1. Introduction of Double Auction
As one of the most popular way to trade, auction has a long history. According to the different kinds of market structure, there are two kinds of auction, one-side auction and two-side auction (i.e. double auction). Compared with one-side auction, the market structure of the double auction is many-to-many, which means that there are more than one seller and more than one buyer, so both the buyers and sellers lose their own comparative advantage which exists in one-side auction. The relationship between them is a kind of equality between demand and supply.
In recent years, with the development of the global economy, there are more challenges for new auction theory and its applications, which are shown in the following aspects. First, with the development of the Internet and communication technologies, e-commerce has become a new business method. At the same time, internet auction has been widely used in the field of e-commerce. Since auction is widely used in the field of trading non-scarcity goods, it changes structure of auction, which used to be based on the buyer’s market or the seller’s market. As double auction can solve the problem of collusion and malignant bid, it has become a widely used way in e-commerce.
For example, both NYSE and Chicago Exchange have put different kinds of double auction into practice. With the growing of financial market and e-commerce, the need for better auction rules is surely increasing. Besides, as enterprise restructuring has become a hot topic, double auction is found to be a way to deal with property transactions, mergers and acquisitions. What’s more, with the upgrade of demand structure, the diversification strategy for enterprise business and influence of the supply chain management, there are diversified production, diversified manufacturing and diversified demand. Many famous car companies such as Ford are using combinatorial double auction to the sale and purchase of auto parts. When auction is used in the above fields with various needs, there is a great need for the development of new auction model.
Compared with other auction mechanisms, double auction can not only solve the problem of monopoly, but also reduce the duration and costs of trade. However, there are still some problems about it, such as trading rules, the way to release information and the formation of transaction price.
According to the above reasons, double auction has attracted much attention. And the core theories about it are Smith’s mystery and Hayek problem, which try to explain how the double auction can reach the equilibrium price and equilibrium quantity predicted by the demand-supply model, under the conditions that there is no complete information and few people in the auction.
2. Design of Experiment
In order to do some further analysis, we conducted an experiment which simplified real double auction situations with the help of several undergraduate students under four assumptions:
First, rational man assumption, which means everyone wants to maximize his/her profit. Second, the goods are homogeneous, so there is no monopoly. Third, there is no cost of sending or getting information, i.e. the transaction cost is zero. Fourth, the participants have no information about the experiment parameter (the value for buyer or cost for seller) before they join the experiment.
(2) Trading rules
a. When the trade begins, the buyers give their quotes first. Who raises his hand first will quote first. Then, the sellers offer the price they ask. b. Both the sellers and the buyers don’t know the information about the other side. c. When quoting, the seller must ask for a price that is lower than the former seller, and the buyer must ask for a price that is higher than the former buyer. When the price of the two sides is equal, a transaction is reached. d. The trade for each group is divided into three periods and each period lasts for 5 minutes.
a. Divide all the participants into two groups, buyer and seller, and each group has 8 people. b. Zero phase trade. In order to make the participants familiar with the rules, there is a zero phase trading, the result of which is not included into the final result. c. Begin official trade.
d. End trade and take down all the results, including every participant’s own profit in each period and their total profits.
Double Auction Experiment
1. Concept Definition and Parameter Settings
If we carefully consider each quoting process, we will find the game process in double auction. There are inside game and outside game in the experiment. Firstly, there is among-group game between buyer and seller and they want to drive down or push up prices to earn as much profit as possible. And based on the theoretical design, a buyer and a seller can be trading at least one unit of commodity in one trading period. Experiment shows that each buyer and seller generally trade exactly once each period.
Secondly, there is within-group game both among buyers and among sellers. Taking buyer as an example, when the seller quotes a price, buyer X can choose to trade or wait. If choosing to trade, he will probably miss a next lower quoted price although obtaining a certain surplus. However, if he chooses to wait, then the other buyers may intervene in transaction, which constitute competition for buyer X. Each buyer and seller should consider the competition within group, acting as an individual in the game. Meanwhile, they should also consider the competition between groups, acting both as an individual and a group in the game.
As can be seen, the core of the complex game process is the relationship of individual-collective interaction. By observing dynamic transaction and competitive process among individual, between the individual and the collective and between the collective, we try to find the interpretation of some problems. Therefore, constructing appropriate parameters, we proceed from rational assumption to define individual rationality and collective rationality respectively.
(1) Constructing quoted price functions of the buyer and the seller Firstly, assuming the quoted price function of the buyer i is bidi = vi(1-πi) where bidi denotes the quoted price of buyer i, vi denotes the willing-to-pay of buyer i, πi denotes expected rate of return.
Secondly, assuming the quoted price function of the seller is: askj = cj(1+πj) where askj denotes the quoted price of seller j, cj denotes the cost of seller j, πj denotes the expected rate of return of seller j.
(2) According to the market parameters and theoretical equilibrium price(7.8) of buyer and seller, we can calculate the standard rate of return of every buyer and seller, πi* and πj*, respectively.
(3) On this basis, we assume that individual rationality index is bri=πi/πi* of the buyer, srj= πj/πj* of the seller. The higher the index, the higher is the degree of the buyer or the seller’s individual rationality, for it indicates more incentive to maximize their interests strongly. The standard individual rationality is 1, indicating that the expected rate of return and the actual rate of return of the buyer or the seller’s are consistent.
(4) Similarly, we can define the buyer’s collective rationality index as sbri=πi/πi*, where πi is the average expected rate of return of all the buyers who participate in a transaction, πi* is the standard average rate of return of the buyers. What’s more, we can define the seller’s collective rationality index as ssrj=πj/πj*. The higher the index, the higher is the degree of collective rationality, because it indicates more strongly motivation to maximize their collective interests. The standard collective rationality is also 1, indicating that the expected collective rate of return and the actual collective rate of return of the buyer or the seller’s are consistent.
2. Analysis of Parameters
In order to study the market inefficiency of the second group, we selected two sets of data as a typical example. Group #1’s transaction price is equal to the theoretical equilibrium price (7.8), while Group #2’s transaction price is far from the theoretical equilibrium price, only 6.3.
Let’s observe the quoted price of the buyers and the sellers in the first group,
We find that in the second offer, S2’s quoted price fell directly from 10 to 7.5, its price is less than the theoretical equilibrium price 7.8, which makes the seller’s quoted price become farther and farther away from the equilibrium price. Instead, the buyer’s offer is steadily rising with smaller fluctuation, which makes the buyers successfully lower price.
Let’s move to the quotes of the buyers and sellers from Group #2.
Obviously, the quote fluctuations of both sides are smaller and the final equilibrium price (Pe) is closer to the ideal value. Therefore, we may infer that in the double auctions where the number of quotations reaches a certain level, if we want the final transaction price to be close to the theoretical equilibrium price, fluctuations in both quotations cannot be too great, otherwise it will be very easy to deviate from Pe, consequently making either buyer side or seller side gains too much surplus.
The reason why we emphasize a certain level of the number of quotations is that if either one side of the bidders or the askers offers a price that is quite close to the expected one, a deal is very likely to be made. Actually, we do see lots of quick transactions in the experiment, especially the 2nd round of Group #1, where there would be expected to exist fluctuation.
But that quick deal is based on that either buyer or seller or both of them have an intuition about Pe or are simply informed of that price. If the information for both sides is so inadequate or even zero, then it will seem quite important to estimate Pe by price fluctuation during the first several quotes. On the other hand, however, substantial quote fluctuations will not only compress the quote space afterwards, but also impair one’s own market power, while increasing the other side’s market power and surplus.
Let’s now look at the changes in individual rate of return and individual rationality index of Group #1 and #2.
A. Expected Rate of Return
The fluctuation of expected rate of return is distinct around the seller S2, where the expected rate of return of buyers is far higher than that of the sellers.
B. Individual Rationality
As can be seen, buyers’ individual rationality fluctuates more than the sellers’, while the degree of buyers’ individual rationality (>1, since round 2) is much higher than that of sellers’ (<1, and decreasing since round 2). Thus, we can assume that there is a collective irrational phenomenon among the sellers.
Usually, both buyer and seller will hope to receive higher surplus and try to make the price more beneficial to themselves. So it’s reasonable to see that individual rationality index fluctuates a bit around the standard line or even higher than it, just to make the final transaction price and surplus closer to the theoretical ones.
But in this case, seller S2 suddenly lowers his expected rate of return (from the normal 2.22 to 0.86), probably because he underestimates Pe or just wants to quickly sell the product. It’s quite hard for the following sellers to reach the ideal level, but can only quote under 1, giving buyers more space for quote and more power to control the market.
If the sellers following S2 used to have higher expectation than 1, then a phenomenon called inconsistency of collective rationality occurs by the influence of inconsistency of individual rationality. When it happens, the irrational collectivity’s power is impaired and Pe and surplus will end up more beneficial to the other collectivity. As can be seen from the experiment results, the collective rationality index of buyers is 2.793, which is much higher than that of sellers, 0.808.
A. Expected Rate of Return
The expected rate of return fluctuates more, comparatively. What’s more important, the values of both sides are quite close.
B. Individual Rationality
This corresponds to our previous analysis. The individual rationality index of both buyers and sellers is always larger and very close to Group #1. The individual rationality equals to the standard individual rationality when the final deal is made. In this situation, the quote range and market power for both sides are similar without any doubt. The price at equilibrium and the surplus are close to the theoretical value. Therefore, we can think that there is no disturbance of individual rationality to collective rationality for both sides. From the perspective of collective rationality index, sbri=1.879，ssrj=1.692，the difference is relatively small and both values are larger than 1.
From the analysis above, the following could be inferred. In certain times of quote, if the market’s transaction price and surplus diverge from the theoretical Pe and surplus, there must exist a negative disturbance of individual rationality to collective rationality for one side which enables the other side to have dominant power of quote. In another way, if the market’s transaction price and surplus are close to theoretical values, there is going to appear two cases.
First, individual rationality and collective rationality are in accordance with each other for both sides and both have the similar market power which results in mutual antagonism; second, disturbances for both sides may be very large and the rationality values are not close at all, which does not show in all the experimental data while it could be thought to exist theoretically. The values are close only at the transaction price and it is based on an obvious premise that the values are larger than 1, since as long as one value is less than 1, it is very easy (not necessarily) to fall to a weak position (which usually happens in the first round of Group #2). The way to analyze these three situations is pretty similar to the one analyzing monopoly and competitive markets. The key element is the effect of the relation between individual rationality and collective rationality to market power.
From the analysis above, we can reach a primary conclusion that the quote from a single seller or buyer can affect the formation and the result of market equilibrium. We can use individual-collective rationality model to explain the process. Individual rationality can cause a destabilization to collective rationality (resulting in remarkable fluctuation), when the destabilization is small, which means that the individual rationality that causes it is not less than one, the collective rationality still tends to be consistent and this will not have a significant effect on the market power of both sides.
But when the destabilization is large, which means that the individual rationality that causes it is less than one, there will be a significant effect on the consistency of the collective rationality, so there will be a change of the two sides’ market power. Then, the equilibrium price and the surplus of both sides also change. So, there is a kind of tension between individual desirability and collective desirability.
There are some points that we need to pay attention to. First, this kind of relationship becomes more obvious with the increasing of quotation times during one period. The deeper reason for this is the lack of information and each side has to continuously probe and tries to release as much information, which can maximize his profit, as possible, no matter whether the information is true. If the deal can be reached within one or two times, there will be no such complex game about the info releasing and collecting. Second, in reality, the destabilization is always negative (i.e. a sudden decrease in bid or expected return), which indicates the bounded rationality. We don’t need to take positive destabilization into account.
So, how can we use this model to explain the non efficiency phenomenon of market in double auction? There are two aspects that we need to pay attention to. First, the relationship of individual rationality and collective rationality has an effect on the market power, and then affects the equilibrium price and the ratio of the surplus of both sides; the price either has a bad effect on the sellers or the buyers (In our experiment, it is the sellers that lose).
Thus, some buyers or sellers that could have entered the deal are kicked off now, which leads to some deadweight loss. Second, when the market power is out of balance, the number of games between buyers and sellers will increase. Let’s take the third group (whose transaction price is averagely 10% less than theoretical Pe) as an example. When the buyers realize their power and advantage, they will wait the sellers to decrease the price. At that time, from the perspective of buyers, the earnings of waiting is larger than the risk. The sellers are forced to decrease the price, but only for a little each time to avoid a worse price. As a result, during the limited trading times, the number of transactions decreases, which leads to further deadweight loss.
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