,
(a) For which value of a, b and c, the strategy profile (B; L) is the result of the
iterated elimination of dominated strategies?
(a) For which value of a, b and c, the strategy profile (B; L) is the unique pure
strategy Nash equilibrium?
- The municipality of Madrid is organizing an operation called “Madrid Verde”. In a
street of Chamartín, each family that has a house there will get two trees. Only two
neighbors live on this street (only two houses). Each of the two neighbors must decide
how many of these trees he will plant in his garden (in which case you cannot see the
trees from the street) and how many he will plant in the entrance of his house (in which
case you can see the trees from the street). The trees that can be seen from the street
- Find the mixed strategy Nash equilibrium of the following normal form game:
L R
T 2,1 0,
B 1,2 3,
- Find the mixed strategy Nash equilibrium of the following normal form game:
L R
T - 2,- 1 0, 0
B 0 , 0 - 1 ,- 2
- Find all the pure and mixed strategy Nash equilibria for the game from Exercise 5.
- In a small island there are only two consumers and two importing cars firms. One
firm imports American cars while the other imports European cars. Right now, both
consumers are without a car, and since there is no public transport services in the island,
each of them is going to buy a car, no matter the price. But they don’t want to expend
too much, so they both are going to buy from the firm with the lowest price. If both
firms charge the same price, each one buys from a different firm.
When a firm sells a car, it has to pay the import cost, which is equal to €10,000 if the
car comes from USA, or €8,000 if from Europe. The firms cannot fix prices with
decimals, since the smaller coin in the island is €1. The firms mainly want to maximize
profits, and if at the same profits, they prefer to sell as much as possible.
(a) For the case where the two firms choose their prices simultaneously, write the
payoff functions for each of them.
(b) If the firm who sells American cars sets a price (𝑝
𝐴
) is € 10 ,000, what is the
best response for the other firm? And if 𝑝
𝐴
Note: If a firm is indifferent among many different prices, we assume that it sets a price
equal to its import cost.
(c) If the firm who sells European cars sets a price (𝑝
𝐸
) is € 15 , 000 , what is the
best response for the other firm? And if 𝑝
𝐸
(d) Find the reaction functions for each firm, for all possible prices from its rival.
Find the only Nash equilibrium in pure strategies. How many cars each firm
sells, and at which price?
Suppose now that the government of the island is worried about the prices of the cars,
and decides to subsidize one (and only one) of the firms, paying €1,000 of its imports
costs for each car it sells.
(e) If the government's objective is that the consumers buy the cars at the lowest
possible price, to which firm should the government give the subsidy?
(f) Who is going to sell now? At which price? And how much the government will
expend in subsidies?
- Consider a market with two firms that sell two slightly different products (for
instance, pastas with different colors or tastes). The two demand functions are
respectively 𝑞 1
1
2
and 1. 000 − 2 𝑝
2
1
. Both firms have access to
the same technology. The cost of one unit for any good produced is 2 (there are no fixed
costs). The strategic variable for both firms is the price. The quantity sold and the
utilities depend on the decisions of both players. Find the Nash equilibrium of this
game, knowing that choices are made simultaneously.
- Two neighbors consider cleaning their street on a Sunday morning. Each neighbor
has one hour that he can use to watch TV or to clean the street. Denote by 𝑐 𝑖
the time
dedicated to cleaning the street and by Neighbor i, and 1 − 𝑐 𝑖
the time spent watching
TV. Each neighbor considers that it is important that the street is clean and likes
watching TV. Their utility functions are:
1
1
2
) = 2 ln ( 1 + 𝑐
1
2
1
2
1
2
) = 2 ln ( 1 +
1
2
2
where 2 ln ( 1 + 𝑐 𝑖
𝑐
𝑗
2
) represents the utility of living in a clean street for Neighbor i
and 1 − 𝑐 𝑖
is the utility from watching TV for Neighbor i. Calculate and represent
graphically the best response functions and the Nash equilibria of the game when both
neighbors take their decision simultaneously.
- Two firms which operate in the car industry, have to decide the quantities they will
produce. Each of them knows the market demand for its product but does not know the
amount its rival will produce. Let us assume that total cost for each firm is 𝑐
𝑖
𝑖
and that aggregate demand is
1
2
1
2
(a) What are the strategies of each firm? Since firms want to maximize profits, what
are their payoffs? Which are the best-response functions?
(b) What is the Nash equilibrium if both firms choose their quantities
simultaneously?
- Consider the Cournot duopoly model with the demand function
= 𝑎 − 𝑞, but
with asymmetric costs: 𝑐 1
for Firm 1 and 𝑐
2
for Firm 2. What is the Nash equilibrium if
𝑖
𝑎
2
? Same question but now 𝑐
1
2
< 𝑎 and 2 𝑐
2
1
? Note: You can
solve this problem with numbers that satisfy this condition.
Extra Exercises
- Consider two firms, one of them operates in a market (Firm 1) and the other one
(Firm 2) wants to enter in that market. Firm 1 is thinking about building a new
production plant. Payoffs for the two firms are given by:
Enter Do not enter
Build 0,- 1 2,
Do not build 2,1 3,
where the first term represents the utility from living in a clean apartment (which is a
result of the time the friends decide cleaning the 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.
(a) Find the Nash equilibria of the game.
(b) Do the Nash equilibria of the game found in the previous question maximize the
joint welfare of the friends: 𝑢 1
1
2
2
1
2
