By Christian Seidl,

University of Kiel, Germany

Contents1.Expected Utility 1.1 Theory1.2 Experimental Design1.3 Pitfalls 1.3.1 Allais’ Paradox 1.3.2 Ellsberg’s Paradox 1.3.2 Preference Reversal 1.3.4 Response-Mode Bias

2. Main Fields of Experimental Economics2.1 General2.2 Own Research 2.2.1 Equitable Income Taxation 2.2.2 Acceptance of Distributional Axioms 2.2.3 Background Context Effects 2.2.4 Own Research Other than Mentioned Above3. How Can We Evaluate Income Distributions?4. Aim of the Experiment

1. EXPECTED UTILITY 1.1 Theory

In their famous book on Game Theory and Economic Behavior, von Neuman and Morgenstern devised a method to represent preferences among possible actions, ai, i=1,…,n, by the expected value of the utilities of the results, πij, i=1,…,n, j=1,…,k, of a decision problem under risk, i.e., with given probabilitiers, pj, j=1,…,k, of the possible states of the world which, together with the chosen action, engender the results of the decision problem.

More precisely, consider a decision matrix:

Provided that some common-sense axioms, which every rational decision-maker would approve, hold, von Neumann and Morgenstern showed the existence of a real-valued utility function u( )∙ defined on the space of results such that

(1)

They also showed that the utility function u( ) ∙ is cardinal, i.e., unique up to a positive linear transformation.

Whereas von Neumann and Morgenstern assumed the probabilities of the states of the world as given, later on, Savage devised a more general theory based on a qualitative probability relation and derived subjective probabilities jointly with the respective utility function. In this introduction, I will not go into details.

To exemplify expected utility in a simple framework, let us restrict the decision problem to three outcomes such that

Obviously, expected utility is:

Substituting p2=1-p1-p3 gives us:

(2)

This formula allows us to arrange this simple lottery in terms of a Marschak-Jensen-Machina triangle:

p1

p3

1.0

1.0

p1

p3

p2

As the utilities of the three results are given, (2) represents an indifference curve for ū as a linear function of p1 in terms of p3 and ū (red line). Let us replace ū by û, û>ū, then the indifference curve moves in a parallel way to the North-West of this triangle (green line). Hence, the family of indifference curves are parallel lines with the North-West preference direction. Note that this applies to expected utility theory.

(2)

1.2 Experimental Design

Experimental elicitation of utility usually works by way of a series of binary lotteries. Suppose we have four results [e.g., pay-offs]:

and two probabilities p and q. then we consider two binary lotteries

(3)with expected utility values

(4)

(5)

Notice that for 2=3 the second lottery reduces to a single result, 2.

Parameter elicitation follows the pattern that one component of the lotteries in (3) is taken as variable, and the values of the other components are fixed. The subject is asked to choose the value of the variable component such that the subject is indifferent between the two lotteries. Then we insert the chosen value of the variable in the expression (4) or (5), equate both expressions, and derive points of the von Neumann-Morgenstern utility function by simple rearrangements of (4) and (5).

(3)(4)

(5)

Consider first 2=3; this means that we look for component values which render the first lottery in (3) indifferent to π2.

This constitutes the group of standard-gamble methods: (i)Certainty equivalence: 2 is the variable component.(ii)Value equivalence: 1 or 4 is the variable component.(iii)Probability equivalence: p is the variable component.

= u(π2)

Consider second ; then again we look for component values which render the lotteries in (3) indifferent. This constitutes the group of paired-gamble methods:

(iv) Value equivalence: exactly one component out of 1,2,3,4 is the variable component.

(v) Probability equivalence: one out of the two probabilities, p or q, is the variable component [lottery equivalence if 3=4=0 for real-valued results].

=

We illustrate utility elicitation using the standard-gamble method which was used originally.

Notice that, as the values j are given, this method provides both the values of the domain j and the range u(j) of the utility function.

In the Mosteller and Nogee experiment, k denoted the loss of 5 Cents, i.e., u(k)=-1, and 1 meant refusal to participate in the lottery, i.e., u(1)=0. Keeping the loss of 5 Cents constant, Mosteller and Nogee repeated this experiment for seven different values of pj, which provided them seven payoffs j and seven utility values u(j) on the von Neumann-Morgenstern utility schedule in addition to u(1)=0 and u(k)=-1.

Proceeding this way further by sequentially using a subject’s certainty equivalents as prizes for lotteries which share the probability p for the better reward of the respective lottery, one can derive arbitrarily many points on the subject’s von Neumann-Morgenstern utility function.

In addition to that, Becker, deGroot, and Marschak (1964) devised an ingenious procedure to elicit subjects’ true certainty equivalents [although it is similar, it was developed independently of Vickrey’s (1961) famous sealed-bid second price option which induces subjects to truthfully reveal their valuation for an object at auction]:

Subjects are asked for their selling price of a binary lottery. Then a buying price is randomly chosen by the experimenter. If it exceeds the stated selling price, then the subject gets the buying price; if not, then the respective lottery is played out. Under the precepts of expected utility theory, it never pays for the subject to cheat with respect to the revelation of his or her true certainty equivalent: if the stated certainty equivalent exceeds the true one, then the subject runs the risk of playing out the lottery, whereas the subject would have preferred the buying price (provided that it is between the stated and the true certainty equivalent). If the certainty equivalent understates the true one, then the subject runs the risk of getting the buying price (provided that it is between the true and the stated certainty equivalent) , whereas the subject would have preferred playing out the lottery.

1.3 Pitfalls

Recall that expected utility theory rests on plausible common-sense axioms which engender the shape of a von Neumann-Morgenstern utility function by mathematical reasoning. Yet experimental research has shown that subjects’ actual behavior deviates from expected utility theory. Subsequently this led to the proposal of other models to capture subjects’ behavior, such as prospect theory, state-dependent utility theory, causal utility theory, regret theory, range-frequency theory, etc. Note, however, that it were the pitfalls of expected utility theory as pointed out by experimental results, which triggered the development of later theories of decisions under risk and uncertainty. In the following, pitfalls of expected utility theory are illustrated by four famous examples.

1.3.1 Allais’ Paradox

Consider four lotteries S’, S”, S*, and S** with at most 3 results in a Marschak triangle such that the slopes of the lines connecting S’ and S” on the one hand and S* and S** on the other are parallel. This means:

which gives us the Marschak triangle:

p1

p3

S’

S”

S*

S**

1.0

1.0

Suppose a subject’s preferences are characterized by the family of red indifference curves. Then this subject must prefer S’ to S” whenever it prefers S* to S**, and vice versa for the green indifference curves.

Let’s now come to the gist of Allais’ Paradox. Nobel laureate Maurice Allais presented the following choice problem to subjects (among them Leonard Savage) and asked them for their preferences for the following pairs of lotteries [all payoffs in (old, i.e., 1952) French Francs]:

S’: Certainty of receiving 100 million Francs 10% chance of winning 500 million FrancsS”: 89% chance of winning 100 million Francs 1% of winning nothing

S*: 11% chance of winning 100 million Francs 89% chance of winning nothing

S** 10% chance of winning 500 million Francs 90% chance of winning nothing

Allais observed that many subjects preferred S’ to S” but also S** to S*, which contradicts expected utility because we have:

Illustrating this in terms of a Marschak triangle shows:

p1

p3

1.0

1.0S’

S”

S*

S**

Preferring S’ to S” and S** to S* violates the condition that the family of utility indifference curves have to be parallel lines under expected utility. Hence, this example shows violation of expected utility.

1.3.2 Ellsberg’s Paradox

Ellsberg’s Paradox concerns ambiguity, i.e. it focuses on the probability structure. Ellsberg (1961) posed the following problem to subjects:

Consider an urn containing 100 balls. You know there are 33 red balls but you do not know the composition of the remaining 67 black and yellow balls. One ball is to be drawn at random from an urn.

Which do you prefer: S’ or S”?S’: Receive $ 1000 if a red ball is drawnS”: Receive $ 1000 if a black ball is drawn

Which do you prefer: S* or S**?S*: Receive $ 1000 if a red or a yellow ball is drawnS**: Receive $ 1000 if a black or a yellow ball is drawn

As drawing a yellow ball in the second pair of lotteries implies that the subject gets $ 1000 anyway, the preferences should be the same across both pairs of lotteries, i.e.:

However, many subjects preferred S’ to S” and S** to S*.

Let us express Ellsberg’s Paradox in terms of a Marschak triangle. As we have only two payoffs, 1=1000 and 3=0, p2=0, all lotteries lie on the diagonal boundary of the Marschak triangle:

p1

p3

1.0

1.0

0.33 S’

S” if less black than red balls

S” if more black than red balls

S* S** if less black than red balls

S** if more black than red balls

McCrimmon and Larsen (1979) found for a probability of 0.33 for red balls a maximum of 70% of Ellsberg-type violations of expected utility, but falling to 20% for a probability of 0.2 for red balls and to 0% for a probability of 0.5 for red balls.

S’: $ 1000 if red ball S”: $ 1000 if black ball

S*: $ 1000 if red or yellow ball S**: $ 1000 if black or yellow ball

1.3.3 Preference Reversal

That is, eliciting subjects’ preferences for lotteries contradicts their preferences when eliciting them in terms of certainty equivalents and vice versa.

The preference reversal phenomenon was discovered by Lindman (1965; 1971) and Slovic and Lichtenstein (1968; 1971). Their experiments showed that this phenomenon can mainly be observed for lottery comparisons between a P-bet, i.e., a binary lottery which accords a high probability of winning a modest amount and a low probability of losing (or winning) an even more modest amount, and a $-bet, i.e., a binary lottery which accords a low probability of winning a high amount and a large probability of losing (or winning) a modest amount.These authors observed that many subjects preferred the P-bet to the $-bet, but indicated a higher certainty equivalent for the $-bet than for the P-bet.

The next table shows the results from three Lichtenstein and Slovic experiments:

Meanwhile, there are hundreds of papers which investigated the preference-reversal phenomenon [for a survey see Seidl (2002)].

The preference-reversal phenomenon was extended to more general lotteries and compared with equally structured income distributions by Camacho, Seidl, and Morone (2005) . They modeled lotteries and income distributions by appropriate distributions of 100 tally marks, where P-bets were represented by negatively skewed and $-bets by positively skewed distributions. In addition to that, bimodal, symmetric, and uniform distributions were tested. Camacho, Seidl, and Morone found four patterns of preference reversals, where two of them were only observed for income distributions. They also observed more preference reversals for income distributions than for lotteries. [For income distributions the equally distributed equivalent income replaced the certainty equivalent.] The transfer principle was violated in more than 50% of the cases. Negatively skewed distributions engendered greater happiness.

1.3.4 Response-Mode Bias

Seidl and Traub (1999) compared von Neumann-Morgenstern utility functions derived according to the certainty-equivalence method with utility functions derived according to the lottery-equivalence method. The observed:

(1) Dependence of the utility functions on the probability used for their generation.(2) Even when the same generating probability was used, the two types of utility functions are different.(3) The correlation of risk attitudes are higher for the same response mode than for different response modes.(4) For risk averse [risk loving] subjects, a utility function derived with a higher generating probability dominates [is dominated by] a utility function derived with a lower generating probability.

2. MAIN FIELDS OF EXPERIMENTAL ECONOMICS

2.1 General

Experimental economics has been applied to behavioral checks of economic theory in many fields. The first comprehensive books on experimental economics were published in the 90’s:

John D. Hey, Experiments in Economics, Blackwell, Oxford and Cambridge MA 1991.Vernon L. Smith, Papers in Experimental Economics, Cambridge University Press, Cambridge 1991.Douglas D. Davis and Charles A. Holt, Experimental Economics, Princeton University Press, Princeton NJ 1993.John H. Kagel and Alvin E. Roth (eds.), Handbook of Experimental Economics, Princeton University Press, Princeton NJ 1995.

The main fields to which experiments were applied, are:

• decision making• bargaining• game theory• market organization• public goods• coordination problems• industrial organization• auctions• fairness• asymmetric information• voting and social choice

In this short introduction I will single out the voluntary provision of public goods and present the most basic experiment in this field:

Each of four subjects in an experiment is given € 5.00 to be used for two purposes. They may put part of it in an envelope to be used for a group project [i.e., a public good] or kept to be used as a private pay-off. The experimenter will collect the contributions for the group project, total them up, double the amount, and then divide this money equally among the subjects of the group. Note that the private benefits resulting from this operation is only half of the total contribution.

What does theory tell us about this situation? Game theory tells us that no one will ever contribute anything. Each subject will “free ride” instead. The dominant strategy is to contribute nothing because each € 1.00 contributed yields only € 0.50 to its contributor, no matter what others do.

In contrast to that, the group as a whole would be best off if all participants contributed € 5.00 because this would give an additional € 5.00 to each participant. However, the best result for an individual subject would be if he or she contributes nothing (i.e., keep all € 5.00) and all the others contribute € 5.00 each. This results in € 12.50 for the first subject and € 7.50 for all others. However, if all subjects follow this reasoning, nobody will contribute and all end up with € 5.00 each. If just one subject contributes all € 5.00, this subject would end up with € 2.50 and the three other subjects would end up with € 7.50 each. So it is better for him or her to contribute nothing. This holds, vice versa, for all subjects.

The set of Pareto-optimal states is: {(10,10,10,10), (12.5,7.5,7.5,7.5), (7.5,12.5,7.5,7.5), (7.5,7.5,12.5,7.5), (7.5,7.5,7.5,12.5)}, the equilibrium resulting from uncoordinated individual actions is (5,5,5,5).

This theoretical reasoning seems to invalidate the model of voluntary contributions for the provision of public goods [which was proposed by Mazzola, de Viti de Marco, Sax, Lindahl, Samuelson and others], as Wicksell (1896) had surmised. Hence, is a system of mandatory taxation the only solution for providing public goods?

Experimental results show that neither theory is right. Some subjects contribute nothing, some contribute their whole budget. Generally, total contributions can be expected to lie between € 8.00 and € 12.00, or 40% and 60% of the group optimum. Moreover, as this experiment is repeated with the same subjects, it was observed that contributions decrease with the number of rounds, which reflects disappointment of the original contributors with the free riders. Gradually free riding gains ground.

Why should we care about public goods experiments? Now, the desired outcome is that anybody contributes € 5.00. Experimental evidence showed that voluntary contributions will not produce the desired outcome. Hence, we have to look for institutions which causes the outcome to be closer to the group optimum. To achieve that we have to anticipate how individual choice will change as the institutions change. Experiments are the only way to perform this task.

[Recently, an anonymous private person in Germany donated € 5 million to buy a ship for an organization of rescuing ship-wrecked people.]

2.2 Own Research

Let me now give you a short account of my own research in experimental economics.

2.2.1 Equitable Income Taxation

Stefan Traub and I investigated the perception of equitable income taxation by means of field data. We polled some 200 employees in firms twice. First, they got information about the current tax burden for singles in Germany. Then we asked them for the equitable taxation for four household types (single, couple without children, couple with one child, couple with two children) and five income levels. Subjects started with the tax burden of singles and were asked for the equitable taxation of other household types. We collected the questionnaires without leaving them a copy. Some ten days later, we told them that a tax reform was envisaged and indicated the taxes for the households with two children. These were precisely the same taxes as proposed by the respective subjects. Then subjects were asked for the equitable taxes of the other household types.

Interestingly enough, subjects indicated lower taxes for the other household types than in the first round. They also disapproved of the German tax-splitting boon. This applied to married couples (the beneficiaries of the splitting boon) as well.

2.2.2 Acceptance of Distributional Axioms

Elizabeth Harrison and I polled hundreds of students by questionnaire whether they obey the main axioms of income inequality measurement. We observed violations amounting to 20% to 40% of the responses. In particular, the transfer axiom was very often violated, but also the population principle.

2.2.3 Background Context Effects

Subjects perceive phenomena in relation to their background. Psychologists observed that the darkness of grey squares is upgraded if they are presented in the context of many lighter grey squares, and downgraded if they are presented in the context of many darker grey squares. In joint work with Stefan Traub and Andrea Morone, we embedded certain income levels in a context of several distributions, viz. bimodal, normal, negatively skewed, and positively skewed. We observed background context effects: an income level received better grades when it was presented in the frame of a positively skewed distribution than when it was presented in the frame of a negatively skewed distribution.

In spite of that, mean income was lowest for the positively skewed distribution and highest for the negatively skewed distribution.

Moreover, we struck another paradox: whereas individual income satisfaction with a certain income level is highest for a background of positively skewed distributions, a Harsanyi-type social welfare function demonstrates higher average social welfare for a background of negatively skewed distributions. This paradox results from the weighting of income satisfaction with the stratified frequency of the involved subjects: although an individual has a bit less satisfaction with his or her income under a background of a negatively skewed distribution, there are more individuals with higher incomes in a negatively skewed distribution. This frequency effect overcompensates the higher income satisfaction under the background of a positively skewed income distribution. [Compare Derek Parfit’s (1984) repugnant conclusion!]

2.2.4 Own Research Other than Mentioned Above

• Testing Decision Rules for Multiattribute Decision Making • Knock-Out for Descriptive Utility or Experimental Design Error? • A New Test of Image Theory• Stochastic Independence of Distributional Attitudes and Social Status. A Comparison of German and Polish Data• Friedman, Harsanyi, Rawls, Boulding – Or Somebody Else? An Experimental Investigation of Distributive Justice•The Performance of Peer Review and a Beauty Contest of Referee Processes of Economics Journals• Lorenz Meets Rating but Misses Valuation• An Experimental Study on Individual Choice, Social Welfare and Social Preferences

Problem 1: Consider two income distributions:

A = [10, 20, 30, 50, 70, 80, 90] B = [50, 50, 50, 50, 50, 50, 50]

Question: Which of the two income distributions is more equally distributed?

3. HOW CAN WE EVALUATE INCOME DISTRIBUTIONS?

Obviously B is more equally distributed than A, although some sub-jects might prefer to live in a society with income distribution A.

Questions:

(i) Which of the two income distributions is more equally distributed?

(ii) Which of the two income distributions generates more income for the economy?

(iii) Which of the two income distributions generates more welfare for the economy?

(iv) Would you rather live in a society with income distributions B or C if your ownincome position will be later on determined by chance?

Problem 2: Consider the income distributions: C = [100, 200, 300, 500, 700, 800, 900] B = [50, 50, 50, 50, 50, 50, 50]

Obviously B

Obviously C

The answer depends on the subject’s social welfare function. For welfarist socialwelfare functions, C generates more welfare. If equality preferences enter the social welfare function, B might also emerge as generating higher welfare.

The answer depends on your distributional preferences and your risk attitude.

Problem 3: Consider three income distributions:

A = [10, 20, 30, 50, 70, 80, 90] D = [10, 20, 30, 50, 70, 70, 90]

E = [10, 10, 30, 50, 70, 80, 90]

Questions:

(i) Is D more equally or more unequally distributed than A?

(ii) Is E more equally or more unequally distributed than A?

In D, as compared with A, the second richest person loses 10 monetaryunits. In E, as compared with A, the second poorest person loses 10 monetary units. There is no right or wrong answer; it is up to the view of the beholder whether D or E is considered more equally or more unequally distributed than A.

Problem 4: Consider three income distributions:

A = [10, 20, 30, 50, 70, 80, 90]

F = [10, 30, 30, 50, 70, 70, 90]

G = [20, 20, 30, 50, 70, 80, 80]

Questions:(i) Is F more equally or more unequally distributed than A?(ii) Is G more equally or more unequally distributed than A?(iii) Is D more equally or more unequally distributed than F?

G can come about from F in two ways: (i) the second poorest transfers 10 monetary units to the poorest income recipient, and the richest transfers 10 monetary units to the second richest income recipient; thus, we have two progressive transfers. (ii) The richest transfers 10 monetary units to the poorest income recipient, and the second poorest transfers 10 monetary units to the second richest income recipient; thus, we have a progressive and a regressive transfer.

F comes about from A if 10 monetary units are transferred from the second richest to the

second poorest income recipient.

G comes about from A if 10 monetary units are transferred from the richest to the poorest

income recipient.

Problem 5: Consider three income distributions:

A = [10, 20, 30, 50, 70, 80, 90]

H = [5, 20, 30, 50, 70, 70, 90] J = [15, 20, 30, 50, 70, 80, 80]

Questions:(i) Is H more equally or more unequally distributed than A?(ii) Is J more equally or more unequally distributed than A?(iii) Is J more equally or more unequally distributed than H?

J can come about from H in two ways: (i) the poorest income recipient receives 10 monetary units and the richest income recipient transfers 10 monetary units to the second richest income recipient. (ii) The richest income recipient transfers 10 monetary units to the poorest income recipient and the second richest income recipient receives 10 monetary units.

H comes about from A by an income loss of 5 monetary units of the poorest income

recipient and by an income loss of 10 monetary units of the second richest income recipient. This depicts income

changes which point in the same direction.

J comes about from A by a gain of 5 monetary units of

the poorest income recipient and a loss of 10

monetary units of the richest income recipient.

Again there is no right or wrong answer to these questions.

4. AIM OF THE EXPERIMENT:

You will be shown a model income distribution in terms of EUROs. The computer will add or subtract EURO 100 to or from one person’s income in this income distribution. Then the computer will determine some other person in this income distribution and will ask you to change this person’s income such that the former degree of income inequality in this income distribution is restored, according to your perception.

Please, note that income changes are caused by external events, such as the correction of errors of monetary transactions which happened in the past, tax refunds or tax payments from past incomes, bonus premiums for last year or re-payment of erroneous too high salary in the last year, etc.

Moreover, your suggestion of changing another person’s income is a purely hypothetical correction which would, according to your perception, restore the former degree of income inequality in this society. It does not mean actually giving income to or taking off income from this person. Just imagine which external income change which occurred to this specified person would restore the former degree of income inequality in this society.