There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. In the first half of the talk I will describe some often used methods and discuss their pros and cons (it’s all in the rounding).
One easy method is called Hamilton’s method (also known as the method of largest reminder or Hare’s method). It suffers from a drawback called the Alabama paradox, which e.g. made USA abandon it for the distribution of seats in the house of representatives between states. It is still in use in many other countries including Sweden.
In the second half of the talk I will describe a joint work with Svante Janson (Uppsala Univ.) where we study the probability that the Alabama paradox will happen. We give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector $p_1,\dots,p_m$ of relative sizes of the states.