| We consider a model of a discrete time "interacting particle system" on the integer line where infinitely many changes are allowed at each instance of time. We describe the model using chameleons of two different colors, say, red ($R$) and blue ($B$). At each instance of time each chameleon performs an independent but identical coin toss experiment with probability $\alpha$ to decide whether to change its color or not. If the coin lands head then the creature retains its color (this is to be interpreted as a "success"), otherwise it observes the colors and coin tosses of its two nearest neighbors and changes its color only if, among its neighbors and including itself, the proportion of successes of the other color is larger than the proportion of successes of its own color. This produces a Markov chain with infinite state space $\left\{R, B\right\}^{\Zbold}$. This model was studied by Chatterjee and Xu (2004) in the context of diffusion of technologies in a set-up of myopic, memoryless agents. In their work they assume different success probabilities of coin tosses according to the color of the chameleon. In this work we consider the "critical" case where the success probability, $\alpha$, is the same irrespective of the color of the chameleon. We show that starting from any initial translation invariant distribution of colors the Markov chain converges to a limit of a single color, i.e., even at the critical case there is no "coexistence" of the two colors at the limit. As a corollary we also characterize the set of all translation invariant stationary laws of this Markov chain. Moreover we show that starting with an i.i.d. color distribution with density $p \in [0,1]$ of one color (say red), the limiting distribution is all red with probability $\pi\left(\alpha, p\right)$ which is continuous in $p$ and for $p$ "small" $\pi(p) > p$. The last result can be interpreted as the model favors the "underdog". |