future expectations. The move of the strategies of individuals in a $ \Lambda $ -individual system is described by $ X^{(\Lambda)}(t) $ as mentioned.

Let us be given Lipschitz continuous functions $ r_{h}^{i}:D\to R_{+} $ and $ p_{h}^{i}:D\to\triangle^{i} $ for $ h\in S^{i},i\in I $ , where $ D\subset R^{m} $ is an open domain containing $ \Theta $ , and the l-th component of $ p_{h}^{i} $ is denoted by $ p_{hl}^{i} $ . Then, when $ X^{(\Lambda)}(t)=H^{(\Lambda)} $ , the reviewing time after t of a h-individual in population i is given by a random exponential time with parameter $ r_{h}^{i}(V(X^{(\Lambda)}(t)))=r_{h}^{i}(V(H^{(\Lambda)})) $ , and the renewal strategy at the reviewing time is chosen according to $ p_{h}^{i}(V(X^{(\Lambda)}(t)))=p_{h}^{i}(V(H^{(\Lambda)})) $ . The random exponential times of individuals are independent, and their strategy revisions, which are independent of the random exponential times, are also independent across individuals and time. $ r_{h}^{i} $ and $ p_{h}^{i} $ are called review rate (for h-individual in population i), and choice probability (for h-individual in population i) respectively.

In the above formulation, which we call the original formulation (of Björnerstedt and Weibull), major feature is that the review rate and the choice probability depend only on the current strategy of each oneself and the current population state. We relax this feature by assuming that the review rate $ \bar{r}_{\lambda}^{i} $ and the choice probability $ \bar{p}_{\lambda}^{i} $ of the $ \lambda $ -th individual in population i are given as follows: For $ i \in I, \lambda \in \bar{\Lambda} $ , the review rate $ \bar{r}_{\lambda}^{i}(X^{(\Lambda)}(t)) = r_{X_{\lambda}^{i}(t)}^{i}(V(X^{(\Lambda)}(t))) + D r_{\lambda}^{i}(X^{(\Lambda)}(t)) $ for a $ R_{+} $ -valued Lipschitz continuous function $ r_{h}^{i} $ on D and a bounded function $ D r_{\lambda}^{i} $ on $ S^{\otimes\Lambda} $ such that $ D r_{\lambda}^{i}(H^{(\Lambda)}) = o(1) $ for any $ H^{(\Lambda)} \in S^{\otimes\Lambda} $ . The choice probability $ \bar{p}_{\lambda}^{i}(X^{(\Lambda)}(t)) = p_{X_{\lambda}^{i}(t)}^{i}(V(X^{(\Lambda)}(t))) + D p_{\lambda}^{i}(X^{(\Lambda)}(t)) $ for a $ \triangle^{i} $ -valued Lipschitz continuous function $ p_{h}^{i} $ on D and a $ R^{m^{i}} $ -valued bounded function $ D p_{h}^{i} $ on $ S^{\otimes\Lambda} $ such that the l-th component $ D p_{\lambda l}^{i}(H^{(\Lambda)}) = o(1) $ for any $ H^{(\Lambda)} \in S^{\otimes\Lambda} $ . Here o stands for Landau's symbol.

By this relaxation, the dynamics of a $ \Lambda $ -individual system is modified to be approximately determined by the current population state in the sense that $ D r_{\lambda}^{i}(X^{(\Lambda)}(t)) $ and $ D p_{\lambda}^{i}(X^{(\Lambda)}(t)) $ diminish and the dynamics is almost determined by the current population state when $ \Lambda $ is sufficiently large. The relaxed formulation, which we call the modified formulation (of Björnerstedt and Weibull), can be typically realized by the following assumptions on models.

(A1) At each instant all individuals in all populations are randomly divided into $ \Lambda $ ordered tuples of n individuals to which only one individual in each population