INTRODUCTION
According to Fama’s (Fama, 1965) definition for
"weak efficient markets", if all prices are able to fully reflect the information
contained in a historical price sequence for a continuous auction market, meaning
the historical information can fully reflect the current market price, then
no arbitrage opportunities will be left untapped and this market is considered
a weak market.
Concerning research to verify whether continuous auction markets exist or not,
there are primarily two foci: first, the verification of random walk and second
the affective relationship between random walk and weak markets. In regard to
the former, Dickey and Fuller (1979, 1981)
put forward the wellknown DF method to verify the random walk which Chinese
scholars Fan and Zhang (1998) used to investigate stocks
in Shenzhen that aligned with the random walk model. In the second, the affective
relationship between random walk and weak markets was considered. Lan
et al. (2005) summarize the random walk model by referring to former
scholars, to reveal a weak efficiency in the manner that continuous auction
markets relate to "random walk" processes. By mining sign patterns and time
sequences from China’s historical stock market information, it can be seen
that China’s stock market has not yet reached weak efficiency.
Zhang and Zhang (2005) considered the trading price
of futures as a research object, by using both the unit root test and the autocorrelation
test. Concurrently they employed the ratio variance test with a multiple variance
ratio test to empirically research the random walk hypothesis. Their results
showed that the logarithms of major futures markets can not yet reject the hypotheses
of the weakform efficient market. Dai et al. (2005)
while using the unit root method to verify the Composite Index of the Shanghai
stock market, displayed that a vast majority of Shanghai stock market composite
indices have a weak effect.
Regarding research into continuous auction simulation strategies, scholars
have made great progress. Gode and Sunder (1993) first
proposed a doublezero trading strategy: Zerointelligence (ZI), ZI Unconstrained
(ZIU) and ZI with Constraint (ZIC). The main difference between these strategies
is the bidding interval: ZIU rests on the premise that bidders give no consideration
to resources, requiring random auction (loss) within the scope of market mechanisms;
ZIC rests on the premise that bidders fully consider their resources, requiring
an auction where they suffer no loss. Duffy and Unver (2006)
look to Gode and Sunder’s research to verify ZI when resources are limited
and use a near zero auction model to explain the emergence of market bubblesthus
claiming the ability to predict market bubbles. Othman (2008)
built a ZI agent model based on market forecasts where, under the premise that
people are rational and desire to maximize profit, multiple transaction prices
in the ZI model compliments the theory that rational participants help predict
the price of the market. Based on the ZI strategy, Cliff
and Bruten (1997) elicit the "Enhanced Zero" bidding strategy (ZIP, ZeroIntelligence
Plus) based on information from historical markets and the ability to forecast
prices. Here, the main idea is that buyers and sellers can continuously adjust
profit margins in accord with historical transactions, to determine a new bid.
Based on the ZI and ZIP policy, in view of the belief function, Gjerstad
and Dickhaut (1998) put forward a model to calculate a maximum expectation
which can adjust bids by relying on historical information to build selfconfidence
formulas and calculate a maximum expected profit. Finally, specific applications
in this field include: The transaction model and the adaptive capacity auction
model (Dawid, 1996; Preist, 1999),
among others.
Xiaobo et al .(2006) proposed a strategy learning
model based on the Particle Swarm Optimization (PSO) for individual learning
behaviors in reference to cost and closed bilateral. By using the PSO algorithm
to first summarize group and individual learning pathways and then to imitate
learning in order to achieve a theoretical balanced level of biddingthey sought
to rectify the oneonone bargaining system of bidding laws. Next, Pengyu
and Yijun (2006) proposed the hybrid theory for dynamic bidding. Xiaoyong
and Qing (2008) sought to tackle the marketclearing mechanism to maximize
combinatorial auctions by employing double auction transactions based on the
shadow price. Wenjie and Jie (2008) then proposed a
trading strategy based on the Markov chain and in turn verified how algorithms
designed for ZIC work alongside Markov tendencies. In situations where transaction
prices for ZIC continuous auction markets are coupled (based on the ZIP strategy),
Shengfeng and Chengjian (2009) present the ZIP2 strategy,
to achieve greater network efficiencies for resource allocation. Xu
and Chengjian (2010) proposed a revised RiskBased strategy from continuous
double auctions and took advantage of the PSO to increase convergence of the
double auction market price.
In summary, even though scholars have done much to test methods for weak continuous
auction markets and the strategies for continuous auction simulations, still
research is lacking to consider the relationship between and the manner in which
simulation results guide continuous auction market participants. Presently,
strategies for continuous auctions focus more upon historical data and transactional
data. Likewise research pertaining to the best price in the relevant group is
rare. By investigating the best price data and setting individual income maximization
as a guideline, this paper aims at: (1) Putting forward a continuous bidding
strategy (EOB strategy) based on the best price in the group; (2) Achieving
near or greater than market equilibrium results for individual gain. The results
of this research seek to provide a valuable theoretical basis and a feasible
solution for those continuous double auction, random walking, market participants.
RANDOM WALKING TEST OF ZIC TRANSACTION PRICE
Continuous auction market trading rules and the ZIC strategy: There
are M buyers and N sellers in the continuous auction market and the market environment
of M:N represents a symmetrical relationship. In order to facilitate this study
a simplified market provides that: (1) Each buyer can only buy a commodity,
each seller can only sell a commodity and the goods held by buyers and sellers
are homogeneous; (2) Each buyer holds a different valuations of goods, noted
by: V1, V2,…, Vm; (3) The costs
to each seller, noted by: C1, C2,…,
Cm, differ from the estimates of buyers and sellers and also differ from costs
shown on the supply and demand curves in the continuous auction market.
ZIC bidding strategy is a scenario where the buyer’s bid is a random
number occurring between the minimum price and the buyer’s valuation while
the seller’s quotation is a random number uniformly distributed between
the seller’s cost and the price ceiling (Gode and Sunder). Assuming the
minimum price in the continuous auction market is 0, the pricecap is 100, so,
a mathematical representation for the ZIC bidding strategy is: BiU(0, Vi),
where Vi is the value of resources obtained by the “i” buyer. The
formula for a ZIC selling bid is:
where, “C”
is the cost for the “j”
seller.
Based on Gode and Sunder, this study uses the continuous auction market trading
rules. Specifically: (Rule 1) a buyer “i”, may bid “Bi”
according to their own valuation “Vi” of goods  while a seller “j”,
auctions “Aj” according to their individual cost “Cj”, under
the rules: BI≤Vi, Aj≥Cj. (Rule 2) At any time “t”, randomly
selected by a buyer or seller, the buyer bidding “Bi” marks the highest
bid “Bbest” while the seller auctions “Aj” and marks the
lowest bid as “Abest”. (Rule 3) At the time Bbest≤Abest, the
corresponding bidding party completes the transaction and the price equals half
of both auctions’ sum (i.e., TP(i, j) = 1(Bbest+Sbest)). Following the
first transaction, the second highest bid of the buyer becomes “Bbest”
while at the same time, the second lowest bid by the seller becomes “Abest”.
This process continues until all bids are taken and finally, the bidding ceases
with both parties leaving the market. (Rule 4) No dealing party reissues the
bid, at the time of “t+1”, repeat rule 2 and rule 3. (Rule 5) Repeat
the above rules 14 as long as all Bi < Aj holds true. End.
Random walking test of transaction price sequence: According to previous
studies, the Runs test and DF testing are the two most preferred methods for
testing transaction price in the ZIC strategy.
Table 1: 
Random walking test results of ZIC transaction price sequence 

The Runs Test ideological states: Let “N” be the total number of
samples in a price sequence {TP_{t}} with n_{i} (i = 1, 2, 3)
to represent the number of price changes in various directions, where “n1”
is the number for positive price changes, “n2” is the number for negative
price changes, “n3” is the number for events with no price change
and “M” is the actual number of runs. So, the {TP_{t}} sequence
expected number of runs is:
with a variance of:
From the standardized normal variable:
where, M>E(m), uses an adjustment coefficient of 0.5 and when M<E(m)
the 0.5 adjustment coefficient should be subtracted. When “N”
is sufficiently large, then “K”
is close to zero and the variance is 1.0 in a normal distribution. Accordingly,
at the 1% significance level, the critical value of “K”
is 2.33, if the value of “K”
lies within the range (2.33, 2.33), then we accept the assumption of randomness
for the stock price. Otherwise, we conclude that stock price is moving with
a trend.
The DF testing method was created by Dickey and Fuller on the premise that
it has: a firstorder autoregressive process for price sequencing:
where ε_{t} is independent and evenly distributed and where E(ε_{t})
= , D(ε_{t}) = E(ε_{t}^{2}) = σ^{2}<∞.
Parameters ρ can be tested, so that if ρ = 1, the price sequence {TP_{t}}
follows a random walk process. If ρ<1, then the price sequence obeys
the stable firstorder autoregressive process and the least squares estimation
of the parameters is:
The estimated value of the standard deviation for
is :

Fig. 1: 
ZIC Strategy Transaction Price 
The t_{T} statistic for ρ is .
Further, ρ = 1 can use
in addition with t_{T} to conduct tests while the critical value of
and t_{T} (Enders et al., 2012) at a 1%
confidence interval issues the critical value 13.7 for
and 2.58 as a critical value for t_{T}.
We selected 100 ZIC strategies from the buyer and seller to perform a "random
walk" experiment of transaction prices in a ZIC simulation. Conducting the
test, adhering to the market rules stated above, we discovered that after 30
simulations, each result was remarkably similar with randomly selected transaction
price data. Figure 1 shows the ZIC Transaction Price Strategy.
“CP” in Fig.
1 corresponds to a balanced equilibrium price that was achieved by the market.
Conducting a Runs test and DF test for the ZIC Transaction Price Sequence
produced the results are shown in Table 1.
Test results revealed that: below the 1% confidence level, the transaction
price of the ZIC strategy passed the Runs test and the DF test. This proves
that the transaction price of the ZIC strategy is in line with the "random
walk" model in the continuous auction market.
EYEONBEST STRATEGY
The main idea of continuous bidding strategy (EOB) based on the best group
best price (EyeOnBest) focuses on the buyer’s
highest bid and the seller’s
minimum bid in the market (i.e., buyers and sellers each auction). In accord
with both sides’ bids it revises
a new bidding strategy, in order to elicit higher individual incomes. Specifically:
assume that at the time “t”
in a continuous auction market, the highest bid of the buyer is “Bbest(t)”
while the seller’s minimum bid
is “Abest(t)”.
At the time of “t+1”,
buyers and sellers bid in accordance with the best price “Bbest(t)”
and “Abest(t)”
at the time of “t”
and concurrently determine their new bids whether the seller and the buyer close
the deal at the time “t”
or not. If at time “t”,
buyers and sellers can trade in the market, indicating that buying and selling
is profitable, the offers can continue according to the current prices at time
“t+1”. This is all taken
into consideration with factors such as: increasing revenue, the buyer reducing
a minimum on the basis of Bbest(t); and the seller increasing a minimum based
on Abest(t). If at time “t”
buyers and sellers cannot trade, indicating that buyers or sellers will not
make a profit, the offer should be adjusted to reissue the at the time “t+1”.
In regard to fixing profit, the buyer may adjust minimal value on the basis
of Bbest(t) while the seller adjusts on the basis of Abest(t) to mutually
achieve a beneficial transaction.
In a scenario where the buyer bids at time t+1 and makes an adjustment to the
bid based upon Bbest(t), at time t the adjustment only yields a minimum Δ;
while at the same time the seller at time t+1 makes an adjustment to the bid
based upon Abest(t), at the time t, the adjustment only yields a minimum Δ.
Meanwhile while constrained by bounded rationality the buyer can not bid higher
than their disposable resources allow (Bi≤Vi) and the seller can not bid
less than the cost (Aj≥Cj). Based upon these assumptions we derive the mathematical
expression for an EOB strategy auction, at the moment t+1:
The buyer bidding strategy for i is:
The seller’s bidding strategy
for i is:
From Eq. 1 and 2 above, buyers and sellers
adjust bids according to the EOB strategy at the time t+1. If buyers and sellers
agree to a transaction at time t, then at the time t+1, the buyer should lower
the bid while the seller should raise the bid, so that, Bbest(t)Δ decreases
the value in the buyer’s bidding
formula and Abest(t)+Δ increases the value in the seller’s
bidding formula. If buyers and sellers do not achieve a trade at time t, then
at the time t+1, the bid should be raised for the buyer and reduced for the
seller: thus the value of Bbest(t)+Δ increases the value in the buyer’s
bidding formula and the value of Abest(t)Δ decreases the value in the
seller’s bidding formula.
EXPERIMENTAL DESIGN
In our experiment 100 buyers and 100 sellers were used for a test market where
each buyer or seller could trade one item under the position substitution principle.
To achieve a random market, as we expect exists in (ZIC) strategy and which
is required for an EOB policy, each supply and demand curve was reproduced 10,000
times to minimize any uncertainty resulting from random factors to the experiment.
Thus, the following individual incomes represent the average of 10,000 experiments.
We used individual gain level as the variable to measure the various impacts
that bidding strategies had upon revenue. Buyer’s
individual gain equals resources owned minus the transaction price. Seller’s
individual gain equals the transaction price minus its cost. The mathematical
expressions for individual gains are:
In Eq. 3 and 4 above, profitbuye(i)r is
the buyer’s individual gain from
the i position, Vi is the resources owned by the buyer at the i position and
TP(i, j) is the transaction price when buyer in the i position and seller in
the j position reach a trade. Here, “i"
= 1, 2,...m and j = 1, 2,...n. The profitbuye(j) is the seller’s
Individual gain from the j position and Cj is the seller’s
costs at the j position.
Meanwhile, based on profit maximization theory, we established a standard to
measure the merits for each policy and judged whether it could achieve an equalized
individual gain or not. If individual gains were close to equilibrium, the individual
could profit; if individual gains were greater than the equilibrium, there was
excessive profit. We expected that the individual gain produced by the strategy
would not be less than a balanced individual gain, meaning that individuals
are able to reap excessive profit. Individual equalized gain equals the buyer’s
individual gain and the seller’s
individual gain at market equilibrium, where the former equals the resources
occupied by the buyer minus the market equilibrium price, the latter equals
the market equilibrium price minus the cost to the seller. The specific mathematical
formula follows:
The CP from formula 5 and 6 above is the theoretical transaction price in market
equilibrium, Competitivebuyer(i). For the buyer this is individual gain at
the i position in market equilibrium (i = 1, 2þ,m) while Competitiveseller(j)
is for the seller individual gain at the j position in market equilibrium (i
= 1, 2þ,n) and CP is the price in market equilibrium.

Fig. 2: 
First supply and demand curve 
In order to avoid influences resultant from bidding strategies by traders in
a continuous auction market, the present study uses a position substitution
method. The position substitution assumes that bidding strategy is fixed at
all positions in the market (in addition to the test position). If the test
position changes with a different bidding strategy then bidding strategy in
other positions remains fixed. Specific experimental procedures follow where
we assume that all policies available in the market are ZIC, respectively,
statistics for individual gains in the ZIC strategy at the position of Vi (i
= 1, 11,...,91) and Cj (j = 1, 11,...,91). Afterward, one follows the other
in the EOB strategy to replace Vi (i = 1, 11,..., 91) and Cj (j = 1, 11,...,
91) so that successive individuals record profits in the EOB strategy in the
test position. And then when comparing levels of gain in the same position of
the ZIC strategy and EOB strategy, one should finally compare individual gains
in the same position under market equilibrium.
Note: Strategy may be affected by a shift in supply and/or demand. Therefore
in order to verify that EOB strategies are more capable than ZIC strategies
in scenarios where supply and demand curves shift, we divided the market into
the following three categories according to buyer and seller abilities to control
the market (we divided the buyer’s
market power MPB and the seller’s
market power MPS). First, a bilateral market: where buyers and sellers share
the same market resources, MPB = MPS; Second, a buyer’s
market where a buyer has superior market resources, MPB>MPS and Third, a
seller’s market where a seller
has superior market resources, MPB<MPS. Equation follow:

Fig. 3: 
Second supply and demand curve 

Fig. 4: 
The third supply and demand curve 
Above in Eq. 7 and 8, CK is the buyer’s
or seller’s corresponding position from the supply and demand curves in
market equilibrium. Three types of supply and demand curves correspond to the
following Fig. 24.
These three supply and demand curves were selected in respective Runs tests
and DF tests. The test results are shown below in Table 2
of the ZIC strategy transaction price sequence random walk under the curves.
Test results revealed that below a 1% confidence level, the transaction price
of the ZIC strategy passes the runs test and DF test in all three curve types.
This shows that in the three categories under the curve selected, the transaction
price of the ZIC strategy continues to comply with the random walk model.
EXPERIMENTAL RESULTS AND ANALYSIS
MATLAB’S ability compute and
map out simulations, makes it a tool increasingly used by a majority of scholars
for continuous auction market research. Likewise, this study used Matlab 2011
to perform tests. Supply and demand curves were selected according to the market
resources of buyers and sellers in the market; which in turn was divided into
three categories. Analysis of these experimental results follows.
In a bilateral market, buyers and sellers share equal market resources (MPB
= MPS). The market structure is shown in Fig. 2 and the simulation
data is shown in Table 3.
Table 3 shows that, buyers and sellers can achieve greater
individual returns by using the EOB strategy when they share equal market resources.
In this scenario, individual returns from the EOB strategy are higher than the
individual returns from the ZIC strategy. Further, the EOB strategy can yield
higher returns in this environment. In a buyer’s
market (MPB>MPS) the buyer has a majority share of market resources. This
market structure is shown in Fig. 3 and the simulation data
for this structure is shown in Table 4.
Table 4 shows that, in the case where the buyer has superior
market resources, the EOB strategy can achieve greater individual returns than
the ZIC strategy for the buyer in dominant position, but it will not reach
market equilibrium. The EOB strategy can achieve both greater individual returns
than the ZIC strategy for the seller of inferior position and more income than
a balanced revenue scenario. Above all, it shows that where resources are unequal,
this strategy is more conducive to achieve excess returns.
In a seller’s market (MPB<MPS),
the seller holds a greater share of market resources. The market structure is
shown in Fig. 4 while the simulation data is shown in Table
5.
Table 5 shows that, in the case where a seller holds more
market resources, the performance runs contrary to when the buyer has a superior
position, with the same benefit now on the side of the sellers.
When comparing Individual earnings from the EOB strategy and the ZIC strategy
in differing market conditions, the following conclusions can be drawn: if buyers
and sellers share equal market resources or whether the buyer or the seller
holds superior resources, then the EOB strategy can achieve about 1523% greater
personal returns than the ZIC strategy.
In different markets, comparing individual returns from the EOB strategy, with
the ZIC strategy and individual balanced income, the following conclusions
can be drawn: first, the person in an inferior resource position can achieve
individual returns from the ZIC strategy to reach approximately 95% of balanced
individual gain.
Table 2: 
Random walking test results of ZIC Transaction price sequence
in three market power 

Table 3: 
Result of first market MPB = MPS 

Table 4: 
Result of second market MPB>MPS 

Table 5: 
Result of third market MPB<MPS 

However, the maximum individual return of the EOB strategy can achieve more
than a balanced individual gain. Second, the resource dominant party can achieve
individual returns from the ZIC strategy to up to 87% of the individual balanced
gain. Finally, the individual income of the EOB strategy can reach up to 97%
of the individual equalized return. In these scenarios the individual returns
from the EOB strategy are still higher than the individual income from the ZIC
strategy.
CONCLUSION
The random walking ZIC strategy was verified by transaction prices. A weak
efficient market environment can be simulated using the ZIC strategy. On this
basis, using group optimal pricing information in the market, a group optimal
bidding strategy (EOB) can be designed to improve individual returns in continuous
auction markets. The experimental results show that: in continuous auction markets:
(1) The EOB strategy is significantly better than the ZIC strategy to achieve
individual returns greater than the ZIC strategy in various market environments;
(2) An EOB strategy can help the individual in an inferior resource position
because the EOB strategy is best suited for bidding when in an inferior position;
and (3) in a continuous auction market, the random walk EOB strategy reveals
that group competitive bidding is a decisive factor.
Therefore, in a continuous auction market, individual returns can be improved
by using all known information to design a bidding strategy. The resource inferior
position can improve individual returns by using the EOB strategy. On this basis,
first of all, bidding strategies can be used to achieve greater returns in a
variety of market positions. This provides both a theoretical basis and feasible
ideas for intelligent algorithmassisted bidding practitioners. Secondly, this
method offers a new tool for use in computer aided financial market simulations.