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The paper presents a complete information model of bidding in second price sealedbid and ascendingbid (English) auctions, in which potential buyers know the unit valuation of other bidders and may spitefully prefer that their rivals earn a lower surplus. Bidders with spiteful preferences should overbid in equilibrium when they know their rival has a higher value than their own, and bidders with a higher value underbid to reciprocate the spiteful overbidding of the lower value bidders. The model also predicts different bidding behavior in second price as compared to ascendingbid auctions. The paper also presents experimental evidence broadly consistent with the model. In the complete information environment, lower value bidders overbid more than higher value bidders, and they overbid more frequently in the second price auction than in the ascending price auction. Overall, the lower value bidder submits bids that exceed value about half the time. These patterns are not found in the incomplete information environment, consistent with the model.
One of the most basic and apparently innocuous assumptions about behavior in games is that players will adopt dominant strategies. One reason why players may avoid dominant strategies, as expressed in monetary payoffs, is because they have social preferences such as spite or conditional Results and Discussion cooperation. Recent laboratory research in public good mechanism design, for example, has documented extensive failure by subjects to follow dominant strategies even in fairly simple environments, perhaps due to a desire to cooperate with others who are also cooperative (Attiyeh
Overbidding is much less pronounced in the English, ascendingbid auction. Especially in the “Japanese” version of ascendingbid auction [
This paper explores the importance of alternative, spiteful preferences as an explanation for overbidding in second price and ascendingbid auctions. A spiteful agent has utility that increases when the earnings of her rivals decrease, and so she may be willing to sacrifice some monetary payoff in order to reduce the other agent's monetary payoff (Saijo and Nakamura [
We construct a twobidder, intentionbased reciprocal decision model which belongs to the class of reciprocity models including Rabin [
Another novel feature of our analysis is that we consider a complete information environment, which strengthens the impact of social preferences such as spite and reciprocity. This is intended to approximate conditions in which bidders have some information about rivals' values or costs, such as in local government procurement settings with repeated competition between the same set of bidders. In the incomplete information environment typically employed in the auction literature, adding spiteful and reciprocal preferences as we have modeled them still results in bids equal to value in the unique (but not dominant strategy) symmetric equilibrium. By contrast, bidders with spiteful and reciprocal preferences should overbid in equilibrium when they have complete information about their rival's value and they know their rival has a higher value than their own.
Spiteful and reciprocal preferences also make the second price and the ascendingbid auction forms nonisomorphic. In an ascendingbid auction, an auctioneer or clock raises a calling price until there remains only one active bidder. A climbing calling price gradually reduces the winner's payoff. Taking this effect into account, in our sequential decision model the bidders are more aware of the extent of the other's spitefulness when they reach each new, higher calling price, because they can infer that their rival did not drop out. This makes the bidder with the higher value willing to retaliate at an earlier stage. Consequently, for the same level of spiteful preferences, in response the lower value bidders should overbid less in the ascendingbid auction than in the second price auction. Thus, the upper bound of the set of equilibria in ascendingbid auctions is likely to be lower.
The second part of the paper presents experimental evidence that provides some qualified support for the predictions of this model. In the complete information environment, lower value bidders overbid more than higher value bidders, and they overbid more frequently in the second price auction than in the ascending price auction. Overall, the lower value bidder submits bids that exceed value about onehalf the time. These patterns are not found in the data we collected for the incomplete information environment, consistent with the model. Similar to most of the literature on incomplete information second price and ascending bid auctions, bids are near values for both lowand highvalue bidders.
Researchers have recently measured and explored the impact of social preferences that include reciprocity and spite in a variety of environments, but often in noncompetitive contexts such as public good provision, twoagent bargaining and simple games. A small amount of research has studied the impact of spite in auctions, starting with Morgan
Our results are also consistent with spiteful bidder preferences, and we observe overbidding and underbidding in a pattern consistent with our model of reciprocal spite. Lower value bidders overbid relative to their values, but in response the higher value bidders underbid to punish this overbidding (or at least make overbidding risky). In equilibrium these spiteful social preferences substantially reduce the size of the set of Nash equilibria. Moreover, this combination of spite and reciprocity is the reason that isomorphism fails for the second price and ascending price auction, and the particular pattern of larger and more frequent overbids in the second price auction predicted by the model is also observed in the experimental data.
Consider, for simplicity, the case of two buyers with unit demand of values {
We consider all values and bids in terms of minimum transaction unit
In the second price auction with two buyers, the winner's payment is equal to the loser's bid. Thus, buyer
It is well known that the second price auction has multiple Nash equilibria.
In the following, we show that introducing spiteful motivations narrows the set of Nash equilibria and that the equilibrium set also differs between when bidders are spiteful with and without reciprocity.
Morgan
The black dotted line in
The main innovation of our model is to incorporate the possibility of retaliation against a bidder's spiteful behavior. We label this
Specifically, consider the case of buyer 1. If
Consider next the case of buyer 2. The size of bid deviation
Segal and Sobel [
A strategy profile
Thus, we can identify the equilibrium set for any (
(i) There exists a unique threshold bid
The equilibrium set with buyers of the spitewithreciprocity type is given by
The boundaries of equilibrium set are defined by the two buyers' threshold bids, and the equilibrium set in
In an ascendingbid auction the calling price rises by unit
Let
We are particularly interested in buyers' behavior when
For each
At each
For a given
There exists
Buyer
The interim equilibrium set
For
Let
A bid strategy profile
Proposition 2 together with Lemma 2 implies that we have an inclusion relation among all nonnull interim equilibrium sets of the following sort:
The bid strategy profile
It is immediate that the equilibrium set
The analysis up to this point boils down to the following testable hypothesis: if the bidders are all selfregarding moneymaximizing preference types, it is known that the prices observed in the ascendingbid auction should coincide in distribution with those in the second price auction. This also is the case when bidders are the type of spitewithoutreciprocity, because the upper bound of corresponding interim equilibrium sets remains the same, which is easy to check. But if some bidders are the type of spitewithreciprocity, it immediately follows from Proposition 3 that higher prices should be less frequent in the ascendingbid auction than in the second price auction.
In the incomplete information case, there are two main differences compared to the complete information case. First, the two players are now perceived as symmetric buyers (
Let
We consider the second price auction as the special case of the ascendingbid auction where the calling price is zero. Thus, our analysis focuses on the ascendingbid auction. Let
Suppose that at a given
Following the same steps in the preceding subsection 2.1B, let us define
Let a function
For each buyer, her ultimate withdrawal decision point is given by
Let a function
Then the symmetric equilibrium bid function is sequentially rational if it also generates symmetric interim equilibrium at each
There exists a unique symmetric interim equilibrium strategy
Proposition 4 asserts that the valuerevealing bid strategy is a unique symmetric equilibrium bid strategy (but not a dominant strategy) in both the second price and ascendingbid auctions. The intuition behind Proposition 4 is simple. In the incomplete information environment, buyers are no longer aware of their relative value position, which is the driving force for their spiteful bids in the complete information case. This result contrasts with Morgan
The theoretical model in the previous section generates a range of empirical implications that we evaluated in a controlled laboratory experiment. The experiment consisted of seven sessions of 12 subjects each (84 total subjects), all conducted with undergraduate econ major students at Shinshu University. Subjects bid in a series of twobidder auctions with one item for sale. Motivated by the differing testable implications derived above, the principal treatment variables were the auction format (ascendingbid versus second price sealedbid) and information conditions (complete versus incomplete). Both of these treatment variables were varied within sessions, and in four sessions all subjects bid in both formats and both information conditions. In the remaining three sessions subjects only bid in complete information, sealedbid auctions. Subjects submitted bids for 6 to 10 consecutive periods within each treatment configuration.
A secondary treatment variable was the matching rule. This was also varied within sessions, so sometimes subjects bid against the same opponent for 6 to 10 periods, and at other times subjects bid against randomlychanging opponents every period. We included fixed pairings in some sessions because the multiple equilibria (cf
In the complete information treatment, the two possible resale values for the two bidders were 700 and 800 yen. These two values were randomly assigned each period, and this was common knowledge. Therefore, after a bidder learned that her resale value was 800 yen, for example, she knew with certainty that the other bidder's resale value was 700 yen. In the incomplete information treatment, resale values were drawn independently for each bidder each period from the discrete uniform distribution between 500 and 800 yen. The uniform distribution is the most commonlyused distribution in the extensive literature on independent private value auctions (Kagel, [
Subjects received the difference between their resale value and their price paid when they won the auction. The price was determined by the lowest bid or the first dropout price, depending on the auction format, with the highest or the remaining bidder winning the auction. (Consistent with the theoretical model, ties were resolved randomly.) Subjects received written instructions to describe the auction rules and procedures, which they first read in silence before the experimenter read them aloud. The instructions included both equation and payoff table explanations describing the relationship between bidder actions, allocations, and payoffs. A translation of the instructions is shown in
In order to orient the reader, we first summarize the data using a series of figures before turning to formal hypothesis testing. Recall that in the complete information environment, the valuations are either 700 or 800 yen.
In all panels of these figures, the modal bid equals the bidder's value. Overbidding by the lowvalue bidder, however, is pronounced in
Careful inspection of the figures should remind the reader that bids were constrained to 10yen intervals, while value draws could correspond to any integer yen amount. Therefore, by design the bidders will typically not be able to bid exactly equal to their drawn value. Overbidding and underbidding appear about equally common on the figures, and on average bids are within one percent of value.
This section reports tests of the hypotheses generated by the complete information reciprocity model presented in Section 2.1.
The regression shown in column 1 determines whether bids relative to values are different between the lowvalue and the highvalue bidders. The difference (Bid – Value) is actually lower for the lowvalue bidder, but this is mainly because of a small number of “throwaway” and overtly collusive bids, which were more common in the periods with fixed pairs of bidders. Although such bids were relatively rare, they add substantial variance and are a major reason that the regression coefficient estimate does not approach statistical significance. By contrast, the random effect probit model in column 2 is more robust to such outliers, and it indicates that the likelihood of overbidding is much higher for lowvalue bidders. Lowvalue bidders overbid 47 percent of the time, whereas highvalue bidders overbid only 20 percent of the time. This difference is highly significant and is consistent with Hypothesis H1.
The theoretical model's predictions are based on agents who have spiteful preferences, which suggests that empirical results might be sharper when the analysis is focused more narrowly on those types of subjects. Therefore, columns 3 and 4 present estimates for the subset of subjects who bid above their value at least half the time when they had the low value draw. These 39 subjects represent roughly half the sample and their bids most clearly reveal spiteful preferences. Conclusions drawn for this spiteful subset of bidders are similar to those drawn for the entire sample.
The figures and the summary statistics presented above provide some suggestive evidence in support of H2. For a formal statistical test, however, we must account for the censoring of the bids in the ascendingbid auction. Recall that for this institution we do not observe the bid of the winning bidder—only the price at which the other bidder drops out. This censoring occurs for 30 of the 305 (10 percent) of the lowvalue bidders' bids. We employ survival analysis to account for this censoring, where “failure” occurs when the rival bidder drops out. The approach we use accounts for differing censoring points since the rival bidder drops out at different prices in different periods.
For all other bids < 800, the survivor function estimates imply that the sealed bid auction has a higher probability of observing bids exceeding all particular bid prices that are higher than 700. For example, if we define large overbid as a bid greater than or equal to 750, large overbidding occurs with probability 0.22 in the ascendingbid auction, and with probability 0.34 in the sealed bid auction. A logrank test rejects the null hypothesis that these survivor functions are equal (
Since large overbids by lowvalue bidders are more common in the sealedbid auction, a natural auxiliary hypothesis is that transaction prices are also higher in the sealedbid auction:
Section 2.2 established that even with spiteful preferences, in the incomplete information (unknown values) environment a unique symmetric equilibrium strategy exists where each bidder bids at her own value.
Support for this hypothesis may be difficult to obtain since in the incomplete information environment subjects do not know when they have the low or high value draw. They may have reasonably confident beliefs when they have very low value draws near 500 or very high value draws near 800, but not when they have intermediate values in the range between 600 and 700.
Because bids are typically not observed for the higher value bidder in the ascendingbid auction, to test this hypothesis we consider only the sealed bid auction where all bids are observed. In order to make the two environments comparable, we normalize all bids by subtracting the associated value draw. We then regress this difference on a dummy variable for the complete information environment, using the same control variables as in the regressions reported above. To be consistent with Hypothesis H4, the dummy variable for the complete information environment should be positive for lowvalue bidders (H4a) and negative for highvalue bidders (H4b). The results, shown in
Another implication of the equilibrium result that bids should equal values in the incomplete information environment is that there should not be significant differences between bidding behavior for low and high value bidders.
This hypothesis is the counterpart of Hypothesis H1 (b), where for the complete information environment the hypothesis was that overbidding
The final hypothesis is the incomplete information counterpart to Hypothesis H2. Recall that with complete information, overbids by the lowvalue bidder are predicted to be larger in the secondprice compared to ascendingbid auction. By contrast, with incomplete information there should be no systematic difference between the bids across auction institutions.
We test Hypothesis H6 in exactly the same way that we tested Hypothesis H2. To account for the bid censoring in the ascendingbid auction, we again employ survival analysis. In the incomplete information environment, this censoring occurs for 30 of the 372 (8 percent) of the lowvalue bidders' bids.
We have investigated bidding behavior in both complete and incomplete information environments for twoperson second price sealedbid auctions and ascendingbid auctions for a single indivisible object with independent private values. Our intentionbased bidding model features individuals who are reciprocally spiteful. A lower value bidder may be spiteful in the sense that he receives a positive psychological payoff when he loses if he reduces the winners' payoff. A highvalue bidder who faces a spiteful bidder's overbidding would reciprocate by underbidding to increase the likelihood that the spiteful bidder wins and incurs a negative monetary payoff. The possibility of this underbidding causes such spiteful bidders to refrain from bold overbidding. Our theoretical analysis in the complete information environment indicates that the equilibrium bidding strategy differs from the Nash equilibrium strategy set generated without spite and reciprocity, in the following three respects. First, the equilibrium strategy set is much smaller and does not contain any inefficient outcomes. Second, although a strategy of “bidding at one's value” is no longer part of an equilibrium strategy profile, the equivalent outcome in which the lower value bidder bids at her own value and loses is one of the equilibrium outcomes. Third, the threat of reciprocal underbidding is more important in ascendingbid auctions than second price sealedbid auctions, since a rising calling price directly reveals the spiteful intention of a lowvalue bidder. This leads to a lower equilibrium spiteful overbidding in ascendingbid auctions, which implies an even smaller equilibrium set with lower price upper bound.
As summarized in
Subjects' decisionmaking seems to be different when they do or do not know each other's values. When they have complete information about all bidders' values, this allows them to evaluate their relative payoffs. A lowvalue bidder in our environment who bids 750, for example, knows that this bid will likely reduce the winning bidder's payoff by half relative to the payoff if all bids equal values. A bidder with the higher value can also perceive the spiteful intentions of her opponent's bid in the complete information environment. It is this spiteful intention that induces underbidding by the higher value bidder. This is the driving force behind our theoretical result that bidders make more timid overbids in the ascendingbid auction, because the rising calling price directly reveals the lower value bidder's spiteful intention. This can be interpreted as an additional evidence of negative reciprocity at work, but here in the context of an auction, consistent with negative reciprocity observed in the context of ultimatum and related games (e.g., see Charness and Rabin [
Example of Nash equilibrium set in second price auction with standard moneymaximizing preferences.
Example of equilibrium set with spiteful motivations but no reciprocalspite motivation (spitewithoutreciprocity).
Example of equilibrium set in the second price auction with spite and reciprocalspite motivations (spitewithreciprocity).
Example of interim equilibrium set in the ascending bid auction with spite and reciprocalspite motivations (spitewithreciprocity).
(
Distribution of second price sealed auction bids for value = 800.
All bid pairs for random groups complete information secondprice sealed bid auctions.
Example fixed pairs sealed bids in complete information environment.
(
Second price sealed auction bids for highest value in incomplete information environment.
Comparison of bid (survivor) functions for complete information with value = 700.
Cumulative distribution function of transaction prices for complete information auctions.
Comparison of bid (survivor) functions for incomplete information lowvalue bidder.
Regression models of bid deviations from value and overbidding: Complete information environment, sealed bid auction.
Model  

 
1 (Random Effects GLS)  2 (Random Effects Probit)  3 (Random Effects GLS)  4 (Random Effects Probit)  
Dependent Variable  Bid − Value  = 1 if Bid > Value  Bid − Value  = 1 if Bid > Value 
Dummy Variable = 1 if  −35.03  1.30 
−6.12  1.54 
Lower Value  (47.56)  (0.12)  (38.10)  (0.15) 
Dummy Variable = 1 for  5.40  0.22 
20.72  0.08 
Fixed Pairings  (5.95)  (0.11)  (26.49)  (0.13) 
1/period  94.99  −0.11  155.03  −0.26 
(85.23)  (0.19)  (170.91)  (0.23)  
Intercept  −21.17  −1.61 
−28.47  −0.45 
(27.18)  (0.22)  (46.29)  (0.20)  
Observations  1150  1150  542  542 
Number of Bidders  84  84  39  39 
R^{2} or Loglikelihood  0.01  −470.0  0.01  −273.5 
Notes: Standard errors (in parentheses) are based on a subjects random effects model and for the GLS regressions in columns 1 and 3 are calculated to be robust to unmodeled correlation of choices within clusters defined by sessions.
denotes significantly different from zero at the fivepercent level, and
denotes significantly different from zero at the onepercent level.
Regression models of transaction prices and likelihood of high prices: Complete information environment.
Model  

 
1 (Random Effects GLS)  2 (Random Effects Probit)  3 (Random Effects GLS)  4 (Random Effects Probit)  
Dependent Variable  Price  = 1 if Price > 740  Price  = 1 if Price > 740 
Dummy Variable = 1 if  −16.73  0.04  11.36 
0.77 
SealedBid Auction  (12.51)  (0.12)  (3.07)  (0.16) 
Dummy Variable = 1  28.73 
−0.05  −4.73  −0.17 
for Fixed Pairings  (10.87)  (0.10)  (2.91)  (0.15) 
1/period  27.14  −0.02  −3.98  −0.20 
(19.21)  (0.17)  (4.92)  (0.25)  
Intercept  697.19 
−0.66 
754.09 
0.22 
(18.18)  (0.21)  (3.59)  (0.18)  
Observations  887  887  312  312 
Number of Sessions  7  7  7  7 
R^{2} or Loglikelihood  0.01  −465.6  0.05  −178.7 
Notes: Standard errors (in parentheses) are based on session random effects models.
denotes significantly different from zero at the fivepercent level, and
denotes significantly different from zero at the onepercent level.
Regression models of bid deviations from value to compare the complete and incomplete information environments.
Model  

 
1 (Lower Value Bidders)  2 (Higher Value Bidders)  3 (Lower Value Bidders)  4 (Higher Value Bidders)  
Dependent Variable  Bid − Value  Bid − Value  Bid − Value  Bid − Value 
Dummy Variable = 1 if  −43.03  6.34  24.86 
27.86 
Complete Info.  (40.96)  (14.48)  (12.07)  (29.76) 
Environment  
Dummy Variable = 1 for  −20.28  21.53  −27.16  51.78 
Fixed Pairings  (17.24)  (12.32)  (18.27)  (42.97) 
1/period  20.57  88.07  20.61  203.74 
(18.21)  (82.36)  (23.37)  (218.76)  
Intercept  27.66  −34.26  15.41 
−90.53 
(20.02)  (36.82)  (4.40)  (101.01)  
Observations  957  956  424  391 
Number of Bidders  84  84  39  39 
R^{2}  0.00  0.01  0. 02  0.02 
Notes: Standard errors (in parentheses) are based on a subjects random effects model and are calculated to be robust to unmodeled correlation of choices within clusters defined by sessions.
denotes significantly different from zero at the fivepercent level,
denotes significantly different from zero at the onepercent level.
Regression models of bid deviations from value and overbidding: Incomplete information environment, sealed bid auction.
Model  

 
1 (Random Effects GLS)  2 (Random Effects Probit)  3 (Random Effects GLS)  4 (Random Effects Probit)  
Dependent Variable  Bid − Value  = 1 if Bid > Value  Bid − Value  = 1 if Bid > Value 
Dummy Variable = 1 if  −4.30  −0.05  −13.69  −0.10 
Lower Value  (6.70)  (0.13)  (7.25)  (0.20) 
Dummy Variable = 1 for  −3.58  0.86 
−15.11  1.11 
Fixed Pairings  (5.40)  (0.13)  (13.10)  (0.20) 
1/period  −2.30  −0.76 
11.07  −0.43 
(5.22)  (0.23)  (10.44)  (0.35)  
Intercept  3.16  −0.85 
17.28 
0.05 
(3.36)  (0.30)  (6.99)  (0.35)  
Observations  763  763  273  273 
Number of Bidders  48  48  17  17 
R^{2} or Loglikelihood  0.00  −325.8  0.02  −136.2 
Notes: Standard errors (in parentheses) are based on a subjects random effects model and for the GLS regressions in columns 1 and 3 are calculated to be robust to unmodeled correlation of choices within clusters defined by sessions.
denotes significantly different from zero at the fivepercent level, and
denotes significantly different from zero at the onepercent level.
Summary of theoretical predictions and hypotheses support.
 

Theoretical Prediction  Consistent with Hypothesis:  Theoretical Prediction  Consistent with Hypothesis:  

 
H1  H2  H3  H4  H5  H6  
Conventional Model  No  No  No  No  Yes  Yes  
Spite without Reciprocity  Yes  No  No  No  Yes  Yes  
Spite with Reciprocity  Yes  Yes 
Yes 
Yes 
Yes  Yes 
Financial support was provided by GrantinAide for Scientific Research by Japanese Education Ministry #14330003 and #19330041. We are grateful for valuable comments provided by Masaki Aoyagi, Takuma Wakayama, conference audiences at the European Regional Meeting of the Economic Science Association, the AsianPacific Regional Meeting of the Economic Science Association, the Annual Conference of Experimental Economics in Japan, and seminar audiences at Osaka University, Tokyo University, and Kobe University. We acknowledge research assistance by H. Fujii, K. Suzuki, K. Kondo, H. Ohki, R. Tanaka, N. Mizuno, and H. Ando. An earlier version of this paper circulated under the title “Spite and CounterSpite in Auctions.”
To identify buyer
Specifying
A solution for
(i) There exists a unique solution
Consider the case of buyer 2 first. The solution
For a given solution
Let us state the best response correspondence for each buyer, which we repeatedly use in the proof of Proposition 1 below. The best response correspondence of buyer 2 for a given buyer 1's bid can be given by
Buyer 1's best response correspondence for a given buyer 2's bid can be stated as
Suppose that a bid profile
Conversely, suppose that
As for the case where
We follow the same steps we used in Section A1. First, consider the difference between buyer
Specifying
For
There exists a unique solution
For all
There is no solution to
(i) We can identify
Note that
(ii) and (iii)
Consider first
(iv) From the first line of (
The second step is to find a number
There exists
There exists a unique threshold bid
For all
There is no threshold bid for buyer 1 for all
The best response correspondence of buyer 2 for a given buyer 1's bid at decision point
Buyer 1's best response correspondence for a given buyer 2's bid at decision point
For
An interim equilibrium set
Consider the case
Consider the case
Consider the case
To sum up,
Since bidding at
(ii) The part of Proof of Lemma 3′ dealing with the case
(i) Suppose that
Conversely, suppose that
Suppose next that
To examine the remaining case of
Second, consider the case
The last case to examine is the case where
Proposition 2 implies that no buyer stops the auction at any
Consider symmetric buyers who employ the same bid strategy
Let Δ be the set of cumulative probability distributions with support
The first term of (
Consider a continuous and continuously differentiable bidding function
Note that the integral of (
It is immediate that a bidding function
To prove uniqueness, suppose that
Next suppose that
Thank you for participating in our experiment. This is a study on auctions. The Ministry of Science and Education has provided funds for our research. The instructions are simple. You are being paid 1,000 yen in cash as startup money. All you have to do is to make a bid in each of the auction situations according to the rules described in this instruction. In each round of auctions, depending on the bid you make and the resolution of the uncertainty, you may receive or pay a specified amount of money as a result of transaction.
Your acts will be recorded and kept only in terms of purely anonymous data for the academic research on microeconomics.
Furthermore, and most importantly, this is not a project to see if you can make a “right” decision, or if you can come up with a “correct” answer.
We will ask you to bid, not for a commodity as in a real auction market, but for a monetary prize. In other words, you are going to compete for the right to earn a monetary prize. In an exchange for the prize, a winner has to pay according to the rules of the auction.
At the beginning of each round of auction game, we will assign each of you a number. This represents a prize value to you. Once you win the auction, then you will receive the prize worth that value number, and you must pay the amount specified by the corresponding auction rules to obtain the prize. Your payoff is the difference between the prize and the amount of payment. If you do not win, you will not receive any prize and pay nothing, that is, your payoff is zero.
You know your prize value for sure once it is assigned to you. But you may or may not know the value assigned to the other participants in this experiment, depending on the experimental design.
We will run two kinds of auctions. One is a sealed bid auction, called the secondprice auction, and the other is an open bid auction, called the ascendingbid auction. The experiment starts with a session of sealed bid auctions, containing eight rounds of auctions each with fresh value assignment, and then proceeds to a session of ascendingbid auctions also containing eight rounds.
Each of you is paired with another anonymous participant. Every auction round starts with the value assignment. You and the other participant receive a number each as your prize value, which we call “assigned value.” After receiving the assigned value, both of you are asked to make a bid. That is, you have to specify a number to submit to the experimenters. We collect those submitted bids, and identify a bidder who submitted the higher bid between the two of you, as a winner. The winner receives the right to obtain her assigned value.
You know that you are paired with someone, but you do not know who is paired with you. There are two treatments as to pairing. In one treatment, you are paired with a different person, randomly determined, in every round. We call this treatment, “Part 1.” In the other treatment, you are paired with a person randomly in a first round, and continue to bid against this same anonymous person in the rest of the rounds. We call this treatment “Part 2.” In Part 1, we start with the second price auction for eight rounds, and then proceed to “Part 2” for eight rounds. After that, we conduct the ascendingbid auction experiments under the treatment of Part 1 and then Part 2.
Upon being instructed to do so, the first thing you have to do is to double click the icon indicated by “Sealed Bid” on your windows screen. Then you will see the dialogue window shown below, popping up in your screen.
Make sure that you see the header “Sealed Bid” on the left of the dialogue window. Your ID number will be shown in a box in the first line of the dialogue window.
Next, the experimenter will send you a private “assigned value,” which will appear in the box of the dialogue window labeled “Value Assignment.” In the event you win, this is the prize you are entitled to earn by making the appropriate payment.
Only after the assigned values are distributed, the face of the “send bid” button turns black and you can then type the amount you decide to bid in the box labeled “Your Bid.”
You will find the winner's ID number shown in the box labeled “Winner's ID” in the middle part of the dialogue window. The further right box under the header “the second highest bid” will show the amount that the winner has to pay, which is the lower bid in the pair. If you are the winner, the further left box under the header “payoff” will show the number that is your payoff obtained by subtracting the payment from your assigned value. On the other hand, if you are not the winner, the number in the box under the “payoff” is zero, since you do not get the prize and do not have to make any payment.
For example, consider the case where your assigned value is 400 yen. Suppose that you bid 300 yen, and the other participant paired with you bids 350 yen. In this case, you are not the winner since your bid is not the highest, and your payoff is zero.
Next, suppose that the other participant paired with you bids 200 yen instead. Your bid, 300 yen, is the highest bid and you become the winner. Then you win the prize of 400 yen but must pay 200 yen, which comes down to the payoff of 200 yen.
If you win,
If you do not win,
If you and the other participant bid the same amount, then a winner will be randomly selected. In this case, you will be a winner with probability of 50%.
In the previous example, suppose that two of you bid 300 yen. Then, you can obtain the prize of 400 yen and make the payment of 300 yen with probability of 50%, and obtain zero payoff otherwise.
In the very bottom of the dialogue window, you find the “comments” box. Please type why and how you have come to a bid decision, when we, the experimenters, ask you to do so. Having finished typing your comments, click the “send comments” button. Your comments won't be sent to the experimenters unless you click the “send comments” button.
In the ascendingbid auction, we, the experimenters, raise a price gradually from a very low level. At the beginning of the ascendingbid auction, all of you are “active” in the sense that you are bidding at that price level. Unless you indicate that you wish to withdraw from bidding, you are considered being active and willing to pay that amount of indicated price in the event that you become the winner at this very moment. As long as two of you are active, we continue to raise the price. At the moment when one bidder withdraws from bidding, then the remaining bidder becomes the winner awarded with the prize, and she has to pay the last price level at which two bidders were active. After you double click the icon named “Ascending Bid” on your screen, you will see the following dialogue window.
Please make sure that your ID number is shown in the box located at the top of the dialogue window.
Similar to the experiment of the secondprice sealed bid auction, you will see your assigned value pop up in the corresponding box. If you become a winner, this is the amount you will get as a prize. Underneath that box, there is a box indicating the “current price”. Once the auction starts, the number shown in that box rises gradually from 0 yen. As the number increases, the price thermometer on the left of the dialogue window grows higher. The price increases by 10 yen.
Let us consider the case where the number in the “current price” box is 100. Suppose that you are willing to pay 100 yen if you win at this moment, but you would not want to pay more than 100 yen. In this case,
As long as the “current price” increases, the other participant paired with you is active. When one of the pair drops, the process of the ascendingbid auction stops. If the price stops increasing before you click the “Drop” button, this means that you win. Then, you will obtain the prize worth your assigned value, and your payment is 100 yen, which is one unit (= 10 yen) lower than the price level at which the process stops. The payment amount will be indicated in the “payment” box. The ID number of the winner will be shown in the “winner” box. Your payoff will be indicated under the large box in the middle under the header of “payoff.” If you win, your payoff is your prize minus the payment. If you do not win, your payoff is zero. Such information will be listed in that box in each round. Each new result enters at the top of the list.
Let us review the above details by some examples. Suppose that your assigned value is 400 yen. Suppose that the number in the “current price” box increases and stops at 350 before you click the “drop” button. This means that you win, and your payment is one step earlier than the 350 level, which is 340 yen. Your payoff is 400 minus 340, which is 60 yen.
Suppose that you click the “drop” button at 360. It means that you do not win, and your payoff is zero.
When two of you simultaneously click the “drop” button, then a winner will be randomly selected, and the winner pays the price at the moment of withdrawal minus 10.
At the bottom of the dialogue window, there is the “comments” box. Describe why and how you figure out the amount of price at which you choose to withdraw, when instructed to do so. To transmit your comments to the experimenter, click the “send” button.
After you sit down in front of the computer terminal, each of you will be given ID number, which you continue to use during the full course of experiment. Do not show that number to any other participant. It is very important for the academic quality of this experiment that you keep your ID number completely private.
Having mentioned elsewhere, you are paired with another person among those participants in this room. You will never be told with whom you are paired. In Part 1, the person you paired with will be determined randomly every round of the auction, while in Part 2, your paired person will be determined randomly in the first round, and maintained the same anonymous person through out all rounds.
There are two ways in which your value is assigned in a round. In one case, which we call “treatment VA1,” your value is selected randomly from a predetermined set of values, which consists of 700 and 800, every round. Your computer screen informs you of your own value only, but if you receive 700, it automatically implies that your paired participant receives 800, and vice versa.
In the other case, which we call “treatment VA2,” your value is a purely random variable from a predetermined range of value of [500,800] with uniform distribution. In every round, a fresh value is drawn independently. Again, you are informed of your own value only, and never be informed of the realized value drawn for your paired person. But the same random procedure is applied to both of you and your paired person, independently.
In order to get familiar with the auction rules, the first four rounds are for your practice. The outcomes from the subsequent rounds are recorded for real prize and payment.
Please make sure that you fully understand the rules and procedure. You will be given a short quiz after all the instruction is completed.
Your payoff is a joint product of your own bid choice and a bid chosen by the other participant paired with you. The other participant's payoff is also a joint product. Though there are numerous combinations of your bid and the other participant's bid, in the treatment VA1, we provide you a payoff table that looks like the figure shown below. The table lists your payoffs and the other participant's payoffs under the various but limited number of bid combinations, because of the space limit.
The figure below shows the example of the payoff table with your assigned value being 350 yen and the other participant's being 400 yen. The further left column lists the possible bids you can choose, and the first row lists the bids available for your paired person. Though those possible bids are listed with 50 yen increments, you are free to bid in 10 yen increments in the auction.
There are two numbers shown in each cell. The number in the upper left of the cell is your payoff and the number in the lower right of the cell is the other participant's payoff.
For example, suppose that your bid is 350 yen and the other participant bids at 300 yen. Then, your bid is higher and you are the winner. You will be awarded 350 yen prize, your assigned value, and you have to pay 300 yen, the other participant's bid level, so that your net payoff is 50. The other participant who loses receives zero net payoff.
Let us consider another case. Suppose that you bid 350 yen and the other participant bids 400 yen. Since her bid is higher, she is the winner and gets 400 yen prize and pays 350 yen equal to your bid. Her net payoff is 50, subtracting 350 from 400, which is shown in the lower right of the cell in the row of your 350 bid and the column of the other participant's bid 400.
Another important case is the event of tie. Suppose that you bid 300 yen and the other participant bids also 300 yen. Then the winner will be randomly selected. That is, you become the winner with 50% chance, receiving your value prize of 350 yen and paying 300 yen. Your net payoff is 50 yen in when you win, and your expected payoff is 25 yen, which is shown in the upper left of the corresponding cell. Since the other participant's assigned value is 400 yen, her expected payoff is 50 = 0.5*(400 – 300), which is shown in the lower right of the corresponding cell.
There are two parts regarding pairing; In Part 1 you are paired with a new randomly selected person among other participants every round. In Part 2 you are paired with a randomly selected person in the first round and maintain the same person in the rest of the rounds.
There are two treatments regarding value assignment. In treatment VA1, you are assigned either 700 or 800 yen randomly, and in treatment VA2, you are assigned with a random number drawn from the range between 500 and 800 yen according to the uniform distribution. Each draw is independent.
There are two types of auctions; one is the secondprice sealed bid auction and the other is the ascendingbid auction, and the sealed bid auction precedes the ascending bid auction.
As a total, there are 2 × 2 × 2 variations in our experimental treatments. We will run on average 8 rounds for each treatment. At the beginning of the first sealedbid auction experiment, there are four rounds set as the practice session. All payoffs generated during the practice session will not be counted. After completing the practice rounds, then we move on to the first set of 8 rounds of auction and start recording the realized payoffs.
Your payoffs will be all recorded. At the end of the experimental session, we will pay you the cumulative amount in cash, on the spot.
Please read this instruction carefully. It is very important that you understand these instructions. Should you have any questions, please feel free to ask us.
Once we start explaining the instructions, you are not allowed to talk to any other participants. You can only talk to us, the experimenters, if necessary. You are not allowed to look at the other participants' PC screens. This no talking and no peeking rule is very important for the validity of our experiments. Not conforming to this rule would jeopardize the quality of our experiments.
The “Japanese” version of English auction is the type of ascendingbid auction we employed in our theoretical and experimental analyses, where all the bidders are active from the start by pushing their buttons as default and they leave the auction by releasing their buttons. Thus, they are not allowed to leave the auction and reenter at the later stage. It is well known that the valuerevealing bidding strategy generates the unique dominant strategy equilibrium and there is no room for any asymmetric equilibrium such as jumpbidding strategy (see Milgrom and Weber [
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Their original paper deals only with the incomplete information environment, which we extend here to the complete information setting.
Their model has the simplest form among various reciprocity models proposed, yet general enough to include such models as Rabin [
We set reference losing payoff to be zero for both buyers. The reciprocity model prescribes utility as a sum of own monetary payoff and psychological payoff. Note that buyer
To be precise, the negative weight means spite motivation.
It is common in reciprocity models to normalize the deviation from some reference by the payoff range defined by the maximum and the minimum payoff available under the relevant consideration. Here, we add
In order to identify
Based on Lemma 1, we can derive the best response correspondences for both buyers. See
The position of buyer's threshold bids depend on
This is not the only case where allowing for the possibility of reciprocity leads to eliminating some of Nash equilibria generated by the standard model. The wellknown example is an ultimatum game which has multiple Nash equilibria under the standard model. Allowing reciprocity eliminates the proposer's strategy of choosing the most selfadvantageous offer expecting the receiver to accept it that is subgame perfect Nash equilibrium under the standard model, but keeps other Nash equilibria that would have failed the subgame perfect refinement.
A change in the weight
In the same manner as described in [
Though we omit the argument regarding beliefs in our analysis, it may be worthwhile to point out the following; First, it is clear from Definition 3 that every equilibrium bidding strategy profile
The psychological dynamics are different in nature from the comparison of Nash equilibrium versus subgame perfect Nash equilibrium. The subgame perfect equilibrium set in the conventional, selfinterested preferences ascendingbid auction is equivalent to the Nash equilibrium set in the second price auction without its inefficient elements. However, with the spite motivation the maximum equilibrium price in the ascending bid auction is
The corresponding tie bid
It is well known that the symmetric equilibrium bidding strategy must be strictly increasing in the buyer type (value in this model) in the auction where the highest bidder wins; for example, see Milgrom [
These figures pool the data from the treatments with fixed groups and randomlyreformed groups of bidders. The subsequent analysis controls for different matching rules in the parametric regression models.
The censoring problem is much greater for the highvalue bidder, since in the ascendingbid auction this bidder wins in 278 of the relevant 309 auctions. Therefore, we do not report a bid distribution for the highvalue bidder for this auction institution, nor do we use such bids in any of the statistical tests that follow.
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