# Markets, Games, and Strategic Behavior/2008

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This page is an experiment in student generated answers to homework questions. In Markets, Games, and Strategic Behavior, we covered a diverse number of topics. The lecturer will pose a question and the students will provide the answer(s). Feel free to improve on other students' answers, put alternate answers and pose new questions. Feel free to clarify the questions as well.

Take from class, the Diamond Dybvig model with $L=.5$ and $R=2$ , two impatient depositors and two patient depositors. Each depositor has $1000 to deposit in the bank. Let us say that deposits are insured up to fraction $f$ . For what values of $f$ is there only one equilibrium and what values are there two equilibria? (For each dollar put in the bank yesterday, early withdrawers are guaranteed to get $1*f$ and late get $2*f$ .) ### Answer 1 The bank expects the two impatient depositors to withdraw today and the two patient depositors to withdraw tomorrow. Hence, yesterday, the bank sets aside$2000 for today and invests $2000 for tomorrow. Now today, the depositors must decide whether to withdraw today or tomorrow. We assume that the impatient depositors withdraw today. Now We can examine this as a game between the two patient depositors. Each has to decide whether or not to withdraw today. When $f=0$ the payoffs as discussed in class.  Today Tomorrow Today$750, $750$1000, $0 Tomorrow$0, $1000$2000, $2000 For a general $f$ , we must calculate again each of the payoffs. If both withdraw today, the bank can pay the first 3 depositors the$1000. The last depositor will receive $f$ . Thus, the expected payoff is $750+(f/4)*1000$ . If one withdraws today and the other withdraws tomorrow, the bank will be able to pay all three today, and the depositor withdrawing tomorrow receives $2000*f$ . Rewriting the game yields.

 Today Tomorrow Today $750+1000*(f/4),$750+1000*(f/4) $1000,$2000*f Tomorrow $2000*f,$1000 $2000,$2000

We see that if one patient depositor withdraws today, the second patient depositor only has incentive to withdraw today if $750+1000*(f/4)>=2000*f$ . Hence, if and only if $f<=3/7$ , there is a possibility of two equilibria.

### Question 2

Part A.

Examine the second treatment of the Beer-Quiche game where there is a 2/3 chance of the proposer being strong.

Payoffs: Proposer, Responder

 Flee Fight Beer (Strong) $1.40,$1.25 $0.60,$0.75 Quiche (Strong) $1.00,$1.25 $0.20,$0.75 Beer (Weak) $1.00,$0.75 $0.20,$1.25 Quiche (Weak) $1.40,$0.75 $0.60,$1.25

Can there be a pooling equilibrium where both proposers choose Quiche and the responder flees? Does this seem reasonable to you? Part B.

 fold call raise (Strong) $1.00, -$1.00 $2.00, -$2.00 fold (Strong) -$1.00,$1.00 -$1.00,$1.00 raise (Weak) $1.00, -$1.00 -$2.00,$2.00 fold (Weak) -$1.00,$1.00 -$1.00,$1.00

Assume the odds of a strong hand is 80%. Find any equilibrium. Is it signalling or pooling? Extra hard: what happens if it is 60%?

### Question 4

There is a Beersheva to Haifa train line. Travellers either go between Haifa and Tel Aviv with demand $12-p$ , Tel Aviv and Beersheva 12-p, Haifa and Beersheva. $18-p$ , Say it is all owned by one profit maximizing monopolist with marginal cost of zero. For simplicity assume that the monopolist must set the price of the Haifa-Beersheva route equal to the sum of the other two. What would he charge for all three routes? Now say the government thinks it needs to add competition to the rail industry. It divides things into two companies. One takes care of the Haifa-Tel Aviv route and the other the Tel Aviv-Beersheva route. The price of the combined trip is the sum of the other two. What are the new prices? Who wins and who loses?

### Question 5

Students like to go to the Haifa Ball depending upon how many other students go there. Tickets cost 32 NIS each. There are 1000 students indexed by i from 1 to 1000. Student i has value vi=i. Student i has utility (in shekels) for going to the Ball of $v_{i}{n \over 5000}$ , where n is the total number of students going to the Ball. (i) If everyone believes $n=500$ , which students will be willing to go to the ball? (ii) What is the threshold number of tickets sold above which it will be a success and below which it will be a failure? (iii) What is the equilibrium of tickets sold if the ball is a success? (iv) What is the equilibrium of tickets sold if the ball is a failure?

(i)n=680

(ii) 200

(iii) 800

(iv) 0

### Question 6

In the takeover game, assume that the value for the seller is uniformly distributed between 50 and 100. Assume that it is still worth to the buyer 3/2 times the seller’s value. What should the buyer offer to the seller?

p=100

### Question 7

A monopoly has marginal cost of 5 and faces a demand of q=20-p. What price should he charge to maximize profits? Let us say it is a vertical market of two firms: supplier and retailer. What would the price would the supplier charge the retailer? What would be the price charged to the end consumer? If the supplier charged a franchise fee in addition to wholesale price, what would they be? Extra: Solve the above problem for the general case of marginal cost of c facing demand of q=A-p where (A>c).

Max П=(20-p)p-5(20-p)=(20-p)(p-5)

,П'/p'=-2p+25=0   ,,Pm=12.5 , Пm=12.5*7.5-5*7.5=56.25


 П retailer=Pc*q-Ps*q=(20-q)q-Ps*q
,Пr'/p'=20-2q-Ps=0   2q=20-Ps          q=10-0.5Ps

,П supplier=Ps*q-5*q= Ps*(10-0.5Ps)-5*(10-0.5Ps)


Пs'/p'=10-Ps+2.5=0 Ps=12.5 q=3.75 Pc=20-3.75=16.25

П retailer=14.0625, П supplier=28.125 summ П=42.188

### Question 8

El Al and British Air are competing for passengers on the Tel Aviv- Heathrow route. Assume marginal cost is 4 and demand is Q = 18 − P. If they choose prices simultaneously, what will be the Bertrand equilibrium? If they can collude together and fix prices, what would they charge. In practice with such competition under what conditions would you expect collusion to be strong and under what conditions would you expect it to be weak. Under what conditions should the introduction of BMI affect prices?

Solve a three stage ultimatum game where in the first stage player A offers player B an offer for a $10 pie. If this offer is rejected, then the pie shrinks to$8 and player B makes the offer. If this offer is rejected, then the pie shrinks to $6 and player A makes the offer. If this final offer is rejected, then the payoffs are 0 to both players. (Assume the possibility of continuous offers.) ### Question 10 You get in a taxi. Should you bargain over the price at the beginning or end of the trip? Why? ÷===Question 11=== Home Box Office is a pay-TV service that is based in the US. After showing only movies they decided to increase subscribers by introducing shows such as Sex and the City and the Sopranos. A person enjoys a show for its quality and whether they can talk about it next to the water cooler the next day at work. Given that someone has seen the show, the probability that one can talk about it f is just the number of people who have seen the show divided by the total number of people. We index the possible viewers by i (from 1 to 1000). Viewer i has parameter vi where vi = i. Viewer i values subscribing to HBO 9+ (vi/10)· f. Note that 9 is the value of HBO from the sheer quality. The price charged for HBO is 30. (i) If everyone believes f = .4, which people will subscribe to HBO? (ii) What is the threshold number of subscribers above which HBO will be a success and below which HBO will be a failure? (iii) What is the equilibrium number of subscribers if HBO is a success? (iv) What is the equilibrium number of subscribers if HBO is a failure? i.475 ii.300 iii.700 iv.0 ### Lab report 2 questions Which price should be higher: monopoly or Bertrand price competition? Which to consumers should be higher: monopoly or Bertrand Complements? When demand is 15-p and mc=6, what should a monopolist charge? When demand is 15-(p1+p2) and mc=3 for both firms, what is the equilibrium price (it is Bertrand Complements)? ### Answers to Lab Report 2 ### Lab report 3 questions If they value R tomorrow less than 1 today, Impatient depositors will always want to withdraw today: True or False? For no matter what values of L and R (L<1, R>1) and any number of depositors, there will always be two equilibria in the Diamond Dybvig model: True or False For 4 depositors (2 impatient and 2 patient), L=.6 and R=1.5 what are the payoffs for (today, today)? For 4 depositors (2 impatient and 2 patient), L=.6 and R=1.5 what are the payoffs for (today, tomorrow)? ### Lab report 4 questions If in the Network Experiment values were drawn from 0,10 and p=2.1, then what is the expected proportion of consumers buying in the success equilibrium. .2 .3 .4 .5 .6 .7yes .8  If in the Network Experiment values were drawn from 0,10 and p=2.1, then what is the threshold/tipping point? .2 .3yes .4 .5 .6 .7 .8  If in the Network Experiment values were drawn from 0,10, then for what prices is failure the only equilibrium? p=25 Give an example of Network Externalities that wasn't mentioned in class. ### Lab report 5 questions Assume the seller has a value distributed uniformly from 50 to 100 and receives an offer of p. What is the expected value of a seller accepting an offer? (50-p)/2 p/2 (100+p)/2 (50+p)/2 yes  Assume the seller has a value distributed uniformly from 50 to 100 and receives an offer of p. What is the probability of the seller accepting an offer? p/50 (p+50)/100 (p-50)/50 yes (p-50)/100  Assume the seller has a value distributed uniformly from 50 to 100. It is worth 3/2 times the value to the buyer. The buyer does not know the value. What is the optimal offer p to make to the seller? p=100 What is the expected profits of the buyer making the optimal offer in the previous question? profit=12.5 ### Lab report 6 questions The profit of a monopolist is higher than the upstream firm in a vertical monopoly. True False  If the upstream firm can charge a franchise fee, how much would it charge per item in addition to this fee if its marginal cost is 3 0 between 0 and 3. 3 twice the marginal cost (i.e., 6). = this  other  Say demand is D=16-p and mc=0. How much would a monopolist charge? p=8 Say demand is D=16-p and mc=0. How much would the consumer pay if there is a vertical market (upstream and downstream)? ### Lab report 7 questions A buyer's investment cost is C. The gains in profits is G. The supplier can raise price by R. If the buyer switches the loss is B. By what amount R (in a one shot game) will it be rational for the buyer to stay with that firm? any R no R. R>G G>R G-C-R>B (tomer)  When will the buyer choose not to invest? G-C-R<0 G-C>0 G-B<0 G-C-B<0  Assume C=2000. Give a case of R, B, G where the subgame equilibrium is inefficient? Assume C=2000. Give a case of R, B, G where the subgame equilibrium is efficient? ### Lab report 8 questions If a team goes up in order of the draft (can select earlier), then the team will always get better players the team will never get better players the team will sometimes get worse players the team will always get the same players  Teams selecting players by a draft will Always be Pareto Efficient Only sometimes be Pareto Effiicent Never be Pareto Efficient Can always be improved by allowing trading afterwards  If Team 1 prefers A to B to C to D (A>B>C>D) and Team 2 prefers B to C to D to A, then what are the sincere choices. If Team 1 prefers A to B to C to D (A>B>C>D) and Team 2 prefers B to C to D to A, then what are the sophisticated choices. ### Lab report 9 questions 1st degree price discrimination always gives (weakly) higher profits than 2nd or 3rd degree price discrimination. true false  2nd degree price discrimination always gives (weakly) lower profits than 3rd degree price discrimination. True False  There are two students. One values one calculator at$20 and two at $20. The other values one at$30 and the other at $40. It costs$7 to produce each calculator. If a seller can charge different prices for different quantities, what would he charge?

There are two students. One values one calculator at $20 and two at$20. The other values one at $30 and the other at$40. It costs \$7 to produce each calculator. If a seller can charge different prices to different students but not for different quantities, what would he charge?

### Lab report 10 questions

The buyer (row player) knows the state. For which state would he buy insurance?

Good


The company (column player) knows the state. For which state would it agree to sell insurance?

Good


### Lab report 11 questions

A peacock's tail is an example of signalling.

True
False


In treatment 2, what is the minimum fraction of strong types needed for a pooling equilibrium?

3/4
2/3
1/2
1/3


What was the reason that Cho Kreps gave for why pooling could not occur having both types each quiche?

Give an example of signalling in the real world.

Why is the payoff to a row player buying insurance 60?

What would the equilibrium (without information) be if this payoff were instead 40?