Voting power and Qualified Majority Voting with a “no vote” option

In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish government suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the “Jagiellonian Compromise”). The Penrose definition of voting power is derived from the citizens’ freedom to vote either “yes” or “no”. This paper defines a corresponding voting power based on “yes”, “no” and “abstain” options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme.