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115ECON 701 MODULE 7 EXERCISES 微观经济练习代写 Exercise 1 (The Monty Hall Game). The example we now discuss is now fairly well known. The name comes from an American television game show, Ex...
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博弈论作业代写 Exercise 1. Draw the following two trees and give the function p by specifying what p(x) is for each x in X for both trees.
Draw the following two trees and give the function p by specifying what p(x) is for each x in X for both trees. [Note: the trees differ in only the initial node.]
X = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}
B = {{x_{1}, x_{2}}, {x_{1}, x_{3}}, {x_{1}, x_{4}}, {x_{4}, x_{5}}, {x_{4}, x_{6}}, {x_{6}, x_{7}}, {x_{6}, x_{8}}, {x_{6}, x_{9}}}
x_{0} = x_{1};
and
X = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}
B = {{x_{1}, x_{2}}, {x_{1}, x_{3}}, {x_{1}, x_{4}}, {x_{4}, x_{5}}, {x_{4}, x_{6}}, {x_{6}, x_{7}}, {x_{6}, x_{8}}, {x_{6}, x_{9}}}
x_{0} = x_{6}.
The following is a formal description of a game tree using the (X, p) notation. Draw the game tree and give the equivalent definition using the (X, B, x_{0}) notation.
X = {x, v, y, z, w, a, b, c, d, e, f}
p(x) = ∅ p(v) = x p(y) = x p(z) = y p(w) = y
p(a) = w p(b) = w p(c) = w
p(d) = z p(e) = z p(f) = z
For the following formal description of an extensive form game draw the extensive form game.
N = {1, 2}
X = {x, v, y, z, w, a, b, c, d, e, f}
p(x) = ∅ p(v) = x p(y) = x p(z) = y p(w) = y
p(a) = w p(b) = w p(c) = w
p(d) = z p(e) = z p(f) = z
α(v) = T α(y) = B α(w) = U α(z) = D
α(a) = L α(b) = C α(c) = R
α(d) = L α(e) = C α(f) = R
H = {H_{1}, H_{2}, H_{3}}
H(x) = H_{1}H(y) = H_{2} H(w) = H_{3} H(z) = H_{3}
n(H_{1}) = 1 n(H_{2}) = 1 n(H_{3}) = 2
And the utility function is given by
T | v | a | b | c | d | e | f |
u_{1} | 5 | 8 | 0 | 6 | 0 | 7 | 6 |
U_{2} | 4 | 5 | 0 | 3 | 0 | 6 | 3 |
Consider the extensive form game in the game shown in Figure 5.
(1) Give a formal description of the game tree, both in the form (X, B, x_{0}) and in the form (X, p).
(2) Give a formal description of this extensive form game.
(3) Find the associated normal form game.
(4) Find all the weakly dominated strategies in the associated normal form game.
Consider the extensive form game shown in Figure 6.
(1) Give the associated normal form of this game.
(2) Find all the equilibria in mixed strategies of this game.
(3) Which of the equilibria are (equivalent to) subgame perfect equilibria of the extensive form game.
Consider the extensive form game shown in Figure 7.
(1) Give the associated normal form game.
(2) Find all the equilibria of this game. What equilibria are (equivalent to) subgame perfect equilibria?
In the game of Figure 8 if we split Player 1’s choice into two binary choices, having him first choose whether to play T and if he chose not to play T then to choose whether to play M or B, we might think that we have not very much changed the game. However in this case subgame perfection will rule out the equilibria in which Player 1 chooses T. This problem asks you to work out some of the details of this example.
Give the associated normal form of the game of Figure 8. and find all the Nash equilibria.
What are the subgames of the game of Figure 8 Find all the subgame perfect equilibria.
Suppose that we split Player 1’s choice into two binary choices, having him first choose whether to play T and if he chose not to play T then to choose whether to play M or B. Draw the extensive form of this game, give the associated normal form, and find all the Nash equilibria of the new game.
Find the subgame perfect equilibria of the new game. Remember that subgame perfect equilibria are defined in terms of behaviour strategies and in the new game Player 1 moves twice, so his mixed strategies and behaviour strategies are different.
Suppose that we change the game again so that Nature moves first and that with probability one half the payoffs are as the were in the original game and with probability one half the payoffs 5, 1 and 4, 1 are reversed with 5, 1 being the payoff if Player 1 chooses B and Player 2 chooses L and 4, 1 being the payoff if Player 1 chooses M and Player 2 chooses L. Suppose that Player 1 observes Nature’s choice before making any moves and Player 2 does not observe Nature’s choice.
Draw the extensive form games for these cases, both when Player 1 chooses between T, M, and B at a single node and when we split Player 1’s choice into two binary choices, having him first choose whether to play T and if he chose not to play T then to choose whether to play M or B. Find the associated normal forms and the Nash equilibria. In each case find the subgames and all the subgame perfect equilibria.
In the game of the game of Figure 8 find the unique sequential equilibrium. Explain why none of the equilibria (in behaviour strategies) in which Player 1 plays T are the strategy parts of sequential equilibria.
Find all the Nash equilibria of the game “Selten’s Horse” given in Figure 18. Which of these are Kuhn equivalent to sequential equilibria? For those that are give a complete description of the equivalent sequential equilibrium.
Draw a signalling game to represent the following simplified Spence educational signalling model. There are two types of the sender, L and H, three possible messages, E_{L}, E_{M}, and E_{H}, and three possible responses, W_{L}, W_{M}, and W_{H}. The payoff to the sender is the sum of two components one depending only on W and equal 10 for W_{H}, 6 for W_{M} and 2 for WL and one depending on the type and the message and being negative (the cost of education) with the cost being 0 for either type for E_{L}, −1 for type H and −2 for type L for E_{M}, and −2 for type H and −12 for type L for E_{H}.
The payoff for the receiver does not depend on the message and is equal to 3 for W_{H}, 2 for W_{M} and 0 for W_{L} if the type is H and −3 for W_{H}, −1 for W_{M} and 0 for W_{L} if the type is L.
What is the response of the receiver to E_{M} in such an equilibrium? What can you say about the probabilistic response of the receiver to the messages E_{L} and E_{H} in such an equilibrium? Pick one set of responses that make both types sending E_{M} an equilibrium. What can you say about what the beliefs of the receiver at each of the information sets must be that equilibrium? An equilibrium like this is called a pooling equilibrium because both types send the same message.
There is another sequential equilibrium in which the L type chooses E_{L} with probability 1 and the H type chooses E_{H} with probability 1. What can you say about the response of the receiver and the beliefs of the receiver at each of his information sets?
Are there any other sequential equilibria? If so find them.
Consider the two-person extensive form game in Figure 19.
(1) Suppose that both types of the sender choose m’ , that is, the behaviour strategy of Player 1 is ((1, 0),(1, 0)). What restrictions, if any, does this put on the beliefs of the receiver, in a consistent assessment, at the information set following the choice m?
(2) There are three (sets of) sequential equilibria: one equilibrium in which type 1a chooses m and type 1b chooses m’ ; one equilibrium in which type 1a chooses m’ and type 1b chooses m; and a set of equilibria in which both types choose m’ . Explicitly construct one of these equilibria.
(3) Explicitly construct all of the other sequential equilibria.
(4) Give the normal form of this game and say which strategy profile is equivalent to the equilibrium you found in part (2).
Consider the following two-person extensive form signaling game in Figure 20.
(1) Suppose that both types of the sender choose L, that is, the behaviour strategy of Player 1 is ((1, 0),(1, 0)). What restrictions, if any, does this put on the beliefs of the receiver, in a consistent assessment, at the information set following the choice R?
(2) There are two (sets of) sequential equilibria: one equilibrium in which type 1a chooses R and type 1b chooses L; and a set of equilibria in which both types choose L. Explicitly construct one of these sets of equilibria.
(3) Explicitly construct the other set of sequential equilibria.
(4) Give the normal form of this game. Are there any pure strategy equilibria other than the sequential equilibria you found above?
Consider the signalling games given in Figures 21 and 22.
(1) In each case give the normal form of the game.
(2) In the first game suppose that both types of the sender choose m’ , that is, the behaviour strategy of Player 1 is ((1, 0),(1, 0)). What restrictions, if any, does this put on the beliefs of the receiver, in a consistent assessment, at the information set following the choice m? Show how your answer follows from the definition of a consistent assessment.
(3) For each game find the sequential equilibria.
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