INSIDE THIS ISSUE:
Stephane Aymard
Department of Economics
University of Montpellier, France
I. Introduction
Contributions to public goods have been studied experimentally by economists and sociologists for a long time. The main result is that subjects free ride, but not as much as game theory predicts. The standard game is as follows. Each member of a group of n players receives an endowment z_{i}. Each player has to choose how much to invest in a public good, i.e., a contribution t_{i}£ z_{i}. The experimenter collects the contributions, multiplies the total T=S t_{i} by a and divides equally the product among the players. Thus, the utility of a player is u_{i}=z_{i}t_{i}+aT/n. The gametheoretic prediction is that no one contributes as long as a/n<1. Experimental results showed that this prediction is not verified: subjects contribute around 40% of their endowments (see Ledyard, 1995, for a survey). The main studies focused on the rate of return of the public good, the number of players, the introduction of thresholds, institutional rules, etc. Some of them also examined subjects' preferences: comparative studies have been done on gender or education (for instance, BrownKruse and Hummels, 1992). In this paper, we propose to test the influence of fairness on subjects' decisions. Previous studies used questionnaires to discriminate among participants those who have stronger senses of fairness. Here, we study it directly in games with unfair redistribution, i.e., with payoffs and endowments heterogeneity. Our experiment, easy to reproduce in the classroom, shows that contribution rates differ largely and proves that fairness play a role in subjects' decisions to contribute.
II. Experimental design
Many authors observed different types in the subject population. For Ledyard, there are 50% Nash players, 40% Nash players if the incentives are high enough (but who also make mistakes) and 10% irrationals. The proportions of these types are not the main issue. What is important is that they are present. As it is well known in game theory or in experiments, a small proportion of altruists may change the behavior of a rational player. Thus, the detection of these behaviors is important. In this paper, we analyse several variations of a public goods experiment, in the same spirit as Hoaas and Madigan (1996). We test different games with more or less inequity to evaluate the influence of nonselfish considerations. In each game, the number of players is n=4, but endowments z_{i} and payoffs u_{i} are different. This leads to the 11 following cases (A to K):
Figure 1. Games structures
Games  Player i's endowment  Player i's payoff (%)  Others' endowments  Others' payoffs (%) 
A  40  25  40 / 40 / 40  25 / 25 / 25 
B  40  15  40 / 40 / 40  15 / 15 / 55 
C  40  55  40 / 40 / 40  15 / 15 / 15 
D  40  30  40 / 40 / 40  30 / 30 / 10 
E  40  10  40 / 40 / 40  30 / 30 / 30 
F  40  25  40 / 40 / 15  25 / 25 / 25 
G  20  25  40 / 40 / 40  25 / 25 / 25 
H  40  30  40 / 40 / 15  30 / 30 / 10 
I  15  10  40 / 40 / 40  30 / 30 / 30 
J  40  15  40 / 40 / 15  15/ 15 / 55 
K  15  55  40 / 40 / 40  15 / 15 / 15 
Payoffs are expressed as the percentage of aT that a player receives (with a = 3/2). Normally, each player consumes the same public good and shares are identical. Thus, some games above do not correspond to a standard (pure) public good. But this is not crucial since in practice we often observe that individual consumptions of a public good are not similar. If we adopt player i’s point of view, we can arrange the following games into five classes: fair, unfair for all players, unfair for player i, unfair for the opponents, and unfair for one of the opponents. Case A is"fair" since the proportions of players' payoffs to their endowments are all equal. Each player receives a share that depends on his means. Cases B, F, and J are "unfair for all players" since player i and all the other players are in the same situation, except one who has a proportion of payoff to endowment largely higher. Cases E and I are "unfair for player i" since he is the only one who has a proportion of payoff to endowment lower than those of the others. Cases C, G, and K are "unfair for the opponents" since player i has a proportion of payoff to endowment largely higher than those of the others. Finally, Cases D and H are "unfair for one of the opponents" since one of the opponents has a lower proportion of payoff to endowment. Figure 2 summarizes these classes:
Figure 2. Games classification
Games types  Games 
Fair  A 
Unfair for all players  B/F/J 
Unfair for player i  E/I 
Unfair for the opponents  C/G/K 
Unfair for one of the opponents  D/H 
Among the four types of unfair games, we can note that the first two types (games B, F, J and E/I) are unfair for player i whereas the last two types (games C, G, K and D/H) are unfair for one or more other players. The purpose of this classroom experiment is to show that there are some differences in subjects' behavior due to their senses of fairness.
III. Experimental results
The experiment can be run during a course in microeconomics or game theory after the concept of Nash equilibrium has been discussed (see Brock, 1996, for the relevance of classroom experiments on public goods). We conducted our experiment at the University of Montpellier with 48 undergraduate subjects from several microeconomics classes. They were separated into 12 groups of 4 subjects. We used additional grade points as rewards. Subjects had to choose a contribution level after discovering the game structure, i.e.: their share of the public good, their opponents’ endowments and shares. Except for this information, instructions were similar to other experiments. Results clearly show that fairness is present because contribution rates differ largely :
Figure 3. Experimental results (player i’s contribution rates)
Games types  Games  Rates 
Fair  A  60% 
Unfair for all players (except one)  B/F/J  51% 
Unfair for player i  E/I  47% 
Unfair for the opponents  C/G/K  70% 
Unfair for one of the opponents  D/H  67% 
The rates indicated above are aggregated contribution rates from players i (see Appendix for detailed results). From this table, we can conclude that player i uses considerations of fairness to choose his contribution rate. When a game structure is unfair to himself as in games B/F/J or in game E/I, he reduces his contribution rate from 60% to 51% (games B/F/J) or 47% (games E/I). We observe that the decrease is higher when he is the only harmed player. Inversely, when a game structure is unfair for the opponents like in games C/G/K or in games D/H, player i increases his contribution rate from 60% to 70% (games C/G/K) or 67% (games D/H). We also observe that the increase is higher when he is the only one who takes advantage of the unfair structure. In other words, player i seems to compensate for the injustice by increasing his contribution and thus diminishing the free riding. This high rate also contradicts the considerations expressed by Stodder (1994) on the use of grade points, suspected to restrain cooperation.
IV. Conclusions
The purpose of our experiment was to show the relevance of players' sense of fairness in voluntary contributions to public goods. We observed that contributions were higher when the opponents (or one of them) was harmed by an unfair redistribution, and lower when the player we study was harmed by an unfair redistribution. These observations show that players' senses of fairness interfere with other preferences in the decision to contribute to a public good.
Finally, we can note that what we call fair or unfair is subjective from a political economy point of view. In effect, when an individual with a low endowment receives a higher share than his opponents, we qualify this as unfair, but many redistribution systems such as taxation precisely follow this for sympathy purposes. Thus, it is possible that in other contexts this situation would be perceived as fair by the subject.
Appendix
Games  A  B  C  D  E  F  G  H  I  J  K 
Contributions  23.9  16.3  29.9  25.1  13.5  20.2  9.3  28.3  9.2  23.7  13.5 
Rates  59.7  40.7  74.7  62.7  33.7  50.5  46.5  70.7  61.3  59.2  90.0 
References
Brock, J. (1991),. "A Public Goods Experiment for the Classroom." Economic Inquiry, 29, 395401.
BrownKruse J. and Hummels D. (1992), "Gender Effects in Public Goods Contribution: Do Individuals Put Their Money Where Their Mouth Is?", Journal of Economic Behavior and Organization 22, 255267.
Ledyard J. (1995), "Public Goods: A Survey of Experimental Research", in The Handbook of Experimental Economics (Kagel and Roth eds), Princeton University Press.
Hoaas D., Madigan L. (1996), "The Alleviation of Free Riding: A Research Program Progress Report", Classroom Expernomics 5, Fall.
Stodder J. (1994), "What is Being Taught in Voluntary Contribution Experiments?", Classroom Expernomics 3, Fall.
In the teaching of college and advanced placement economics, some of the characteristics of the production possibility frontier (PPF) are as difficult to convey as they are important to understand. John Neral and Margaret Ray (1995) suggest a useful and instructive classroom experiment in which two products, "widgets" and "whajamas," are produced to study tradeoffs between outputs. Tearing a piece of paper in half, folding it twice, and stapling it creates a widget; folding the paper three times makes a whajama. We have designed the links and smiles experiment to incorporate one of the most challenging concepts to grasp in relation to the PPFthe specialization of inputs. Of course, it is this crucial factor that results in the increasing opportunity cost of production and the concave shape of the production frontier.
This experiment has been run numerous times by at least five instructors. All report that their students benefited from the opportunity to work with and transform specialized resources from one use to another. Having acted as producers and derived PPFs themselves, students come away with a better understanding of resource specialization, increasing opportunity costs, and the tradeoffs incumbent in our every decision. In subsequent classes, instructors were able to refer back to this experiment as a reminder and reinforcement of the inherent concepts.
Time Required: Approximately 30 minutes
Materials for each student:
2 sheets of 8 1/2 x 11 paper 
1 roll tape 
1 pair scissors 
1 pencil or pen 
Objective:
This game allows students to derive and get a feel for production possibilities frontiers. After experimenting with different allocations of resources, students can discuss the reasons for increasing opportunity costs based on personal insight.
Setup:
There are two paper inputs used in this experiment: 5 ½" x 11/16" strips, and 2 ¾" x 11/8" rectangles. To obtain enough of each paper input for the whole experiment, have each student stack two sheets of paper, and ask them to: 1) fold the two mostdistant ends together; 2) fold the new mostdistant ends together; 3) undo the last fold and fold each of the mostdistant ends in so that they touch the center line; 4) without doing any unfolding, fold one side in once more so that it touches the center line. Now if they unfold their papers, cut along the creases, and cut the four wider strips in half as indicated by the dotted lines, they will have 16 strips and 16 rectangles.
Outputs defined:
Student producers will be producing links and smiles. A link is a 5 ½" x 11/16" strip of paper wrapped into a circle and taped in place. Subsequent links are put through the previous link and taped to interconnect the links into a "paper chain."
A smile is produced by using scissors to round the four edges of a 2 ¾" x 11/8" rectangle and drawing two eyes and a smile on one side of the circle.
Note to students that although strips are best for making links, and rectangles are best for making smiles, creative cutting and taping will permit strips to be made into regulation smiles and rectangles to be made into regulation links. (That is, if they cut the strips/rectangles in half and tape the halves together appropriately, they can make a rectangle out of a strip and viceversa.)
Conducting the experiment:
Round
1: Make four smiles and as many links as you can.
Round 2: Make only links.
Round 3: Make only smiles.
Round 4: Make one smile and as many links as you can.
Discussion:
Have the students draw their PPF’s on the board. The following are graphs from a typical links and smiles experiment:
In class, or for homework, ask the students:
"What was the opportunity cost of the first smile?"
(One link in the left graph, zero links in the right graph.)
"What was the opportunity cost of the last 1 or 2 smiles?"
(Three links for one smile in the left graph, four links for two smiles in the right graph.)
"Why did the opportunity cost increase?"
(Due to the specialization of resources. To make the first few smiles, we use resources that are better for making smiles than making links, so we give up a small number of links for a relatively large number of smiles. As we make more smiles, we must use resources that are more specialized for making links, so we lose a lot of links for relatively few smiles.)
"How does this relate to realworld production possibilities?"
(Resources are specialized in the real world as well. For example, the inputs to butter production are less useful for making guns. If we wanted to produce only guns, after using the resources best suited for gun production, we would have to melt down the steel vats used to make butter and mold them into guns. This is analogous to altering linkspecialized strips for use (however inefficiently) in smile production. Similar specialization of resources, and the increasing opportunity costs of production that results, exists for many of the goods we consider producing.
"What would the PPF look like if our two products were smiles and frowns (frowns are just like smiles except for the shape of the mouth)?"
(It would be a straight line with a slope of –1 because every smile that we made would result in one fewer frown.)
Reference
Neral, John, and Margaret Ray (1995), "Experiential Learning in the Undergraduate Classroom: Two Exercises." Economic Inquiry, January, 170174.
Most models of economic behavior are based on the assumption of rationality of economic agents and common knowledge of rationality. This means that an agent selects a strategy that maximizes his utility believing that all others do the same (are equally rational) and that all agents believe that all others believe that all agents are rational etc.
The pbeauty contest game is an appropriate game to test the assumption of this kind of reasoning. In this game a player has to guess what the average choice is going to be and the player will win if his choice is closest to some fraction of the average choice. Think of a seller in the stock market. He wants to sell his shares just before the average person is selling, thus when the price of the share is at its peak. Therefore, he does not want to sell it too early. As a consequence if everybody thinks like him, the selling time is unraveling. Unraveling can be seen in many real world markets, such as entrylevel medical labor markets or clinical psychology internships as documented in Roth and Xing (1994). The name of the game is due to Keynes (1936, 156) who compared a clever investor to a participant in a newspaper beautycontest where the aim was to guess the average preferred face among 100 photographs.
This experiment can be introduced in many different courses and at all levels of teaching. For example, it can be used in game theory in order to discuss the problems of iterated elimination of (weakly) dominated strategies and the issue of common knowledge of rationality; in macroeconomics to discuss rational expectations; and in microeconomics to discuss strategic interaction between players.
I. The rules of the basic beautycontest game.
The rules of the game are straight forward. Elements of the rules (indicated by bold/italic) can be varied in many ways (below I give some suggestions about different treatments).
Each person of Nplayers is asked to choose a (real or integer) number from the interval 0 to 100.
The winner is the person whose choice is closest to p times the mean of the choices of all players (where p is, for example, 2/3). The winner gets a fixed prize of $20. In case of a tie the prize is split amongst those who tie.
The same game may be repeated several periods. Subjects are informed of the mean, 2/3 mean and all choices after each period.
The students should write down a brief comment how they came to their choice.
Time to think: about 5 minutes or as a take home task.
II. The game theoretic solution and the contrast to a bounded rationality model using the basic game.
In equilibrium all players have to choose zero. Figure 1a describes the process of iterated elimination of weakly dominated strategies for p=2/3. A rational player does not simply choose a random number or his favorite number, nor does he choose a number above 100p, since it is dominated by 100p. Moreover, if he believes that the others are rational as well, he will not pick a number above 100p^{2}, and if he believes that the others are rational and that they also believe that all are rational, he will not pick a number above 100p^{3} and so on, until all numbers but zero are eliminated. If the number of players is 2, 0 is a dominant strategy. There are no other equilibria in the (in)finitely repeated game. If p>1 then the upper bound of the interval is also an equilibrium. For p=1 any number chosen by all players can be an equilibrium.
Equilibrium ¬ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ITERATION
¬ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾


Figure 1a: Infinite process of iterated elimination of dominated strategies for p=2/3. E(0) is the area of dominated choices, E(1) is the area of one iteration of elimination and so on. Adapted with permission from Ho et al. (1998). 
In contrast to the iterated elimination of dominated strategies, Figure 1b shows another process, the process of iterated best reply, which better explains actual behavior. The elimination process does not start at 100 but instead at 50, because a player will, for insufficient reasoning, think that any number is equally likely; therefore the mean should be 50. Thus best reply is 2/3*50=33.33. If everybody thinks that way best reply should be 22.22 and so on.
Equilibrium ¬ ¾ ¾ ¾ ¾¾ ¾ ¾ ¾ ITERATION


Figure 1b: Infinite process of iterated elimination of best replies, starting at 50. 
III. Why is a study of human behavior with this game interesting?
Each single aspect can be also found in other games but the combination of all three are not easily met at once in other games. Furthermore it is very easy to change the rules of the game in such a way that the iteration process leads to a few or even infinitely many eliminations to equilibrium.
IV. Results of actual behavior.
a) Results in the first period:
Figure 2ac shows the relative frequencies of choices in the first period of a) labexperiments with Bonn undergrad students; b) experiments with game theorists run in several conferences; and c) experiments run in different newspapers. Note that in all treatments there are high picks near or at 22 and 33. Game theorists and newspaper readers exhibit the largest modal frequency of behavior at or near the equilibrium zero. Dominated choices are rarely chosen. The following comment by a high school class (submitted to one of the newspaper studies, the Spektrum der Wissenschaft) summarizes the most important thought processes:
I would like to submit the proposal of a class grade 8e of the FelixKleinGymnasium Goettingen for your game: 0.0228623. How did this value come up? Johanna …asked in the mathclass whether we should not participate in this contest. The idea was accepted with great enthusiasm and lots of suggestions were made immediately. About half of the class wanted to submit their favorite numbers. To send one number for all, maybe one could take the average of all these numbers.
A first concern came from Ulfert, who stated that numbers greater than 66 2/3 had no chance to win. Sonja suggested to take 2/3 of the average. At that point it got too complicated to some students and the finding of the decision was postponed. In the next class Helena proposed to multiply 33 1/3 with 2/3 and again with 2/3. However, Ulfert disagreed, because starting like that one could multiply it again with 2/3. Others agreed with him that this process then could be continued. They tried and realized that the numbers became smaller and smaller. A lot of students gave up at that point, thinking that this way a solution could not be found. Other believed to have found the path of the solution: one just has to submit a very small number.
However, one could not agree how many of the people who participated realized this process. Johanna supposed that the people who read this newspaper are quite sophisticated. At the end of the class 7 to 8 students heatedly continued to discuss this problem. The next day the math teacher received the following message: We think it best to submit number 0.0228623.
Figure 2a: Labdata. With permission from Nagel (1995) 
Figure 2b: Game theorist data. 
Figure 2c: Newspaper readers data. 
b) Results over time:
Figure 3 shows the behavior over time in some selected treatments. In general behavior converges to equilibrium. The speed of convergence depends on the parameter used. Typically the reasoning process from period to period is anchored to the mean of the previous period. Level 1reasoning is then 2/3 times the mean of the previous period, level 2reasoning is 4/9 times the mean of previous period and so on. The average level of reasoning typically applied by subjects does not increase over time.
Figure 3: Mean behavior over time for different variations. 
V. Further reading
Nagel (1995) discusses the basic game with parameters p=2/3, ½ and 4/3 played for 4 periods with about 15 students. Ho et al. (1998) discuss the game with group size 3 and 7, and intervals from [0,100] or [100,200] and several parameters p. Thaler (1998) discusses his newspaper results in connection with financial economics. Nagel et al (1999) compare the behavior in experiments run in three newspapers, run with game theorists and lab experiments. Nagel (1998) presents a survey of beautycontest experiments and analyzes the case where the prize to the winner is his choice paid in dollars. Rubinstein (1999) lets his students play the basic game among other games on his webpage.
References
Ho, T. , Camerer, C. & Weigelt, K. (1998). "Iterated Dominance and Iterated BestResponse in Experimental ‘PBeauty Contests’," American Economic Review 88, 4, 947969.
Keynes, J.M. (1936). The General Theory of Interest, Employment and Money. London: Macmillan.
Nagel, R. (1995), "Unraveling in Guessing Games: An Experimental Study", American Economic Review, 85 (5),13131326
Nagel, R. (1998) "A Survey on Experimental "BeautyContest Games:" Bounded Rationality and Learning, in Games and Human Behavior, Essays in Honor of Amnon Rapoport." Eds. D. Budescu, I. Erev, and R.Zwick. Publisher: Lawrence Erlbaum Associates, Inc., New Jersey (1998), p.105142.
Nagel, R. A. BoschDomènech, A. Satorra, and J. GarcíaMontalvo (1999). "One, Two, (Three), Infinity: Newspaper and Lab BeautyContest Experiments" Univerisitat Pompeu Fabra Working Paper 438 (download http://www.econ.upf.es/cgibin/onepaper?438 )
Roth, A. and X. Xing (1994). "Jumping the gun: Imperfections and Institutions Related to the Timing of Market Transactions." American Economic Review, 84, 9921044.
Rubinstein, A. (1999). "Experiments from a Course in Game Theory: Pre and Postclass Problem Sets as a Didactic Device." Games and Economic Behavior, 28 (1999), 155170. (See also http://www.princeton.edu/~ariel/99/gt100.html)
Thaler, R. (1997). "Giving Markets a Human Dimension." Financial Times, section Mastering Finance 6, June 16, 1997.