Multiagent model and mean field theory of complex auction dynamics

The data sets used in this paper were downloaded from two websites: www.auction-air.com and www.uniquebidhomes.com. The auctions on the two websites were performed in British Pounds and US Dollars, respectively. Moreover, the minimum unit of bid price on the first website is one Pound, while the second website requires the minimum unit to be cent. The historical information of the auctions recorded on these websites includes two categories: (1) general information such as the value V of the item, entry fee c, minimum bid price $\underline{k}$, number N of bids (or participants), and (2) the bid price k offered by each participant and the winner's bid ${k}^{*}$.

To be as general as possible, we chose six different types of items varying from mobile phone to digital camera of different values V. For each item, the auction was performed for R rounds. The corresponding six data sets are labeled as (a)–(f), respectively, and the related information is listed in table 1, where (a)–(d) are from the first website and (e), (f) are from the second.

Due to the various sources of randomness involved in the decision making process, the bids offered by different agents participating in the same LUBA will in general be different. If the number of agents is statistically significant, the bid price can effectively be regarded as a random variable, with its probability distribution depending on the game setting/parameters. Figure 1 shows the bid distributions, denoted by f_k, obtained from six data sets of LUBAs (black bars).

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In spite of the fluctuations caused by the finite number of bids, the distributions exhibit some general features that are independent of the particular details of the games such as the actual item values and the number of participants in the auctions. Interestingly, the lowest bid is not selected with the highest probability, and all distributions exhibit an inverted J-shaped curves, i.e., the bid probability increases non-monotonously for small bid price, reaches maximum for some medium bid price, and decays to zero in the large price region. For example, in the data sets (a), (e), (f), the most probable bid price is £ 7, $0.17, and $0.11, respectively. For the data sets (b), (c), (d), the most probable bid price is about £17.

Intuitively, the inverted J-shaped distribution can be attributed to users' efforts to bid less but to avoid biding the same as others [13], as driven by the rule that the lowest unique bid wins. Each participant thus seeks to bid at price as low as possible, so a bid originally regarded as low by some agent may in fact not be 'low' enough to win the game, as the bid prices offered by some other agents could be lower. As a result, the collective behaviors of the agents tend to self-organize into a non-monotonous distribution as exemplified in figure 1. The exponential decay of the probability in the large bid price region is in fact a general feature induced by the LUBA rule [16]. The tail region of each distribution can be well fit by the exponential function ${f}_{k}\sim {{\rm{e}}}^{-{ak}}$, represented by the black solid curves in figure 1, where the values of the exponential parameter a for the data sets are listed in table 2.

Table 2.  Fitting parameters in the bid distributions from the six data sets.

Data sets	p (std.)	KS	R2	$\hat{a}$	$a^{\prime} (p)=\mathrm{ln}(\frac{1+2p}{1-p})$
(a)	0.046 (0.013)	0.059	0.895	0.13	0.136
(b)	0.034 (0.007)	0.029	0.963	0.10	0.100
(c)	0.025 (0.004)	0.045	0.903	0.09	0.089
(d)	0.058 (0.012)	0.030	0.943	0.16	0.170
(e)	0.015 (0.001)	0.038	0.912	0.04	0.045
(f)	0.006 (0.001)	0.055	0.904	0.02	0.018

The attracting probability p is obtained through the minimum KS value in fitting the model generated with real bid distributions. The coefficient of determination R², and the KS coefficient are listed. The exponent $\hat{a}$ is obtained from fitting with the tail region of the real bid distributions, and the exponent $a^{\prime} (p)$ as a function of p is obtained analytically.

In spite of some subtle differences, the online settings of LUBAs (or LUPIs) that determine the game outcomes are the following. (1) The value V of an item and the target maximum number of bids, N, are known to all agents, where N is the total number of players if each bids once. (2) The bid price, denoted by k, is discrete and constrained within the interval $[\underline{k},\bar{k}]$, where $\underline{k}\gt 0$. (3) The outcome of the game is calculated when all N bids are placed, and the participant who makes the lowest bid among the distinct ones (denoted by ${k}^{*}$) wins the game. Additionally, a constant entry fee c is required for each bid ($c\geqslant 0$), which is identical for all players and thus will not affect the outcome of the game. Assuming that each participant bids once in each round of the game, we have that the payoff of the exclusive winner is $V-{k}^{*}-c$ (with the bid ${k}^{*}$ subtracted off), while all the other participants get the identical payoff $-c$. The rational upper bound of bid price k is $\bar{k}\lt V-c$ , ensuring positive payoff for the winner.

Our construction of a computational model for the LUBA auction process benefits from the fact that a host of social and economical behaviors can be described by the minority game model [21–37], a multi-agent model characterizing a population of selfish individuals competing for limited common resources. The similarity between LUBA auction process and those described by the minority game model suggests strongly the suitability of using multi-agent models to understand the LUBA auction dynamics. Without loss of generality, we discuss the case where the bid interval is normalized to unity and each agent adapts his/her bid k step by step in the discrete space of bid price. Our model is schematically shown in figure 2, with the following detailed setting: the auction is performed iteratively and, in each round, the agent who bids the lowest unique ${k}^{*}$ wins the game. Each of the N agents acts based on the historical information with probability p (which we name as the attracting probability), or acts without any information with the probability $1-p$. In the former case, the winner's bid ${k}^{*}$ in the last round forms a field that attracts the agents towards ${k}^{*}$, i.e., agents adapt their bid towards the last ${k}^{*}$ for one bid unit. In particular, those with bid $k\lt {k}^{*}$ choose $k+1$, the agents with $k\gt {k}^{*}$ bid $k-1$, and the winner simply keeps his/her bid. In the latter case, agents bid randomly by simply keeping the bid k in the last round, or moving one bit unit away from it (i.e., choosing $k-1$, k, or $k+1$ with equal probability). The zero-flux boundary condition is adopted in the bid space $[1,V-c]$ at the left hand side, subject to the requirement of positive bids. At the right hand side, the boundary condition is that the payoff of the winning agent must be positive. In simulations, the auction process is conducted iteratively to yield a stable bid distribution.

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The bid distributions obtained from simulations of our multi-agent model are also plotted in figure 1 (red open circles), where the number of agents N is the same as the number of participants recorded in the LUBA data, and the attracting probability p is chosen to best fit the real bid distribution. Table 2 lists the values of the parameter p obtained from the minimum Kolmogorov–Smirnov (KS) test for fitting the real data sets. The corresponding KS coefficients denoted by KS, and the coefficient of determination denoted by R², are also listed for reference. The remarkable agreement between the model generated and real bid distributions indicates that the processes articulated in our model capture the main dynamics of LUBA systems. In addition, we observe that small values of p lead to the best fit. From the perspective of multi-agent interactions, this implies that the observed inverted J-shaped bid distributions are result of small movements of participants to adapt their bid towards the previous winning bid ${k}^{*}$.

The three parameters in our multi-agent model are the value of item V, the number of agent N, and the attracting probability p. The respective effects of these parameters on the auction dynamics are illustrated in figure 3. The parameter V represents the upper bound for rational bids in the real cases and in our model, which is rarely observed due to the auction rule that the 'lowest unique bid wins.' As shown in figure 3(a), the value V has little influence on the bid distribution. In contrast, the parameters N and p both play a significant role in the emergence of the bid distribution. As N is increased, the location of the peak of the distribution, i.e., the most probable bid price, moves rightward and the height of the peak is reduced, as shown in figure 3(b) where, the variance of the bid is enhanced accordingly. We see that larger values of p give rise to a more concentrated bid price distribution with higher peak and narrower width. Distinct from the effects of N, the most probable bid price changes in a relatively small region of p.

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The stable bid distribution f_k can be regarded as the probability for each agent to bid at k. Suppose that the number N of participants and the bid probability distribution f_k are known. We can calculate the probability w_k for one given bid k to win the auction, i.e., the lowest unique bid ${k}^{*}=k$. The quantity w_k denotes the distribution function of the winning price. In contrast, since the winner's bid ${k}^{*}$ attracts all other participants, the winning probability distribution w_k provides effectively a mean field with respect to which all the agents adapt their bids. That is, w_k determines, probabilistically, the new bid distribution f_k. The new f_k will again result in an updated w_k. These observations suggest that the approach of self-consistency in the mean-field framework can be used to analyze the dynamical process of online auction.

Our detailed analysis of the iterative, self-consistent process between f_k and w_k is as follows. Firstly, for a given bid distribution f_k, we calculate the winning probability distribution w_k. The simplest case is ${k}^{*}=1$, i.e., there is only one single bid at k = 1. The probability for this case is

$$\begin{eqnarray}&&{w}_{1}=\displaystyle (\genfrac{}{}{0em}{}{{n}_{1}}{1}){f}_{1}{(1-{f}_{1})}^{{n}_{1}-1},\end{eqnarray} \tag{ 1 }$$

where f₁ is the probability for each player to bid at k = 1, and ${n}_{1}=N$ is the number of potential agents who may bid at 1.

The case of ${k}^{*}=2$ is critical for the analysis, which is slightly more complicated. In particular, the agents bidding at 1 unit will be excluded from the number of potential players who may bid at 2 units (denoted by n₂). We have, on average, ${n}_{2}=N(1-{f}_{1})$. The probability for ${k}^{*}=2$ is then

$$\begin{eqnarray}&&{w}_{2}=\displaystyle (\genfrac{}{}{0em}{}{{n}_{2}}{1})f{^{\prime} }_{2}{(1-f{^{\prime} }_{2})}^{{n}_{2}-1}(1-{w}_{1}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&=\;{n}_{2}f{^{\prime} }_{2}{(1-f{^{\prime} }_{2})}^{{n}_{2}-1}[1-{n}_{1}{f}_{1}{(1-{f}_{1})}^{{n}_{1}-1}],\end{eqnarray} \tag{ 3 }$$

where ${f}_{2}^{\prime }$ is the renormalized bid probability: ${f}_{2}^{\prime }={f}_{2}/{\displaystyle \sum }_{j=2}^{\bar{k}}{f}_{j}$, which is the probability of bidding 2 for all the agents except those who bid 1. For larger ${k}^{*}$ values, those agents who bid lower prices are excluded gradually. We have

$$\begin{eqnarray}&&{w}_{k}={n}_{k}{f}_{k}^{\prime }{(1-{f}_{k}^{\prime })}^{{n}_{k}-1}\displaystyle \prod _{j=1}^{k-1}[1-{n}_{j}{f}_{j}^{\prime }{(1-{f}_{j}^{\prime })}^{{n}_{j}-1}],\end{eqnarray} \tag{ 4 }$$

where the number of potential agents to bid at k is ${n}_{k}\doteq N(1-{\displaystyle \sum }_{j=1}^{k-1}{f}_{j})$ and the renormalized bid probability is ${f}_{k}^{\prime }={f}_{k}/{\displaystyle \sum }_{j=k}^{\bar{k}}{f}_{j}$. For a given bid distribution f_k, the winning probability distribution w_k can be uniquely determined from equation (4).

We next consider the process to determine the bid distribution f_k in the field created by a given winning probability distribution w_k. In our multi-agent model, agents move their bids one unit towards the winning bid ${k}^{*}$. Given w_k, without specific knowledge of ${k}^{*}$, the winner's influence can be regarded as probabilistic, since each agent makes decision in the mean field determined by w_k. Specifically, each agent decreases (or increases) its bid by one unit with probability ${W}_{\lt k}$ (or ${W}_{\gt k}$), or keep his/her original bid with probability ${W}_{=k}$. The quantities ${W}_{\lt k}={\displaystyle \sum }_{j=1}^{k-1}{w}_{j}$, ${W}_{=k}={w}_{k}$, and ${W}_{\gt k}={\displaystyle \sum }_{j=k+1}^{\bar{k}}{w}_{j}$ are the probabilities that the winner is located on the left side of k, is exactly at k, and is on the right side of k, respectively. That is, these probabilities are for the cases where the winning price ${k}^{*}$ is less than, equal to, or larger than k, respectively. We have ${W}_{\lt k}+{W}_{=k}+{W}_{\gt k}=1$. In our model, the two probable actions of each agent are attraction towards the winner's bid with probability p, and random adoption of its bid with probability $1-p$, which correspond to drift and diffusion in a Markov stochastic process, respectively. The corresponding transition matrix can be written as

$$\begin{eqnarray}{\bf{M}} & & =p\cdot {{\bf{M}}}_{\mathrm{drift}}+(1-p)\cdot {{\bf{M}}}_{\mathrm{diffusion}}\\ & = & p\cdot [\begin{array}{ccccccc}{W}_{=1} & {W}_{\lt 2} & & & & & \\ {W}_{\gt 1} & {W}_{=2} & \ddots & & & 0 & \\ & {W}_{\gt 2} & \ddots & {W}_{\lt k} & & & \\ & & \ddots & {W}_{=k} & \ddots & & \\ & & & {W}_{\gt k} & \ddots & {W}_{\lt \bar{k}-1} & \\ & 0 & & & \ddots & {W}_{=\bar{k}-1} & {W}_{\lt \bar{k}}\\ & & & & & {W}_{\gt \bar{k}-1} & {W}_{=\bar{k}}\end{array}]\\ & & +(1-p)\cdot [\begin{array}{ccccccc}2/3 & 1/3 & & & & & \\ 1/3 & 1/3 & \ddots & & & 0 & \\ & 1/3 & \ddots & 1/3 & & & \\ & & \ddots & 1/3 & \ddots & & \\ & & & 1/3 & \ddots & 1/3 & \\ & 0 & & & \ddots & 1/3 & 1/3\\ & & & & & 1/3 & 2/3\end{array}],\end{eqnarray} \tag{ 5 }$$

where both ${{\bf{M}}}_{\mathrm{drift}}$ and ${{\bf{M}}}_{\mathrm{diffusion}}$ are tridiagonal matrices, and the elements of ${{\bf{M}}}_{\mathrm{drift}}$ are determined by the winning probability distribution w_k. According to the ergodic theorem of Markov chain [38], every irreducible Markov chain of finite states is ergodic and admits a unique stationary distribution. The stable state f_k can then be obtained from the iterative operation of ${\bf{M}}$ until the following condition is met:

$$\begin{eqnarray}&&{f}_{k}={\bf{M}}{f}_{k}.\end{eqnarray} \tag{ 6 }$$

We see that equations (4)–(6) represent a self-consistency equation set, which can be solved iteratively. Figure 1 shows the stable distribution f_k obtained from the self-consistency equations (the blue dashed curves labeled as 'mean-field theory'), with the same parameters N and p as in the simulations. We observe a good agreement between the stable f_k from the self-consistency equations and those from direct model simulations (red open circles) and from the real data (black bars). Take, for example, the data sets (a) and (f) in figure 1. Figure 4 shows the stable w_k and f_k distributions obtained from the self-consistency equations, and the inserts show the corresponding probabilities ${W}_{\lt k}$, ${W}_{=k}$ and ${W}_{\gt k}$ versus k. Through extensive simulations, we find that the winning probability distribution w_k has two main features: monotonic decreasing trend and bimodal distribution, as shown in figure 4.

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Comparing the predicted bid distribution f_k to the real data, we see that our model successfully reproduces the whole curve of f_k, insofar as the value of p is chosen properly (as listed in table 2). Especially, both the non-monotonous feature in the small k region and the exponential decay in the large k region are predicted. Here we demonstrate that, in our model, the exponential decay of the bid distribution f_k can be calculated analytically based on the concept of detailed balance.

For a large arbitrary bid price k, majority bids of the winner's bid price ${k}^{*}$ appear on its left-hand side. For the action towards ${k}^{*}$ for certain value of the attracting probability p, the agents bidding about k tend to reduce their bids. If the action is random with probability $1-p$, an agent chooses a nearby bid, leading to a chance to increase their bids. A stable bid distribution f_k implies a detailed balance, i.e., the probability of flowing out of k is equilibrated by that into k. The corresponding equation can be written as

$$\begin{eqnarray}&&{{pf}}_{k}+\displaystyle \frac{2(1-p){f}_{k}}{3}=\displaystyle \frac{(1-p)({f}_{k-1}+{f}_{k+1})}{3}+{{pf}}_{k+1}.\end{eqnarray} \tag{ 7 }$$

We have

$$\begin{eqnarray}&&({f}_{k-1}-{f}_{k})/({f}_{k}-{f}_{k+1})=(\displaystyle \frac{1+2p}{1-p}).\end{eqnarray} \tag{ 8 }$$

The solution of equation (8) subject to the constrain ${\mathrm{lim}}_{k\to \infty }{f}_{k}=0$ is the exponential function

$$\begin{eqnarray*}&&{f}_{k}\sim \mathrm{exp}\;[-\mathrm{ln}(\displaystyle \frac{1+2p}{1-p})k]\equiv \mathrm{exp}[-a^{\prime} (p)k].\end{eqnarray*}$$

Table 2 shows the corresponding analytical result of the exponent $a^{\prime} (p)=\mathrm{ln}(\frac{1+2p}{1-p})$ for each data set, with the parameter p satisfying the the best fitting of f_k from multi-agent simulation (red circles in figure 1) with the whole curve of f_k from real data (black bars in figure 1). A remarkable result is that the exponent a' with the best fitted p is approximately equal to the exponent $\hat{a}$ obtained directly from the exponential fitting (black solid curves in figure 1) to the tails of f_k from real data. This means that, our multi-agent model is self-consistent in predicting the local decaying feature of f_k in the large k region and in predicting the whole curve of f_k, both being associated with the same value of parameter p.