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The effect of beliefs about risk aversion levels on bidding
behavior in first price sealed bid auctions
Marjon Dirksen 10673121
First supervisor: Dr. Audrey Hu
Second supervisor: Prof. dr. Joep Sonnemans.
Date: 6-12-2014
Abstract
Most articles that discuss the effect of risk aversion, look at every individual separately. They try to answer the question: what is the effect of a person’s level of risk aversion on the optimal bid they should make? The answer they found was that this approach to risk aversion does not fully explain the overbidding phenomena found in first price sealed bid auctions. To find a better fit to the data, a different approach is used in this article. Not only the personal level of risk aversion of a bidder be taken into account, but also their ideas about their opponent’s level of risk aversion. Based on the data gather during class experiments, the results are that people bid according to their personal level of risk aversion if they believe others are equally risk averse and bid higher than what they should according to their personal level of risk aversion when they believe others are more or less risk averse.
Keywords: first price sealed bid auction, risk aversion, beliefs, Holt-Laury risk aversion test
1. Introduction
In the general theory on first price sealed bid auctions, people are usually assumed to be risk neutral. This implies that they are payoff maximizers. Also, in auction experiments values are often independently and identically drawn from a uniform distribution.1 In other words, bidders are symmetric and they have the same bid functions. (Vickrey, 1961) In first price sealed bid auctions, people submit private bids and have to pay their own bid if they win (have submitted the highest bid). For the bidders to bid optimally, they have to consider the balance between risk and profit. In order to make a profit in first price auctions, you have to bid lower than your value. But the lower you bid, the less change you have at winning the auctions and so the risk increases.
In the model by Vickrey (1961), the bidders receive their private values from a uniform distribution. They have three options: bid higher, equal or lower than your received value. Bidding higher than your value, or bidding your value, is always worse than bidding lower than your value. This because by bidding higher or equal, you will never make a profit. Therefore no one should bid higher than their value or bid their exact value in a first price auction. This leads to the conclusion that in order to make a profit, people should bid lower than their value. The question is, how much lower?
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The expected payoff consists of two parts: the profit and the probability of winning. The profit you get with value 𝑣 and bid 𝑏, is 𝑣 − 𝑏. The probability that you win, is the probability that your bid is the highest bid : 𝑃𝑟(𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑏𝑖𝑑 𝑖𝑠 𝑏). This is equal to the probability of everybody else bidding lower than you. The total number of bidders in an auction is n, so everybody else but you is n-1. The expected payoff then becomes: (𝑣 − 𝑏)𝑃𝑟( 𝑛 − 1 𝑏𝑖𝑑𝑑𝑒𝑟𝑠 𝑏𝑖𝑑 ≤ 𝑏). This is the bid function is denoted as 𝛽(𝑣) for a value of 𝑣.2
(Osborne, Chapter 9: Bayesian games, 2009)
Since the values that people have for an object are normally never observed, but the bids are, an inverse bidding function is used to determine the probability of winning. So the probability that the valuations of the n-1 players have a value of at most 𝑏, is 𝛽−1(𝑏). The expected payoff then becomes (𝑣 − 𝑏)𝐻(𝛽−1(𝑏)), where 𝐻 denotes the winning probability. To maximize your expected utility, the first derivative with respect to b has to be taken and this has to be set equal to zero (Osborne, Chapter 9: Bayesian games, 2009):
−𝐻(𝛽−1(𝑏)) +(𝑣 − 𝑏)𝐻′(𝛽−1(𝑏)) 𝛽′(𝛽−1(𝑏)) = 0
By simplifying and rearranging you end up with (see 9. Appendix: Calculations bid function for complete calculations):
𝛽∗(𝑣) =∫ 𝑥𝐻′(𝑥)𝑑𝑥
𝑣 𝑣
𝐻(𝑣) for 𝑣 < 𝑣 ≤ 𝑣
Another way of writing this formula down is 𝛽(𝑣) = (𝑛 − 1/𝑛)𝑣. As you can see, the bidding a person makes should depend on their personal value but also on the number of total bidders. The more people join, the higher you should bid.3 (Osborne, Chapter 9: Bayesian games, 2009)
When people started doing auction experiments, they soon realized that bidding behavior did not match the behavior predicted by theory. One of the first experiments to research the actual behavior of people during auctions was done by Coppinger, Smith and Titus (1980). By doing lab research consisting of many sessions, they found that their outcomes did not met the existing theory. One finding was that the biddings in first price sealed bid auctions were significantly above the optimal biddings. Two years later, the article “Theory and Behavior of Single Object Auctions” was published (Cox, Roberson and Smith, 1982). In this article the finding of another lab experiment is presented in which they also find that the bidding for first price sealed bid auctions do not match the predicted biddings but are significantly higher.
Soon after these findings, many explanations for overbidding in first price sealed bid auctions were given. Risk aversion is the best known explanation for overbidding. Cox, Roberson and Smith (1982) already found that a model that includes risk aversion fitted the data better than the model that assumes risk neutrality. After that, many different models of
2 This bidding function must be an increasing function and differentiable. (Osborne, Chapter 9: Bayesian games, 2009)
3 The reason for this is straight forward. If there are values from 0 to 1, and you have a value of 0.8, you have 80 percent change that your value is highest. However, when you play against two other people, this chance becomes 0.82= 0.64. The more people enter, the smaller this change gets and thus higher you have to bid.
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risk aversion were developed4, for example the constant relative risk aversion (CRRA) model by Cox, Smith and Walker (1983). Another example is the increasing relative risk aversion (IRRA) model (Kagel et al. (1987), Smith & Walker 1993). Of both models evidence is found that they fit the data better than the model than assumes risk neutrality.
Another often mentioned reason for overbidding is ‘joy of winning’(Cox, Smith, & Walker (1992), Holt & Sherman (1994) and Andreoni, Che & Kim (2007)), This theory says that people bid higher than the risk neutral Nash equilibrium (RNNE) because they not only get utility from the profit they make, but also get utility from winning. Also the non-linear probability weighting model could help us better understand the overbidding phenomena (Dorsey & Razzolini, 2003 and Armantier & Treich, 2009). According to this model people overestimate small probabilities and underestimate large probabilities. Finally, the feedback given to the participant could be of influence. People could anticipate losers regret (loosing while your value was higher than the highest bid) and therefor bid higher (Filiz-Ozbay & Ozbay, 2007).
In all of these given explanations, the focus lies solely on the person making the biddings. Is he/she risk averse? Does he/she like winning? Is he/she making miscalculations when calculating the probabilities? For a large part, winning the object or not depends indeed on your own actions. However, you are competing against others al well, so it is not only your actions alone that matter, but also the actions of your opponents. In standard theory, it is assumed that the person you are playing against, acts as a risk neutral, value maximizing agent. However, this is probably not the case. Auctions are not static, but dynamic. Winning does not only depend on your own actions, but also on what your opponents do. When a person is bidding in an auction, he or she should form beliefs about the other players and incorporate this in the chosen strategy. There has been some research done with respect to beliefs. In Griesner, Levitan & Shubik (1967) and Maskin & Riley (1983), the assumption of symmetric beliefs is relaxed and heterogeneous beliefs with respect to values are allowed for, since they find it unconvincing that players know each other’s true values and cost. However, they did not look at beliefs about the level of risk aversion.
To better understand the bidding behavior in (first price sealed bid) auctions, a game theoretical approach should be taken: you form beliefs about the opponents and then form a best reply based on those beliefs. Risk aversion is still believed to be one of the most prominent causes for overbidding. However, not only could you be risk averse, but also the
4 In auction literature, six types of risk aversion are known. First there is absolute risk aversion The Arrow-Pratt measure of absolute risk aversion is 𝐴(𝑐) = −𝑢′′(𝑐)
𝑢′(𝑐), where c is income. There are three types of absolute risk
aversion: CARA, IARA and DARA. CARA means constant absolute risk aversion. With this type of risk aversion, the level of risk aversion is unaffected by the income. With IARA, increasing absolute risk aversions, the level of risk aversion increases as the amounts increase. So when larger sums of money are involved, people want to take less risk. Finally, DARA stand for decreasing absolute risk aversion and is the opposite of IARA: when the income level gets higher, you want to take more risk.
Besides absolute risk aversion, there is also relative risk aversion. The Arrow-Pratt-De Finetti measure of relative risk aversion is𝑅(𝑐) = 𝑐 𝐴(𝑐) = −𝑐𝑢′′(𝑐)𝑢′(𝑐) . Again, there are three types of relative risk aversion: CRRA, IRRA and DRRA. Here it is assumed that people have diminishing marginal utility of income. Now it is about percentages instead of amounts of money.
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people you are playing against. So when deciding on what to bid, people should look at their own level of risk aversion relative to the other players, not only at their own level on risk aversion. This is what this paper will focus on, by trying to answer the question: What is the effect of the beliefs about the level of risk aversion of the opponents on the biddings in a first price sealed bid auction?
The answer to this question has not yet been given. I therefor believe it is an underexposed aspect of the effect of risk aversion in auctions. Answering this question could help to better understand the biddings made in first price sealed bid auctions and dissolve an extra factor in the overbidding phenomena. This will be done by examining the data collected from class experiments, where the participants take part in a first price sealed bid auction. Their bids will are compared to that what they should have bid according to the optimal bid function, assuming CRRA. Also, they are ask to fill in a Holt-Laury risk aversion test5(Holt & Laury, 2002) to measure their level of risk aversion and are asked if they believe their opponents are more, equal or less risk averse than they themselves are.
The results show that the bids made when people believe others to be equally risk averse, are almost equal the bids they should have made according to their own level of risk aversion. However, this result is not significant. If people believe others are more or if others are less risk averse, they bid significantly more than they should have according to their personal level of risk aversion. This is what was expected in the case where people believed others are more risk averse. However, it was not expected for when people believed others to be less risk averse.
I will now first give an overview of the existing literate on auction theory, with the focus on risk aversion and the theory on beliefs. Then the model that is used for this research will be presented. Chapter 4 will elaborate on the hypotheses and in chapter 5 the experimental design will be explained, followed by a data description in chapter 6. Then the results will be presented in chapter 7. Finally, a conclusion will be given in chapter 9.
2. Theoretical overview
Auction theory has had the interest of many researchers for many years. Therefore, there is much literature to find on this subject. There has been overviews written about the most important articles, for example “Auctions: a survey of experimental research” by John Kagel (1995), (Kagel, Auctions: A Survey Of Experimental Research, 1995) which has been updated over the years with the latest version “Auctions: A Survey of Experimental Research, 1995 – 2010” by John H. Kagel and Dan Levin (2011). Also theoretical overviews are written, for example “Auction Theory: A Guide to the Literature“, by P Klemperer (2004).
In these overviews all four main type of auctions are discussed: English (open ascending) auction, Dutch (open descending) auction, first price sealed bid auction and second price sealed bid auction. A distinction is made between private value auctions, where everybody
5 5 In a Holt-Laury test, people have to make 10 decisions. At every decision, they have to choose between two lotteries: a lottery with a low risk and a lottery with a high risk. In the low risk lottery, the difference between payoffs is low. In the high risk lottery, this difference is larger. However, in the high risk lottery, you can earn more money. For an example of the Holt-Laury test, see 14. Appendix: Questionnaires.
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values the object different and the value other people attach to it has no influence, and common value auctions, where the valuation of others do influence your personal value of the object. Finally, a distinction is also made between single unit auctions, where there is one object for sale, and multi-unit auctions, where several object are for sale.
This paper focusses on first price sealed bid; private value auctions where there is only one unit for sale. Therefore, only the theory relevant for this type of auction will be presented.
Someone who is very important for the theory on first price sealed bid auctions is William Vickrey. In his article “Counterspeculation, Auctions, and Competitive Sealed Tenders” (1961) , he derived the very first auction model for the optimal bidding strategies and found the RNNE. He found this under the assumption that the values were uniformly distributed. He also found evidence for the revenue equivalence theorem.6 About twenty years later, new contributions were made with respect to the revenue equivalence theory. Before that time, the revenue equivalence theory was proved by Vickrey only for the case with two uniformly distributed buyers. Riley and Samualson (1981) found that the revenue equivalence holds more generally, but still under the assumption of risk neutrality. This was also found by the Myerson (1981).
Riley and Samualson (1981) did not only look at the revenue equivalence theorem, but also looked at the best auction design a seller can use with respect to profit optimization. They proved that the seller could use either two of the auctions (Dutch or English), but with a reserve price. However, in 1997, McAfee and Vincent looked beyond the static case and found that this is not always true. They noticed that it is not that easy for a seller to make a ‘Take it or leave it’ statement credible. There will always be an option to resell. Based on this idea, they find that when the time in between the auctions goes to zero, the seller will make the same profit as when he would be in a static situation with an auction without a reserve price.
Even though the auctions should produce the same revenue, sometimes different type of auctions are used for different type of products. For example English auctions are used for art, but job contracts are distributed through sealed bid auctions. In Maskin and Riley (2000a) an explanation is given for this fact. Originally revenue equivalence was proved between the English and Dutch auctions and first and second price sealed bid auctions. In Maskin and Riley (2000a) the reason for the use of different auctions for different products is explained by asymmetry7, which causes the revenue equivalence to no longer hold. Cheng (2006) discusses another reason for the difference in usage between a sealed bid and open auction. He states that the Getty effect could be the reason.8
After Vickrey published his article in 1961, there were no other large contributions for a long time. Only in the end of 1970 auction theory really became a subject that was widely researched. Many articles with many different subjects were published; for instance asymmetric auctions. In his well-known article ‘Counterspeculation, Auctions and Competitive Sealed Tenders’ (1961), Vickrey started to look at what would happen if you
6The revenue equivalence theorem states that the revenue of an English (second price) auction and a Dutch (first price) auction are the same if you assume people are risk neutral.
7 They believed these ex ante asymmetries are caused by the budget constraints of the bidders. If differences in budgets are common knowledge, beliefs become asymmetric.
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would have an asymmetric situation. He ask himself the question what would happen in a two bidder situation, if one person knew that the other persons value was between 0 and 1, but the other player knew the exact value of his opponent? He found that in this case it is sometimes advantageous and sometimes disadvantageous to choose for a first price auction for the seller (Vickrey, 1961). Griesmer et al. (1967) continued with the work by Vickrey (1961) and derived the equilibrium bidding strategies in case of an asymmetric situation (Griesmer, Levitan, & Shubik, 1967). Plum (1992) used weaker assumptions and derived bidding strategies for an auction with more than two bidders. Marshall et al. (1994) looked at what would happen if bidders were to form a coalition (and therefore creating heterogeneous bidders) and formed an algorithm for this situation. First they looked at the case where a few buyers form a coalition and the other a counter coalition, which resembles an asymmetric two player situation. Then he also looked at what would happen if no counter coalition was made. Their algorithm was supported by their data. Maskin and Riley (2000b) found a monotonic equilibrium for first price sealed bid auctions where the types are asymmetrically distributed. Before then the equilibrium was only found for when types are symmetrically distributed.
Another area of auction theory is the effect of budget constraints. Che and Gale (1998) looked at the effect of budget constraints on the revenue of the first and second price auction. They conclude that first price auctions provide more revenue and larger social surplus when budget constraints are present. This result holds for multiple types of auctions, for instance all pay or auctions with an entry fee.
Another main branch of auction theory is the inclusion of the option to resale. As already mentioned, McAfee and Vincent (1997) showed that profit goes down when you include an option for resale. The effect of resale for the buyers is analyzed in Gupta and Lebrun (1999). They look at a case with asymmetric buyers and find the equilibrium bidding strategies. (Gupta & Leburn, 1999) Haile (2003) also looked at the effect of resale. He stated that the resale option made the valuations no longer solely private, but that there was now also a common value effect. Revenue equivalence could still hold in an auction with resale, only it can fail as well if there is information asymmetry in the secondary market.
The existence of an option to resale also had another effect. People whose only desire it is to make a profit, now bid on an object, only with the intention to resale it with a profit. These speculators have no private value for the object themselves. The effect of these speculators is discussed in Garrat and Troger (2006). They find that in English and second price auctions the speculators can make profits. However, for first price and Dutch auctions, speculators will not make profits. Halifar and Krishna (2008) formulate a two bidder model with resale and independent values (may or may not be asymmetric) with which they can compute the revenue ranking of first price and second price auctions.
Risk aversion
One of the main assumptions of standard auction theory is risk neutrality. However, research has shown that people tend to be risk averse. In the 1980’s this was introduced in auction theory. Coppinger, Smith and Titus (1980) were one of the first to find that people bid higher than the RNNE prediction in first price sealed bid auctions. This finding was supported by another lab experiment done by Cox, Roberson and Smith (1982)
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Holt (1980) showed mathematically that the revenue equivalence theorem does not hold when people are risk averse, but that in this case the profit of the seller is larger with a Dutch or first price auction. Riley and Samualson (1981) also come to this conclusion in their paper ‘Optimal Auctions’.9
Maskin and Riley (1984) use this result and design an optimal auction for (risk neutral) sellers which maximized their revenue when they are facing risk averse bidders.
Matthews (1987) looked at the effect of risk aversion for a buyer, instead of for the seller. He wanted to know what type of auction buyers prefer if they have a specific type of risk aversion. He looked at first price auctions, second price auctions and to first price auction where the number of competitors would be revealed. His finding is that in the case of increasing absolute risk aversion, first price auction is preferred to the revealed first price auction to the second price auction. When a person has decreasing absolute risk aversion it is the opposite. In the case of constant absolute risk aversion, players should be indifferent between all the auctions (Matthews, 1987).
In 1983, Cox, Smith and Walker, developed a model that allows for bidders to have individual levels of risk preferences. So one person could be more risk averse than the other. They assumed that people have a constant relative risk averse ulility function with respect to money and that bidders have the same probability expectations. They called this model the constant relative risk aversion model (CRRA model). With CRRA, the utility people get from money decreases with a constant rate as they get more money. In the article the focus lies on first price auctions. To test their model, they run multiple auction rounds and triple the payoff (by tripling the conversion rate). They find evidence that supports their model and it provides a better fit than the RNNE. In 1988, they published another paper about the CRRA model. On many aspects, the results again match their model better than the RNNE model. First, they found that many people bid as if risk averse, but there were also some that bid as if risk neutral or risk loving. Second, in many cases, the hypothesis that there was a common bid function was rejected. They did find a linear relation between the bidding and the values. Both of these results are in favor of the CRRA model. Tripling payoff again had no effect on bidding behavior. So they found some evidence that supported their model (Cox, Smith, & Walker, 1988).
Not everybody agreed with the success of the CRRA model. Harrison (1989) believed that the deviations from the RNNE were so small, that the player lost very little by deviating. Another point of critique came from that fact that other researchers found evidence of IRRA instead of CRRA. Based on a 5% significance, the CRRA for individual bidding function is rejected for 43 percent of the sample used by Kagel and Levin (1985). A reason they give is that instead of only multiplying the profits, they also stretched the values. When only the profits were enlarged, people did not have to change their strategy in order to remain faithful
9 The reasoning of this finding is quite intuitive. In an English auction, you have a lot of information about the values and bidding of the other players and you just bid a little bit more than the one with the second highest value. For a second price sealed bid auction, you know that your personal bid has no effect on the price you have to pay. However, for a Dutch auction or a first price auction, there is much more uncertainty. In a Dutch auction, the longer you wait the more profit you will get but the risk of losing also increase. For a first price sealed bid auction holds the same. The lower you bid, the more profit you make but the change someone else bids more that you also increase. So because the uncertainties are higher for those auctions, risk averse people will tend to bid higher in those auctions to minimize the risk.
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to the CRRA model. But now, when the values were stretched they did had to change their bidding behavior. They give two reasons for why the model could fail: People have IRRA instead of CRRA, or difference between objective and subjective estimation of the distribution of the private values of the other participants are the actual cause of what is usually assigned to the failing of the risk neutrality hypothesis.
Two years later, when researching affiliated private value auctions, Kagel et al. (1987) found that the CRRA model could explain the biddings for higher values well, only not for the lower values. In order to let the model explain all the biddings, an IRRA model should be used.
Kagel and Roth (1992) and Smith and Walker (1993) follow up on the critique made by Harrison (1989) and found that the bidding behavior changes when the stakes get higher. When they increased the profits people would earn if they won the auctions, there were fewer errors and the outcomes were closer to the rational models (RNNE model). This supports the notion that small costs of deviating leads to more errors.
When you sum up the respondes the CRRA model has received, you can see it got much critique. However, it is the most commonly used utility fucntion.(Holt & Laury, 2002)
Beliefs
Compared to the literature on risk aversion, there is a lot less literature on beliefs. Only few articles have discussed the effect of beliefs on behavior in auctions. Even though the subject is not heavily researched, the first time beliefs were discussed were rather early.
In 1976, Griesner, Levitan and Shubik were the first to relax the assumption of symmetric beliefs. They thought it was not probable that player knew the true costs and values of other bidders. They solved the two player model for first price sealed bid auctions while relaxing the symmetric beliefs assumption. (Griesner, Levitan, & Shubik, 1976)
In 1983, Maskin and Riley formed a model with weak assumptions about preferences and applied that to the n-player case only then with symmetric beliefs (Maskin & Riley, 1983). In ‘Auctions with asymmetric beliefs’, Maskin and Riley (1983) continued with their research, but now relaxing the assumption of symmetric beliefs (with respect to values). They developed the optimal bidding functions and the expected profits for first price sealed bid and second price sealed bid auctions. They found that in most cases, the profit is higher with the first price sealed bid auctions compared to second price when beliefs are asymmetric and uniform.
Crawford and Iriberry (2007) use a level-k model and drop the equilibrium assumption. The k-level model is based upon the following idea: a zero level individual has no particular strategy and chooses a bid at random, a one level player best responds to his beliefs, if he believes everybody else is a zero level player. So a level k player believes the rest is K-1 level. A level-k model allows for heterogeneous behavior, but with behavior drawn from a common distribution. Their result is that their model explains the winner’s curse (the winning bid is always to high) for common value auctions and overbidding in first price auctions with non-uniform value distribution better than other models. For the first price auctions with uniform values their model does not add anything to the existing explanation for overbidding. Ivanov and Levin (2010) use the standard game theoretical approach to common auctions: you form beliefs about the other player, you find a best reply to those beliefs and the beliefs of
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a person matches their actions. If people would do this correctly, you should have the predicted Nash equilibrium. Since this does not happen in common auctions, they relax the first assumption and allow for inconsistent beliefs. They find that it is not just inconsistent beliefs is what causes the Winner’s curse, since the winner’s curse still exist when allowing for inconsistent beliefs.
3. Model
The auction design used for this research is a first price sealed bid auction with a single object for sale. In this auction people make anonymous bids on one single object.10 The auction is won by the person who submitted the highest bid. He/she then has to pay his/her personal bid. The values are independently and identically drawn from a uniform distribution. If the assumption of risk neutrality is also made, the optimal bidding function is:
𝐵(𝑣) =𝑛 − 1 𝑛 𝑣
(1) In this article the assumption of risk neutrality is dropped. This because previous research has shown that the risk neutral model does not explain the observed behavior in experiments. Here I will assume CRRA because this is the model that is chosen most often. (Holt & Laury, 2002)
To incorporate risk aversion to the model, it has to be adapted. The utility function that describes the risk neutral model is linear. In this case the expected utility obtained from a certain bid increases constant as the biddings increase. The expected utility function for the risk neutral case is: (𝑣 − 𝑏)𝑃𝑟( 𝑛 − 1 𝑏𝑖𝑑𝑑𝑒𝑟𝑠 𝑏𝑖𝑑 ≤ 𝑏). This is however a specific case of a more general function. Normally the expected utility function is a power function, and the only situation where it is linear is when people are risk neutral. The power function which allows for various levels of risk aversion, has an expected utility of (𝑣 − 𝑏)1−𝑟 𝑃𝑟( 𝑛 − 1 𝑏𝑖𝑑𝑑𝑒𝑟𝑠 𝑏𝑖𝑑 ≤ 𝑏). The higher the value of r, the higher the level of risk aversion.11
When the values are drawn from a uniform distribution the probability of winning is the probability your bid is higher than all the other bids. If 𝑚 would be the highest possible value and you are playing against one other person, the probability your bid is the highest bid is 𝑏/𝑚. When more people bid, the formula changes. To generalize this formula to a multiple player case, 𝑏/𝑚 must be raised to the power of 𝑛 − 1. So if 𝑃𝑟( 𝑛 − 1 𝑏𝑖𝑑𝑑𝑒𝑟𝑠 𝑏𝑖𝑑 ≤ 𝑏) is replaced by (𝑏/𝑚)𝑛−1, the expected utility function becomes (𝑣 − 𝑏)1−𝑟(𝑏/𝑚)𝑛−1. In order to find the optimal bid, the first derivative should be taken with respect to b, just like the risk neutral case: (𝑣 − 𝑏)1−𝑟(𝑛 − 1)(𝑏 𝑚)𝑛−2 1 𝑚− (1 − 𝑟)(𝑣 − 𝑏)−𝑟( 𝑏 𝑚)𝑛−1 = 0 Rewriting this gives:
10 They do this by submitting their bids on a piece of paper and then handing it over to the auctioneer. This way the bidders only know what their personal bid is and not the bids of the other participants.
11 When 𝑟 = 0 (risk aversion level is zero), you get the same formula as we did before and you thus get the risk neutral model.
10 (𝑣 − 𝑏)1−𝑟(𝑛 − 1)1
𝑚= (1 − 𝑟)(𝑣 − 𝑏)−𝑟 𝑏 𝑚 Multipling both sides by 𝑚 and dividing by (𝑣 − 𝑏)−𝑟 gives:
(1 − 𝑟)𝑏 =(𝑣 − 𝑏)1−𝑟
(𝑣 − 𝑏)−𝑟 (𝑛 − 1) Dividing by 𝑏 and 𝑛 − 1 and simplifying gives
1 − 𝑟 𝑛 − 1=
𝑣 − 𝑏 𝑏
Solving this for b then gives the optimal bid function (see 10. Appendix: Calculations bid function with risk aversion for the extensive calculations):
𝑏 =𝑛 − 1 𝑛 − 𝑟𝑣
(2)
4. Hypotheses
Since not much research has been done on this subject, the hypotheses are based on common sense rather than previous evidence. Risk aversion has been named as an explanation for the found behavior in first price sealed bid auctions, which is that people bid more as compared to their optimal bidding. The intuition behind this is that people have to choose between higher profits and a higher change of winning. When people are risk averse, they would want to avoid the risk of losing. This is done by bidding higher, with the results that they earn less. If people believe that their opponents are just as risk averse as they are, then there does not change anything compared to the static case. Then the biddings should be the same as when homogeneity of risk attitudes is assumed. In other words, if people believe everybody has the same level of risk aversion, they should just bid according to their personal level of risk aversion.
Hypothesis 1: If people believe they are equally risk averse as their competitors, they should bid according to their personal level of risk aversion.
When people do not expect the other to have the same level of risk aversion, it could either be that they believe theirs is lower or theirs is higher. In the first case, they thus believe that others are more risk averse than they themselves are. If this is the case, then they should expect the others to bid higher than you yourself would for a specific value. This means that the chance of winning decreases for the same bid, so:
Hypothesis 2: If people believe they are less risk averse than their competitors, they should bid more than they should have based on their personal level of risk aversion.
When people expect others to be less risk averse than they themselves are, they should do the opposite. Now you think other will bid lower, so you yourself can also bid lower and keep the same chance of winning, so:
11
Hypothesis 3: If people believe they are more risk averse than their competitors, they should bid less than they should have based on their personal level of risk aversion.
5. Experimental design
To collect the data, classroom experiments were run. In these experiments, the bidders were in groups of three. This means that they were bidding against two other persons. Their optimal bidding function then becomes:
𝑏 = 2
3 − 𝑟𝑣
(3)
To make every round independent of each other, the composition of bid-groups differs between rounds. To accomplish this, the classroom was split up in groups of 6 so that two bid-groups of three could be made in every round.12
To make sure that the only thing that is different between observations, is the level of risk aversion of the participants, 6 different values set were drawn from a random distribution. Every group of six thus had the same value set. For this reason the groups are small. This way I have the most observation per value set.
To raise the students’ awareness level of their bidding behavior, each of them received an endowment in terms of points.13 One point equaled 2 euro cents.
In total there were six rounds, of which two were paid out. To make sure people could not end up having to pay the auctioneer money, the endowment had to equal the maximum amount people could bid in two rounds. This is why everybody started with an endowment of 200 point and values were drawn from a random distribution from 0 to 100 points. Also, they were not allowed to bid more than the maximum amount of points, so 100 points.
In every round, the participants received a note which stated their group number and their value. The participants then had to write down their personnel letter and their bidding.
Before the auction rounds started, the participants first had to fill in a questionnaire in which they had to do a Holt-Laury risk aversion test (Holt & Laury, 2002) and state if they believed their classmates were more, equal or less risk averse then they themselves are. (See 14. Appendix: Questionnaires). By letting the students take this test, their levels of risk aversion and also their beliefs become known.
After the questionnaire, the auction was held. Then when all auction rounds were played, the winning bid for every group and for every round was announced. This way they knew if they had won an auction. Finally, they had to fill in a questionnaire where they could say if they still had the same beliefs about the level of risk aversion of their opponents.
12 If it was the case that the class could not be split up in groups of six, then I chose another grouping so that I could still rotate the people that were in a group together.
13 The reason people get an endowment, is to make sure people do not bid extremely high. Now they would lose their endowment if they were to bid higher than their value. Usually the endowment is an amount of money. However, there is a downside to giving the participants an endowment. It could be that the participants now feel richer and this can alter their bidding behavior. To avoid this negative effect of giving an endowment, the participants received an endowment expressed in point rather than money.
12
To make sure the participants took the questionnaires as well as the auctions serious, some students were selected for payment. For the payout of the auction round three students per class were selected and for the questionnaire two students (not the same as who were already selected for payment of the auction round) were selected. These students were randomly selected by drawing letters from a bag. These letters resembles the personnel letters the students receives at the beginning of the auction.
The participants received instructions saying what was expected of them during the experiment. These instructions are in 12. Appendix: General instructions.
Discussion of experimental design
This experiment was conducted at a high school with students aging from 14 to 17. Normally in experiments, University students are used. This difference in the sample has both positive and negative sides.
First of all, the students had no prior knowledge of auction theory. This leaves them with a more open mind. If this type of research is conducted at a university, the chance exists they have had some auction theory during a course. This can influence their behavior while they are in the experiment because now they want to give the ‘right’ answers rather than what they actually want to answer. The negative side of this, is that high school students found it difficult to understand the process; for example the Holt-Laury test. Furthermore they were very unfamiliar with the concept of auctions.
The experiment was conducted in class rooms instead of an experimental lab. This made it harder to make sure the students did not talk at all. Also, because they were not used to participating in an experiment, they probably were more excited than normally is the case.
To get the subjects to take the auction serious, payments were handed out. However due to financial constraints these were not handed out to everybody. A standard for the amount of payment is usually what they could earn if they spent the time to work instead of the experiment. Because these students were rather young, the average payment of 4 euro’s should have been enough.
To make sure students could not walk away with less money than with what they came in, the endowment was just as much as the total amount they could bid. This enabled students to earn a total of 2 euros of profit per round if their value was 100 and their offer 0. So even though it looked like a lot when it is expressed in points, the profits per round were very often not more than a few cents. This could make them take the auction less seriously.
The experiment was conducted in 4 different classes. For each class there was a total of 50 minutes available. In every case I was barely able to finish the experiment within this time frame. So a final positive point is that as much rounds as possible were played.
6. Data
The participants were students from a high school in Dronten. The students were in the classes 4HAVO, 4VWO and 5VWO.14 These are the two highest levels of education. The
14 Because the research was conducted during the final exam period, 5HAVO and 6VWO could not participate. See 15. Appendix: Dutch high school system for an overview of the Dutch schooling system.
13
total number of participants was 77. Little more than half (50,67 %) of the sample is female. The age of the participants ranged from 14 to 17, with the mean around 16.
The Holt-Laury test was not understood by everybody. In a Holt-Laury test, the chance to win the highest payoff of the lottery increases with every decision. In the first decision, the chance to win the highest payoff is only 10%. At the final decision, decision 10, you win the highest payoff for sure. So somewhere people should switch from choosing the saver to the riskier lottery. The point where they switch determines their level of risk aversion. In my sample, some students switched several times instead of only once. Some even chose the saver lottery when you would win the highest payoff for sure. In this case they chose a €2,- for sure over a €3,85 for sure.People who switched more often are dropped from the data set. A total 53 observations are left. 15
Of the 53 students who are left, most of them chose 4, 5 or 6 safe choices. This means that most of them are either risk neutral or (slightly) risk averse. In Table 1 below all the percentages are presented. 16
Table 1: Risk aversion classification
Number of safe choices
Range of relative
risk aversion Value of r
Risk preference
classification Number Percentage
0 and 1 r ≤ -0,95 -0,95 highly risk loving 1 1,9%
2 -0,94 ≤ r ≥ -0,49 -0,72 very risk loving 2 3,8%
3 -0,48 ≤ r ≥ -0,15 -0,32 risk loving 2 3,8%
4 -0,14 ≤ r ≥ 0,14 0 risk neutral 12 22,6%
5 0,15 ≤ r ≥ 0,41 0,28 slightly risk averse 14 26,4%
6 0,42 ≤ r ≥ 0,67 0,55 risk averse 13 24,5%
7 0,68 ≤ r ≥ 0,97 0,83 very risk averse 5 9,4%
8 0,98 ≤ r ≥ 1,36 1,17 highly risk averse 2 3,8%
9 and 10 1,37 ≤ r 1,37 stay in bed 2 3,8%
Total 53 100%
In order to use the level of risk aversion of the participants given by the Holt-Laury test, To be able to fill in a level of risk aversion, in the bid function, one value of r per number of safe choices is needed instead of a range of values. For this reason the value in the middle is the value that will be used for the analyses. For the cases where people chose 0 or 1 and 9 or 10 safe choices, the largest and smallest value will be used respectively.
15 In their research, Holt and Laury (2002) did keep in those who switched more than once. They assumed these were errors and they found a clear division line between clusters of safe and risky choices. They therefore decided to also include these observations and only look at the total number of safe choices. Also, their results would not have differed much if these observations were dropped. I however believe that this behavior indicates the students did not understand the Holt-Laury test. I am not able to see a clear switching point. This is why the results for students who switched more than once will not be used. This will leave too much room for error. For this reason I decided to drop these observations.
16 In the article by Holt and Laury (2002), they do not assume people exhibit constant relative risk aversion. Here however, this is assumed. For this reason, the values of r that match the number of safe choices are recalculated. This way the value of the r’s represent the range of CRRA for U= x1-r, instead of the range of RRA
14
The participants received their personal value from the auctioneer. These values were drawn from a random distribution ranging from 0 to 100 (http://www.random.org/). In Table 2 the drawn values are presented. Most biddings made by the students were below the value that was given. Only in two cases the average lies above the value, this is for group 4 in round 4 and 5. Of the first three groups, there is a total of 9 observations per group. For group 4 and 5 there are 8 observations and the final group is the largest group with a total of 10 observations.
Table 2: Overview of the values and average biddings
Round 1 Round2 Round 3 Round 4 Round 5 Round 6
Value Actual bid Value Actual bid Value Actual bid Value Actual bid Value Actual bid Value Actual bid 1 88 72,88 81 68,56 3 1,75 66 53,78 68 57,89 42 37 2 2 1 54 41,11 68 42,33 97 62,11 91 61,55 88 60,56 3 32 27,78 89 63,13 96 76,44 58 43,89 80 65,33 21 16 4 67 46,25 74 58,63 34 22,5 31 35 2 12,8 36 37,43 5 54 45,75 87 73,25 73 57,13 72 61,75 100 78,25 8 5,75 6 18 13,5 58 46,6 68 60,8 98 77,9 31 24,8 63 53,8
To the question if you believe your classmates are more, equal or less risk averse than you are, 13 students answered more, 11 answered less and most of the students, 28, said that they believed that others are just as risk averse as they themselves are.
7. Results
In this section the answers to the hypotheses will be presented by analyzing the statistical results from the experiment. First I will present the results for the entire sample, after which I will discuss the results for every group separately.
When analyzing the results, the Bonferroni correction will be used. The Bonferroni correction is necessary when you have multiple hypotheses for the same data set. Here I have three different hypotheses so this correction is necessary. The problem with multiple hypotheses, is that there is a chance of finding a significant result while it is in fact not significant. In this case, when using a 5 % significance level and with three hypotheses, this chance becomes: 𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑟𝑒𝑠𝑢𝑙𝑡 = 1 – 𝑃(𝑛𝑜 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑟𝑒𝑠𝑢𝑙𝑡𝑠 = 1 – (1 − 0.05)3 = 14.26%. To correct for the multiple hypotheses effect, the critical p-values are divided by the number of hypotheses. So results are significant if the p p-values are smaller than 0.05/3= 0.017 (based on the 5% significance level)17 (Altman & Bland, 1995). Total group analyses
For the first hypothesis, where people believe others are equally risk averse, the actual bid should be equal to the optimal bid according to their personal level of risk aversion. Looking at Table 3, you can see that the difference between the two means is quite small, 0.18 point.
17 For a significance level of 1 % the value becomes 0.01/3≈0.003. For a significance level of 10% the value becomes 0.1/3≈0.033
15
However, when it is tested in Stata the p value of mean=0 is 0.5557. So hypothesis 1 is not supported by the data.
With the second hypothesis, people believe others are more risk averse. With this hypothesis people should bid higher as compared to the optimal bid according to their personal level of risk aversion. Now the actual bid is almost 6 points higher than the optimal bid. And with a p-value of 0.0058 this result is significant on a 5 % level (0.0058<0.017).
The third hypothesis states that people believe that others are less risk averse. Now people should bid lower as compared to their optimal bid. This is however not the case. The actual bids are again higher than the optimal bid. It is even significantly larger with a p value of 0.0281 on a 10 percent level. So no support is found for the third hypothesis.
Table 3: Average actual bid and optimal bid
Average value actual bid Average value optimal bid Difference
Hypothesis 1: Actual bid = Optimal bid 45.30723 45.48940 -0.18217 Hypothesis 2: Actual bid > Optimal bid 50.42466 44.43397 5.99069** Hypothesis 3: Actual bid < Optimal bid 47.29231 43.98439 3.30792
* significant at 10% level ** significant at 5% level *** significant at 1% level
Separate group analysis
With the separate group analyses, all that is different between observations is their risk aversion and their ideas about other’s level of risk aversion. In Table 4 the average actual bids and optimal bids per group are presented.
With the first group, two people said that they believed others were equally risk averse. So for the first hypothesis, there are 12 observations. With a p value of 0.7682, the first hypothesis is rejected for group 1. For the second hypothesis, the p value that corresponds to the mean being larger than 0, that is the actual bid is larger than the optimal bid is 0.0000. Therefore, the second hypothesis is supported by group 1 (based on 23 observations18). For the third hypothesis, there are 17 observations. Just like with the group as a whole, people bid higher as compared to their optimal bid based on their personal level of risk aversion. Also, the p value for actual bid is smaller than the optimal bid is 0.986. So here hypothesis 3 is rejected.
The second group has made very different biddings. Here the average actual bids are smaller than the optimal actual bids for every hypothesis. The first hypothesis is supported based on a 10 percent significance level with a p value of 0.0258 and 42 observations. For the second hypothesis however, the p value is 0.9974. So the second hypothesis is not supported. The third hypothesis has a p value 0.0042 and thus is supported on a 5% significance level. The second and third hypothesis sample for group 2 both consisted of 6 observations.
16
Table 4: Average true bid and optimal bid per group
Group Average value actual bid
Average value optimal bid
Difference Hypothesis 1: Actual bid =
Optimal bid 1 2 49.17 46.86 47.28 53.94 1.88 -7.08*
3 43.83 46.34 -2.50
4 32.41 31.23 1.18
5 50.67 46.92 3.75
6 45.47 42.69 2.78
Hypothesis 2: Actual bid >
Optimal bid 1 2 51.13 51.83 39.03 61.44 12.10*** -9.61
3 53.18 63.12 -9.94
4 41.76 35.72 6.04
5 62.50 53.61 8.90**
6 59.50 37.33 22.17**
Hypothesis 3: Actual bid <
Optimal bid 1 2 46.24 23.17 40.32 49.02 5.92 -25.85** 3 49.83 41.61 8.22 4a . . . 5 62.67 60.52 2.15 6 48.08 43.45 4.64 a
There were no observations for this group
* significant at 10% level ** significant at 5% level *** significant at 1% level
The third group seems to have bid somewhat the opposite of the entire group. For the first hypothesis the difference is -2.5 and the p value is 0.4482. Therefore the first hypothesis is rejected. With the second hypothesis the average actual bids are smaller than the optimal bids and for the third hypothesis the actual bids are larger. This is the opposite of what the hypotheses tells us. And with p values of 0.997 and 0.9948 respectively, both hypothesis are rejected. The sample size is 18 for the first, 11 for the second and 24 for the third hypothesis. With the fourth group, only the first and second hypothesis can be tested. With a sample of 22, and a p value of 0.6091 the first hypothesis is rejected. The second hypothesis is also rejected with a p-value of 0.2444 and an sample size of 21.
With the fifth group, the average actual biddings are higher with every hypothesis. Therefore only the second hypothesis is supported with a p value of 0.0066. The first hypothesis is rejected with a p value of 0.1017 and the third with a p value of 0.977. The sample sizes are 36.6 and 6 for the first, second and third hypothesis respectively.
Just like with the fifth group, in the sixth group only for the second hypothesis support is found on a 5% significance level ( with a sample of 6 observations). The first hypothesis has a p value of 0.0414 and a sample size of 36. The third hypothesis has 12 observations and a p value of 0.9771.
17
8. Conclusion
The main question of this paper is: What is the effect of the beliefs about the level of risk aversion of the opponents on the biddings in a first price sealed bid auction? To answer this question, a class experiment was conducted. With this experiment I was able to test three different hypotheses: 1) If people believe they are equally risk averse as their competitors, they should bid according to their personal level of risk aversion, 2) If people believe they are less risk averse than their competitors, they should bid more than they should have based on their personal level of risk aversion and 3) If people believe they are more risk averse than their competitors, they should bid less than they should have based on their personal level of risk aversion.
With the first hypothesis, the difference between the actual bid and the optimal bid was small, but the hypothesis is not supported by the data based on a 5% significance level, and also not on a 10% significance level. The second hypothesis does find support, at the entire group level and also in some groups in the separate group analysis. The final hypothesis is only supported by one group in the separate group analysis.
Based on these results, I can state that there is an effect of beliefs on the bidding behavior. The effect of believing others to be more risk averse, gives the expected result, the effect of believing others to be less risk averse does not. This final result will need some further research before it can be explained .
The subject discussed in this paper is something that is not widely discussed in the existing literature so far, so future research should focus on the subject discussed in this paper. When this happens, it is important to conduct the experiment in a proper lab and make sure that the profit they can make is larger than it was during this experiment. Also it could be wise to use students who have some knowledge of auctions.
18
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9. Appendix: Calculations bid function
Bid function of player 𝑖 with value 𝑣 is 𝛽𝑖(𝑣). Because of risk neutrality assumption (or symmetric Nash equilibrium) everybody has the same bid function, so 𝛽𝑖(𝑣) = 𝛽(𝑣)
The expected payoff consist of the profit you get when you win and the probability of winning, which is the probability that your bid id the highest bid, which is equal to the probability the rest bids lower
(𝑣 − 𝑏) Pr(𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑏𝑖𝑑 𝑖𝑠 𝑏) = (𝑣 − 𝑏) Pr(𝐴𝑙𝑙 𝑛 − 1 𝑜𝑡ℎ𝑒𝑟 𝑏𝑖𝑑𝑠 ≤ 𝑏) Which is
(𝑣 − 𝑏)𝐻(𝛽−1(𝑏)) if 𝐵(𝑣) ≤ 𝑏 ≤ 𝐵(𝑣) To optimize the function, the first derivative is taken with respect to b:
−𝐻(𝛽−1(𝑏)) +(𝑣 − 𝑏)𝐻′(𝛽−1(𝑏)) 𝛽′(𝛽−1(𝑏)) = 0
It is always optimal to bid according to your value, so 𝑏 = 𝛽(𝑣). Because you do not get to see the value of the other players, but you do get to see their bids, you can use the inverse function to calculate the value based on the bid they give. So 𝛽−1(𝑏) = 𝑣
If 𝛽−1(𝑏) is replaced by 𝑣 you get:
−𝐻(𝑣) +(𝑣 − 𝑏)𝐻′(𝑣) 𝛽′(𝑣) = 0 Times 𝛽′(𝑣)
−𝐻(𝑣)𝛽′(𝑣) + (𝑣 − 𝑏)𝐻′(𝑣) = 0 As mentioned 𝑏 = 𝛽(𝑣), so replacing 𝑏 by 𝛽(𝑣) gives:
−𝐻(𝑣)𝛽′(𝑣) + (𝑣 − 𝛽(𝑣))𝐻′(𝑣) = 0 −𝐻(𝑣)𝛽′(𝑣) + 𝑣𝐻′(𝑣) − 𝛽(𝑣)𝐻′(𝑣) = 0 𝑣𝐻′(𝑣) = 𝐻(𝑣)𝛽′(𝑣) + 𝛽(𝑣)𝐻′(𝑣) for 𝑣 < 𝑣 ≤ 𝑣
The right-hand side of this equation is the derivative of 𝛽(𝑣)𝐻(𝑣) with respect to 𝑣. So for a constant 𝐶 you get:
𝛽(𝑣)𝐻(𝑣) = ∫ 𝑥𝑣𝑣 𝐻′(𝑥)𝑑𝑥 + 𝐶 for 𝑣 < 𝑣 ≤ 𝑣
The bid function is a bounded function, so as 𝑣 approaches 𝑣, you can deduce that 𝐶 = 0. This leads to a bid function:
𝛽∗(𝑣) =∫ 𝑥 𝑣
𝑣 𝐻′(𝑥)𝑑𝑥 𝐻(𝑣)
(3)
22
10. Appendix: Calculations bid function with risk aversion
For an auction with n players and where there is allowed for risk aversion, the expected utility is:
(𝑣 − 𝑏)1−𝑟(𝑏 𝑚)𝑛−1 To optimize this function, the first derivative has to be taken:
(1 − 𝑟)(𝑣 − 𝑏)−𝑟(𝑏
𝑚)𝑛−1+ (𝑣 − 𝑏)1−𝑟(𝑛 − 1)( 𝑏
𝑚)𝑛−2 = 0 Rewriting this gives:
−(1 − 𝑟)(𝑣 − 𝑏)−𝑟(𝑏 𝑚)𝑛−1+ (𝑣 − 𝑏)1−𝑟(𝑛 − 1)( 𝑏 𝑚)𝑛−1𝑏−1= 0 −(1 − 𝑟)(𝑣 − 𝑏)−𝑟+ (𝑣 − 𝑏)1−𝑟(𝑛 − 1)𝑏−1= 0 −(1 − 𝑟) + (𝑣 − 𝑏)(𝑛 − 1)𝑏−1 = 0 𝑟 − 1 + (𝑣𝑛 − 𝑏𝑛 − 𝑣 + 𝑏)𝑏−1= 0 𝑟 − 1 + (𝑣𝑛 𝑏 − 𝑏𝑛 𝑏 − 𝑣 𝑏+ 𝑏 𝑏) = 0 𝑟 − 1 +𝑣𝑛 𝑏 − 𝑛 − 𝑣 𝑏+ 1 = 0 Solving for 𝑏 gives:
𝑣𝑛 𝑏 − 𝑣 𝑏= 𝑛 − 𝑟 𝑣(𝑛 − 1) = (𝑛 − 𝑟)𝑏 𝑏 =𝑛 − 1 𝑛 − 𝑟𝑣 (4)
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11. Appendix: General instructions
Welcome to this experiment. You are not allowed to communicate with each other and make sure your mobile phone is switched of. Please read the instructions carefully.
You will participate in a “first price sealed bid auction”, consisting of 6 rounds. In these auctions, you will play in a group of 3 people (yourself and 2 others). Every round the composition of the groups will differ, so you do not play against two fixed players. No one will know against whom he/she will be bidding against, so all your decisions will only be known to yourself and the experimenter.
If the number of people who showed up today is not dividable by 3, then it could be that in some rounds you are in a group smaller than three. However, we will not say who so always assume you are playing against two other people.
First I will distribute a questionnaire. You have to fill in the entire questionnaire. With one of the question you can win money. On the questionnaire it will say which question this is and how it will be paid out. You will have 4 minutes to fill in this questionnaire. After these 4 minutes, I will ask if someone needs more time. If so, everyone gets 1 more minute.
After all of you have completed the questionnaire, the auction will commence by handing out your auction sheets. On your sheet will be your assigned “value”. Each player will randomly receive a random, different valuation from a uniform distribution (this means, every value has the same change of being selected) between 0 and 100 points. Every point equals 2 euro cents. Your value will be different in each round.
In every auction round, each player will submit a bid that is only known to him/herself. The person with the highest bid per group wins the auction and receives their given value and pays his/her personal bid. If you win the auction, you thus receive the difference between your bid and your value. This is your profit per round. In case there is a tie, so two or three people bid the same highest bid, I will throw a dice to decide who wins.
Everyone starts with an endowment of 200 points. This endowment has nothing to do with your biddings during the auctions, only when your bid is higher that your value and you win. Then then your loss incurred in that round will be subtracted from your total endowment. The initial endowment is not part of the profit per round, but will be added at the end of the game if no bids were higher than the values. Again, one point equals 2 euro cents.
Please round your bid to whole points. Bids that do not comply to this will be rounded to the nearest whole point by me. Bids that are higher than the maximum possible value, 100 points, will not be accepted. You can choose not to bid by indicating an “X” for your bid.
After each round the bids will be collected and we will start a new auction round.. When all rounds have been played, I will announce the winning bids for each round and for each group. We will randomly pick 3 of you who will actually get paid The payment will consist of your profit of 2 randomly chosen rounds plus your endowment. The payment will be rounded to 5 cents.
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When all the winning bids are announced, each player will fill in another questionnaire, during which we will arrange the pay-out. If you get selected for payment for the auction rounds, you cannot be selected for the payment of the question of the questionnaire. In other words, you cannot get paid double.
On the back of this paper is your personnel letter, which you should keep private at all times. Please write this number on the top of every paper we give you.
There an extra paper on your table where you can write down all your bids for yourself. At the end when all the winning bids are announced, you can see if you won the auction or what your bid was compared to the winning one.
You have one minute per round to decide what bid you want to make. After that minute, I ask if someone needs more time. If so, everyone gets 1 more minute.
If you have any questions please raise your hand and I will come by to answer your question privately.
Finally, here is an example of an first price sealed bid auction experiment (expressed in terms of money instead of points).
Value round 1 €8,-
Bid round 1 €7,-
Did you win? No ____________
Profit round 1 €0,-
Value round 2 €4,-
Bid round 2 €1,-
Did you win? Yes ____________
Profit round 2 €3,-
Value round 3 €5,5
Bid round 3 €6,-
Did you win? Yes ____________
Profit round 3 - €0,5
Value round 4 €3,5
Bid round 4 €5,-
Did you win? No _____________
Profit round 4 €0,-
Randomly chosen rounds: 1 and 3 Endowment: €3,-
Total payment: €3,- + €0,- - €0,5= €2,5
