|Classroom Expernomics: Volume 10 (Fall 2001)|
A Simple Investment Game Experiment for the Classroom
Department of Economics
Washington State University
We present a simple way of carrying out the Investment Game, introduced by Berg, Dickhaut and McCabe (1995) inside the classroom for instructional purposes. This game is a handy way of illustrating the principle of backward induction in sequential move games. In a slight deviation from the original design we allow each subject to play both as a Sender as well as a Receiver.
The Investment Game, first proposed by Berg, Dickhaut and McCabe (1995), provides an excellent way of illustrating (1) how the principle of backward induction works in sequential move games and (2) how behavior often differs from that predicted by backward induction. The Investment Game proceeds like this. Subjects are paired up with one person called the Sender (alternatively Proposer or Allocator) and the other person called the Receiver (alternatively Respondent or Recipient). Each Sender is given $10. Each Sender is told that she is free to keep the entire $10 or she can split it with an anonymous Receiver (who is in another room). However any amount that the Sender offers the Receiver will be tripled by the experimenter and given to the latter. The Receiver will then decide whether to keep the entire amount offered or to send some back to the anonymous Sender who made the offer in the first place. This latter amount is not tripled. The game ends at that point. To take an example, suppose the Sender decides to keep $5 out of the initial $10 and offers $5 to the anonymous Receiver. Then the experimenter triples the $5 offered and gives the Receiver $15. The Receiver can then decide to keep the entire $15 or send part or all of it back to the Sender.
The solution to this game using backward induction goes like this. Consider the Receivers decision. Since the game ends after this point, the Receiver has no incentive to send any money back to the Sender. Knowing this the Sender should not send any money to the Receiver in the first place since she should not expect to get anything back. The principle of backward induction dictates that the Sender should keep the entire $10. This way the Sender gets $10 and the Receiver gets $0. However there is an alternative way of looking at this. Suppose the Sender decides to "trust" the Receiver and sends her the entire $10. The Receiver then will receive $30. If the Receiver "reciprocates" the Senders "trust" then there are numerous possible splits of this $30 (say $15 each) which makes both the Sender and the Receiver better off than if the Sender had sent nothing in the first place. However if the Receiver does not "reciprocate" the Senders "trust" then the Sender is worse off since she loses all or part of the $10 that she could have kept.
This game then provides a handy way of discussing backward induction as well as documenting behavior that deviates from the game theoretic prediction.
I used this experiment in my class on Behavioral Economics. Students play the game for extra-credit points rather than money. This however posed a problem at the very outset. When carrying out experiments with extra-credit points, it is important to avoid any appearances of "unfairness". See Stodder (1998). But in this experiment the Sender is in a more advantageous position. So we modify the original experiment to allow every subject to play as both a Sender and a Receiver.
Each subject was given a copy of the instructions (see Appendix). The instructions are also read aloud. Each student gets an initial endowment of 50 extra-credit points that she could keep or split with an anonymous partner who would be in another room. There were 14 students who were assigned ID numbers, #1 through #14. They are told that each of them would make both a Sender decision as well as a Receiver decision. They know that they would always be paired with someone who would be in the other room. So while they knew who the people in the other room were, no one (except the Experimenter) knew who she was paired with. They were also told that they would not be interacting with the same person in the two roles. For instance, subject #1 (as Sender) offers a split to subject #8 (as Receiver), while subject #1 (as Receiver) receives a split from subject #14 (as Sender), while subject #8 (as Sender) offers a split to subject #2 (as Receiver) and so on. The following scheme illustrates this point.
Subjects 1 through 7 were asked to stay in the same room (Room A) while 8 through 14 went into the next (empty) classroom (Room B). Each subject, at this point, was asked to fill out Boxes B and C on the record sheet. Box A already had 50 points written in it. Each subject, as Sender, decided how much she wished to keep and how much she wished to offer to the anonymous Receiver. Let us look at subject #1. Suppose Subject #1 decided to keep 25 points and offer 25 points to the Receiver she is paired with (which happens to be subject #8). At this stage the Record Sheet looks like the following.
|B||Amount you wish to KEEP||25|
|C||Amount you wish to SEND (A B)||25|
Then this page of the Record Sheet is carried to the other room and given to subject #8. Except, for subject #8, box D is filled in and reads as 75 points. Subject #8 then is asked to decide how much she wants to keep and fill up Boxes E and F accordingly. Suppose subject #8 decides to keep 50 points (out of the 75 offered) and send back 25. Boxes D-F then look like as follows:
|D||Amount you have been sent (3 times the amount in Box C)||75|
|E||Amount you wish to KEEP||50|
|F||Amount you wish to SEND BACK (D E)||25|
Subject #8 is also told (since this sheet will go back to subject #1, the Sender) to copy the information from D-F onto Boxes G-I on Page 2 of the instructions. This way subject #8 will have a record of what happened to her in the role of the Receiver. At this point subject #1 has earned 50 points 25 points that she kept as Sender and another 25 points that are sent back by the Receiver (subject #8). But subject #1 is the Receiver in the (subject #14, subject #1) pair. So as Receiver subject #1 receives a split from subject #14. Let us say that sheet looks like this: (filled in by subject #14)
|B||Amount you wish to KEEP||30|
|C||Amount you wish to SEND (A B)||20|
So subject #14, the Sender, has offered 20 points (which gets tripled to 60) to subject #1, the Receiver. Say subject #1 keeps 30 (Box E) and returns 30 (Box F). Boxes D-F then look as follows:
|D||Amount you have been sent (3 times the amount in Box C)||60|
|E||Amount you wish to KEEP||30|
|F||Amount you wish to SEND BACK (D E)||30|
Then subject #1 notes down the same information on Boxes G-I which appear as follows:
|G||Amount you have been Offered||60|
|H||Amount you wish to KEEP||30|
|I||Amount you wish to SEND back (D E)||30|
Subject #1s total earnings in the experiment are 80 points. 25 points she kept back as Sender (Box B), 25 points she got back from the Receiver, subject #8 (Box F) and finally the 30 points she kept as the Receiver (Box H) out of the split offered by subject #14.
Results of the Experiment
Looking at the decisions made by the Senders we find that out of the initial endowment of 50 points, on average, the Sender keeps 31 points, i.e. 62%, and sends 19 points (38%) to the Receivers. Figure 1 shows the distribution of the amounts sent by the Senders to the Receivers.
In order to look at Receiver decisions we need to look at percentages since Receivers receive different amounts. On average Receivers get 57 points (3 times 19) and out of those they keep 40 points (70%) of the amount they receive and send back 17 points (30%). Figure 2 shows the distribution of the amounts sent back by the Receivers. Note that no Receiver sent back more than 50% of the amount that she received.
Since each subject plays both as a Sender and a Receiver, each subject, on average, makes 88 points: 31 points kept back as Sender, 17 points sent back by the anonymous Receiver she is paired with, and finally 40 points that she keeps back as Receiver out of the 57 points sent by the Sender she is paired with. Each subject then does better than if they had kept back all of the initial 50 points as Sender since that would give each a maximum of 50 points while this way each subject gets 88 points.
Let us compare this behavior with Berg, Dickhaut and McCabe (1995). In their experiment the initial endowment is $10 out of which Senders keep $4.82 (48%) and send $5.16 (52%). On average Receivers Get $15.48 and keep $10.71 (70%) and send back $4.77 (30%). So the Senders in our experiment are much more parsimonious keeping back 62% of the initial endowment compared to 48% in Berg et al. But the behavior of the Receivers in both experiments in remarkably similar.
There is no correlation between the amounts that the receivers receive and the amounts they send back. So it is not the case that those who get more send more back
Finally out of 14 Senders, only 3 (21%) sent all of their initial endowment as opposed to 5 out of 32 (15.6%) in Berg et al.
In this paper I have presented a simple way of carrying out the Investment Game of Berg, Dickhaut and McCabe (1995) for instructional purposes. I do not intend to present these findings as research. My aim is to provide other instructors with a simple way of conducting the Investment Game in the classroom since this is a good game to illustrate the principles of backward induction as well as deviations from that principle. The experiment described above has the added advantage that the instructor does not run the risk of appearing to be "unfair" since the experiment allows for each subject to play as both a Sender and a Receiver. With 14 students it takes me at most 10 minutes to read the instructions and then at most another 10 minutes to conduct the experiment. If the instructor has the pairing scheme made up then the design can be easily extended to classes that have many more students. Except one must make sure that there is another empty classroom available close by and preferably right next door. Also I carry out the experiment at the end of the class period so that I can collate the data and report the results the next class period.
Instructions and Record Sheets
Berg, Joyce, John Dickhaut and Kevin McCabe (1995), "Trust, Reciprocity and Social History", Games and Economic Behavior, 10, pp. 122-42.
Bolton, Gary, Elena Katok and Rami Zwick (1998), "Dictator Game Giving: Rules of Fairness Versus Acts of Kindness", International Journal of Game Theory, 27, pp. 269-99.
Cox, James (2000), "Trust and Reciprocity: Implications of Game Triads and Social Contexts", Mimeo, University of Arizona.
Roth, Alvin (1995),"Bargaining Experiments", Ch. 4 in John Kagel and Alvin Roth (eds), "Handbook of Experimental Economics", Princeton University Press, Princeton, NJ.
Stodder, James (1998), "Experimental Moralities: Ethics in Classroom Experiments", Journal of Economic Education, Spring 1998, pp. 127-138.
1. I have put the words "trust" and "reciprocity" within quotes. The behavior of the subjects in this game, which deviates quite sharply from game theoretic predictions, is usually explained by appealing to the above concepts. But the real motivation behind such behavior is still open to debate and the subject of research by many. See Cox (2000) for one.
2. We actually had 15 students in the class while the experiment requires an even number of subjects. So after explaining the instructions we announced that we needed one student to sit out this particular experiment. In return we offered a fixed amount of extra-credit points. We started the bidding at 50 points fully expecting to have to go higher than that for a student to accept our offer to opt out. But a student immediately raised her hand. She was asked very specifically and more than once, if she was sure she wanted to opt out for 50 points, i.e. she would be awarded 50 points but would not take part in the experiment and forego whatever she could have earned there. She replied emphatically each time that she understood the offer and was willing to exclude herself for 50 points. We then proceeded with the remaining 14 students.
3. The original Berg et al experiment followed a complex double-blind procedure where even the experimenter was unaware of which subject made which decision. However introducing double-blind procedures in this classroom experiment will complicate things and increase the duration of the experiment. Also, it is debatable whether a double-blind procedure is absolutely essential. Bolton, Katok and Zwick (1998) comment "We find no basis for the anonymity hypothesis..." referring to double-blind procedures. Roth (1995, pp. 301) comments "... there is no evidence to the effect that observation by the experimenter inhibits player 1 in ultimatum games, nor that it is the cause of extreme demands in dictator and impunity games."
4.This parsimonious behavior may reflect the fact that the students had just gone through my lecture on sequential games and backward induction the previous week and the material was still fresh in their mind. As a result their behavior is more in accordance with the game theoretic prediction than that of the average experimental subject who are recruited from widely divergent backgrounds as was the case in Berg et al's experiment.
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