EC403
Game Theory and Applications
Problem Set 13
Due Tue, April 1
Your name: ________________________________________
1. Recall the simplest version of the centipede game that was previously discussed in the text:
The game begins with $1 on the table. There are two players. Player 1, who is moving first, can either
take the dollar or wait.
If player 1 takes the dollar, the game is over, and she gets to keep the dollar. Player 2 in this case gets
nothing.
If player 1 waits, the dollar on the table quadruples to $4, after which it is player 2’s turn. Player 2 can
either take the entire $4 or split the $4 evenly with player 1.
Player 2 belongs to one of the two types. He can be selfish, in which case he receives utility strictly equal
to his monetary payoff from the game.
Or, he can be nice, in which case he experiences a strong sense of guilt if he takes all the money at his
turn. His utility in this case (if he takes all the money) equals the amount of money he gets minus the
burden of guilt equivalent to $4. As a result, his overall payoff from taking the money is 4 – 4 = 0.
Player 2 knows his own type but has no means of credibly signaling his type to player 1.
a. Present this as a game of imperfect information in extensive form, which involves Nature as another
(third) player. Make player 1 move first, Nature – second, and player 2 – third.
Denote player 1’s belief of player 2 being nice as p.
b. Practice gradual transformation of this game to the reduced form by
- eliminating dominated strategies (branches) for player 2 as we did in class on slide 129;
- replacing the uncertainty with the expected payoffs resulting from the Nature’s moves, similar to
the Conscription game (Figure 7.8 on page 164 in the text).
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EC403
Game Theory and Applications
Problem Set 13
Due Tue, April 1
c. Player 1 in her beliefs places a 30% probability on player 2 being “nice” (so p=0.3).
Under these circumstances, what is the best strategy for player 1 to play? Carefully explain why.
d. State the Bayesian Nash equilibrium you have found. Redraw the complete game tree and clearly show
your solution by highlighting the branches that are parts of the equilibrium.
e. Go back to the version of the game when p is unknown. In what range of values for p player 1 will
prefer to wait?
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EC403
Game Theory and Applications
Problem Set 13
Due Tue, April 1
2. Find and state all the Nash equilibria (including the mixed-strategy one) for the game presented on
slide 139.
Bird 2
(H, H)
(D, H)
Hawk
-1, 0
1.5, 0.5
Dove
0, 4
1, 3
Bird 1
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EC403
Game Theory and Applications
Problem Set 13
Due Tue, April 1
3. The following problem will help you practice using the Bayes’ theorem.
Stephen is shopping for a used car. He knows that there are some good cars and some bad cars in the
market. More specifically, he knows that one in five cars is of very low quality (we call such a car a
“lemon”) while the rest are of high quality. There is no way to perfectly tell a lemon from a good car but
there are some tell-tale signs, such as rust around the doors, some signs that the car was re-painted, oil
leaks, etc. For simplicity, let us focus on one such sign, an oil leak.
80 percent of lemons have oil leaks (other 20 percent do not but they have other problems that still make
them lemons). At the same time, 30 percent of good cars have oil leaks (after all, these are all used cars!)
but are otherwise very good.
Let us assign letter A to the event “this car is good” and B to “oil leak spotted”.
a. Given this notation, state the following probabilities. The first two can be obtained from the statement
of the problem.
Probability that a car picked at random is a good car, P(A) =
Probability that a good car has an oil leak, P(B|A) =
The next two require calculations. Show your work!
Probability that a car picked at random leaks oil,
P(B) =
Probability that a car with an oil leak is a good car,
P(A|B) =
b. Stephen is willing to pay $8000 for a good car and $2000 for a lemon.
Given his prior beliefs, what is the expected value to him of a car picked at random?
c. Stephen opens the hood and sees an oil leak. What is the expected value of this car to him?
d. For extra credit:
Suppose Stephen opens the hood and sees no oil leak. What is the expected value of such a car to him?
(Use the back of this page or attach an extra sheet if needed.)
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EC403
Game Theory and Applications
Problem Set 13
Due Tue, April 1
4. Another optional question for extra credit.
Invent a game in which
- There are two players,
- Each of the players has two pure strategies to choose from,
- One of the players (say, player 2) can be of one of two types,
- Player 1 does not know player 2’s type,
- Player 2’s type affects some of the game payoffs,
- As a result, the payoff matrix of the game has two versions depending on player 2’s type,
- One version of the game has a unique pure strategy Nash equilibrium and the other version has no
pure strategy Nash equilibrium.
Provide a story and the set of payoffs that supports that story. Present each of the two versions of the
game in the normal form and solve for equilibria.
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