Micro II Question sets

Note: 2019, 2020 seems different.

General equilibrium

2022 Aug Prelim Part 2 Q2

Tom Sawyer, Joe Harper, and Huckleberry Finn each have received a ticket after performing a difficult task. A ticket, by itself, is not worth anything. But if one of the boys has three tickets, he can exchange them for a prize valued at $\mathrm{\$}100$$\$ 100$. Tom decides that rather than work hard to earn more tickets, he's going to try to trick Joe and Huck into giving him theirs. a. What would the Walrasian equilibrium (if there is one) of this economy be? b. What is the core of this economy?

2021 Aug Prelim Part 2 Q3

(35 points) Consider an economy with two consumption goods 1 and 2 and two individuals Mr A and Mr B. The utility functions are given as $U_A(x_1^A,x_2^A)=x_1^A-\frac{1}{8}{\left(x_2^A\right)}^{-8}$$U_A\left(x_1^A, x_2^A\right)=x_1^A-\frac{1}{8}\left(x_2^A\right)^{-8}$ and $U_B(x_1^B,x_2^B)=-\frac{1}{8}{\left(x_1^B\right)}^{-8}+x_2^B$$U_B\left(x_1^B, x_2^B\right)=-\frac{1}{8}\left(x_1^B\right)^{-8}+x_2^B$. Here, $x_k^A$$x_k^A$ and $x_k^B$$x_k^B$ are the good $k\in \{1,2\}$$k \in\{1,2\}$ consumptions of Mr A and Mr B, respectively. The initial endowments of agents are $e_A=(2,r)$$e_A=(2, r)$ and $e_B=(r,2)$$e_B=(r, 2)$ where $r=2^{8/9}-2^{1/9}$$r=2^{8 / 9}-2^{1 / 9}$. (a) Write down each individual's optimization problem. (5 points) (b) Find each individual's demand for the consumption goods. (5 points) (c) Find the market clearing prices (10 points) (d) Now consider a general equilibrium model with initial endowments, $M$$M$ goods and $I$$I$ individuals. Each individual's utility only depends on her consumption and it increases with the consumption of each good. Prove that any competitive equilibrium outcome is Pareto efficient.(15 points)

2020 Aug Prelim Part 2 Q3

(10 points) Consider the following exchange economy. There are 2 consumers with identical preferences over two consumption goods, $x_1$$x_1$ and $x_2$$x_2$, given by:

Consumer 1 has endowment $w_1=(1,0)$$w_1=(1,0)$ and consumer 2 has endowment $w_2=(0,1)$$w_2=(0,1)$. Calculate the set of Pareto efficient allocations.

2019 June Prelim Part 2 Q2

Consider the following exchange economy. There are 2 consumers with preferences over two consumption goods, $x$$x$ and $y$$y$, given by:

Consumer 1 has endowment $(1,0)$$(1,0)$ and consumer 2 has endowment $(0,1)$$(0,1)$​. Calculate the set of Pareto efficient allocations.

2018 June Prelim Part 2 Q2

(40 percent) Consider an economy with two consumption goods 1 and 2 and two individuals $\mathrm{Mr}\mathrm{A}$$\mathrm{Mr} \mathrm{A}$ and Mr B. The utility functions are given as $U_A(x_1^A,x_2^A)=x_1^A{x}_2^A$$U_A\left(x_1^A, x_2^A\right)=x_1^A x_2^A$ and $U_B(x_1^B,x_2^B)=$$U_B\left(x_1^B, x_2^B\right)=$ $x_1^B{x}_2^B$$x_1^B x_2^B$. Here, $x_k^A$$x_k^A$ and $x_k^B$$x_k^B$ are the good $k\in \{1,2\}$$k \in\{1,2\}$ consumptions of Mr A and Mr B, respectively. The initial endowments of agents are $e_A=(2,1)$$e_A=(2,1)$ and $e_B=(1,5)$$e_B=(1,5)$. (a) Find each individual's demand for the consumption goods and market clearing prices.(10 percent) (b) Suppose Mr A knows Mr B's preferences and can make a take-it-or-leave-it offer in terms of good 1 and good 2 . If Mr B rejects the offer, there is no trade. What will the resulting allocation be? Will it be efficient?(10 percent) (c) Assume that Mr A still knows Mr B's preferences and can make a take-it-or-leave-it offer, but now he can only offer a price, and Mr B is free to choose the quantities he wants to exchange. What will the resulting allocation be? Will it be efficient?(10 percent) (d) Suppose the government considers imposing a $50\mathrm{\%}$$50 \%$ tax on the consumption of good 1 by both Mr A and Mr B. Compute the equilibrium price and the equilibrium allocation. What happens to the welfare of $\mathrm{Mr}\mathrm{A}$$\mathrm{Mr} \mathrm{A}$ and $\mathrm{Mr}\mathrm{B}$$\mathrm{Mr} \mathrm{B}$ ? Is the equilibrium allocations efficient? Hint: The consumption tax is paid from all good 1 they consume, not from what they trade.(10 percent)

2018 Aug Prelim Part 2 Q1/ 2023 Final Q3

Consider the following exchange economy. There are 2 agents, Robinson and Friday, and 2 consumption goods, $x_1$$x_1$ and $x_2$$x_2$. Robinson has utility function

and Friday has utility function

Here, $x_{f,k}$$x_{f, k}$ and $x_{r,k}$$x_{r, k}$ denote the consumption of good $k\in \{1,2\}$$k \in\{1,2\}$ for Friday and Robinson, respectively. Robinson has endowment $w_r=(w_{r,1},w_{r,2})=$$w_r=\left(w_{r, 1}, w_{r, 2}\right)=$ $(2,4)$$(2,4)$ and Friday has endowment $w_f=(w_{f,1},w_{f,2})=(4,2)$$w_f=\left(w_{f, 1}, w_{f, 2}\right)=(4,2)$. (a) Calculate the set of Pareto efficient allocations.(Hint: Robinson's utility depends on the total consumption of good 2.) (10 points)

The PEA allocation is $x_{r,2}=0,x_{f,2}=6,x_{r,1}+x_{f,1}=6$$x_{r, 2} = 0, x_{f, 2} = 6, x_{r, 1} + x_{f, 1} = 6$. It shows a horizonal line on the graph.

(b) Compute Walrasian equilibria of this economy. (10 points)

For WEA (competitive equilibrium), we need to find a price $p$$p$ and corresponding allocation $x$$x$ such that both agents maximize their utility and market clearing.

From the previous question we know that the PEA is $x_{r,2}=0,x_{f,2}=6,x_{r,1}+x_{f,1}=6$$x_{r, 2} = 0, x_{f, 2} = 6, x_{r, 1} + x_{f, 1} = 6$, the WEA must be a corner solution that has a price range. And, we know $W(e)\subseteq C(e)\subseteq PEA(e)$$W(e) \subseteq C(e) \subseteq PEA(e)$, which implies that $p>\frac{3}{2}$$p>\frac32$ (derived from core). Under this price (ratio), both UMP and market clearing conditions are satisfied. The corresponding allocation is

$$\begin{array}{}\text{(5)}& \begin{array}{c}{x}_{f,1}=\frac{4p-4}{p}\\ x_{r,1}=\frac{2p+4}{p}\end{array}\end{array}$$

2021 Final Q1 - [General Equilibrium]

An exchange economy consists of two consumers, $A$$A$ and $B$$B$, with utility function

So the first commodity is a good for each consumer, whereas the second commodity is a bad for each consumer. Their initial endowments are $e^A=(4,3)$$e^A=(4,3)$ and $e^B=(1,0)$$e^B=(1,0)$​ (a) (8 points) Find the consumers' Walrasian demand functions. (Hint: Setup individual maximization problem where budget constraint holds with equality and find demand for commodities in terms of prices.)

For agent A, the consumer's problem is,

$$\begin{array}{}\text{(7)}& \underset{x_1^A,x_2^A}{max}{x}_1^A(4-x_2^A)\end{array}$$

s.t. (normalize the price of $x_2$$x_2$ to be 1)

$$\begin{array}{}\text{(8)}& p{x}_1^A+x_2^A=4p+3\end{array}$$

FOCs:

• $4-x_2^A-\lambda p=0$$4 - x_2^A - \lambda p = 0$​

• $-x_1^A-\lambda =0$$-x_1^A - \lambda = 0$

• $p{x}_1^A+x_2^A=4p+3$$px_1^A + x_2^A = 4p + 3$​

Then, the demand for commodities are,

$$\begin{array}{}\text{(9)}& \begin{array}{c}{x}_1^A=\frac{4p-1}{2p},x_2^A=\frac{4p+7}{2}\\ x_1^B=\frac{p-4}{2p},x_2^B=\frac{p+4}{2}\end{array}\end{array}$$

(b) (7 points) Show that an allocation is Pareto efficient if and only if $x_1^A+x_2^A=4$$x_1^A+x_2^A=4$​​​. (Hint: Think about the contract curve.)

(c) (10 points) Find the competitive equilibria in this economy.(Hint: Price of one good can be negative.)

According to the market clearing condtion, $p=-1$$p = -1$

 

2021 June Prelim Part 2 Q1 - [General Equilibrium]

Suppose Umut and Will are trapped in a remote island and they can only consume banana $(B)$$(B)$ and coconut $(C)$$(C)$. Umut has nine bananas and one coconut. Will has one banana and nine coconuts. Their utility functions are:

(a) Show this economy in an Edgeworth box. (2 points)

Omit.

(b) Find the set of Pareto efficient allocations and depict these in the Edgeworth box. (5 points)

Under this utility function, there is only interior solution. So we pins down the PEA using their MRS.

$$\begin{array}{}\text{(11)}& \frac{C_u}{B_u}=\frac{C_w}{B_w}\end{array}$$

substitute the market clearing condition into this equation,

$$\begin{array}{}\text{(12)}& \begin{array}{c}\frac{C_u}{B_u}=\frac{10-C_u}{10-B_u}\\ \iff C_u=B_u\end{array}\end{array}$$

which describes the PEA (contract curve).

(c) Find the Walrasian equilibrium allocation and price ratio. (5 points)

To find the WEA, we need to solve

$$\begin{array}{}\text{(13)}& \underset{}{max}{B}_i{C}_{i}s.t.p{B}_i+C+i=p{e}_{i,B}+e_{i,C}\end{array}$$

For Umut, the solution is

$$\begin{array}{}\text{(14)}& \begin{array}{c}{B}_u=\frac{9p+1}{2p}\\ C_u=\frac{9p+1}{2}\end{array}\end{array}$$

For Will, the solution is,

$$\begin{array}{}\text{(15)}& \begin{array}{c}{B}_w=\frac{p+9}{2p}\\ C_w=\frac{p+9}{2}\end{array}\end{array}$$

And according to the market clearing condition,

$$\begin{array}{}\text{(16)}& \frac{9p+1}{2}+\frac{p+9}{2}=10\Rightarrow p=1\end{array}$$

Now suppose that, Thayer has arrived to the island to rescue them. Unfortunately, his boat is damaged and it is not working. Luckily, he brings five bananas and five coconuts with him. His utility function is $U_T(B,C)=BC$$U_T(B, C)=B C$​​.

(d) Find the set of Pareto efficient allocations in this new economy with three individuals. (7 points)

To solve this problem rigourously, we need to solve the following maximization problem (suppose it's Umut solve this problem)

$$\begin{array}{}\text{(17)}& \underset{B_u,C_u,B_w,C_w,C_t,B_t}{max}{U}_u=B_u{C}_u\end{array}$$

Subject to,

$$\begin{array}{}\text{(18)}& \begin{array}{c}{B}_w{C}_w\ge {\overline{u}}_w\\ B_t{C}_t\ge {\overline{u}}_t\\ B_u+B_w+B_t=15\\ C_u+C_w+C_t=15\end{array}\end{array}$$

(e) Find the Walrasian equilibrium allocation and price ratio in this new economy with three individuals. (8 points)

Intuitively, the price ratio is still $p=1$$p = 1$

(f) Do Umut and Will become better off due to arrival of Thayer? (3 points)

No.

Game Theory

2020 Oct Prelim Part 2 Q2 - [PBE]

(20 Points) Solve for the Perfect Bayesian equilibria of the game below.

2020 Aug Prelim Part 2 Q1 - [PBE]

(20 points) My department head does not know if I need a new computer or not, but he believes that there is a $50\mathrm{\%}$$50 \%$ chance that I do. If I need a new computer, I will ask him for one. But even if I don't need one, I would like one. The payoff for me if I need and get a computer is $\mathrm{\$}3,000$$\$ 3,000$. The payoff to me if I do not need but get a computer is $\mathrm{\$}500$$\$ 500$. If my department head turns me down, this embarrasses me and "costs" me $\mathrm{\$}100$$\$ 100$ worth of psychic pain. It costs my department head $\mathrm{\$}2,000$$\$ 2,000$​ to get me a new computer, and he cares about the expected "profit" (benefit minus cost) for the department. Solve for the perfect Bayesian equilibrium.

2021 Aug Prelim Part2 Q1 -

(25 points) Consider the following extensive form game with 3 players.

(a) Represent this game as normal form. (5 points) (b) Find all Nash equilibria (including the mixed strategy). (7 points) (c) Find all (pure) subgame perfect Nash equilibria. (5 points) (d) Find all (pure) perfect Bayesian equilibria. (8 points)

2020 Oct Prelim Part 2 Q3

(15 points) Suppose there are two firms and that the market price is given by $P=100-$$P=100-$ $q_1-q_2$$q_1-q_2$ where $q_i$$q_i$ is the quantity firm $i$$i$ chooses. Both firm have zero marginal costs. Firm 1 wishes to maximize profits. Firm 2 is unconcerned about profits and instead wishes to minimize the distance between her choice and firm 1's choice. Specifically, firm 2's payoff is given by ${\pi }_2(q_1,q_2)=-{(q_1-q_2)}^2$$\pi_2\left(q_1, q_2\right)=-\left(q_1-q_2\right)^2$. (a) Suppose the firms choose simultaneously. Solve for a pure strategy equilibrium. (b) Suppose firm 1 chooses first, firm 2 learns 1's choice, and then firm 2 chooses. Solve for the subgame perfect equilibrium. (c) Suppose firm 2 chooses first, firm 1 learns 2's choice, and then firm 1 chooses. Solve for the subgame perfect equilibrium.

2020 Aug Prelim Part 2 Q2

(20 points) Ben and Jerry are going to have a meeting to write a paper together. Before the meeting, each must simultaneously decide how much reading $(R_J$$\left(R_J\right.$ and $R_B)$$\left.R_B\right)$ to do for the project. If a person reads an amount $R$$R$, it costs them $c\ast R$$c * R$ of effort. At the start of the meeting, they talk, and both observe how much the other person has read. At that point, they each (simultaneously) decide how much effort to exert $(e_B$$\left(e_B\right.$ and $e_J)$$\left.e_J\right)$. The degree of success of the project is determined by how much each has read and how much exert each exerts. Specifically, the payoffs to each person is:

Solve for the Nash equilibrium.

2022 Aug Prelim Part 2 Q3

Consider the following game. Player 1 chooses $x$$x$ and Player 2 chooses $y$$y$. The payoff they receive from these choices are:

(a) If the two players choose simultaneously, what is the Nash Equilibrium? (b) If Player 1 first chooses $x$$x$, Player 2 observes $x$$x$, and Player 2 then chooses $y$$y$, what is the Nash Equilibrium?

2019 June Prelim Part 2 Q1

Students are of three types: High, Medium, or Low. The cost of getting a college degree to a student is 2 if High, 4 if Medium, and 6 if Low. The proportion of students of High, Medium, or Low type are $\frac{1}{6},\frac{1}{2}$$\frac{1}{6}, \frac{1}{2}$, and $\frac{1}{3}$$\frac{1}{3}$​, respectively. The salaries for managers are 15 and 10 for clerks. The Firm's profits (net of paying wages) are 7 from hiring anyone as a clerk, 4 from hiring a low type as a manager, 6 from hiring a medium type as a manager, and 14 from hiring a high type as a manager. Students must choose whether or not to go to college. The firm observes whether or not a student goes to college and must decide what position to give the student. Find a Perfect Bayesian Nash Equilirbium of this game.

2021 June Prelim Part 2 Q2

Suppose Mr. Ford wants to sell his Mustang. There is only one buyer in the market, Mr Buyer. Mr Ford knows what is the quality of his car, but Mr. Buyer does not. Mr Buyer knows only that the car could be a good quality with probability 0.8 and poor quality with probability 0.2 . If the car is good, Mr. Buyer's valuation for it is $\mathrm{\$}20,000$$\$ 20,000$ and Mr. Ford's is $\mathrm{\$}10,000$$\$ 10,000$. If the car is poor quality, the valuations of both Mr. Ford and Mr Buyer are $\mathrm{\$}0$$\$ 0$. Mr. Ford can make two offers: $\mathrm{\$}5,000$$\$ 5,000$ and $\mathrm{\$}15,000$$\$ 15,000$. Then, Mr. Buyer can accept the offer or reject the offer. (a) Draw the extensive form of this game. (5 points) (b) Find all the separating perfect Bayesian equilibria, if there exists any. (5 points) (c) Find all the pooling perfect Bayesian equilibria, if there exists any. (10 points)

Now consider the following variation in which Mr. Ford has an option of passing an inspection. If the inspection finds the car in good shape, Mr. Ford is charged $\mathrm{\$}200$$\$ 200$ to get the proof of inspection. Otherwise, he needs to pay $\mathrm{\$}15,200$$\$ 15,200$ to have his car serviced and get the proof of inspection. In this case, the Mr Buyer's valuation for the serviced car is $\mathrm{\$}20,000$$\$ 20,000$ and Mr. Ford's is $\mathrm{\$}10,000$$\$ 10,000$ (Notice that once a car is serviced it will be in good quality). Now, Mr. Ford has four choices: $\mathrm{\$}5,000$$\$ 5,000$ with inspection, $\mathrm{\$}15,000$$\$ 15,000$ with inspection, $\mathrm{\$}5,000$$\$ 5,000$ without inspection, and $\mathrm{\$}15,000$$\$ 15,000$ without inspection. (d) Draw the extensive form of this game. (5 points) (e) Find a separating perfect Bayesian equilibrium. (10 points)

2021 June Prelim Part 2 Q3

Consider the following game between two rational, expected-utility maximizing primates, Little Lemur and Big Baboon. There are two bananas, a delicious yellow one and a not-quite-delicious green one. First, Little Lemur picks one of the two bananas. Then, Big Baboon after observing Little Lemur's choice, chooses whether to whack Little Lemur on the head and eat the banana that he picked (in which case Little Lemur is left to eat the other banana), or eat the other banana and leave Little Lemur alone to eat the banana he picked. Eating the yellow banana gives either primate 2 units of utility and eating the green one gives 1 unit of utility. Whacking Little Lemur on the head given Big Baboon an extra $X$$X$ units of utility, while being whacked on the head costs Little Lemur $Y$$Y$ unit of utility; both $X$$X$ and $Y$$Y$ are strictly positive. The structure of the game is common knowledge. (a) Draw the extensive form of this game. (5 points) (b) Describe all the subgame perfect equilibria of this game, assuming $X>1$$X>1$. (7 points) (c) Describe all the subgame perfect equilibria of this game, assuming $X<1$$X<1$. (8 points) (d) Is Little Lemur better off when Big Baboon really likes whacking him $(X>1)$$(X>1)$ or when she does not like it that much $(X<1)$$(X<1)$ ? Explain. (5 points) (e) Now suppose that it is common knowledge that Big Baboon is green-yellow colorblind, and so CANNOT TELL which banana Little Lemur picked. Draw the extensive form of this new game and find all the perfect Bayesian equilibria, assuming that $X<1$$X<1$. (10 points)

2018 June Prelim Part 2 Q1 - [Signaling]

(30 percent) Consider a signaling game under incomplete information between a worker (W) and employer (E). The worker (W) can be two types and it is determined by nature. W is high-type with probability $1/3$$1 / 3$ and low-type with probability $2/3$$2 / 3$. W privately observes his own type, i.e., E does not know E's type. W then decides whether to pursue more education that he might use as a signal about his type. The cost of education is 4 for the high type and 7 for the low type. E can hire $\mathrm{W}$$\mathrm{W}$ as a manager or as a cashier, observing whether $\mathrm{W}$$\mathrm{W}$ acquired an additional college degree, but not observing W's real productivity. The utility $\mathrm{W}$$\mathrm{W}$ gets from being manager is 10 and being cashier is 4 . If $\mathrm{E}$$\mathrm{E}$ hires high type worker as manager he gets the utility of 10 . If $\mathrm{E}$$\mathrm{E}$ hires low type worker as manager he gets the utility of 0 . If $\mathrm{E}$$\mathrm{E}$ hires $\mathrm{W}$$\mathrm{W}$ as a cashier then he gets the utility of 4 independent of the type of W. (a) What are the pure strategies of $\mathrm{W}$$\mathrm{W}$​ and E. (5 percent) (b) Find all pooling perfect Bayesian Nash equilibria (if any). (12.5 percent) (c) Find all separating perfect Bayesian Nash equilibria (if any) (12.5 percent)

2021 August Prelim Part 2 Q2

(40 points) A Turkish producer has an Automated Doughnut Filler (ADF) worth $V>0$$V>0$ if sold at home and worth $V+s$$V+s$ if sold abroad, $s>0$$s>0$. To ship ADF abroad, the producer use a shipper. If the shipper ships the ADF, they choose the probability $q$$q$ that the ADF arrives undamaged. This costs the shipper $c(q)=100q^3$$c(q)=100 q^3$, $q\in [0,1]$$q \in[0,1]$. Any damage to the ADF is total and it reduces its value to 0 . Both the producer and the shipper are risk neutral expected profit maximizers

For parts a-e, assume that the values of $V$$V$ and $s$$s$ are common knowledge. Further, assume that the shipper makes a legally binding take-it-or-leave-it offer $(p,r)$$(p, r)$ to the producer. The producer accepts or rejects the offer, selling at home if the offer is rejected, paying $p$$p$ to the shipper if the offer is accepted, and being reimbursed $r$$r$ if the ADF is damaged in transit. If the offer is accepted the shipper chooses $q$$q$. (a) If an offer $(p,r)$$(p, r)$ is accepted, find the shipper's choice of $q$$q$ (depending on $(p,r))$$(p, r))$. (5 points) (b) Find the efficient level (for producer and shipper together) of $q$$q$. (5 points) (c) Find the producers expected profits of accepting an offer $(p,r)$$(p, r)$. ( 7 points) (d) If $(p,r)$$(p, r)$ is accepted by the producer, find the shipper's expected profits. (8 points) (e) Find the subgame perfect equilibrium of this game. (7 points)

For the rest of this problem, suppose that $s$$s$ is a random variable that takes on one of two possible values, 72 or 143, each with equal probability. Further assume that the producer knows the realization of $s$$s$, but that the shipper does not. The shipper now offers take-it-or-leave-it menu of shipping options $(P_H,r_H)$$\left(P_H, r_H\right)$ and $(p_L,r_L)$$\left(p_L, r_L\right)$, to the producer. The producer either rejects both menu choices and sells at home, or accepts one of the offer and ships. If an offer is accepted, the shipper chooses $q$$q$. (f) Find, or explain how to find, a perfect Bayesian equilibrium of this modified game. (8 points)

2018 Aug Prelim Part 2 Q2

(40 percent) Adam and Bob are two investors and they have each deposited $\mathrm{\$}D$$\$ D$ with a bank. They are the only depositers and the bank has invested these deposits in a long term project. The project matures in January 2020. If the bank is forced to liquidate its investment before the project matures, a total of $2r$$2 r$ can be recovered, where $D>r>D/2$$D>r>D / 2$. If the bank allows the investment to reach maturity, the project will pay out a total of $2R$$2 R$, where $R>D$$R>D$. The bank will not get any commision from its investors.

There are two dates at which the investors can make with drawals from the bank: February 2019 (before the maturity date) and February 2020 (after the maturity date). If both investors make withdrawal in February 2019 then each receives $r$$r$ and the game ends. If only one investor makes an early withdrawal in February 2019, then that investor receives $D$$D$, the other receives $2r-D$$2 r-D$, and the game ends. Finally, if neither investor makes an early withdrawal in February 2019 then the project matures and the bank will get $2R$$2 R$ from the investment. If both investors make withdrawals in February 2020, then each receives $R$$R$. If only one investors makes a withdrawal then that investor receives $D$$D$ and the other investor receives $2R-D$$2 R-D$. If neither investors makes withdrawal in February 2020 then each receives $R$$R$.

(a) Write down the complete strategies of Adam and Bob. Represent the game in normal form. (10 points) (b) Find all pure strategy Nash equilibrium of this game.(10 points) (c) Find all pure strategy Subgame Perfect Nash equilibrium of this game.(10 points) (d) Now suppose Adam and Bob can decide to invest or not. Write down the complete strategies of Adam and Bob. Find all pure strategy Subgame Perfect Nash equilibrium of this game. (10 points)

2018 August Prelim Part 2 Q3

(20 percent)The game matrix below gives Player 1's payoffs

3. 		Player 2
   		$\mathrm{S}$$\mathrm{S}$	$\mathrm{D}$$\mathrm{D}$
   	$\mathrm{U}$$\mathrm{U}$	15	90
   Player 1	$\mathrm{M}$$\mathrm{M}$	35	75
   	$\mathrm{D}$$\mathrm{D}$	55	40

Let $q$$q$ be the probability with which Player 1 believes that Player 2 will play $S$$S$. (a) Find three ranges of values of $q$$q$ for which $U,M$$U, M$ and $D$$D$ are optimal, respectively. points) (b) Is any action strictly dominated for Player 1 , and if so, by what mixed action?(10 points)

2023 June Prelim Part 2 Q1 - [Repeated Game]

(20 points) Consider the following infinitely repeated game with common discount factor $\delta \in (0,1):$$\delta \in(0,1):$

Let $(a_1,a_2)$$\left(a_1, a_2\right)$ be a pair of actions different from $(F_1,F_2)$$\left(F_1, F_2\right)$. Consider the following strategy profile: Player $i$$i$ :

• Conform: Play $a_i$$a_i$ until nobody else deviates from $(a_1,a_2)$$\left(a_1, a_2\right)$. If somebody deviates from $(a_1,a_2)$$\left(a_1, a_2\right)$ then go to $T$$T$-period punishment part.

• T-Period Punishment: Play $F_i$$F_i$ for $T$$T$ periods. If somebody deviates from $(F_1,F_2)$$\left(F_1, F_2\right)$ in these $T$$T$ periods go to the beginning of $T$$T$-period punishment part, and do it again. If nobody deviates, then after $T$$T$ periods go back to Conform.

(a) If $T=1$$T=1$ what is the critical value of discount factor ${\delta }^{\ast }$$\delta^*$ such that for all $\delta \ge {\delta }^{\ast }$$\delta \geq \delta^*$ the above strategy profile is SPE for $(a_1,a_2)=(M_1,M_2).(5$$\left(a_1, a_2\right)=\left(M_1, M_2\right) .(5$ points $)$$)$​

If nobody deviates, their payoff should be

$$\begin{array}{}\text{(21)}& 4+4\delta +4{\delta }^2+\cdots =\frac{4}{1-\delta }\end{array}$$

If one player deviates (let's say at the first round), then his/her payoff becomes

$$\begin{array}{}\text{(22)}& 5+\delta +4{\delta }^2+\cdots =5+\delta +\frac{4{\delta }^2}{1-\delta }\end{array}$$

Since the penalty only persists for one period, if we want this player has no incentive to deviate, we have to keep

$$\begin{array}{}\text{(23)}& 4+4\delta \ge 5+\delta \iff \delta \ge \frac{1}{3}\end{array}$$

This makes sure that any one of the players have no incentive to deviate at anytime.

(b) If $T=2$$T=2$ what is the critical value of discount factor ${\delta }^{\ast }$$\delta^*$ such that for all $\delta \ge {\delta }^{\ast }$$\delta \geq \delta^*$ the above strategy profile is SPE for $(a_1,a_2)=(M_1,M_2)$$\left(a_1, a_2\right)=\left(M_1, M_2\right)$​. (5 points)

If the player deviates, then the payoff becomes,

$$\begin{array}{}\text{(24)}& 5+\delta +{\delta }^2+4{\delta }^3+\cdots =5+\delta +{\delta }^2+\frac{4{\delta }^3}{1-\delta }\end{array}$$

To make sure the player has no incentive to deviate, we need

$$\begin{array}{}\text{(25)}& \begin{array}{c}5+\delta +{\delta }^2\le 4+4\delta +4{\delta }^2\\ \iff 3{\delta }^2+3\delta -1\ge 0\\ \delta \ge 0.2638\end{array}\end{array}$$

(c) If $T=\mathrm{\infty }$$T=\infty$ what is the critical value of discount factor ${\delta }^{\ast }$$\delta^*$ such that for all $\delta \ge {\delta }^{\ast }$$\delta \geq \delta^*$ the above strategy profile is SPE for $(a_1,a_2)=(M_1,M_2)$$\left(a_1, a_2\right)=\left(M_1, M_2\right)$​​​. (5 points)

If the player deviates, then the payoff becomes,

$$\begin{array}{}\text{(26)}& 5+\delta +{\delta }^2+\cdots =5+\frac{\delta }{1-\delta }\end{array}$$

To make sure the player has no incentive to deviate, we need

$$\begin{array}{}\text{(27)}& \begin{array}{c}\frac{4}{1-\delta }\ge 5+\frac{\delta }{1-\delta }\\ \iff \delta \ge \frac{1}{4}\end{array}\end{array}$$

(d) Why are these critical values different explain the intuition behind this? (5 points)

The longer the penalty period is, the smaller ${\delta }^{\ast }$$\delta^*$ we have. $\delta$$\delta$​ has the interpretation of the discount factor, which becomes less if the payoff in the future becomes less important. Therefore, if the payoff in the future becomes less important, we presumbly need to increase the length of our punishment period to reach an indifferent punishment that guarantee players not deviate.

Auction

2022 Aug Prelim Part 2 Q1

The government decides to auction off the oil rights under Sulphur Mountain. Bob and Umut decide to participate in the auction, and since they do not know how much oil is under the mountain, each hires a consultant to estimate the amount of oil. Consultants are expensive, and Bob and Umut can each only afford to estimate the amount of oil under one side of the mountain. Bob's consultant estimates that there is $e_B$$e_B$ worth of oil under the north side and Umut's consultant estimates there is $e_U$$e_U$ worth of oil under the south side, where $e_B$$e_B$ and $e_U$$e_U$ are both uniformly drawn from the interval $[0,1]$$[0,1]$. Thus, the total value of the oil is $V=e_B+e_U$$V=e_B+e_U$ but each person only knows their own estimate and that the other estimate is uniformly drawn from $[0,1]$$[0,1]$. Suppose the government holds a second-price auction, and suppose that Umut's strategy is to bid twice his estimate $\left(2e_U\right)$$\left(2 e_U\right)$. (a) What is the probability Bob wins from bid $b$$b$ (given Umut's strategy). (b) What is the expected price Bob will pay if he wins? (c) What is Bob's estimate of Umut's value if Bob wins? (d) What is Bob's expected profit from bid $b$$b$ given his estimate $e_B$$e_B$ ? (e) What is Bob's best response to Umut's strategy of bidding $2e_U$$2 e_U$ ?

2020 Oct Prelim Part 2 Q3

(15 points) In an all-pay auction, each bidder must pay her bid whether she wins or loses the auction. The highest bidder wins the good. Suppose there are three bidders. Bidder $i$$i$ has valuation $v_i^2$$v_i^2$ for the good where $v_i$$v_i$ is uniformly (and independently) distributed between 0 and 1 . Solve for the equilibrium bidding strategy.

2019 June Prelim Part 2 Q3

Consider a first-price auction with two bidders, in which the bidders' types are i.i.d. random variables uniformly distributed on the interval $[0,100]$$[0,100]$. Each bidder knows her own type but only the distribution of her opponent's type., and each bidder is risk-neutral. The bidders simultaneously and independently submit sealed bids, and the high bidder wins the item and pays the amount of her bid. When bidder i's type equals $t_i$$t_i$, her valuation from winning the item is given by $\sqrt{t_i}-2$$\sqrt{t_i}-2$​. Determine the Bayesian-Nash equilibrium of the first-price auction.

2018 June Prelim Part 2 Q3

(30 percent) In all pay auction, each player places a bid and pays the value of their bid whether they win or lose. The player with the maximum bid wins and receives utility equal to their valuation. (a) Consider the complete information version of the all pay auction. There are two bidders, and each bidder values the object at $\nu >0$$\nu>0$. Show that there cannot be a pure strategy Nash equilibrium in this game.(10 percent) (b) Let $I=\{i_1,i_2\}$$I=\left\{i_1, i_2\right\}$. We have incomplete information, i.e. each agent's valuation is private. For each $i\in I{\nu }_i$$i \in I \nu_i$ is independent and identically distributed according to a uniform distribution on $[0,1]$$[0,1]$ and this is known by both agents. Find a monotone increasing bidding function, $b(v)$$b(v)$, that constitutes a symmetric Nash equilibrium.(10 percent) (c) Under the complete information assumption, suppose that we have two bidders $i_1$$i_1$ and $i_2$$i_2$ and their values of the object are ${\nu }_{i_1}={\nu }_{i_2}=1$$\nu_{i_1}=\nu_{i_2}=1$. Show that there exists a mixed strategy Nash equilibrium such that each bidder bids according to a uniform $[0,1]$$[0,1]$ distribution. (10 percent)

2023 Final Q4 - [Auction]

An object is being auctioned by first price sealed-bid auction and there are two bidders. If there is a tie for the highest bid, the object is awarded with probability $1/2$$1 / 2$ to either bidder. Each bidder's value of the object is either $\mathrm{\$}1$$\$ 1$ with some commonly known probability $p\in (0,1)$$p \in(0,1)$ or $\mathrm{\$}0$$\$ 0$ with probability $1-p$$1-p$. The value of each bidder is independently distributed and privately observed by her. The utility of each bidder is 0 if she loses the auction and her value minus the bid she placed if she wins the auction. (a) Define the symmetric stragies of the players depending their value. Will any player bid more than 1 ? What will a player with 0 value bid? (7 Points)

Answer:

Strategy: For each $i$$i$ their bid is $b_i(v_i)$$b_i(v_i)$where $i=\{0,1\}$$i = \{0,1\}$.

No players will bid more than 1,

The expected payoff function of player is

$$\begin{array}{}\text{(28)}& E(b_i(v_i))=Pr(b_{-i}<b_i)(v_i-b)+\frac{1}{2}Pr(b_{-i}=b_i)(v_i-b_i)\end{array}$$

We know that the max value is one, so in this case whenever player bid more than one, the expected payoff will be negative no matter what their value are. So no players will bid more than one.

(b) Write down the expected payoff of a player with value 1 under the symmetric strategies. (8 Points)

The expected payoff of a player with value 1 is

$$\begin{array}{}\text{(29)}& E(b_i(v_i))=Pr(b_{-i}<b_i)(1-b_i)+\frac{1}{2}Pr(b_{-i}=b_i)(1-b_i)\end{array}$$

Case 1: player with value 1 will bid 0 as well, then the expected payoff will be

$$\begin{array}{}\text{(30)}& E(b_i)=\frac{1}{2}\end{array}$$

Case 2: player with value 1 will bid $b_i(1)>0$$b_i(1) > 0$, then the expected payoff is,

(c) Show that this game does not have a symmetric pure strategy Bayesian Nash equilibrium for all $p\in (0,1)$$p \in(0,1)$.(10 Points)

Answer: player with value 0 will always bid 0.

for any $p\in (0,1)$$p\in (0,1)$, the payoff of bid more than 0 is greater than 0.5, which means player with value 1 always have incentive to deviate to bid more than 0, and always have incentive to deviate to a smaller bid that approach 0, so there is no pure strategy BNE for all$p\in (0,1)$$p\in (0,1)$

Mechism Design