Dynamics of competitive learning: The role of updates and memory

We examine the effects of memory and different updating paradigms in a game-theoretic model of competitive learning, where agents are influenced in their choice of strategy by both the choices made by, and the consequent success rates of, their immediate neighbors. We apply parallel and sequential updates in all possible combinations to the two competing rules and find, typically, that the phase diagram of the model consists of a disordered phase separating two ordered phases at coexistence. A major result is that the corresponding critical exponents belong to the generalized universality class of the voter model. When the two strategies are distinct but not too different, we find the expected linear-response behavior as a function of their difference. Finally, we look at the extreme situation when a superior strategy, accompanied by a short memory of earlier outcomes, is pitted against its inverse; interestingly, we find that a long memory of earlier outcomes can occasionally compensate for the choice of a globally inferior strategy.

Figure 1

(SS update) Phase diagram of the model with an SS update. Plot of the inverse energy 1/$E\left(t\right)$ at time $t=512$ for a square lattice of size $N={64}^2$ in the $p$-$\varepsilon$ plane. The black region shows the disordered phase and the yellowish (light gray) region shows the frozen phase.

Figure 2

(SS update) Snapshots of the dynamics of the SS-updated model. Each plot is a portion (of size ${100}^2$) of a square lattice of $N={256}^2$ for $p=0.72$ and $\varepsilon =1.0$ at times $t=8$ (top left), $t=64$ (top right), and $t=512$ (bottom).

Figure 3

(SS update) Plot of the inverse energy $1/E\left(t\right)$ vs the natural logarithm of time ln $t$ for different values of $p$ close to $p_{c2}=0.70$, with $\varepsilon$ set to 1.0. The lattice size $N={256}^2$ and the $p$ values are indicated on the curves. The curve corresponding to $p=p_{c2}=0.70$ (shown in red) has slope $2/\pi$ approximately [see Eq. (11)].

Figure 4

(PP update) Plot of the absolute value of the magnetization $\left|M\right|$ (top) and the absolute value of the staggered magnetization $|M_{\text{stag}}|$ (bottom) at time $t=512$ for a lattice of $N={100}^2$ in the $p$-$\varepsilon$ plane. In the top figure, the yellowish (light gray) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP) and the disordered region. In the bottom figure, the black region represents the disordered phase characterized by very low $|M_{\text{stag}}|$.

Figure 5

(PP update) Snapshots of the dynamics of the PP-updated model on a square lattice, for $p=0.41$ (left), $p=0.50$ (center), and $p=0.59$ (right) at time $t=512$, with $\varepsilon$ $=$ 1.0. The yellow (light gray) and black colors represent the two strategies, while the grayish grid corresponds to antiparallel arrangements of yellow (light gray) and black. The left picture represents the PFP (see text), the center picture represents the disordered phase, and the right picture represents the AFP (see text).

Figure 6

(PP update) Plot of the absolute value of magnetization $\left|M\right|$, the absolute value of staggered magnetization $|M_{\text{stag}}|$, and energy $E$ against $p$, with $\varepsilon =1$ and lattice size $N={80}^2$. Each curve is drawn using symbols (and color) as indicated in the legend.

Figure 7

(PP update) Plot of the inverse energy $1/E\left(t\right)$ vs ln $t$ for different values of $p$ close to $p_{c1}=0.43$ and $\varepsilon =1$, for a lattice of size $N={100}^2$. The $p$ values are indicated on the curves. The straight line corresponding to $p_{c1}=0.43$ (shown in red) has slope $1/2\pi$ approximately.

Figure 8

(PP update) Plot of the inverse energy $1/E\left(t\right)$ vs ln $t$ for different values of $p$ close to $p_{c2}=0.57$, with $\varepsilon =1$, for a lattice of size $N={100}^2$. The $p$ values are indicated on the curves. The straight line corresponding to $p_{c2}=0.57$ (shown in red) has slope $-1/5\pi$ approximately.

Figure 9

(PS update) Phase diagram in the $p$-$\varepsilon$ plane of the PS-updated model, with a plot of the absolute value of staggered magnetization $|M_{\text{stag}}|$ at time $t=512$, for a lattice size of $N={100}^2$. The black region represents the disordered phase (very low $|M_{\text{stag}}|$), while the yellowish (light gray) region represents frozen phases with high $|M_{\text{stag}}|$.

Figure 10

(PS update) Plot of the absolute value of magnetization $\left|M\right|$, energy $E$, and the absolute value of staggered magnetization $|M_{\text{stag}}|$ against $p$ with $\varepsilon$ set to 1.0 for a lattice size $N={100}^2$. Each curve is drawn using symbols (and color) as indicated in the legend.

Figure 11

(PS update) Plot of the inverse energy $1/E\left(t\right)$ vs ln $t$ for different values of $p$ close to $p_{c1}=0.31$, with $\varepsilon$ set to 1.0 for a lattice size $N={256}^2$. The $p$ values are indicated on the curves. The straight line corresponding to $p_{c1}=0.31$ (shown in red) has a slope of approximately $4/3\pi$.

Figure 12

(PS update) Plot of the inverse energy $1/E\left(t\right)$ vs ln $t$ for different values of $p$ close to $p_{c2}=0.69$, with $\varepsilon$ set to 1.0, for lattice size $N={256}^2$. The $p$ values are indicated on the curves. The straight line corresponding to $p_{c2}=0.69$ (shown in red) has slope $-4/15\pi$ approximately.

Figure 13

Transition probabilities [$1-w_{+}(+2)$] drawn as a solid line [in green (gray)] and $w_{-}(+2)$ drawn as a dashed line (in black) against $p$ [from Eq. (8)].

Figure 14

The difference in the transition probabilities [$1-w_{+}(+2)$], with $w_{-}(+2)$ plotted against $p$ [see Eq. (8)].

Figure 15

(SP update) Phase diagram of the SP-updated model. Plot of the absolute value of magnetization $\left|M\right|$ at time $t=512$ for lattice size $N={64}^2$ in the p-$\varepsilon$ plane. No phase transition is visible.

Figure 16

(SP update) Plot of the inverse energy $1/E\left(t\right)$ vs ln $\left(t\right)$ for different values of $p$ and $\varepsilon =1$, for a lattice size $N={256}^2$. The value of $p$ for each curve is given by a different color as indicated. No phase transition is visible.

Figure 17

(SS update) Plot of magnetization $M$ against biasing field $H$ for different values of $p$. Each curve is drawn using different symbols (and color) as shown in the legend, at time $t=2000$ with $N={100}^2$ and $\varepsilon =1.0$.

Figure 18

(SS update) Plot of magnetization $M$ against field $H$ for $p=0.58$ and $\varepsilon =1.0$ at time $t=2000$ with $N={100}^2$. The fit of a $\tanh \left(bH\right)$ curve almost completely overlaps with our numerical results.

Figure 19

(SS update) Plot of magnetization $M$ in the $p_{+}$-$p_{-}$ plane at time $t=7000$, for a lattice of size $N={100}^2$ with $\varepsilon =1.0$. The $+$ strategy dominates in the yellow (light gray) region, while the $-$ strategy dominates in the brown (black) region.

Figure 20

(PP update) Plot of magnetization $M$ against field $H$ for different values of $p$, each curve indicated by a different symbol (and color) as shown in the legend, at time $t=2000$ with $N={100}^2$ and $\varepsilon =1.0$.

Figure 21

(PP update) Plot of the absolute value of staggered magnetization $|M_{\text{stag}}|$ against field $H$ for different values of $p$, each curve indicated by a different symbol (and color) as shown in the legend, at time $t=2000$ with $N={64}^2$ and $\varepsilon =1.0$.

Figure 22

(PP update) Plot of magnetization $M$ in the $p_{+}$-$p_{-}$ plane for a lattice of size $N={100}^2$ at time $t=7000$, with $\varepsilon =1.0$. As before, the regions of $+$ and $-$ strategy domination are colored yellow (light gray) and brown (black); the orange (gray) region corresponds to both the paramagnetic and the AFP region (see text).

Figure 23

(PP update) Plot of staggered magnetization $|M_{\text{stag}}|$ in the $p_{+}$-$p_{-}$ plane, for a lattice of size $N={100}^2$ at time $t=7000$, with $\varepsilon =1.0$. Here, yellow (light gray) represents the region of parallel ordering, black represents the disordered phase, and the light brown (gray) represents AFP order.

Figure 24

(PP update) A three-dimensional plot of magnetization $M$ with parameters ${\varepsilon }_{+}$ (along $x$), ${\varepsilon }_{-}$ (along $y$) and $p_{-}$ (along $z$), setting $p_{+}=0.50$ for a lattice of size $N={64}^2$ at time $T=2000$. Green (gray) denotes the dominance of the $+$'s, while blue (black) denotes that of the $-$'s. The other colors represent cases of intermediate ordering.

Figure 25

(PP update) A three-dimensional plot of magnetization $M$ for the parameters ${\varepsilon }_{+}$ (lying between $0.7$ and $0.8$), ${\varepsilon }_{-}$ and $p_{-}$, with $p_{+}=0.70$, for a lattice of size $N={64}^2$ at time $T=2000$. The red (lower black) region represents the dominance of the $+$ strategy and the blue (upper black) region represents the dominance of the $-$ strategy. The green (light gray band) region represents AFP.

Figure 26

(PP update) A three-dimensional plot of absolute staggered magnetization $|M_{\text{stag}}|$ with parameters ${\varepsilon }_{+}$(along $x$), ${\varepsilon }_{-}$(along $y$), and $p_{-}$(along $z$) for $p_{+}=0.50$ and lattice size $N={64}^2$, at time $T=2000$. The graph delineates the region of AFP for these parameter values.