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Games and decisions, 2016/2017, list 2
List 2
1. Compute all equilibria pairs (Nash equilibria in pure strategies) in the following game.
α1
α2
α3
α4
β1
(2, 1)
(4, 0)
(1, 3)
(4, 3)
β2
(4, 3)
(5, 4)
(5, 3)
(2, 5)
β3
(7, 2)
(1, 6)
(3, 2)
(4, 0)
β4
(7, 4)
(0, 4)
(4, 1)
(1, 0)
β5
(0, 5)
(0, 3)
(1, 0)
(1, 5)
β6
(3, 2)
(5, 1)
(4, 3)
(2, 1)
2. Compute all Nash equilibria in the following two person nonzero sum games:
α1
α2
β1
(2,2)
(1,1)
β2
(0,1)
(3,3)
α1
α2
β1
(3,2)
(-1,3)
β2
(-1,1)
(0,0)
3. In each of the following situations, design the corresponding game and find all Nash equilibria:
(a) Two firms (Smith and Brown) decide whether to design the computers they sell to use
large or small floppy disks. Both players will sell more computers if their disk drives are
compatible. If they both choose for large disks the payoffs will be 2 for each. If they
both choose for small disks the payoffs will be 1 for each. If they choose different sizes
the payoffs will be -1 for each.
(b) Two firms sell a similar product. Each percent of market share yields a net payoff of 1.
Without advertising both firms have 50% of the market. The cost of advertising is equal
to 10 but leads to an increase in market share of 20% at the expense of the other firm.
The firms make their advertising decisions simultaneously and independently. The total
market for the product is of fixed size.
(c) Each of two firms has one job opening. Suppose that firm 1 offers wage 2 and firm 2
offers wage 3. Imagine that there are two workers, each of whom can apply to only
one firm. The workers simultaneously decide whether to apply to firm 1 or firm 2. If
only one worker applies to a given firm, that worker gets the job; if both workers apply
to one firm, the firm hires one worker at random (with probability 0.5) and the other
worker is unemployed (and has a payoff of zero).
(d) Each one of two bars charges its own price for a beer, either $2, $4, or $5. The cost
of obtaining and serving the beer can be neglected. It is expected that 6000 beers per
month are drunk in a bar by tourists, who choose one of the two bars randomly, and
4000 beers per month are drunk by natives who go to the bar with the lowest price, and
split evenly in case both bars offer the same price. What prices would the bars select?
(e) Ann and Beth are not on speaking terms, but have a lot of common friends. Both want
to invite them to a dinner party this weekend, either Friday or Saturday evening. Both
slightly prefer Saturday. If both set the party at the same time, this will be considered
a disaster with a payoff of -10 for both. If one plans the party on Friday and the other
on Saturday, the one having the Saturday party gets a payoff of 5, and the other of 4.
4. Consider the following game. Nature flips a coin, and it comes out H with probability 0.8 and
T with probability 0.2. Player 1 sees the result but player 2 does not. Player 1 announces to
player 2 that the coin came out either H or T (he can tell the truth or not). Then player 2
tries to guess the correct result. The payoffs are as follows: player 1 receives 1$ for telling the
truth and additional 2$ for inducing player 2 to choose H. Player 2 receives 1$ for making a
correct guess and 0$ otherwise. Draw the extensive form of this game, find the normal form
and compute a solution.
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Games and decisions, 2016/2017, list 2
5. A shopper encounter a fish merchant. The shopper looks at a price of fish and asks the
merchant ”is this fish fresh?”. Suppose the fish merchant knows whether the fish is fresh
or not, and the shopper only knows that about 50% of fishes are fresh. The merchant can
then answer the question either ”yes” or ”no”. The shopper, upon hearing the response, can
either buy the fish or wander on. Suppose the price of the fish is 1$, the value of a fresh fish
to the shopper it 2$ (i.e. this is the maximum the shopper would pay for the fresh fish), and
the value of fish that is not fresh is 0$. Suppose that fish merchant must throw out the fish
if it is not sold, but keeps 1$ profit if he sells the fish. Finally, suppose the merchant has a
reputation to uphold, and he loses 0.5$ when he lies, regardless of the shopper action. Draw
the extensive form of this game, find the normal form and compute a solution.
6. Reduce the 2-person zero-sum games given below via dominance considerations:
α1
(a) α2
α3
α4
β1
2
4
-2
-4
β2
4
8
0
-2
β3
0
2
4
-2
β4
-2
6
2
0
α1
(b) α2
α3
α4
β1
2
6
4
2
β2
-3
-4
3
-3
β3
1
1
3
2
β4
-4
-5
2
-4
β3
1
1
3
2
β4
-4
-5
2
-4
7. Find equilibria pairs in the following 2-person zero-sum games:
α1
(a) α2
α3
α4
β1
4
3
4
3
β2
3
2
2
3
β3
1
2
2
1
β4
1
2
2
2
α1
(b) α2
α3
α4
β1
2
6
4
2
β2
-3
-4
3
-3
8. Using the graphical method, find Nash equilibria in the following 2-person zero-sum games.
In the case (c) use a dominance argument to simplify the game.
(a) α1
α2
β1
5
3
β2
1
4
(b) α1
α2
β1
6
0
β2
0
5
α1
(c)
α2
α3
β1
2
2
-1
β2
1
0
3
β3
0
3
-3
9. Player 1 holds a black Ace and a red 8. Player 1 holds a red 2 and a black 7. The players
simultaneously choose a card to play. If the chosen cards are of the same color, Player I wins.
Player II wins if the cards are of different colors. The amount won is a number of dollars
equal to the number on the winners card (Ace counts as 1.) Set up the payoff function, find
the value of the game and the optimal mixed strategies of the players. Give an interpretation
of the obtained solution.
10. After dinner, Alan and Barbara go to a bar for some drinks. I know a game, says Barbara.
We point fingers at each other; either one finger or two fingers. If we match with one finger,
you buy me one drink. If we match with two fingers, you buy me two drinks. If we don’t
match, I will give you a dollar.
(a) Write the payoff matrices for Alan and Barbara of this game (a drink costs $5,50).
(b) What are the optimal strategies for both players?
(c) After a minute Alan answers: I will only play this game, if you pay me $2 before each
game! Do you think Alan ever had a game theory class?
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Games and decisions, 2016/2017, list 2
11. Players 1 (name: Odd) and 2 (name: Even) simultaneously call out one of the numbers 1
or 2. Odd wins if the sum of the numbers is odd, Even wins if the sum of numbers is even.
The amount paid to the winner by the loser is the sum of numbers in Dollars. Would you be
rather Even or Odd?
12. Consider the following 2-person zero-sum game:
α1
α2
α3
α4
β1
-3
-1
3
2
β2
-3
3
-1
2
β3
2
-2
-2
-3
Construct linear programming problems to compute a Nash equilibrium in this game (do not
solve them).
13. Consider the following game in extensive form:
1
O
E
2
C
0,100
F
40,50
-10,0
(a) Represent this game in normal form.
(b) Compute all Nash equilibria in this game.
(c) Using the backward induction compute the subgame perfect equilibrium.
14. Three objects O1 , O2 , O3 have different worths for two players 1 and 2, given by the following
table:
Worth for player 1
Worth for player 2
O1
1
2
O2
2
3
O3
3
4
Player 1 starts with choosing an object. After him player 2 chooses an object, then player 1
takes his second object, and finally player 2 gets the object that is left.
(a) Draw the game tree for this extensive form game.
(b) Determine the subgame perfect equilibria (in pure strategies).
15. Adam, Bill, and Cindy are registering for a foreign language class. The available classes are
ITA100 and FRE100. They do not not care much which, but they care with whom they
share the class. Bill and Cindy want to be in the same class, but want to avoid Adam. Adam
wants to be in the same class as Bill or Cindy, or even better, both. Analyze two cases: 1)
Adam, Bill and Cindy chose the classes independently and simultaneously 2) They choose
their classes in the order Adam, Bill, Cindy and they can observe the choices.
Games and decisions, 2016/2017, list 2
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16. All thousand residents of a certain town live, equally spaced out, on Main Street, which is
the only road in the town. There are no business in the town, so two residents independently
decide to set up some stores. They can each locate at any point between the beginning of
Main Street, which we will label 0, and the end, which we will label 1 (if they locate at the
same point, they move to opposite sites of the street). Each store will get all the costumers
who are closest to it, and they share equally customers who are equidistant between the two.
Each customer spends 1$ each day.
(a) Find the unique pure strategy Nash equilibrium of this game.
(b) Consider a situation in which three customers decide to open stores (they all can open
a store in the same location). Show that there is no pure strategy Nash equilibrium.