The VP and the Power of Tie Breaking
On the face of it, the power to break a tie isn't worth much, since votes are rarely ties. But that way of looking at it is misleading. An ordinary vote is only decisive, after all, if it either breaks a tie or creates one, which is also, in most contexts, a rare situation. The question is how much less power a vote that can be used only to break a tie provides.
Consider a motion in an assembly. For simplicity assume that, absent my vote, the probabilities that the motion will fail by one vote, tie, or pass by one vote—the situations in which my vote might affect the outcome—are equal. Call the common probability p. Assume there is a chairman with the power to break ties, and assume, again for simplicity, that he is equally likely to vote either way on the motion and never abstains. Finally, assume that I want the motion to pass.
If without my vote the motion would fail by one, then I can vote for it and make it a tie. That affects the outcome only if the tie breaker also votes for it. So the probability that I will be able to make it pass by making it into a tie is p/2. If without my vote it would be a tie, then I can vote for it making it pass, but that only affects the outcome if the tie breaker would not have himself voted for it. That gives us another probability of p/2. If without my vote it would pass by one, then my support is not needed and has no effect on the outcome. So the total probability that my support for the bill will affect the outcome is p.
What about the situation from the point of view of the tie breaker? Assuming that a motion fails on a tied vote, he can only affect the outcome if the vote is a tie and he supports the motion. Probability p/2.
What this simple analysis suggests is that the voting power of the VP, if he chooses to exercise it, is half that of an ordinary senator. Whether that plus chairing the Senate, as I gather VP's used to routinely do before LBJ, adds up to more or less than the equivalent of one senator is a question I will leave to others who know more about the subject.