Appendix to “Vickrey Auction vs BDM: Difference in bidding behaviour and the impact of other-regarding motives”

In an experiment we first elicit the distributional preferences of subjects and then let them bid for a lottery, either in a Becker-DeGroot-Marschak (BDM) mechanism or a Vickrey auction (VA). Standard theory predicts that altruistic subjects underbid in the VA – compared to the BDM – while spiteful subjects overbid in the VA. The data do not confirm those predictions. While we observe aggregate underbidding in the VA, the result is not driven by the choices of altruistic subjects. ISSN 1993-4378 (Print) ISSN 1993-6885 (Online)


A Appendix: Proofs and Technicalities
In this appendix, we provide a proof of Proposition 1 which gives the theoretical predictions we make about treatment differences. The proof will be organised in a sequence of lemmata.
Lemma 1. In the BDM treatment, the unique admissible bid b * i is implicitly defined by 1 2 Proof. Trivial, hence omitted.
Recall that (x i , x j ), j ≠ i, denotes an allocation of money. For selfish subjects the payoff is of the form U self i (x i , x j ) = f (x i ). Given the experimental design, the functional specification takes the form where b j is the bid of player j ≠ i, and I (·) is the indicator function taking values 0 and 1, respectively, depending on i winning, losing or entering the tie-break in the auction. Proof. Trivial, hence omitted.
For the next two lemmata, we need an additional assumption. For two strictly increasing functions f i and g i , say that f is more concave than g, written f i ≻ conc g i , if and only if g −1 i (f ) is a concave function. As outlined in Section 2 in the paper, we shall maintain the following assumption: Assumption 1. Let f i and g i be strictly increasing functions where f i evaluates lotteries with consequences affecting monetary payoffs for i, and g i evaluates lotteries with consequences affecting monetary payoffs for j ≠ i. Then f i ≻ conc g i .
Assumption 1 guarantees that the certainty equivalent for a lottery evaluated with function f i (·) is weakly lower than the certainty equivalent under g i (·) and rules out "altruistic overbidding". (A.9) If f i = g i , the r.h.s. in (A.5) is strictly positive, whereas the l.h.s. is strictly negative -a contradiction. To see this, note that by Assumption 1, we haveb > b * i for thê b solving the expression 1 2 g i e + w −b + 1 2 g i e −b = g i (e), and when setting up inequality (A.8), we stipulated that i loses at Consider now the case f i ≠ g i . Note that by the same reasoning, the expression on the l.h.s. of inequality (A.9) is strictly negative: Rearranging inequality (A.5) gives The numerator of this expression is strictly positive by the definition of b * i and b j < b * i , while the denominator is strictly negative as b ′ i < b * i . Thus, the r.h.s. of inequality (A.10) is strictly negative, which establishes the desired contradiction. (ii) altruistic subjects never have an incentive to overbid relative to b * i , and might have an incentive to underbid; (iii) spiteful subjects never have an incentive to underbid relative to b * i , and might have an incentive to overbid.
Proof. Proposition 1 follows from Lemmata 1-4 combined with the following observation. Let µ i ∈ ∆ B j be a belief (i.e., a probability distribution) of i about j's behaviour. !

Instructions:
Welcome to the experiment and thank you for agreeing to participate. This experiment is run by the Department of Economics and is funded by the Austrian Science Foundation. In a moment I will outline the decisions you'll face but first I have some preliminary announcements.
Please can you switch off your mobile phone. Throughout the duration of the experiment you are not allowed to use you mobile phone or to read any other material except that which is provided by the experiment. Please do not talk or attempt to communicate with any other participants. If, at any point, you have any questions regarding the experiment, raise your hand and someone will assist you. Failure to adhere by these rules will result in you being asked to leave and you will not be paid.
Any data collected from you will be used for research purposes only. Any reporting of this research makes no reference to your individual identity. In the data set created from this experiment you will exist solely as an I.D. number and not by name.
You will be paid at the end of the experiment in cash. The level of this payment depends on a combination of the decisions that you and others make.
This experiment consists of two parts: part A and part B. Part A involves answering a number of survey questions, and making a series of decisions about distributing money between yourself and another person. Part B involves bidding for a lottery in an auction. It is important to note that all of you will only be paid for one part of the experiment -either part A or part B. Which part will be paid for real will be determined randomly at the end of the experiment once all decisions have been made.
At the end of the experiment we will flip a coin. If it is heads, then you will all be paid for the decisions made in part A, but not for the decisions made in part B. If it is tails, then you will all be paid for the decisions made in part B, but not for the decisions made in part A. Hence, only one part of the experiment is ever paid for real.

6
! Part Ai: From this point onwards we will refer to this room as "Room 1" and the other room as "Room 2". The individuals in Room 2 are given the same instructions and face exactly the same decisions as all of you do in Room 1. This part of the experiment involves decisions made about the allocation of money between two individuals. These two individuals are referred to as pairs. Every pair always contains one person from Room 1 and one person from Room 2. Every pair contains one active person, who makes binding decisions, and one passive person, who makes no decisions.
You are a member of two different pairs.
• Pair 1: In the first pair you are a member of you are an active person. You are required to make a series of ten decisions about the allocation of money between yourself and the passive person of this pair, who is situated in Room 2 -this person is referred to as your passive person. • Pair 2: In the second pair you are a member of you are a passive person. Here, the active person of this pair, who is situated in Room 2 and is referred to as your active person, is required to make the same ten decisions that you made in Pair 1 about the allocation of money between themselves and you.
Your active person and your passive person are two separate individuals who have both been chosen randomly. At no point will you ever discover their identities. Similarly, they will never discover your identity.
Your choices as an active person: As an active person in Pair 1 your task is to make 10 separate decisions. Each decision is a choice between two options: left or right. Each option specifies an amount of money for you and an amount of money for your passive person in Room 2. The option right always pays €6.00 to you and €6.00 to your passive person in Room 2. The option left pays different amounts in each of the 10 decisions. Consider the following example. Here you have the choice between option right which pays you €6 and your passive person €6, or option left which pays you €5 and your passive person €1. Such a choice is illustrated below. If you wish to choose option left then you would be required to place an X in the left circle. If you wish to choose option right then you would be required to place an X in the right circle. The table below outlines the ten decisions that you have to take.
Your payment: You will receive two separate payments from this part of the experiment: one as an active person and one as a passive person.
• Payment as an active person: of the ten decisions you make as an active person, one of them will be chosen at random. For this single randomly chosen decision you will be paid the corresponding amount determined by your decision, as will your passive person in Room 2. • Payment as a passive person: of the ten decisions your active person in Room 2 makes, one of them will be chosen at random. For this single randomly chosen decision you will be paid the corresponding amount determined by their decision, as will your active person in Room 2.
The two paid decisions are chosen independently. They are randomly drawn from a uniform distribution. This means as an active person each decision number has the same 10% chance of being paid for real. Similarly, as a passive person each of the ten decision numbers have the same 10% chance of being paid for real. In order to check your understanding of these instructions, please answer the following questions. You will not be able to proceed with the decision task until you have answered all the questions correctly. If there is any part of the instructions that need clarification then please raise your hand and someone will assist you.
Question 1: Imagine that you chose right for decision number 4 and that this decision is the one randomly chosen to be paid.
How much will you receive as an active person from this situation?______________________ How much will your passive person receive from this situation?_________________________ Question 2: Imagine that you chose left for decision number 7 and that this decision is the one randomly chosen to be paid. How much will you receive as an active person from this situation?______________________ How much will your passive person receive from this situation?_________________________ Question 3: Imagine that as an active person you chose left for decision number 1 and that this decision is the one randomly chosen to be paid. Imagine that as a passive person your active person chose left for decision number 10 and that this is the decision randomly chosen to be paid How much will you receive as an active person from this situation?______________________ How much will you receive as a passive person from this situation?______________________ How much will your total payment be?_____________________________________________ 9 !

Part Ai: Decision Sheet
Please write your experimental ID number here: ____________________ If your experimental ID number is missing or wrong then we will not be able to calculate your payment.
The following table outlines the 10 decisions you are required to make. For each row of the table please indicate with an X whether you prefer option left or option right. Please consider your answers and take your time. You have 5 minutes within which to make all ten decisions.
Once you have made all ten choices, please place this decision sheet in the envelope provided. Imagine a ladder with 10 rungs. There are approximately 5 million Austrians of working age -that is aged between 16 and 65. These 5 million Austrians are uniformly distributed across the ladder. This means that there are 500,000 people on each rung of the ladder. They are arranged on the ladder by annual income, starting with the poorest. This means that on the first rung are the poorest 500,000 people -i.e. those with the lowest annual income. On the 10 th rung stand the richest 500,000 people -i.e. those with the highest annual income.

Question 14a: (paid)
What is the net (i.e. after tax) annual income of the median person? The median person is the individual that stands exactly in the middle of the ladder, in between rungs 5 and 6. They have 2.5 million people richer than them and 2.5 million people poorer than them. (Remember, we are asking for annual income (after tax), not monthly income) €_______________________ ** if your answer is within plus or minus 10% of the correct answer then you will earn €1 ** Question 14b: What rung are you currently on? Please circle your answer. For part B of the experiment you are all required to make just one decision. That decision is to decide how much you would like to bid for a Lottery in an auction. As in Part A, all participants in both Room 1 and Room 2 are given the same instructions and face exactly the same decision. The type of auction we will use is called a two-bidder second-price auction. The important point of this type of auction is that: the highest bidder wins, but they only have to pay a price equal to the lowest bid. -hence the name second-price auction.
The exact setup is as follows: You each begin this part of the experiment by being given €10. This €10 is referred to as your The Lottery: What you are bidding for is the opportunity to play a Lottery. The Lottery pays one of two possible prizes, a green prize or a yellow prize. Each prize has the same 50% probability of occurrence. The green prize pays €15.00 and occurs with probability one half. The yellow prize pays €0.00, and occurs with probability one half. The way the Lottery is played in practice is as follows: There is a bag containing two balls: one green ball and one yellow ball. If you have won the auction, then you will be asked to place your hand inside the bag and pull out one of the balls. If this ball is green then you have won €15, if this ball is yellow then you have won €0. This draw will be conducted separately for each person that wins the auction. All draws are conducted in private.
The Auction: Every auction consists of two bidders. Both you and the other bidder submit a bid in the range of €0 to €10. The rules of the auction are as follows: • A winning bid is the highest of the two bids.
• A losing bid is the lowest of the two bids.
• A winning bidder buys the right to play the Lottery.
• The price a winning bidder pays in order to play the Lottery is not their own bid. The winning bidder pays a price equal to the losing bid -i.e. the lower of the two bids. • The losing bidder does not get to play the Lottery and keeps all of their endowment.

16
! Hence, the bid you make defines the maximum you will ever have to pay for the right to play the

Lottery.
What happens if both bids are the same? If both bids are identical then the winning bidder will be determined by a coin toss. Hence, if both bids are identical then you have a 50% chance of being a winning bidder and a 50% chance of being a losing bidder.
The Other Bidder: The other bidder is not an actual person, it is just a random number. The random number is drawn from a uniform distribution over the range €0 to €10, in increments of €0.10. This means that the random number can take any one of 101 possible values and that each of the 101 possible values are equally likely to occur -each value occurs with approximately 1% chance.
The way the random number is generated in practice is as follows: After you have submitted your bid, you will be asked to choose two cards: one white card and one yellow card. The cards all have numbers on them. There are ten white cards which have numbers of euros printed on them of: €0; €1; €2; €3; €4; €5; €6; €7; €8; and €9. There are eleven yellow cards which have numbers of cents printed on them of: 0c; 10c; 20c; 30c; 40c; 50c; 60c; 70c; 80c; 90c and 100c. The cards are placed face down so you cannot see the numbers. The sum of the white card you pick plus the yellow card you pick is the random number, which determines the value of the bid of the other bidder.

The Other Bidder:
The other bidder is an individual situated in Room 2. The other bidder has been given the same endowment and the same instructions as you and they face exactly the same decision. They have been chosen randomly from a uniform distribution. This means every individual in Room 2 has the same probability of being paired with you for the auction. At no point will you ever discover their identity. Similarly, they will never discover your identity. In order to check your understanding of these instructions, please answer the following questions. You will not be able to proceed with the decision task until you have answered all the questions correctly. If there is any part of the instructions that need clarification then please raise your hand and someone will assist you.