$ keeps getting wider with increasing $b$. When separatrix reaches the other fixed point the latter disappears resulting a unique fixed point in the large $b$ region. Turning to the SH side, the dynamical cluster approximation predicts increasing $K^{\star}$ fixed point as we leave $b=1$, which is again in nice agreement with the MC results. Further decreasing $b$, however, the approximation predicts a small two-attractor region and finally the $K^{\star}(b)$ function coincides with the phase boundary where $\rho$ becomes 1. 

Before evaluating the predictions of the present cluster mean-field approach, we should stress that the simpler (two-state) version of the three-site approximation, which is valid for homogeneous $K$ case, cannot describe correctly the functions $\rho(K)$ as well as $b_{c1}(K)$ in the low $K$ limit. Namely, the border of all C phase tends to $b=0.75$ instead of $b=1$ when $K \to 0$. (According to this, defectors can survive the deterministic limit when $b<1$.) Such a qualitative failure of the approximation might have been the consequence of the small number of independent variables. In other words, the restricted freedom prevents the approximation to find the valid solution. At the same time, if we increase the number of independent variables by letting different $K_i$ values for players, the approximation is already capable to find the relevant solution (at least in the vicinity of $b=1$). Our argument is supported by the fact that the five-site approximation for the homogeneous $K$ model can also describe the behavior $b_{c1} \to 1$ in the $K \to 0$ limit. The relevant increase in $b_{c1}$ can remove the artifact(s) occurred at low $b$ values. Summing up, the extension of the three-site approximation by letting different $K_i$ values was capable to indicate the correct results of the more sophisticated approximation based on larger cluster of sites.

\section{Discussion and outlook}

We studied evolutionary Prisoner's Dilemma and Stag Hunt games on two 
representative two-dimensional lattices. The underlying structures were the square lattice and the kagome lattice exemplifying spatial connectivity structures without or with triangle percolation. We analyzed the 
simultaneous evolutions of strategy and a player specific noise parameter
used by individuals in the strategy imitation processes. It turned out that players prefer to use the same distinguished noise level, namely the same way of strategy adoption
and the fixed points of the adoption parameter are systematically close to those parameter values which are the most favorable for cooperation for the PD games. It implies that the evolution of the adoption parameter drives the population to a state which assures substantial cooperation within the coexistence region. In the region of SH game the maximum average payoff is achieved at the peripheries of the coexistence region. It is shown that in the latter case the Darwinian selection favors the edge of coexistence region where the noise level ($K$) is lower. The preference of the lower noise level can also be observed within the PD region because the distinguished rule $K^{\star}$ was always smaller than that where the maximum average payoff (or cooperator density) occurs for homogeneous dynamical rules at fixed payoff. The above results raise many general questions about the main features of states (including many aspects of the model itself) favored by the Darwinian selection. Here we emphasize that the Darwinian selection seems to be more efficient within the coexistence region where the simultaneous evolution of strategies accelerates the evolution of adoption rules, too.
The latter observation is confirmed by simulations where the coexistence in maintained artificially by introducing rare mutations in the systems.

The above investigations have required to improve the accuracy particularly in the low noise limit where the relaxation time diverges. The systematic investigation of the cooperator's density versus $b$ (temptation to choose defection) and $K$ have indicated different types of non-analytical behaviors in the limit $b \to 1$ and $K \to 0$ as Figs.~\ref{sqr_mc} and \ref{kag_mc} show. 
At the same time we found qualitatively similar behavior on both structures in the region of SH game. Although most of these features can be reproduced qualitatively well by the dynamical cluster methods on sufficiently large cluster of sites, we think that further analysis is required to clarify the effects of the sucker's payoff $S$, topology of connectivity network, and dynamics (e.g., when irrational choices are forbidden) on the non-analytical behavior appeared here at $b=1$ and $K=0$.

The present investigations have expanded the research of coevolutionary game theory by applying Darwinian selection among a continuous set of noise levels in dynamical rules used by the individuals while other relevant ingredients of the model were fixed.  In most of the previous studies of the coevolutionary games only two ingredients of the system were allowed to evolve simultaneously. Recently, Van Segbroeck {\it et al.} have studied a model where the players could modify three quantities: their strategy, connections, and the way how a new partner is chosen \cite{vansegbroeck_prl09}. In the light of the latter model one can ask what happens if the payoff parameter $b$ is also considered as an individual property ($b_x$) and it can be adopted from the neighbors as well as the strategy $s_x$ and dynamical rule $K_x$. The preliminary MC results have indicated that within the strategy coexistence region of the $b-K$ plane the system evolves toward $K^{\star}(b)$ with favoring smaller $b$ values. It is found that the system evolves fast toward a state where players use game of the lower $b$ value and subsequently the homogenization in $K_x$ will be done as described above. This means that the Darwinian selection prefers SH to PD game, that is, the system develops an environment where the mutual cooperation (providing the maximum average payoff) can be achieved more conveniently. Similar results were reported by Worden and Levin \cite{worden_jtb07} and also by Fort \cite{fort_epl08} who studied models with different adoptions of payoff parameters. Evidently, other results can be obtained if 
the simultaneous evolution of the connectivity structure (interaction and learning networks \cite{ohtsuki_prl07}) is also possible.

In principle, all the ingredients of the multi-agent coevolutionary games can be the subject of Darwinian selection if we assume that these features are determined by the participants. In that case the system can evolve towards a strategy distribution with a proper connectivity structure, payoff parameter, adoption rule(s), mutation, {\it etc.}, that are preferred by the Darwinian selection. As a consequence the given Darwinian selection will show us the preferred features (or parameter values) we can fix when exploring the effect of other properties \cite{ho_jet07}. To be  more precise, the $K^{\star}$ value(s) can be suggested in numerical simulations if one wish to fix the noise level.

Finally we would like to mention that in many real systems besides the evolving individual's features there are external conditions affecting the system behavior. For example, the noise itself can arise from external sources as it was investigated by Traulsen \cite{traulsen_prl04} and Perc \cite{perc_njp06a}. Further systematic investigations are required to clarify the effect of the external noise or any other questions mentioned above.

\section*{Acknowledgments}

This work was supported by the Hungarian National Research Fund (Grant No. K-73449), Bolyai Research Grant and FCT Portugal.

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