Participation costs in a second-price sealed bid auction
Written by: Stanley Hoff Student number: 10339221 Study: Economics
Track: Behavioral economics and Game Theory ECTS: 15
Supervisor: Aljaz Ule
Abstract: In this thesis theoretical and experimental research has been done on the effects of participation costs in a second-price sealed bid auction. A participation cost denotes a player has to pay a fixed value every time he submits a bid. In this thesis, a formula has been created to calculate the cutoff values for n players with a cost of c. In the experiment, it is found that players do not learn to bid according to the prediction of the symmetric Nash equilibrium and players show no other significant signs of learning. It is also found that risk aversion has no significant effect on the cutoff value of the players.
Statement of Originality
This document is written by student Stanley Hoff who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
1. Introduction
When resources are allocated, this is often done through an open market with buyers and sellers. These markets are assumed to determine the price and quantity of goods sold.
However, in non-market environments the use of auctions are increasingly used to determine the price and allocation of goods (Samuelson, 1984). These auctions are a good way to increase competition between buyers of the goods and help to allocate the goods to the player that has the highest valuation a certain good. Auctions are commonly used in the private sector, for instance in the allocation of pollution rights. Auctions are also increasingly used on the internet where goods are auctioned to both consumers and companies. The auction mechanism is not only used in the private sector, but also by governments. Governments occasionally use auctions to purchase goods, sell assets or fund national debt (Kagel & Levin, 2002). However, in the experimental setting of these auctions, the costs of preparing the bids is often overlooked. When someone decides to propose a bid outside of an experiment, costs will have to be made before the person or firm will be able to propose a bid. Samuelson states that ‘Each firm typically devotes significant resources to the bidding competition but has a relatively small chance of winning the contract. These economic facts are variously blamed for saddling losing firms with financial losses, inhibiting competing bids in the first place, or for elevating contract prices (when bidders pass forward these costs to the buyer). The basic question then is whether the benefits of increased competition via a formal bidding institution are worth the costs.’ (1984, p.53). Stegeman also states it is good to assume players incur costs before bidding, which could be anything from gaining approval to propose a bid to driving to the auction (1996).
In this thesis, the Nash equilibria of a second-price sealed bid auction with a flat cost to proposing a bid will be investigated. Section 2 contains a literature review to get a better understanding of the different kinds of auctions and the unknowns in the literature. Section 3 contains a theoretical setup in which one symmetric and one asymmetric Nash equilibrium is derived for a second price sealed bid auction with participation costs. The literature review
and theoretical setup will be used to form hypotheses for an experiment in section 4. These hypotheses will be about individual learning in groups and the influence of risk aversion. Section 4 will also contain the experimental setup. The results of the experiment will be analyzed in section 5. In the results it is found there are few groups in which individuals learn to play according to the symmetric or asymmetric Nash equilibria. However, most groups do show an insignificant trend towards learning to play according to the symmetric Nash
equilibrium in most groups. the results and the validity of the results will be discussed in the Section 6. Finally some concluding remarks are made in section 7.
2. Literature review
An auction is a bidding mechanism that is described by a set of bidding rules. These rules specify the winner of the auction and how much the winner has to pay for the object. These bidding rules can potentially restrict participation and impose certain behavior (Wolfstetter, 1996). The most commonly used auction is the second-price sealed bid auction or English auction. These auctions are followed by the first-price sealed bid auction or Dutch auction (Wolfstetter, 1996). In the first-price sealed bid auction and Dutch auction the highest bidder gets the object for the highest bid. There is a difference between the first-price sealed bid auction and Dutch auction. In a sealed-bid auction, every player submits one bid and the highest bidder gets the object. In the Dutch auction, The objects starts at a predetermined price and this price decreases until one player accepts the bid.
In the second-price sealed bid auction or English auction, the highest bidder gets the object for the price of the second highest bid. There are also differences between the English auction and the second-price sealed bid auction. In the English auction, players can propose multiple bids, where every bid has to be higher than the previous bids. In the second-price sealed bid auction, every player submits one bid. In both auctions the highest bidder gets the object for the second highest bid.
A lot of literature, including Vickrey (1961), Boulding (1948) and Coppinger et. al. (1980) all agree that, in theory, in the second-price sealed bid auctions and English auctions the buyer that places the highest value on the item will get the item at the price of the second highest value among the values of all other players. This can be concluded because bidding your own value is Pareto-optimal (Vickrey, 1961). Bidding your own value is a weakly dominant strategy.
This, however, does not hold for the first price sealed bid auction. When bidding in the first- and second-priced sealed bid auctions, strategies in which you bid above your own
value are weakly dominated. Because of this, it is a weakly dominant strategy for the player with the highest valuation of the good to bid marginally above the valuation of the player with the second highest valuation. To conclude, the first- and second-priced sealed bid auctions should get prices with very similar means. However, since the player with the highest valuation does not know the second highest valuation, variance of prices in the second-prices sealed bid auctions is much larger (Coppinger et. al., 1980). Also, first price sealed bid auctions need not be Pareto optimal. If the bidder with the highest valuation of the good misjudges the other players, the bid could fall below the valuation of the second highest bidder, in which case the second highest bidder could obtain the good (Coppinger et. al., 1980). Because of the variance in prices and the efficiency of the auctions, the second-price sealed bid auction shall be examined more closely.
Even when only looking at the second-price sealed bid auctions, a multitude of different rules can still apply. One of these auctions with a different set of rules is the second price sealed bid all-pay auction. In these auctions, every player submits and forfeits a bid. The highest bidder gets the object for the second highest bid and all other bidders lose the money they bid (Baye, Kovenock & de Vries, 1995). These auctions are used in economics because it captures some elements of contests. For instance, in political campaigns like the presidential campaigns can be modelled in this way because both parties pay for the
campaign regardless of winning. Since the players that do not win the object still have to pay their bids, bidding their own values is no longer an ex-post Nash equilibrium. In these types of auctions, there is one symmetric Nash equilibrium and a continuum of asymmetric Nash equilibria (Baye et. al., 1995). However, the game as described by Baye et. al. (1995) assumes perfect information, which is not given in a sealed bid auction.
Naussair and Silver (2005) use the assumption of imperfect information in an all-pay auction. In the theoretical predictions, Naussair and Silver now find a Bayesian Nash
equilibrium instead of pure or mixed strategies, since the value of other players is unknown for any player. The auction in which this is used is a first-price sealed bid auction in which the valuation of the players are drawn uniformly. A cutoff value is found below which any player should never bid. The all-pay auction without perfect information Naussir and Silver use is, however, a first price sealed bid auction instead of a second-price sealed bid auction that is researched in this thesis (2005).
Tan and Yilankaya (2005) use a slightly different set of rules regarding payment. Players in this theoretical setup have to pay a flat participation cost. In the all-pay auction every bidder has to forfeit their bid. In the auction with participation costs pay a flat
participation cost C which is the same for all players (Tan and Yilankaya, 2005). Tan and Yilankaya (2005) have already done extensive research towards these Bayesian Nash equilibria. They assume that players are identical, i.e. the valuation of all players are drawn from the same distribution functions and players will know the cost of bidding before posing a bid. All of the players are assumed to be rational and will therefore make the same
assumptions and predictions as any other player given their value for the object. Tan and Yilankaya look at the equilibria both when the distribution function of the valuations is convex and the distribution function is concave (2005).
Not a lot of literature has been written about auctions with a flat participation cost. Tan and Yilankaya (2005) used a convex and concave distribution function for the valuation distribution but no research has been done on the uniform distribution function of valuations, even though the uniform distribution of valuations is commonly used in other literature on auctions (Coppinger et. al., 1980). The paper of Tan and Yilankaya also only give a
theoretical setup, but no experiment has been conducted to test how these participation costs influence players of the auction. Though not a lot of literature has been written on auctions with a flat participation cost, this does not imply these types of auctions are uncommon. As has been argued in the introduction, auctions with flat participation costs are very common and should therefore be studied more thoroughly.
To set up an experiment of a second-price sealed bid auction with participation costs, other literature regarding auction experiments are interesting to derive some knowledge regarding the behavior of players. Noussair and Silver note that, in an all-pay first-price sealed bid auction with a uniform valuation distribution, players endure considerable losses during the first few rounds of the experiment, but quickly learn to adapt their strategies to bid lower than they originally did, though still above equilibrium values (2005). This indicates players learn over the course of an experiment. Kagel, Harstad and Levin find in their
experimental study that no learning is observed in a standard second-price sealed bid auction as opposed to a first-price sealed bid auction (1987). This leaves the question whether players would learn in an all-pay second-price sealed bid auction.
Finally, In the theoretical approach of the Symmetric Nash equilibrium, It is often assumed that players are risk neutral. According to Holt & Laury (2002) a majority of the people is risk averse. A Second-price sealed bid auction with flat participation costs confronts players with risk since they do not know the valuation of the other players and players could potentially lose money. For these players, the marginal utility gain of winning money will probably not be the same as the marginal utility lost when losing money (Maskin and Riley,
1984).
In this thesis, a second price sealed bid auction with flat participation costs and uniform valuation distribution will be considered. Both a theoretical and experimental setup will be proposed. Learning and risk aversion will also be looked at during the experiment.
3. Theoretical Setup
As with Tan and Yilankaya, I consider an independent private value environment. There will be n≥2 risk neutral potential bidders. The valuation of player i is . The distribution of will be random and uniform with . There will be a flat cost to bidding c which is the same for all players and will have a value of . To propose a bid, players will have to incur this cost c.
The auction will be a second-price sealed bid auction. Players are assumed to be rational and know their valuation before they decide whether to participate in the auction. For simplicity, the value of every player will be an integer between 0 and 100. Players do know the total amount of players but do not know the participation decisions of the other players before making their own. The players can choose to either abstain from bidding (which will be referred to as ‘abstain’) or pose a bid and incur the cost c to have a chance to obtain the item. In a second-price sealed bid auction without participation costs it is always optimal to bid your own value. This property still holds when bidding in a second-price auction with participation costs (Tan and Yilankaya, 2005). Therefore, it is assumed that players, when proposing a bid, will bid their value. This is formalized by saying . Because , the symmetric Nash equilibrium should be efficient because the highest bid comes from the person with the highest value for the item. Stegeman also concludes that second-price sealed bid auctions can be efficient, i.e. the highest bidder gets the item (1996).
Proposition 1: It is weakly dominant to bid your own value when bidding.
Three outcomes are possible with the choice set of the players. They can choose to either abstain or participate, and can obtain the item when participating, depending on the bid. The profit function of the player can be described as follows:
Where is the payoff of each of the decisions, is the highest bid proposed by any player other than player i.. The formula assumes there is always one bidder other than player i, which would be a close approximation for setups with a large amount of players.
Rational risk-neutral players will only propose a bid when their expected payoff will be higher compared to their payoff when abstaining, which is 0. These players will only pose a bid when:
(2)
Where is the probability of winning the auction and is the value of losing the auction, where . When making the decision whether or not to propose a bid, the only unknown value to the player is . When incorporating this into the formula and with some rewriting you get:
(3)
Which gives a very intuitive equation stating your earnings have to exceed the costs of bidding.
To calculate the expected value of the highest other bidder an assumption made by Harstad, Kegel and Levin (1989) is used. In their paper, they state a bid will be the weighted average of bids that would be chosen by every player in case this player would play when bidding in an auction where there is uncertainty. Since the values of the n-i-1 players that do not have the highest or second highest bid do not matter for player i, it is important to
calculate the expected value of the second highest bidder. To do this, the probability of every integer to be the highest value of any of the other players is calculated. The probability of x being the highest integer of the other players, where x is an integer between 0 and 100, has the following formula:
(4)
Where n is the total amount of players and x is an integer with 0 < x ≤ 100.The derivation of this formula will be explained in Appendix 1. To calculate the expected value of the highest bidder, take the sum of all values times their probability:
This can be rewritten as:
(5)
The probability of player i winning the item is dependent on the amount of people
participating and the value of player i, .The formula for the probability of winning the item is:
(6)
Which will be explained in Appendix 2.
When substituting (5) and (6) into equation (3) you get the following formula:
(7)
With this formula it is possible to calculate for which value of it is rational to pose a bid, given the amount of players that can have an integer value of and the cost of bidding. This value, as also proposed by Tan and Yilankaya (2005), will be called the cutoff value and shall be denoted by . Since every player is equal, this cutoff value is the same for all players. This formula, however, only works when n has a high value, because the formula assumes that there will always be at least two bidders. During the experiment, which will be explained later, only 4 players will participate at the same time. Therefore, it is irrational to assume there will always be two players who pose a bid at the same time. To calculate the cutoff value with only a few players, another method needs to be used. This method is explained in Appendix 3. This method shows that the cutoff value that should be used during the experiment will be 57.
Proposition 2: Players should never pose a bid as long as < in the symmetric Nash equilibrium
These propositions hold for the Symmetric Nash equilibrium that has been found, which has the following strategy: Do not bid when and bid when . There is,
however, another Nash equilibrium, which is the asymmetric equilibrium. It is also a Nash equilibrium when any of the player i poses a bid that is above any other players value minus the participation cost, , and all other players abstain in every round. This can be seen by looking at formula three. If the expected bid of the highest other player is always above 90, the cutoff value will be above 100 and therefore no bids should be posed. Since the highest bidder will incur high losses when the second highest bidder bids a high bid as well, this strategy is will very likely be unstable. Also, any player can be the highest bidder to be in equilibrium and therefore the Nash equilibrium is inefficient. The strategy of this Nash equilibrium is: one player always bids above 90, regardless of value. All other players
abstain. During the experiment, it will be tested whether this strategy is used by the players to see whether this strategy can sustain during the repeated trials.
4. Experimental setup
The experimental design reflects both theoretical and technical considerations. First, the experiment will be explained in the experimental design. Secondly, the experimental procedure will explain the experiment in detail and everything the experimenter adhered to during the experiment. Finally the hypotheses will be explained together with the way to test for these hypotheses in the experimental analysis.
3.1 Experimental design
An experiment will be conducted with 32 players, 4 players in each group. These groups will participate in a virtual second-price sealed bid auction where they have to incur a cost of 10 in order to participate. Every group will play 10 rounds in this setup.
Because it is not possible to let the players pay the experimenter if their earnings happen to be negative, Kagel & Levin (1993) decided it would be best to endow players with money before the experiment. Therefore, every player will be endowed with 5 euros and will be paid according to their performance during the experiment. An exchange rate of 100 tokens for every euro will be used during the experiment. Due to budget constraints, only three players will be paid out. The selection of these players will be decided through a lottery. The information of the economic environments in the first part of the experiment have been summarized in table 1.
Table 1.
Summary of economic environments
Groups
No. Players per group
Exchange rate (tokens for euros)
Total no.
Players Rounds
Endownment (tokens)
8 4 100 32 10 500
After the players are finished with playing in the auction for 10 rounds, the players will receive a sheet to play an 11th round. This round will be played differently. Players have to write down a strategy for every possible value and the experimenter plays the round for them. Finally, the players are handed another sheet on which they have to make decisions between risk and riskless options. The sheets that are handed to the players can be seen in Appendix 4. 3.2 Experimental procedure
At the beginning of each session, Instructions for the setup and a pen will be handed to the players by the experimenter. The players will be given time to read the instructions and can raise their hand if any questions arise. Once every player has read the instructions, every player will receive control questions to test their understanding of the instructions. After all questions have been answered correctly the experimenter shall start the experiment. Each round consists of the following stages:
(1) Every player will be handed a sheet on which they can see their value and fill in their bid. This sheet will also have information on all previous rounds which will be the number of bidders, the highest bid and the second highest bid.
(2) Every player will be given time to fill in their answer and will be asked to raise their hand when they have finished. General questions about the experiment will be allowed but any questions relating to strategies will not be allowed at this point. (3) Once everyone has raised their hand, the experimenter will come and collect the
answer sheet. The experimenter will check the bids and write down the number of bids, highest bid, second highest bid and the players payoff on the sheets. The experimenter will also write down the values for the next round.
There are two reasons to announce this information. The first reason is that players can use this information to possibly adjust their bidding behavior. The second reason is that this is also the information that players would most likely be able to obtain in real life. Therefore this information would maximize the external validity of the results.
After the experiment, the players will be handed a sheet of paper on which they can announce their strategy for the 11th round, after which the experimenter will play the round for them. The players will be told that they have to fill in a strategy so the experimenter knows what to do for every possible value. Sonnemans (1998 & 2000) used a sheet that had every strategy written down beforehand and players had to simply select the strategy and choose for which values this strategy would be implemented. In this thesis it is chosen not to give this set of strategies since knowing whether players would think of a cutoff value is important information and these pre-existing strategies might influence their own strategy. This is why there are also no examples given to the players since this might bias their
strategy. When players filled in their strategy, the experimenter checks whether their strategy could be implemented at every value and the players are only told for which values the experimenter is still unclear what to do. Once this sheet is filled in, the players will be handed the final sheet of paper. This paper contains 10 decisions where players have to choose
between the left and the right option. The decisions sheet can also be found in Appendix 4. 3.3 Experimental analysis.
Kagel, Harstad and Levin (1987) find in their experimental study that, unlike in English auctions, no learning is observed in Vickrey auctions. However, in this experiment, there are more possibilities of learning. Players can both learn when to bid and what to bid. Players are now also punished for making bad bidding decisions since they have to pay the participation cost. It is therefore interesting to test whether groups converge towards the Symmetric Nash equilibrium over time.
Hypothesis 1: Players in a group learn to converge towards the symmetric Nash equilibrium over time, which is bidding their value when their value exceeds the cutoff value.
Kagel, Harstad & Levin (1987) players show no learning in the second-price auction, as has been mentioned before. They do, however, show a learning curve after a few rounds in the first-price sealed bid auction have been played. This learning curve flattens before the end of the trial. If players would show learning in the second-price sealed bid auction with
participation costs that has been used in this thesis, it would be reasonable to assume a similar learning curve to take place. Therefore, a comparison will be made between the first three rounds and the last three rounds of the experiment to test whether players learn to play according a Nash equilibrium. The analysis of the first hypothesis will be twofold, because there are two different types of learning in play. First of all, as has been mentioned in
proposition 2, players should never bid when their value is below the cutoff value when they are playing according to the symmetric Nash equilibrium. Because it is nearly impossible to calculate the exact cutoff value for players, the participation decision will be disregarded every time a player had a value close to the exact cutoff value. Only when players gets a value lower than 47 or higher than 67 the results will be used to test for learning. For every player the amount of correct participation decisions divided by correct participation
opportunities will be measured in the first three rounds and the last three rounds. The correct participation decisions is divided by the opportunities because players with a lot of values that would be ommitted, because their values were between 47 and 67, would have less possibilities to play according to the cutoff value. These values will be tested for every group using a Mann-Whitney U test. A t-test can not be used because a normal distribution can not be assumed. The H0-hypothesis will be that players do not improve their cutoff learning
behavior. The H1-hypothesis will be that players improve their cutoff learning behavior
Except for bidding according to the cutoff value, players could also learn to bid their value when playing according to the symmetric Nash equilibrium. Therefore another test will be conducted to test whether groups will start to bid closer to their value. To test this, the difference between the value and the bid of every player will be used to see whether the players start bidding closer to their value. In this test, both the bids when values are below the cutoff value and the bids when values are above the cutoff value will be taken into account since this test will not be about the cutoff value. Another Mann-Whitney U test will be used to see whether groups start bidding closer to their value in the last three rounds compared to the first three rounds. Because the Mann-Whitney U test uses a ranking system, it is very insignificant to square the differences before using the test. The H0-hypothesis will
be that players in a group do not learn to decrease the difference between their value and their bids. The H1-hypothesis will be that players in a group do learn to decrease the difference
between their value and their bids
However, groups could also converge towards the asymmetric Nash equilibrium. Because any player that bids excessively could be punished by any of the other players, it is to be expected that players that bid according to this equilibrium will disappear over time. Hypothesis 2: The amount of asymmetric Nash equilibria will decrease over time.
When the asymmetric Nash equilibrium occurs, one player has to bid an escessive bid and all other players have to abstain. To test whether the asymmetric Nash equilibrium occurs, Another Mann-Whitney U test will be conducted to test whether one player with an excessive
bid and three abstaining players in any group happens more often in the first three rounds compared to the last three rounds. An excessive bid would be any bid that would make any player want to abstain whatever their value, which is a bid of 90. However, since any bid from 90 to 100 can also be used in the symmetric Nash equilibrium, only bids higher than 100 shall be denoted as excessive bids. The amount of times groups reach the asymmetric Nash equilibrium will be measured in the first and last three rounds for each group. The Mann-Whitney test will be used to compare the bids in the first three rounds with last three rounds for every group. If the asymmetric Nash equilibrium occurs significantly more in later rounds, this would be evidence that players did learn to play according to the asymmetric Nash equilibrium. The H0-hypothesis will be that players in groups do not learn to converge
to the asymmetric Nash equilibrium. The H1-hypothesis will be that players do learn to
converge to the asymmetric Nash equilibrium
In the theoretical approach of the Nash equilibria, it is assumed that players are risk neutral. However, as has been argued in the literature review, a majority of the people is risk averse according to Holt & Laury (2002). When players are risk averse, It can be expected these players will have a higher cutoff value which implies these players need a higher value before they start bidding.
Hypothesis 3: Risk averse players abstain more often when their value is around the cutoff value than risk-neutral and risk-loving players.
To test this hypothesis, first we need to measure the risk aversion level of the players. The level of risk aversion will be measured through a small test that is used by Holt and Laury (2002). The test will be used to distinguish between risk averse and risk loving players. To test whether players abstain more often, only the first round of every player can be used because the bidding behavior of players will be influenced by the bids of other players in later rounds. There are also three categories to distinguish for players. Values below 47, in which it will most likely be in the best interest of every player to abstain from bidding, values above 67, in which it would most likely be in the best interest of every player to participate, and values between 47 and 67, in which it is uncertain for players what they should do. In the categories below 47 and above 67, players should show the same kind of behavior, regardless of their risk aversion. Between 47 and 67, players who are more risk averse might be more likely to abstain. All three categories will be looked at in the results section. It will be measured when risk averse and risk loving players played either risk averse and risk loving and these results will be used for a Fisher’s exact test. For category 1 and 3, the H0
-hypothesis will be that risk averse and risk taking players will not be significantly different in their cutoff decisions and the H1-hypothesis will be that risk averse and risk taking players
are will make significantly different cutoff decisions. For category 1, the H0-hypothesis will
be that risk averse players do not abstain more than risk taking players. The H1-hypothesis
will be that risk averse players do abstain more than risk taking players.
Furthermore, It could happen that none of the Nash equilibria will be reached by a group. Even though it could be that players did not learn to play according to any Nash equilibria, it does not mean players did not learn anything during the experiment. If players learned to anticipate the decisions of the other players, their adjustment to the other players could help them play better during the experiment. Because it is very hard to test every players expected bid of every other players during every round, a good approximation is to test whether groups increased their earnings over time.
Hypothesis 4: Groups learn to increase their profits over time
To test this, the profits of every player during the first three rounds will be compared to the last three rounds. Another Mann-Whitney U test will be used to compare the first and last rounds and see whether group profit significantly increases.
Finally, players will play an 11th round in which they have to describe their strategy so the experimenter can play the round for them. Even though no tests will be conducted with this information, it might be useful for further research since it shows what players were thinking after playing the game. The strategies will be categorized and listed in the results. The catgegories will be:
1. Players used a single cutoff value that was close to the Symmetric Nash equilibrium 2. Players used a strategy that used excessive bidding strategies
3. Players used a strategy in which they always abstain 4. Others
5. Results
The first Mann-Whitney U tests were used to measure whether groups would learn to behave according to the symmetric Nash equilibrium over time. Table 2 shows the results of the Mann-Whitney test on the cutoff value as has been discussed in the experimental analysis. Table 2:
Mann-Whitney test on cutoff decisions
rGroup z-statistic Prob > |z|
1 -1,4 0,1615 2 0,2 0,8415 3 0,2 0,8415 4 0,2 0,8415 5 -1,4 0,1615 6 1,686 0,0918* 7 -1,4 0,1615 8 1,4 0,1615 .
* Significant at the 10% level.
A negative z-statistic indicates players in groups improved their cutoff decisions in the last three rounds compared to the first three rounds. As can be seen in table 2, three groups improved their cutoff decisions, however not significantly. The players of the other groups made worse cutoff decisions over time according to the symmetric Nash equilibrium theory. One of the groups even performed significantly worse at the 10 percent level.
To test the second part of the hypothesis, another Mann-Whitney U test was used. The results can be found in Table 3.
Table 3:
Mann-Whitney test on difference between value and bids.
Group z-statistic Prob > |z|
1 1,604 0,1088 2 1,342 0,1797 3 -1,633 0,1025 4 -1,387 0,1655 5 -0,674 0,5002 6 -1,826 0,0679* 7 -2,366 0,018** 8 0,714 0,475
**=Significant at 5% level, *=Significant at 10% level
In table 3, a positive z-statistic shows that players in a group decreased the difference
between their value and their bid over time. None of the groups show a significant decrease at any level of significance. Three out of eight groups have a decreased difference between values and bids, but not significant. More interesting, however, is the fact that five out of eight groups actually had an increased difference between the values and bids over time. Group six is significant at the 10% level and group seven is significant at the 5% level. This is a clear diversion of the symmetric Nash equilibrium but could be some evidence towards the asymmetric Nash equilibrium.
Because there is not enough evidence to both reject the H0-hypothesis that players in a
group do not learn to play according to the cutoff value and the H0-hypothesis that players in
a group do not learn to bid their values over time in any group, it can be concluded that players do not learn to play according to the symmetric Nash equilibrium.
To test whether the asymmetric Nash equilibrium would arise, the amount of
occurrences of the asymmetric Nash equilibria first have to be counted. However, since this did not occur once during the experiment, it can be assumed this strategy does not occur, and neither do players learn to act according to the asymmetric Nash equilibrium. Excessive bids occurred 69 times, but this was never combined with 3 other players abstaining in this round. Therefore, there is no evidence to reject the H0-hypothesis.
To test whether risk averse player were more likely to abstain during the experiment a Fisher’s exact test has been used on all categories mentioned in the experimental analysis section. Two players were omitted from the results because the risk aversion test could not measure their level of risk aversion. The results of the first category, which is players with a value below 47, can be seen in Table 4.
Table 4:
2x2 matrix of risk aversion and abstain in category 1 Abstain
Risk Averse 0 1 Total
0 5 6 11
1 0 1 1
Total 5 7 12
The row variable is risk aversion. A zero indicates players were risk-taking and a one
indicates players were risk averse. The column variable is about abstaining. A zero indicates players did not abstain in the first round and a 1 indicates players abstained during the first round. When using Fisher’s exact test, a two-sided Fisher’s exact p value is 1. Therefore, there is insufficient evidence to reject the H0-hypothesis.
For the second category, which is all players with a value between 47 and 67, the results can be found in Table 5.
Table 5:
2x2 matrix of risk aversion and abstain in category 2
Abstain
Risk Averse 0 1 Total
0 3 0 3
1 4 1 5
Total 7 1 8
When using a one-sided Fisher’s exact test, the one-sided Fisher’s exact test p-value is 0.625. Therefore, there is not enough evidence to reject the H0-hypothesis.
For the third category, which is all players with a value above 67, the results can be found in Table 6.
Table 6:
2x2 matrix of risk aversion and abstain in category 3
Abstain
Risk Averse 0 1 Total
0 7 2 9
1 1 0 1
Total 8 2 10
Once again, when a two-sided Fisher’s exact test is used, the two-sided Fisher’s exact test p-value is 1. Therefore, there is not enough evidence to reject the H0-hypothesis
The final Mann-Whitney U test has been done to measure whether groups would improve their earnings in the last three rounds compared to the first three rounds. The results can be found in table 7.
Table 7.
Mann-Whitney U test on the increase of profits
Group z-statistic Prob>|z|
1 -2,859 0,0042* 2 1,99 0,0465 3 0,451 0,6519 4 -1,538 0,124 5 -0,169 0,8656 6 2,103 0,0354** 7 1,77 0,0767*** 8 -1,396 0,1626
*: significant at the 1% level. **: significant at the 5% level ***: significant at the 10% level
In table 7, one out of 8 groups shows learning to increase profits that is highly significant, at the 1% level. When looking at the data, the biggest part of this effect is because one player, who was bidding very excessively (400+) learned to stop using this strategy in later rounds, thus significantly improving the players payoff. Three other groups showed increased payoffs
over time, however not significant. The other four groups showed a decreased group payoff, two of these groups significantly at a 5- and 10%, respectively. When looking at the data, this could have happened because of a very high competition between the players. The objective of the players did not seem to be maximizing their respective payoffs, but to decrease the other players payoffs to such a degree that their own payoff would be the least negative. Finally every player has listed a strategy that they would use during their 11th round. This strategy would be used against the other players of their group and was also part of the payoff process. Players were given no guidelines as to what strategies they could use but if the players strategy was vague, i.e. they would have a strategy stating they would bid ‘high’ at a certain value, the experimenter would explain to the player why this strategy could not be implemented and the player had to write down a new strategy. The strategies have been divided into four categories and all strategies are listed in table 7.
1. Players used a single cutoff value that was close to the Symmetric Nash equilibrium 2. Players used a strategy that used excessive bidding strategies
3. Players used a strategy in which they always abstain 4. Others
Table 8.
List of strategies of all players for the 11th round.
Player Category Strategy
1 1 3 1 7 1&2 10 1 12 1 13 1 14 1
17 1 19 1 20 1 23 1 27 1 32 1&2 8 2 18 2 22 2 24 2 26 2 2 3 Always abstain 5 3 Always abstain 6 3 Always abstain 31 3 Always abstain
4 4 9 4 11 4 14 4 16 4 21 4 bid v+10 25 4 28 4 29 4 bid v+10 30 4
When looking at the table, the categorization does not do a perfect job in separating the different types of players but it does get close. In category 1, a lot of players had a strategy that came close to the symmetric Nash equilibrium. However, out of the twelve players being categorized in category 1, two players seemed to have a strategy that did not come close to
the symmetric Nash equilibrium. Player 7 had an almost perfect cutoff value, but decided to bid excessively when a value above the cutoff value was drawn. Player 12 also had a cutoff value that was very close to the actual cutoff value but the player only wanted to pose a bid when his value was below 50. Because this strategy would most likely result in low payoffs for the player, this strategy shows little understanding of the auction.
Category 2 shows all players that posed an excessive strategy during the 11th round. It is common for players that do not fall into category 1 to have multiple cutoff values.
However, player 8 had a cutoff value in which he would bid for low values, not bid for middle values and bid again for high values. This is once again a strategy that shows little understanding of the auction.
Category 3 shows all players that always want to abstain in the 11th round. This strategy has multiple reasons. The first reason is the player is afraid of an excessive bid of one of the other players and therefore chooses not to join the auction. The second reason could be some players were not able to formulate a proper strategy and always abstain is an easy strategy to formulate. Finally the strategy could have been formulated because the 11th round felt different to the players and therefore the players wanted to hold back on their strategy, fearing another strategy would give them significant losses.
Finally category 4 shows all other strategies. These strategies consist mostly of multiple cutoff values or no cutoff values. Only player 25 had a strategy that was similar to player 8. However, the strategy of player 25 seems to be more extreme. The player bids highest for the lowest values, abstains for middle values and once again bids high for high values.
6. Discussion
The experiment has been done with 32 players, divided in eight groups. Every group has done the experiments in almost similar conditions. Groups were isolated from anyone else by performing the experiment in rooms in which there were no other people, the only exception being group 5, 6 and 7. These three groups did the experiment at the same time in the same room. However, the groups were separated to corners of a large room so communication between these groups was not possible. The biggest disadvantage of testing three groups at the same time was the waiting time between each round, since all profits had to be calculated by the experimenter. This might have bored the players, biasing the results.
The biggest influence on the results is most likely the monetary constraint. Without the monetary constraint, it would have been easier to get players to do the experiment at the
same time in a room in which they could be separated. Because of the monetary constraint, players were less inclined to show up, making it impossible to test all the groups at once. The monetary constraint might have also decreased the learning curve of the players. If a player lost a significant portion of their earnings, this would only amount to up to a few euros, provided the player would even win the lottery. Players seem to have taken gambles that will be unlikely to happen if more money was involved. This might have biased the internal validity of the experiment.
A second big influence on the results, which might also be the result of the monetary constraint, is the amount of players. For most of the analyses the amount of groups was sufficient to test the hypotheses. However, for the Fisher’s exact test the players were categorized into three different groups. The amount of players per group was so small that sometimes only one or two persons in a category abstained. Therefore the internal validity of the results is heavily biased.
However, in every group the order was maintained throughout the whole experiment. None of the players has discussed their results with any of the other players during the experiment. Cellphones were also turned off. All control questions were eventually filled in correctly by all of the players without giving information regarding strategies.
The hardest part of the experiment seemed to be the induced value the players
received. Players had a hard time understanding the concept of an induced value. Some of the players might have learned to work with the value without understanding the true meaning of the value. This explains why some players have used irregular strategies in the 11th round. Even though most players showed a clear understanding of the value, a better explanation might have been clearer for the remaining players. Because this understanding of value would never happen in an actual auction, since a value is innate instead of induced, this has probably biased the external validity of the experiment.
7. Conclusions and further research
In this thesis it has been discussed that participation costs have important effects on the behavior of bidders. These participation costs occur at most auctions and are often overlooked in an experimental setting. Because of participation costs, players should not always bid their values in a second-price sealed bid auction but should also decide on a cutoff value when playing according to a symmetric Nash equilibrium. In this thesis a model has been developed that can be used to calculate the symmetric Nash equilibrium for any amount of players for any costs. With this model it is possible to derive the cutoff value for any
player. An experiment has also been conducted in which players played in an experimental second-price sealed bid auction with participation costs. The experiment fails to reject all hypotheses regarding Nash equilibria. However, these results are most likely biased due to a monetary constraint and the amount of players. It has also not been found that people learn to anticipate the other players when making their own decision. The experiment also fails to reject the hypothesis that risk-averse players are more likely to abstain in the second category during the experiment.
For further research I would recommend to run the experiment with larger monetary incentives. Higher monetary incentives might decrease the amount of overbidders in the auction which might enable the players to learn to bid according to the symmetric Nash equilibrium. It would also be interesting to compare a standard second-price auction to the auction with participation costs so the effect of the difference in bidding behavior on the price of the item sold can be measured. This test would be most interesting when subjects are frequent auctioneers, so some light can be shed on actual auctions. Finally, it could be interesting to test whether uncertainty about the number of players has any effect on the bidding behavior of players in a sealed bid auction with participation costs. This could not be performed in this thesis because of a lack of subjects and the monetary restraint that would be able to pay for this second group of players.
Appendix
Appendix 1. Derivation of the expected highest bid ).
To calculate the expectation of any number being the highest number of any opponent, I reasoned that out of all possible integers (1 through 100) the highest number cannot be exceeded by any other number, i.e. if the highest number is one, there is only one possible combination of values that ensure the highest number is 1, which is when all other players get one. The probability of any number is linear, so the probability of someone getting 1 is . Besides player
I, there are n-1 players, all of whom need to get 1. This means the probability everyone gets 1 is . The same holds for the probability of the highest person getting the number 2, which is the probability of everyone getting either 1 or 2, minus the probability everyone gets 1. This means the probability at least one player gets a 2 and no other player gets higher than a 2 is . To generalize, the probability at least one player gets x is the sum of all probabilities that players get either 1, 2, 3… x, minus all probabilities the highest player gets lower than x, starting with x-1. Therefore, the probability of the highest value of any other player is x is , which can be rewritten as
, where 0<X≤100. To derive the average number, I use the fact that the sum of the probability of all numbers times that number will derive the average. This can
be written as . This can be simplified to
. The cumulative probabilities of any x happening when N=4 is shown in table
Table 2. The cumulative probabilities of x=X.
In table 2 it can be seen that the probability of x=X increases when X increases. Therefore, at N=4, the probability of the highest opponent having a rather high number is higher than the probability of the highest opponent having a low number.
Appendix 2. Derivation of the probability of player i winning
The probability of winning is the probability that your value is higher than the value of any other player, which is the sum of the probabilities any of n-i players has a value higher than
. This means that, just as in appendix 1, you calculate the sum of all probabilities of . This can be formulated as . When rewritten this becomes
Appendix 3. Derivation of the cutoff value when n=4.
To calculate the cutoff value, some assumptions have to be made. The first assumption is that players are fully rational. The players are also expected to be risk-neutral. This implies that players are indifferent between bidding and abstaining when they get a value that is equal to the cutoff value. When their value is below the cutoff value, player will not bid.
When a player i gets a value that is equal to the cutoff value ( ) their value of bidding will be equal to their value of abstaining, which is 0. Since there are no players that bid when their value is below the cutoff value, player i Will only win the bid when no other players bid. Player i will always lose if any other player has a higher value. If player i wins, he gets his value minus the costs. This translates into the following formula:
The probability of winning is determined by the value of player i which is equal to the cutoff value. It is the probability of every other player abstaining.
When there are 4 players and there is a participation cost of 10, you get the following equation:
Using the method of Newton-Raphson, which uses an iterative process to approach the root of a function , this equation can be solved. The method of Newton-Ralphson has 3 estimations, of which only one is an integer between 0 and 100 which is:
Appendix 4. sheets that are handed out to players.
Welcome to the experiment!
Please refrain from communication for the duration of the experiment and make sure your cell-phone is switched off. Read and follow the instructions carefully; they contain everything you need to know to participate.
In this experiment, you may earn money. A random lottery will decide whether you will win the prize. The amount of money you can earn will be dependent on the decisions you make during the experiment and the decisions of others. In the experiment, you can earn tokens. The more tokens you earn during the experiment, the higher the payoff will be. During the experiment, 100 tokens will be equal to 1 euro.
The experiment consists of two parts. In the first part of the experiment, every player will play independently and anonymously. The instructions will be the same for every player but the values that will be used during the experiment can be different for every player. Your values and your decisions will determine the amount of tokens you have at the end of the experiment. In the second part of the experiment, some smaller questions will be asked which will be explained to you later. In the first part of the experiment, you will participate in an auction where virtual goods are
auctioned. You will start with 500 tokens (=5 euros). With these tokens you will be able to bid in the auction. The auction will be a second price sealed bid auction. This means that the highest bidder of the good pays the price of the second highest bidder of the good. You can pose one bid every round, where the highest bidder gets the virtual item for the price of the second highest bid. You can bid any amount of tokens as long as it does not exceed the total amount of tokens you have. When your total amount of tokens reaches 0, you will no longer be able to participate in the experiment. The bid every player submits will be anonymous. Every player will also be given a value, which can be any number between 1 and 100, each number with the same probability. With this value you can decide whether you want to bid on the virtual item. However, every time you submit a bid you will pay a
flat participation cost of 10 tokens. You can also choose not to participate, in which case you do not
have to pay the participation cost and this will results in a payoff of 0. If you are the only player that submitted a bid during a round, you will get the item for free and you will only pay the participation cost of 10 tokens. Your possible payoffs will be:
1. Your value minus the second highest bid minus 10 if you submitted the highest bid 2. Minus 10 if you did not submit the highest bid
3. 0 if you decided not to submit a bid that round
The experiment will consist of 10 rounds, where each round you will be given a random value between 1 and 100 and each round you will be able to pose an anonymous bid. At the bottom of the instruction pages, you can find a quick summary of the important information. You can keep this information sheet with you at all times.
During the experiment, the experimenter will hand out a piece of paper every round on which you can fill in whether you want to bid and how much you want to bid and will look like this:
Player number: ………
Round Value Bid No. Players Highest bid Second bid Payoff
1 2 3 4 5 6 7 8 9 10
Where round indicates the round you are playing in, value indicates the value that the virtual item is worth to you (will be given to you by the experimenter every round). In the column of bid, you can write down your bid if you choose to submit a bid. If you choose not to submit a bid, please fill in either ‘X’ or ‘NO’. Filling in 0 will result in a bid of 0 and you will then have to pay the participation cost. No. Players, Highest bid and second highest bid will then be filled in by the experimenter which will give you information about the round. Finally your payoff will be calculated and your value of the next round will be given, so you can start with the next round. When you have filled in your bid, please turn your sheet upside down so the experimenter will know you are done.
Summary of information:
Auction: Second-price sealed bid Amount of players: 4
Participation cost: 10 tokens Exchange rate: 100 tokens = 1 euro
Before we proceed to the experiment, you will first be asked to fill in some control questions to check whether you understood the instructions. Please raise your hand when you are done reading the instructions and the experimenter will come to you to give you the control questions. If you have any questions, please ask them now or during the control questions and the experimenter will come to you in private. The experimenter cannot answer all your questions after the control questions.
IF YOU HAVE FINISHED READING, PLEASE RAISE YOUR HAND SO THE EXPERIMENTER CAN GIVE YOU THE CONTROL QUESTIONS
Control questions:
Question one: Player number: 2 Your value: 35
You have chosen to participate in the auction and you chose to bid 25. Player one has been given a value of 5 and bid 5, player three had a value of 65 and bid 75 and player four had a value of 15 and chose not to participate.
1. What did player 1 earn this round? ………… 2. What did you earn this round? ………… 3. What did player 3 earn this round? ………… 4. What did player 4 earn this round? …………
Question two: Player number: 2 Your value: 25
You have chosen to participate in the auction. and you chose to bid 40. Player one had a value of 30 chose not to participate, player three had a value of 65 and chose not to participate and player four had a value of 55 and chose not to participate.
1. What did player 1 earn this round? ………… 2. What did you earn this round? ………… 3. What did player 3 earn this round? ………… 4. What did player 4 earn this round? …………
YOU ARE NOW DONE WITH THE CONTROL QUESTIONS. PLEASE WAIT UNTIL EVERYONE HAS FINISHED BEFORE WE START WITH THE EXPERIMENT.
You are now done with the first part of the experiment. For the second part of the experiment, the same auction rules will apply. However, this time you will not actually participate in the auction. Instead, you have to write down a strategy the experimenter can use when participating in the experiment for you. The strategy has to be very precise, so the experimenter will know what to do at every value you might get. The experimenter will check your answer to see whether he can use it in every situation before the sheet is handed over to the experimenter. The experimenter will then play one more round with your strategy and the strategies of the other players and your payoffs will be calculated afterwards. These payoffs also count towards your total payoff.
Please describe your strategy here:
For the last part of the experiment, you have to do a small test that will also add to your final payoff. Below there are 10 small exercises. In every row, you can fill In whether you would rather want the left option or the right option. After the experiment one of the ten exercises will be chosen at random and you will be informed of the results. These earnings will be added to your total earnings. please choose the options you would want to have most:
☐ 10% chance of €4, 90% chance of €3,60 Or ☐ 10% chance of €7,70, 90% chance of €0,20 ☐ 20% chance of €4, 80% chance of €3,60 Or ☐ 20% chance of €7,70, 80% chance of €0,20 ☐ 30% chance of €4, 70% chance of €3,60 Or ☐ 30% chance of €7,70, 70% chance of €0,20 ☐ 40% chance of €4, 60% chance of €3,60 Or ☐ 40% chance of €7,70, 60% chance of €0,20 ☐ 50% chance of €4, 50% chance of €3,60 Or ☐ 50% chance of €7,70, 50% chance of €0,20 ☐ 60% chance of €4, 40% chance of €3,60 Or ☐ 60% chance of €7,70, 40% chance of €0,20 ☐ 70% chance of €4, 30% chance of €3,60 Or ☐ 70% chance of €7,70, 30% chance of €0,20 ☐ 80% chance of €4, 20% chance of €3,60 Or ☐ 80% chance of €7,70, 20% chance of €0,20 ☐ 90% chance of €4, 10% chance of €3,60 Or ☐ 90% chance of €7,70, 10% chance of €0,20 ☐ 100% chance of €4, 0% chance of €3,60 Or ☐ 100% chance of €7,70, 0% chance of €0,20
Player number: ……
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