Figure 1

In the main panel, we plot $d_t$ the absolute value of the difference between the two consecutive bid values $b_{t+1}$ and $b_t$. $t$ indicates the rank position of each bid in a ranking in which bids have been sorted according to their time stamp. In the inset, we plot the PDF $P\left(d\right)$ of the bid change amounts $d$ (red circles). The PDF is fitted with a power-law function and the best estimate of the decay exponent is $1.5\pm 0.1$ (black dashed line). This figure refers to the exploration of the bid space performed by a single player in a single auction. A complete analysis of the movement patterns of hundreds of players was performed in our previous work [14].Reuse & Permissions

Figure 2

Probability density function of the winning strategy $w\left(\alpha \right)$ in the case in which players' strategies are randomly extracted from a uniform PDF $g\left(\alpha \right)$ defined over the interval $(0,5)$. We set $N=100$ and $M=1000$ (parameter values similar to those observed in real auctions [14]), and consider for simplicity only the case of a single bid $T=1$ and the case in which all players perform $T=10$ bids. We show in both cases the comparison between the results obtained with numerical simulations ($T=1$, filled circles; $T=10$, empty squares) and those obtained with the numerical integrations of the various equations [$T=1$ full (lower) line, $T=10$ dashed (upper) line]. Note that the optimal value of $\alpha$ depends on $T$ and also on the parameters values $N$ and $M$. However, this is a weak dependence and the peak value of the probability density function of the winning strategy is in the range $1.2$–$1.7$ for a wide range of possible values of $N$ and $M$ and a moderately broad range of values of $T$.Reuse & Permissions

Figure 3

Probability distribution function $w^{\left(e\right)}\left(\alpha \right)$ of the winning strategies at generation $e$. Here we set $M=1000$, $N=100$, and $T=1$. The mutation $\xi$ is randomly extracted from a uniform distribution centered in zero and width $2\mu$: we consider the cases $\mu =0.0$ (panels a and b, absence of mutation) and $\mu =0.5$ (panels c and d). In a and c, the starting distribution for the exponents is $g^{\left(1\right)}\left(\alpha \right)=1/5$ if $\alpha \in (0,5)$ and zero otherwise. Already after one generation ($e=1$, lower black line), the peak of the distribution is around $\alpha \simeq 1.5$. As the number of generations increases, the distribution becomes more and more peaked around a specific value of $\alpha$, and reaches a stationary distribution. The asymptotic distribution has finite width for $\mu >0$, while it is a $\delta$ function for $\mu =0$. For comparison, we show also the results obtained numerically after $e=20$ generations (black circles). In b and d, we consider initial distributions for the exponents of the type $g^{\left(1\right)}\left(\alpha \right)\sim {\alpha }^{-\lambda }$ if $\alpha \in (0,5)$ and $g^{\left(1\right)}\left(\alpha \right)=0$ otherwise. We show the results for the three cases: $\lambda =0$ (uniform), $\lambda =2$ (power law), and $\lambda =5$ (exponential). The asymptotic distribution does not depend on the initial distribution, and the peak value of the asymptotic PDF does not depend on $\mu$. Independently of the value of $\lambda$, as the number of generations $e$ increases, the average value $\langle \alpha \rangle$ and the standard deviation $\sigma$ approach the same stationary values: $\langle \alpha \rangle \simeq 1.3$ for any $\mu$, while $\sigma =0$ (as $1/\sqrt{e}$, thin black line in panel b) for $\mu =0.0$ and $\sigma \simeq 0.4$ for $\mu =0.5$.Reuse & Permissions

Figure 4

Same as Fig. 3, but with different parameter values. Here we set $M=1000$, $N=100$, and $T=10$, and consider $\mu =0.0$ in panels a and b, while $\mu =0.5$ in panels c and d. The asymptotic PDF of the winning strategies have average values $\langle \alpha \rangle \simeq 1.5$ for any value of $\mu$. The standard deviation is $\sigma =0$ (as $1/\sqrt{e}$) for $\mu =0.0$, and $\sigma =0.3$ for $\mu =0.5$. For any value of $\mu$, the best strategy (i.e., the peak of the PDF) is placed at ${\alpha }^{*}\simeq 1.5$.Reuse & Permissions