Silviase/jgov_v1.4_full_ann huggingface dataset:
## 1 Introduction

In recent years, many authors have studied imitation or learning models in which individuals choose actions by imitating or learning others'. For example, Börgers and Sarin (1997) and (2000), Cabrales (2000), Dixon (2000), and Gale et al. (1995) work with aspiration-based reinforcement learning models. Björnerstedt and Schlag (1996) and Schlag (1998) explore imitation models which are formulated by a multi-armed bandit. Weibull (1995) and Björnerstedt and Weibull (1996) formulate social evolution by myopic imitation and give a few specific examples of imitation dynamics.

Among these works, the formulation of Weibull (1995) and Björnerstedt and Weibull (1996) seems more generic in the sense that a model is not specifically prescribed in it. In the works, they imagine multiple populations of individuals who live infinitely and strategically interact forever in a continuous time horizon. Each individual holds some pure strategy in a set of a finite number of strategies and occasionally reviews the strategy. The reviewing times are given by the arrival times of a Poisson process. Then the strategy of each individual is ruled by a review rate, which is the arrival rate of the Poisson process, and choice probability, by which the renewal strategy is chosen when the reviewing time comes to an individual. The review rate and the choice probability are assumed to be Lipschitz continuous functions of the current population state, i.e., the current state of the average frequencies of strategies in populations.

Under the independence among the strategy switches of all individuals, the aggregate stochastic process of individuals is modeled as deterministic dynamics by the law of large numbers for infinite individuals. The deterministic dynamics is reduced to replicator dynamics in a special case. The derivation shows that the aggregate behavior of individuals in finite but large populations can be approximated by deterministic dynamics as replicator dynamics.

The main feature of their formulation is that the stochastic dynamics of an individual is ruled by the review rate and the choice probability that depend only on the average frequencies of strategies in populations. Then the derived deterministic dynamics is described by the average frequencies. However, since their formulation of a model and derivation of the deterministic dynamics is given by
intuitive sketch, important points as the following do not seem clear when we consider the approximation of aggregate behavior of large but finite populations.

I. What models of finite populations can be contained in their formulation? More specifically, how can stochastic dynamics depend only on the current average frequencies of strategies in finite populations?

II. How does the law of large numbers hold in spite of mutual interactions through the current average frequencies of strategies?

In this paper we clarify these points and rigorously justify the approximation of the aggregate imitational behavior in large but finite populations by the deterministic dynamics given by Björnerstedt and Weibull. For addressing the task, we proceed with the following two steps.

In the first step we reformulate the imitational behavior presented by Björnerstedt and Weibull and build up assumptions for models to fit the formulation. Then we explain that under the independence among Poisson processes that give individuals reviewing opportunities, the imitational behavior based on the payoffs of the game by randomly grouped individuals can be approximately described by the current average frequencies. In particular, when the imitational behavior of an individual is based only on his/her payoff, not dependent on the payoff of another individual, models are contained in the original formulation of Björnerstedt and Weibull, in which the corresponding dynamics can be completely described by the current average frequencies.

In the second step we construct a stochastic process that describes imitational behaviors in finite populations. We then prove the law of large numbers for interacting individuals, which implies the average frequencies of strategies in populations come to represent the deterministic dynamics of Björnerstedt and Weibull.

A key for the proof is propagation of chaos. If a finite number of particles are independent and identically distributed time t = 0, then this independence propagates at time t > 0 in symmetric interactions, and the probability law of the finite particle system converges to the infinite product of the probability law of a common stochastic process as the number of particles goes to infinity. This phe-
nomenon was first discovered by Kac and called propagation of chaos by McKean (1966). The common stochastic process is now called a McKean process. When the propagation of chaos holds, particles become mutually independent and the law of large numbers also holds at the same time. We show that for stochastic dynamics that are approximately described by the average frequencies, if the strategies of individuals are initially independent and identically distributed, the propagation of chaos and the law of large numbers for interacting individuals holds in a given finite time interval. This justifies the approximation of the aggregate dynamics in large populations by the deterministic dynamics of Björnerstedt and Weibull.

The rest of this paper is organized as follows. Section 2 presents the formulation of the imitational behavior given by Björnerstedt and Weibull and clarifies assumptions for models to fit it. Some specific examples are given. Section 3 presents the result for the propagation of chaos and discusses the restrictions on it. Section 4 concludes, and the Appendix contains all proofs.

## 2 Model

In this section we reformulate the model of imitational behavior given by Björnerstedt and Weibull. Then we generally show what assumptions on models support the formulation. We also give some specific examples that fit the formulation.

### 2.1 Setup

We begin with a notational setup. We set a n-player game as follows. $ I = \{1, \ldots, n\} $ is the set of players, and $ S^{i} = \{1, \ldots, m^{i}\} $ with $ m^{i} \geq 2 $ is the pure-strategy set of player $ i \in I $ . We use character i, j for a player and h, k, l for a pure strategy. $ \triangle^{i} $ is the mixed-strategy set on $ S^{i} $ , and $ v_{l}^{i} $ is the l-th component of $ v^{i} \in \triangle^{i} $ . A pure strategy $ h \in S^{i} $ is identified with a unit vector $ e_{h}^{i} = (0, \ldots, 0, \frac{1}{h-th}, 0, \ldots, 0) $ in $ \triangle^{i} $ . We denote the set of pure strategy profile $ \times^{i} S^{i} $ by S and the polyhedron of mixed-strategy profiles in $ R^{m} $ by $ \Theta $ , where $ m = m^{1} + \cdots + m^{n} $ . We use $ H = (h^{1}, \ldots, h^{n}) $ and $ v = (v^{1}, \ldots, v^{n}) $ for generic elements of S and $ \Theta $ respectively. $ \pi^{i}(v) $ is the payoff to player $ i \in I $ when $ v \in \Theta $ is played. $ \pi^{i}(v) $ is also denoted by $ \pi^{i}(v^{i}, v^{-i}) $ when player
i uses strategy $ v^{i} $ and the other players use strategy $ v^{-i} $

As the usual setting of population games, I represents a set of populations with a large number of individuals. We call an individual who uses a strategy $ h \in S^{i} $ , a h-individual. $ v \in \Theta $ is identified with a population state (profile), i.e., $ v_{h}^{i} $ is the share of h-individuals in population i. Then $ \pi^{i}(v) $ represents the average payoff of population i when the population state is $ v = (v^{1}, \ldots, v^{n}) \in \Theta $ .

In the following we consider stochastic dynamics of imitational behavior in n populations with finite individuals in a continuous time horizon. We denote the cardinality of populations by $ \Lambda $ and set $ \bar{\Lambda} = \{1, \ldots, \Lambda\} $ . Then we call n populations with $ \Lambda $ individuals suffixed by $ \lambda \in \bar{\Lambda} $ , a $ \Lambda $ -individual system.

Let $ X_{\lambda}^{i}(t) $ be the strategy of the $ \lambda $ -th individual in population i at time t, i.e., $ X_{\lambda}^{i}(t) = e_{h}^{i} $ means that the $ \lambda $ -th individual in population i holds a strategy h at time t. Then we denote the configuration of a $ \Lambda $ -individual system by $ X^{(\Lambda)}(t) = (X_{1}^{1}(t), \ldots, X_{1}^{n}(t), \ldots, X_{\Lambda}^{1}(t), \ldots, X_{\Lambda}^{n}(t)) \in \Theta^{\otimes\Lambda} $ , which describes a stochastic process on $ S^{\otimes\Lambda} $ .

Further, for $ X^{(\Lambda)}(t) $ set an empirical distribution on $ S^{i} $ and S by $ V^{i}(X^{(\Lambda)}(t)) = \frac{1}{\Lambda} \sum_{\lambda=1}^{\Lambda} X_{\lambda}^{i}(t) \in \triangle^{i} $ and $ V(X^{(\Lambda)}(t)) = (V^{1}(X^{(\Lambda)}(t)), \ldots, V^{n}(X^{(\Lambda)}(t))) \in \Theta $ respectively. Then, for $ h \in S^{i} $ , $ V_{h}^{i}(X^{(\Lambda)}(t)) $ represents the average frequency of h-individuals in population i at time t, and $ V(X^{(\Lambda)}(t)) $ is the population state of a $ \Lambda $ -individual system at time t. For $ H^{(\Lambda)} = (h_{1}^{1}, \ldots, h_{1}^{n}, \ldots, h_{\Lambda}^{1}, \ldots, h_{\Lambda}^{n}) \in S^{\otimes \Lambda} $ , $ V^{i}(H^{(\Lambda)}) $ and $ V(H^{(\Lambda)}) $ are similarly defined.

### 2.2 Finite Individual System

We now formulate the process of imitational behavior given by Björnerstedt and Weibull. Although we formulate it in multiple populations $ (n \geq 2) $ , we do not have any difficulties in doing it in a single population $ (n = 1) $ . We consider a $ \Lambda $ -individual system that consists of myopic and memoryless individuals. Each individual in each population holds some pure strategy for some time interval and occasionally reviews the strategy in the process. Since each individual is myopic and memoryless, his/her decision on whether to revise the strategy and which strategy to choose for the renewal strategy can be thought to depend only on the information based on the current strategy profile, not on the past experience and
belongs with equal probability $ ^{1} $ . Then each individual obtains a payoff dependent on the strategy profile of the ordered tuple to which he/she belongs.

(A2) The opportunities of all individuals for reviewing the strategy are given by the arrival times of Poisson processes with a common constant arrival rate, which are independent across individuals. Here we note that only the arrival of an reviewing opportunity to an individual does not necessarily mean that the individual revises his/her strategy at the time, as suggested by (A3).

(A3) When a reviewing opportunity comes to an individual, the individual makes the decision whether to revise his/her strategy based on the value of some function of payoffs that he/she and an individual, who he/she randomly samples in his/her population with equal probability $ ^{2} $ , obtain at the time. Here the function can also depend on Lipschitz continuously on the population state.

(A4) When an individual revises the strategy, he/she chooses the renewal strategy depending on the strategy profile of the ordered tuple to which he/she belongs at the time or on the strategy of an individual who he/she randomly samples (with equal probability) in his/her population.

(A5) The random grouping in (A1), the Poisson processes in (A2), the random sampling in (A3), and the random sampling in (A4) are mutually independent across individuals and time.

(A1) shows that each individual repeatedly obtains a payoff, for example, by playing the n-person game with randomly grouped individuals. (A2) means that the reviewing opportunities of an individual never coincide with those of another individual. This is crucial because, based on this fact and the independence in
(A5), (A3) indicates that the decision of an individual on whether to revise the strategy can be represented to be approximately determined based on a function of payoffs to the current average frequencies of strategies in populations $ ^{3} $ . Further the decisions are independent across individuals and time.

(A4) implies that the renewal strategy of an individual is chosen depending on the current average frequencies of strategies in populations. Finally (A5) ensures that the strategy revisions are independent across individuals and time.

Thus it can be shown that the stochastic process of imitational behaviors based on the above assumptions is approximately determined by the current population state and fits the modified formulation. Especially, the stochastic process matches the original formulation when an individual makes the decision whether to revise his/her strategy depending only on his/her payoff. We will later give specific examples that satisfy the above assumptions and that fit the original and modified formulation of Björnerstedt and Weibull.

### 2.3 Limit of Finite Individual System

For a sufficiently large $ \Lambda $ , individuals almost symmetrically interact with each other only through the current average frequencies in the modified formulation. Therefore, if $ \{(X_{\lambda}^{1}(0),\ldots,X_{\lambda}^{n}(0)),\lambda\in\bar{\Lambda}\} $ are independently and identically distributed, the propagation of chaos holds as $ \Lambda\to\infty $ . That is, $ \{(X_{\lambda}^{1}(t),\ldots,X_{\lambda}^{n}(t)),\lambda\in\bar{\Lambda}\} $ become mutually independent when $ \Lambda $ goes to infinity. Then $ V(X^{(\Lambda)}(t)) $ converges to some $ v(t)=(v^{1}(t),\ldots,v^{n}(t))\in\Theta $ in probability, and $ (X_{\lambda}^{1}(t),\ldots,X_{\lambda}^{n}(t)) $ comes to represent a common McKean process $ (X^{1}(t),\ldots,X^{n}(t)) $ , which is constructed by review rate $ r^{i}(v(t)) $ and choice probability $ p^{i}(v(t))\;(i\in I) $ and has $ v(t) $ as the distribution at time t.

As stated in Section 3, $ v(t) $ represents the following imitation dynamics given by Björnersted and Weibull :

$$ \dot{v_{h}^{i}}=\sum_{l\in S^{i}}r_{l}^{i}(\nu)p_{l h}^{i}(\nu)v_{l}^{i}-r_{h}^{i}(\nu)v_{h}^{i},h\in S^{i},i\in I. $$
Then the above argument suggests that when $ \Lambda $ is sufficiently large, the stochastic process of a $ \Lambda $ -individual system can be approximately represented by $ \overset{\Lambda}{\otimes}_{i=1}^{n}X^{i} $ , a product of independent copies of the McKean process, in a given finite time interval. Further the stochastic process of the average frequencies of strategies in populations can be approximated by the deterministic dynamics (2.1) given by Björnersted and Weibull. It is readily shown that when $ (-r_{1}^{i}(\nu),\ldots,-r_{m^{i}}^{i}(\nu)) $ is payoff monotonic and $ p_{h}^{i}(\nu)=\nu_{h}^{i} $ , (2.1) turns into payoff monotonic dynamics $ ^{4} $ , which is a generalization of replicator dynamics.

### 2.4 Examples

Here we specifically show how models that satisfy the assumptions in Section 2.2 match the original and modified formulation of Björnersted and Weibull. We use four examples.

Simple reinforcement learning. This example, which is a slight modification of the model of Gale et al. (1995) and Section 4.3.3.2 in Vega-Redondo (1996) $ ^{5} $ , fits the original formulation.

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 belongs with equal probability. When the strategies of the tuple to which an $ h^{i} $ -individual in population i belongs is $ H = (h^{1}, \ldots, h^{n}) $ , the $ h^{i} $ -individual receives $ \pi^{i}(H) $ as a payoff by playing the game.

The reviewing opportunities of individuals are given by the arrival times of independent Poisson processes with arrival rate one, which are also independent of the random grouping. When a reviewing opportunity comes to an $ h^{i} $ -individual in population i, he compares the payoff to some target level of satisfaction or aspiration $ \mu $ . Here the satisfaction level is uniformly distributed $ ^{6} $ on $ [\underline{\omega}^{i},\overline{\omega}^{i}] $ with
$ \underline{\omega}^{i} < \min_{H \in S} \pi^{i}(H) < \max_{H \in S} \pi(H) < \overline{\omega}^{i} $ and independent across individuals, their choice of individuals and time. If the payoff is more than $ \mu $ , he retains his current strategy. Otherwise he/she randomly chooses an individual in population i with equal probability and imitates the strategy of the individual.

Thus $ X^{(\Lambda)}(t) $ is reduced to a continuous Markov chain given by the review rate and the choice probability that are taken as $ r_{h}^{i}(V(X^{(\Lambda)}(t)))=\frac{\overline{\omega}^{i}-\pi^{i}(h,V^{-i}(X^{(\Lambda)}(t)))}{\overline{\omega}^{i}-\underline{\omega}^{i}} $ and $ p_{h}^{i}(V(X^{(\Lambda)}(t)))=V^{i}(X^{(\Lambda)}(t)) $ , because of the mutual independence among reviewing opportunities, random grouping, random choice, and satisfaction level. Then, for (2.1) we have

$$ \dot{v}_{h}^{i}=\frac{1}{\overline{\omega}^{i}-\underline{\omega}^{i}}(\pi^{i}(h,v^{-i})-\pi^{i}(v))v_{h}^{i},h\in S^{i}. $$

(2.2) is not only payoff monotonic but payoff positive dynamics $ {}^{7} $ and especially represents replicator dynamics in the case where $ \overline{\omega}^{i}-\underline{\omega}^{i} $ is common among all populations.

Switch below noisy average payoff. This is a modification of a learning model in Dixon (2000) by adding noise. Under the same setting as above, consider the case where the satisfaction level $ \mu $ is taken to be about the average payoff. That is, when the current population state of the $ \Lambda $ -individual system is v, for individuals in population i, $ \mu $ is uniformly distributed on $ [\pi^{i}(v)-\varepsilon,\pi^{i}(v)+\varepsilon] $ with a small $ \varepsilon>0 $ .

For $ H = (h^{1}, \ldots, h^{n}) \in S $ and $ v \in \Theta $ , let us set

$$ \begin{array}{c c}{1,}&{\pi^{i}(H)>\pi^{i}(v)+\varepsilon}\\ {f^{i}(H:v)=(\pi^{i}(v)-\pi^{i}(H)+\varepsilon)/2\varepsilon,}&{o t h e r w i s e}\\ {0,}&{\pi^{i}(H)<\pi^{i}(v)-\varepsilon}\\ \end{array}. $$

Then, in this case the review rate is taken as $ r_{h}^{i}(v) = \int f^{i}(h, H^{-i} : v) v^{-i} (dH^{-i}) $ and it is Lipschitz continuous in v. Therefore, this case also matches the original formulation, if we take, $ p_{h}^{i}(v) = v^{i} $ as above. Then, for (2.1) we have

$$ \dot{v}_{h}^{i}=(\int f^{i}(H:v)v(d H)-\int f^{i}(h,H^{-i}:v)v^{-i}(d H^{-i}))v_{h}^{i},h\in S^{i}. $$

while the utility obtained from the payoff is perturbed by random shock (Gale et al. (1995)).

$ ^{7} $ Population dynamics $ \dot{v}_{h}^{i}=G_{h}^{i}(v)v_{h}^{i}, h\in S^{i} $ is called payoff positive if $ \mathrm{sgn}[G_{h}^{i}(v)] $ coincides with $ \mathrm{sgn}[\pi^{i}(h,v^{-i})-\pi^{i}(v)] $ for all $ v\in\Theta $ . Payoff positive dynamics is also a generalization of replicator dynamics.
Switch to best response. This is also taken from Dixon (2000). Just like the above example, the satisfaction level $ \mu $ is taken to be about the average payoff. However, when individuals revise their strategies, they switch to the best response strategy to the strategy profile of individuals with whom they have played the game.

Let us denote by $ B(H^{-i}) $ the set of best response pure strategies to strategy profile $ H^{-i} $ and by $ I_{B(H^{-i})} $ the indicator function of $ B(H^{-i}) $ on $ S^{i} $ , i.e., $ I_{B(H^{-i})}(h) = 1 $ if and only if $ h \in B(H^{-i}) $ . Then, by using $ f^{i} $ defined in the above, we can take $ r_{h}^{i}(v) = 1 $ and $ p_{hh'}^{i}(v) = \int (f^{i}(h, H^{-i} : v) I_{B(H^{-i})}(h') / |B(H^{-i})|) v^{-i} (dH^{-i}) $ , where $ |B(H^{-i})| $ stands for the cardinality of $ B(H^{-i}) $ . For (2.1) we have

$$ \dot{v}_{h}^{i}=\int\frac{1}{\left|B(H^{-i})\right|}f^{i}(l,H^{-i}:v)I_{B(H^{-i})}(h)v(dldH^{-i})-v_{h}^{i},h\in S^{i}. $$

Clearly not only this case but any case where $ r_{h}^{i} $ is taken to be Lipschitz continuous is contained in the original formulation.

Proportional imitation. This example is based on a multi-armed bandit approach (Björnerstedt and Schlag (1996) and Schlag (1998)). For $ H = (h^{1}, \ldots, h^{n}) \in S $ let $ P_{h^{i}}^{i}(H) $ be a probability distribution on $ R_{+} $ such that it is supported with interval $ [\underline{\omega}^{i}, \overline{\omega}^{i}](0 \leq \underline{\omega}^{i} < \overline{\omega}^{i}) $ and has mean $ \pi^{i}(H) $ . Further, for $ v \in \Theta $ let us denote by $ P_{h^{i}}^{i}(v) $ the randomization of $ P_{h^{i}}^{i}(H) $ by $ v^{-i} $ , i.e.,

$$ P_{h^{i}}^{i}(v)(d x)=\sum_{h^{1},\ldots,\check{h}^{i},\ldots,h^{n}}v^{1}(h^{1})\cdots\check{v}^{i}(h^{i})\cdots v^{n}(h^{n})P_{h^{i}}^{i}(h^{1},\ldots,h^{n})(d x), $$

where $ \check{h}^{i} $ means $ h^{i} $ is dropped out. $ P_{h^{i}}^{i}(u) $ is also supported with $ [\underline{\omega}^{i},\overline{\omega}^{i}] $ and has mean $ \pi^{i}(h^{i},v^{-i}) $ .

At each instant all individuals in all populations are randomly divided into $ \Lambda $ ordered tuples of n individuals as before. When the strategies of the tuple to which an $ h^{i} $ -individual in population i belongs is $ H = (h^{1}, \ldots, h^{n}) $ , he/she observes an uncertain payoff drawn from $ P_{h^{i}}^{i}(H) $ . The payoff is independently realized across individuals and time.
The reviewing opportunities of individuals are also given by the arrival times of Poisson processes with arrival rate one, as in the previous examples. When a reviewing opportunity comes to an $ h^{i} $ -individual, he/she randomly samples one individual in the population i with equal probability. Now suppose his/her current payoff is $ x_{h^{i}} $ and the one of the sampled individual is $ x_{h^{i^{\prime}}} $ . Then he switches his strategy from $ h^{i} $ to $ h^{i^{\prime}} $ with probability $ \frac{x_{h^{i^{\prime}}}-x_{h^{i}}}{\overline{\omega}^{i}-\underline{\omega}^{i}} $ only if $ x_{h^{i^{\prime}}} > x_{h^{i}} $ . This is a proportional imitation and $ \frac{1}{\overline{\omega}^{i}-\underline{\omega}^{i}} $ is called a switching rate $ ^{8} $ (Björnerstedt and Schlag(1996) and Schlag (1998)).

Let us set $ \alpha_{hh'}^{i}(v)=\frac{1}{\overline{\omega}^{i}-\underline{\omega}^{i}}\int_{y>x}(y-x)P_{h}^{i}(v)(dx)P_{h'}^{i}(v)(dy),h'\neq h\in S^{i} $ for $ v\in\Theta^{9} $ and $ p_{hh'}^{i}(V(X^{(\Lambda)}(t)))=\alpha_{hh'}^{i}(V(X^{(\Lambda)}(t)))V_{h'}^{i}(X^{(\Lambda)}(t)) $ for $ h'\neq h\in S^{i\ 10} $ . Then it is easy to see that this model can be reduced to a model in the modified formulation with $ \bar{r}_{\lambda}^{i}(X^{(\Lambda)}(t))=1 $ and $ \bar{p}_{\lambda h'}^{i}(X^{(\Lambda)}(t))=p_{X_{\lambda(t)}^{i}h'}^{i}(V(X^{(\Lambda)}(t)))+Dp_{\lambda h'}^{i}(X^{(\Lambda)}(t)) $ , where $ Dp_{\lambda h'}^{i}(H^{(\Lambda)})=\frac{V_{h'}^{i}(H^{(\Lambda)})}{(\overline{\omega}^{i}-\omega^{i})\Lambda^{n-1}} $ .

$$ \begin{align*}&[\frac{1}{(\Lambda-1)^{n-1}}\sum_{\lambda_{1}^{1}\neq\lambda_{2}^{1},\cdots\lambda_{1}^{n}\neq\lambda_{2}^{n}}\int_{y>x}(y-x)P_{h}^{i}(h_{\lambda_{1}^{1}}^{1},\cdot,h_{\lambda}^{i},\cdot,h_{\lambda_{1}^{n}}^{n})(dx)P_{h^{\prime}}^{i}(h_{\lambda_{1}^{1}}^{1},\cdot,h^{\prime},\cdot,h_{\lambda_{1}^{n}}^{n})(dy)\\&-\frac{1}{\Lambda^{n-1}}\sum_{\lambda_{1}^{1},\lambda_{2}^{1},\cdots\lambda_{1}^{n},\lambda_{2}^{n}}\int_{y>x}(y-x)P_{h}^{i}(h_{\lambda_{1}^{1}}^{1},\cdot,h_{\lambda}^{i},\cdot,h_{\lambda_{1}^{n}}^{n})(dx)P_{h^{\prime}}^{i}(h_{\lambda_{1}^{1}}^{1},\cdot,h^{\prime},\cdot,h_{\lambda_{1}^{n}}^{n})(dy)],\end{align*} $$

and $ Dp_{\lambda h'}^{i}(H^{(\Lambda)})=O(\frac{1}{\Lambda}),H^{(\Lambda)}\in S^{\otimes\Lambda}. $

Noting $ \alpha_{hh'}^{i}(v)-\alpha_{h'h}^{i}(v)=\frac{1}{\overline{\omega}^{i}-\underline{\omega}^{i}}(\pi(h',v^{-i})-\pi(h,v^{-i})),h'\neq h, $

for (2.1) we have again

$$ \dot{v}_{h}^{i}=\frac{1}{\overline{\omega}^{i}-\underline{\omega}^{i}}(\pi^{i}(h,v^{-i})-\pi^{i}(v))v_{h}^{i},h\in S^{i}, $$

which turns into replicator dynamics when $ \overline{\omega}^{i}-\omega^{i} $ is constant for all $ i\in I $ .