Mario Cart and figure skating | Golden Skate

# Mario Cart and figure skating

#### Mathman

OK, so evidently “Mario Cart” is some kind of Nintendo game where players assemble and race go-carts. I happened upon a serious piece of mathematics that uses something called the Pareto Front to figure out the optimal combination of super-chargers, etc., that you can put on your cart to maximize your chances of winning.

Vilfredo Pareto was a 19th - early 20th century Italian economist who also made contributions to social choice theory.

How is this relevant to figure skating? The knock against the old 6.0 ordinal scoring system, and the case for the IJS, is summed up in a result called Arrow’s Impossibility Theorem (Nobel Laureate Kenneth Arrow, 1951). This states that no system of ordinal or ranked voting can possibly work every time. By “work” we mean that certain “obviously” desirable conditions should be met. For instance, no “flip-flops.” A flip-flop occurs when skater A is beating the pants off skater B on a majority of judges’ score cards, but then skater C skates and suddenly skater B vaults ahead of skater A. This is unsettling, especially if you are skater A.

Another (related) condition is called the Pareto Condition and was first examined by this guy Pareto in a economic setting: If a proposed change to economic policy will benefit some members of society while not hurting any members of society, then society is better off to adopt the new policy. Pareto showed that paradoxically sometimes society as a whole might be better off keeping the old policy even though every individual member of society is either better off or no change with the new. That is, if every individual judge thinks that skater A was better than skater B, then the scoring system as whole should give the prize to skater A. The Impossibility Theorem says that it not possible to invent a ranked voting system (that has at least two judges and at least three skaters) that will simultaneously satisfy all the desirable conditions in every conceivable contest. – not “majority of ordinals,” not “one-by-one,” not any variation on ordinal judging.

But the Code of Points does. If skater A gets more points than skater B on every judges card, then skater A will alwaysplace above skater B, no matter what skater C does.

#### Mathman

This is the kind of thing that can happen. Suppose we try a scoring system with these rules: If a skater receives a clear majority of votes (first place ordinals) then that skater wins. If no one receives an outright majority then there is an run-off between the two skaters that receive the most votes. Sounds easonable, right? So here we go.

Skater A goes first, then skater B. The five judges rank them

A A A B B
B B B A A

Skater A is winning so far, 3 votes to 2. Now skater C skates. The judges rank them

C C A B B
A A B A A
B B C C C

The majority of judges put C last. Wh o wins the gold medal?

Well, no one jas the outright majority of first-place ordinals. C has 2 votes, A has 1 vote and B has two votes. There is a run-off between the top two top vote-getters, C and B. Slkater A is eliminated. With A eliminated, the run-off goes

C C B B B
B B C C C

B wins the gold medal, by a vote of 3 to 2, even though she was beaten by A..

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