First of all they guess simultaneously so they can't wait for the first person to guess his own. Second of all, how can someone say if there is even number of something when there are infinity of them.

obviously if there's someone who will tell incorrectly it is the first person, because he doesn't have any information/hints about his hat, so the first person should try to give hint for the other people to help them. so the strategy is that first person will say "Red" if the number of "Red" hats that he can see is odd, otherwise he will say "Blue" (just as if "Red" was 1 and "Blue" was 0 and the first person is telling the XOR of all hats)

now, if the parity of the number of "Red" hats which the second person can see is different from parity of the number of "Red" hats which the first person can see then second person will know he is wearing "Red" hat, otherwise he is wearing "Blue" hat. the third person heard what the first person said and knows what is the color of second person's hat, using similar way he can tell the color of his own hat and so on for the following people.

I think I have a proof that this strategy doesn't exist O_o, It should be wrong but let me say it :)...

Consider two different ways of coloring that the first person has white hat in them but our strategy says it's black...Now for the others our algorithm should say the same thing for both states of coloring, but these states have at least one difference and our strategy won't work in one of the states.(cause the algorithm said the same thing for all the persons behind this person and we don't have any information to recognize these states from each other).

Also we know that we don't have anything to talk about the first persons' hat so we can consider that we always say it's white, now if we consider two states with black hat for the first person we can't do anything as said in the previous paragraph...

Maybe I had some little mistakes but tell if there is some mistake that can't be fixed in my proof...

:)

It's not correct