There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ($A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?
1. **Initial Setup and Strategy**:
   - There are two pizzerias, each with 2010 pizzas.
   - Scientists \( A \) and \( B \) take turns, with \( A \) going first.
   - On each turn, a player can either:
     - Eat any number of pizzas from one pizzeria, or
     - Eat exactly one pizza from each pizzeria.
   - The player who eats the last pizza wins.

2. **Claim**:
   - Scientist \( B \) wins if both use the best possible strategy.

3. **Position Representation**:
   - Let the number of pizzas in the two pizzerias be represented as \((a, b)\), where \(a\) and \(b\) are the number of pizzas in the first and second pizzeria, respectively.
   - Initially, \((a, b) = (2010, 2010)\).

4. **Case Analysis Based on \( b \pmod{3} \)**:
   - We analyze the game based on the value of \( b \mod 3 \).

5. **Case 1: \( b \equiv 0 \pmod{3} \)**:
   - If \( b \equiv 0 \pmod{3} \), then \( B \) will eat pizzas until both pizzerias have the same number of pizzas, i.e., \( a = b \).
   - If \( a = b \), \( B \) will eat one pizza from each pizzeria, maintaining the equality.

6. **Case 2: \( b \equiv 1 \pmod{3} \)**:
   - If \( b \equiv 1 \pmod{3} \), then \( B \) will eat pizzas until \( a = b + 1 \).
   - If \( a = b \), \( B \) will eat one pizza from each pizzeria, making \( a = b + 1 \).

7. **Case 3: \( b \equiv 2 \pmod{3} \)**:
   - If \( b \equiv 2 \pmod{3} \), then \( B \) will eat pizzas until \( a = b - 1 \).

8. **Induction Proof**:
   - We need to show that it is not possible for \( a = b \) or \( a = b + 1 \) to occur initially in cases 1 and 2, respectively.
   - By induction, we can show that \( B \) can always force the game into one of the above cases, ensuring that \( B \) will always have a winning strategy.

9. **Conclusion**:
   - Since \( B \) can always force the game into a position where \( A \) cannot win, \( B \) will always be the one to eat the last pizza and thus win the game.

The final answer is \( \boxed{ B } \) wins.