Contest based on a directed polymer in a random medium

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At its turn, each of the two players picks a move among two alternatives in order to maximize its final score, and minimize the opponent’s return. For a game of length $n$, we find that the probability distribution of the final score $S_n$ develops a traveling wave form, $\mathrm{Prob}(S_n=m)=f(m-vn)$, with the wave profile $f\left(z\right)$ decaying unusually as a double exponential for large positive and negative $z$. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get its maximum theoretical score. By extending this model, we suggest that the front velocity $v$ is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies.

Figure 1

(Color online) $S_{n+1}$ can be obtained recursively from four optimized scores $S_n^{\left(k\right)}$ $(k=1,2,3,4)$, and finding the next optimized move from player $B$ and then from player $A$.

Figure 2

(Color online) We plot the score velocity front $v\left(p\right)$ (full line), the lower bound $v_0\left(p\right)=2p$ (lower dashed line), and $v_1\left(p\right)$, obtained when the players adopt a depth-1 strategy (upper dashed line). For $p<1∕2$, $v\left(p\right)$ is obtained by symmetry, using Eq. (8). The heuristic expression of Eq. (30) cannot be distinguished from the numerical data at this scale.

Figure 3

(Color online) Hull function $f\left(z\right)$ for $p=0.94495>p_c$, for which $v\left(p\right)=2$, and which is defined on negative integer values of $z$ (dots linked by a dotted line). For $p=0.9449$ slightly below $p_c$ $[v\left(p\right)=1.99684\dots ]$, the hull function is continuous but exhibits smooth steps at integer values of $z$ (corresponding thin line). Finally, for $p=0.75$ $(v\left(p\right)=1.53146\dots )$, we plot the continuous hull function (thick line) along with the predicted asymptotics of Eqs. (21, 23) (dashed lines).

Figure 4

(Color online) We plot the exact $p_c\left(\epsilon \right)$ above which $v_A(p,\epsilon )=2$ (full line), whose LMS and NLMS expressions are respectively given by Eqs. (42, 46). We also plot the numerical estimate of ${\epsilon }_c\left(p\right)$ (thick dashed line), the boundary between the LMS and NLMS domains of application. These two curves cross at $({\epsilon }_c,p_c)=(4∕5,5^{1∕3}∕2)$ (dot). We also identify three domains according to the relation between the observed front velocity $v$ and the LMS velocity $v_{\min }$.

Figure 5

(Color online) We plot the velocities $v_A(p,\epsilon )$ (three upper dashed lines, $\epsilon =1,4∕5,1∕4$ from top) and $v_B(p,\epsilon )$ (three lower dashed lines, $\epsilon =1,4∕5,1∕4$ from bottom). The full line corresponds to $v\left(p\right)$ for the original model $(\epsilon =0)$.