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The meaning of the phrase “equivalent up to ties” might become a little clearer if you continue to read the excerpt that you have provided here. It becomes clear that this phrase means that the two voting rules are the same as one another except when the election ends up tied. They are equivalent (the same as one another) up to (at any result except) ties.
In the paragraph under the one that has the phrase “equivalent up to ties” we see a discussion of how the two voting rule break ties. The text says that they each break ties in a different fashion. That makes clear to us that the only difference between the two rules is the way in which they break ties. Once this is clear, we can understand the phrase “equivalent up to ties.”
Thank you so much - I now understand the definion with equivalence! You have described it very well. Chapeau!
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