# Exam 1, Summer term Experimental and Behavioral Economics. Dorothea Kübler & Roel van Veldhuizen. Exercise 1 [32 points]

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1 Exam 1, Summer term 2015 Experimental and Behavioral Economics Dorothea Kübler & Roel van Veldhuizen Exercise 1 [32 points] a) One manifestation of the representativeness heuristic is that people believe in the Law of Small Numbers. Define the Law of Small Numbers, and explain why it is an error. Describe some evidence for the Law of Small Numbers. ANSWER: The Law of Small Numbers generates the expectation that even small samples are fully representative of the underlying distribution. E.g., people expect that after tossing a coin twice with heads appearing, tails will come up with a higher probability than ½ next. (7 points) b) Suppose that basketball players are, during any given game, in one of three states: Hot (they make 75% of their shots), Normal (they make 50% of their shots), or Cold (they make only 25% of their shots). Suppose basketball player Dirk Nowitzki is Hot. What is the probability that he will make three baskets in a row? What if he is Normal? Cold? If you have no idea what state he ll be in before the game (that is, each state is equally likely), what would you believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? ANSWER: If Dirk Nowitzki is hot, Pr(3 baskets in a row/ H)= 27/64. Similarly, Pr (3 baskets in a row/ N)=1/8 and Pr (3 baskets in a row/ C)=1/64. Therefore, Pr(H/ 3 baskets in a row)=(27/64)/(27/64+1/8+1/64). (7 points) c) One of the fans at this game believes in the Law of Small Numbers. She has the wrong model of how likely Dirk Nowitzki is to make a basket. Here s how her model works. The fan imagines that there is a deck of 4 cards. When Dirk is Hot, 3 of these cards say hit on them, and only 1 says miss. Every time Dirk takes a shot, one of these cards is drawn randomly without replacement from the deck, and the outcome is whatever the card says. Therefore, when Dirk is Hot, he always makes 3 out of every 4 shots he takes. (When the deck is used up, the 4 cards are replaced, the deck is shuffled, and the process begins again but that isn t important for this problem.) Similarly, when Dirk is Normal or Cold, the outcome of every shot is determined by the draw of a card without replacement from a deck of 4 cards. When Dirk is Normal, the deck has 2 hit cards and 2 miss cards. When Dirk is Cold, the deck has 1 hit card and 3 miss cards. Suppose Dirk Nowitzki is Hot. According to the fan who believes in the Law of Small Numbers, what is the probability that Dirk will make his first basket? After Dirk makes his first basket, what does the fan believe about the probability that Dirk will make his next basket? Explain why it is lower than the fan s belief about the probability that Dirk will make his first basket.

2 ANSWER: Pr(1 st basket/ H)=3/4 and Pr(2 nd basket/ H)=2/3. The second probability is lower according to the Law of Small Numbers because he misses exactly one basket in four and this probability increases the more baskets he has made already. (7 points) d) According to the fan, what is the probability that Dirk will make 3 baskets in a row? What if he is Normal? Cold? If the fan has no idea what state Dirk will be in before the game (that is, each state is equally likely), what would she believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? Explain intuitively why the fan s beliefs differ from the normatively correct probability that you calculated in part (b). ANSWER: Pr(3 baskets in a row/ H)=(3/4)(2/3)(1/2)=1/4 (7 points) Pr(3 baskets in a row/ N)=0 Pr(3 baskets in a row/c)=0 Pr(H/3 baskets in a row)=1 This last probability differs from the correct number ¾ because according to the fan`s model, Dirk cannot be N or C. e) In this example, the fan s misunderstanding of probability leads her to believe (falsely) in hot hands and cold hands. Something similar may be going on in the mutual fund industry. Even if mutual fund returns are almost entirely due to luck, there will be some mutual funds that have done exceptionally well and others that have done exceptionally poorly in recent years due entirely to chance. Explain how the Law of Small Numbers would lead some investors to conclude (falsely) that mutual fund managers differ widely in skill. ANSWER: If a fund does well for some time, people may think that according to the Law of Small Numbers, this must be due to a good fund manager and is less likely to result, e.g., from a normal fund manager who has simply been lucky. (4 points) Exercise 2 [9 points] For the following statements, write down or FALSE. If the statement is, you don't have to write any justification. But if the statement is FALSE, write down the correct statement. a) External validity means that what is observed in a laboratory/field experiment can be expected to generalize to similar situations in the naturally occurring world; while internal validity requires that the experiment allows for causal inferences to be made in absence of any confounding factors. b) A between-subject experimental design allows each subject to experience different treatments. FALSE, this is a within-subjects design. c) Good instructions should always state the purpose of running an experiment at the beginning, so that subjects have a clear idea what the experiment aims to test. FALSE. This would create a demand effect.

3 d) The Mann-Whitney (Wilcoxon) ranksum test is a test often used for experimental data analysis if the number of independent observations is quite large (say, more than 100). FALSE. This test is used for small samples. e)sometimes experiments with many repetitions might be very expensive. The only way to solve this problem is to run experiments in developing countries, where hourly wages are lower. FALSE. You can for example only pay one round that is randomly selected at the end. f)the biggest difference between surveys and experiments is that surveys focus on individual decisions while experiments focus on strategic decisions. FALSE. There are experiments on individual decisions. g) Suppose you compare two market mechanisms but fail to find any treatment differences. What could be the potential reason(s), i.e. which of the following claims could be true and which are always false? - The instructions are too confusing for the subjects many of them just choose to bid randomly; - The number of independent observations is not enough to generate high statistical power; - The incentives provided are not big enough. Exercise 3 [7 points] Your friend Sarah told you about one of her experimental projects the other day. She attempts to study how responders in an ultimatum game react to money sent by proposers. In the ultimatum game, the proposer starts with a certain amount of money (the endowment). As a second question, Sarah is also interested in whether economics- and business students behave differently than students majoring in other subjects. To investigate this, she ran an experiment with 40 pairs of subjects (20 proposers and 20 responders). Of the responders, 16 were economics/business majors and 4 major in other fields. So far, she found that 18 proposers sent less than 1/4th of their endowment. a) Based on the proposers decisions, does she encounter any trouble in studying her first research question, i.e., how responders react to money sent by proposers? If yes, what is the problem? If yes, how could Sarah add another treatment to help solve this problem? ANSWER: Yes, the problem is that there are not enough proposers who sent more than ¼ of their endowment. This makes it hard to conclude anything about how the amount of money sent by proposers affects the responders behavior. This can be solved by running a treatment where responders answer according to the strategy method. (4 points)

4 NOTES: (1) Several people solved the problem by suggesting that the proposer s offers are manipulated in a certain way (e.g., by letting the offer come from a computer, or forcing the proposer to send a higher amount). Though such measures would solve the problem, they might lead to new problems, such as requiring deception (if the responder isn t told that the money was actually sent by a computer, for example). (2) Another way to answer the question is to point to the large number of economists. This would require arguing why this is a problem. For example, economists are too rational (as suggested by some previous experiments), and will therefore always accept money sent, however little. Hence any data obtained from economists is unlikely to generalize to other populations; a problem, given that for this question Sarah is interested in behavior in general, not just for economists. b) Does Sarah encounter any problem in answering her second research question, i.e., whether responders majoring in economics/business respond differently than responders with other majors? If yes, what is the problem and how can it be solved? ANSWER: the problem is that there are too few other majors. This makes it difficult to obtain statistically significant results. A solution would be to selectively invite more non-economists, run additional sessions in which economists are not allowed to participate, etc. Exercise 4 [18 points] Ashley and Brigitte like going on weekend trips together. This weekend, they intend to go to either Flensburg or Kiel. Ashley prefers Flensburg to Kiel, but would prefer to go with Brigitte to Kiel than go to Flensburg herself. By contrast, Brigitte prefers Kiel to Flensburg, though she would also rather go to Flensburg together than go to Kiel by herself. Their payoffs can be summarized as follows (the first number in each cell represents Ashley s payoff, the second Brigitte s): Ashley: \ Brigitte: Flensburg Kiel Flensburg 4,2 1,1 Kiel 0,0 2,4 a) What are the Nash equilibria of this game in pure and in mixed strategies? ANSWER: Both (F,F) and (K,K) are pure strategy equilibria (2 points each). To compute the mixed equilibria, set U(K)=U(F) for both players. This yields p+4(1-p)=2p or p=1/5 for both players. (6 points) b) Now suppose that Ashley has lost her phone and Brigitte is unable to contact her. Brigitte is still desperate to go. However, she has trouble anticipating what Ashley will do, and hence assumes that Ashley will go to either city with equal probability. (In other words, Brigitte is a level-1 player in terms of the level-k model.) What is Brigitte s expected utility of going to Flensburg or Kiel, respectively, given these beliefs? Which city will she go to? ANSWER: Given the stated assumptions, U(F)=0.5*2+0.5*0=1, U(K)=0.5*1+0.5*4=2.5. U(K)>U(F), hence Brigitte will go to Kiel. (4 points)

5 c) Meanwhile, Ashley is contemplating which city to actually visit. Her k-level is equal to 2. What are her expected utilities from visiting each city? Which city will she visit? ANSWER: Given that level 1 Brigitte goes to Kiel (as per exercise b), level 2 Ashley s utilities are U(F)=1 and U(K)=2. Therefore, she will also go to Kiel. NOTE: In case the answer to the question (b) was Flensburg, an answer of Flensburg can also be counted as correct, if properly supported. Exercise 5 [16 points] In an experiment, there are 10 periods, in which an asset is traded in a double auction. At the end of every period, the asset pays a dividend of 1, 2, or 6, each with probability 1/3. In every period only one trade is possible. a) What is the fundamental value of the asset? ANSWER: The expected value each period equals 1/3*(1+2+6)=3. (2 points) The fundamental value therefore equals 3*10=30. (2 points) b) What can we conclude compellingly from the following observations about X and Y with respect to their risk preferences (risk aversion, risk neutrality, risk lovingness) and rationality? Please write this down separately for every case described below, both for subject X and Y. 1) In period 10, subject X sells an asset to subject Y at a price of 2. ANSWER: Subject X is risk averse. Subject Y can be risk seeking, neutral or slightly risk averse. Nothing can be concluded about the players rationality. (2 points) 2) In period 10, subject X sells an asset to subject Y at a price of 3. ANSWER: X cannot be risk seeking. Y cannot be risk averse. (2 points) 3) In period 10, subject X sells an asset to subject Y at a price of 0. ANSWER: X sells the asset for less than its minimum outcome (1). Therefore X is irrational. Nothing can be concluded about risk preferences, or player Y s rationality. (3 points) 4) In period 6, subject X sells an asset to subject Y at a price of 15. ANSWER: The expected value of the asset is 15. Hence, X cannot be risk seeking, Y cannot be risk averse (2 points). 5) In period 5, subject X sells an asset to subject Y at a price of 100. ANSWER: Y buys an asset for a price that exceeds its maximum possible outcome of 30. Hence Y is irrational. Nothing can be concluded about player X. (3 points)

6 NOTES: (1) Strictly speaking, if one of the players made a trade that seemed rational, we can nevertheless not compellingly conclude that either player is rational. Because even irrational players make good trades sometimes (by luck or accident). However, since almost no one noticed this, no points are deducted for calling players rational except when they are clearly not (in 3 and 5). (2) Many of you were also overly optimistic about what can be concluded about risk preferences. For example, many of you wrote in exercise 2 that both players were risk neutral. This is indeed possible, but X can also be risk averse and Y risk seeking. No points were deducted for this either. (3) Risk aversion or risk seekingness is a matter of preference, not rationality. Exercise 6 [12 points] Usually, subjects are paid by exchanging the points earned with a fixed exchange rate x into real money. One could also use different, non-standard, procedures. Discuss whether these methods have an influence on behavior from a theoretical point of view. Discuss in particular, in which kind of experiments and how behavior differs when non-standard payment is used compared to standard payment. Consider the following non-standard payment procedures (Notation: There are S subjects who play P periods.): a. The subjects get a fixed payment. EXAMPLES OF GOOD ANSWERS: (1) A fixed payment is like a show-up fee and not salient, and therefore fails to remove the influence of homegrown preferences, leading to a loss of control over the participants preferences. (2) Participants cannot increase their payoffs through their actions in the experiment, and may therefore stop paying attention or putting in effort. (3) Participants may behave more altruistically/less competitively given that no real money is at stake. (3 points) b. The subject with the highest payoff is selected and gets a fixed payment. The other subjects receive nothing. EXAMPLES OF GOOD ANSWERS: (1) These incentives will increase the output of competitive/confident/money-maximizing subjects. The incentives will decrease the output of noncompetitive/less confident/altruistic subjects. (2) In experiments with multiple rounds, participants who fail to do well in early rounds may stop providing effort altogether in later rounds. (3 points) c. A fixed sum of money is distributed proportionally to the number of points earned by the subjects. EXAMPLES OF GOOD ANSWERS: (1) Altruistic participants will reduce effort, since high effort indirectly decreases the payoffs of all other participants. (2) Since payoffs for the whole session are fixed, this turns the experiment into a situation where it is socially optimal for all participants to provide minimum effort (provided effort is costly). (3 points) d. The exchange rate is determined at the end of the session and apart from that, the subjects are paid according to the standard procedure.

7 EXAMPLES OF GOOD ANSWERS: (1) Risk averse participants may respond to increased uncertainty about their payoff by reducing their effort. (2) If the exchange rate is totally unknown, participants need to form beliefs about what they expect the exchange rate to be, potentially reducing or increasing effort depending on how optimistic the beliefs are. (3 points) NOTES: For all questions, there are several other possible answers as well, some of which also earn full points. References to specific types of experiments are not always necessary to obtain all the points. Exercise 7 [6 points] The planet Brutopia is inhabited by a society of doctors and soldiers consisting of two alien races, the Asari and Turians. The vast majority of doctors are Asari, whereas the vast majority of soldiers are Turians. a) Give at least four reasons why the Asari may have sorted into becoming doctors rather than soldiers. ANSWER: (1) Asari have more natural aptitude for being doctors. (2) Asari are discriminated against for military positions. (3) Asari have had a better education for doctors. (4) Asari don t like the risks involved with the military profession. (5) Asari are socialized into becoming doctors. (3 points) NOTES: Many other answers are possible as well. Giving arguments why Turians are not doctors is okay as well. b) Now suppose that a local hiring committee has to decide whom to appoint into a vacant doctor position. Two candidates have applied: one Asari and one Turian. Further, the committee assumes that there are only two types of doctors in the population: good doctors and bad doctors. Asaris on average have a higher probability of being a good doctor (p) than Turians (i.e., p A > p T ). If this is the only available information, which candidate should the hiring committee hire to maximize the probability of hiring a good doctor? Why? ANSWER: The committee should hire the Asari. All other things equal, the Asari is more likely to be a good doctor. (3 points) NOTE: This is an example of statistical discrimination. As some of you noted, statistical discrimination can be prevented if the committee put in the effort to find out more about which of the candidates is better. Lacking this information, however, picking the Asari is the optimal thing for the committee to do.

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