i suggest you to save the best absolute value of fitness in any generation, the convergence is satisfied when the absolute value of fitness is constant across generation

I agree with the previous comment from Ali Jamali but if your fitness function is truly relative (i.e. based only on a pairwise comparison of some sort within a generation) then you don't really have an objective function and your evolution strategy has no clear goal. If this is the case you need to redefine your fitness function to ensure that it delivers a clear measure of the distance between your individuals and the goal of your search.

Dear Fabrizio: If you are able to compare the fitness of any two individuals, and you can select the best individual in each generation, it means that you are able to associate to each individual a fitness value? Assuming that you have at least one objective function well developed in your optimization problem, you can calculate the average fitness of the population in each generation and then you can observe if the average fitness of the successive populations is improving as desired along the generations, until a convergence criterion is reached.

Do you know your optimum solution? (i.e. maximum or minimum). Then I would compare against this optimum. You will be able to measure the convergence in a much more efficient way.

Dear Fabrizio Nunnari,

From your description, it seems that you are about dynamic search problem where the fitness function varies from, even, one generation to the next generation!!! (and if this is the case), then  I think that neither GA nor any other search algorithm can evolve normally to check its convergence (it just like random walk inside dynamic search space). if you have a clear objective function, and thus clear fitness value (clear goal or point inside this dynamic search space) that can be valid for a period of time (say for several number of generations) then you can check population's optimal convergence, premature convergence, and/or stagnation using simple statistics (e.g. commented by António Manuel Abreu Freire Diogo ).

The normalisation of the fitness scores within a generation feels somewhat like a tournament selection mechanism - there are winners (better solutions) and losers each round and really the winner score isn't important. Perhaps looking at the variation in genetic diversity of the population as a function of time might provide some indication that the evolution has stagnated (or converged). Good luck - sounds like a challenging problem.

The population has converged if all the individuals are similar. Some type of distance measure between the best with some arbitrary individuals will determine if most of the individuals are similar. Then if its local convergence, you need to create more diverse individuals, usually through parameter settings in your crossover and mutation. 

First of all, thanks everybody for the great answers.

Let me clarify two points that I should have mentioned in the description. (I will update it accordingly)

First, the approach that I am taking is a case of Interactive Genetic Algorithm. It means that the fitness value is determined by people voting/scoring/comparing items. There is not really an objective function. The objective is to increase the level of appreciation/agreement among people.

Second, the fitness value within each generation is normalised between 0-1. Plus, the whole 0-1 range is used. Meaning that in each generation there will be the best individual with fitness value at 1 and the worst individual with fitness value at 0.

Now, I thought of checking the convergence using the following strategy.

I assume that, if the specie is improving from a generation to the next, the elected individuals are surpassed by ones with better rank.

In generation n (containing N individuals), I will sort all the individuals from the best (i=1 fit=1) to the worst (i=N fit=0), where i is the ranking position. The k best individuals will be elected to pass to generation n+1. After collecting users votes, I will sort also the individuals of the generation n+1. Now, I would compare the k individuals which migrated from generation n to generation n+1. Their fitness value will be completely different between the two generations. However, if the elected individuals are "surpassed" by better individuals, their fitness value should decrease, as well as their rank position after sorting. By checking the average of the difference between the fitness values (or rank positions) I should be able to deduce if the specie is improving.

Sounds it good?