An application of fractions to figure skating's team event

1. An application of fractions to
figure skating’s team event
Dr. Diana Cheng (Towson University) &
Dr. Peter Coughlin (University of Maryland)

2. Common Core State Standards
Addressed
 Modeling links classroom mathematics & statistics to everyday life, work,
& decision-making.
 Modeling is the process of choosing & using appropriate mathematics &
statistics to analyze empirical situations, to understand them better, and to
improve decisions.
 Quantities & their relationships in physical, economic, public policy, social, &
everyday situations can be modeled…
 6.RP.A.1 Understand the concept of a ratio & use ratio language to
describe a ratio relationship between two quantities.
 7.SP.C.8b Represent sample spaces for compound events using
methods such as organized lists, tables & tree diagrams. For an event
described in everyday language (e.g., "rolling double sixes"), identify the
outcomes in the sample space which compose the event. [Factorials
included here]
 HSS.CP.A.1 Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes.
 HSS.CP.B.9 Use permutations & combinations to compute probabilities
of compound events & solve problems.
http://www.corestandards.org/

3. Outline
 Application of Banzhaf & Shapley-Shubik power indices
to n = 3 scenarios:
 Youth Olympic Games 2012
 World Team Trophy 2009
 Comparison of Banzhaf & Shapley-Shubik power indices
for n = 3
 Application of Banzhaf & Shapley-Shubik power indices
to n = 4 scenarios:
 Winter Olympic Games 2010
 Winter Olympic Games 2014
 Potential uses of power indices

4. Distribution of power vs Distribution of votes
 Number of votes does not provide an effective
measure of power
 Example #1
 Quota: Majority (5) out of 9 needed
 (quota 5; weights 5, 1, 1, 1, 1)
 No combination of other representatives can defeat a bill
 Example #2
 Quota: Majority (17) out of 33 needed
 (quota 17; weights 8, 8, 8, 8, 1)
 The votes of any three voters pass a bill and it makes no
difference which three combine to form a majority

5. Application of power indices to figure
skating’s team event
 Measure extent to which a member of a voting body
is able to control the outcome from a vote
 Context for voting body: team of skaters
 Voters’ weights (# of votes that each voter has)
 Voters: athletes / “entries” of country’s figure skating
team
 Weights: # points each entrant has earned towards
team’s total
 Quotas
 # points for a team to earn a medal
 # points for a team to beat a rival
 # points for a team to not come in last

6. Figure Skating Team Event
 Similar to relay events (speed skating, track & field,
swimming)
 Four categories contested:
 Mens singles
 Ladies singles
 Pairs
 Ice dancing
 Each team represents one country
 Countries can demonstrate depth of athletes’ skills
 Points earned by placement in each discipline (if there
are x entries in that discipline, the 1st place winner
receives x points, the 2nd place winner receives x-1
points, … the last place finisher in the event receives 1
point)
 Sum of points earned by each country is ranked – the
team with the highest # of points is the winner

7. 2010 Vancouver Olympics results:
Why we would want a figure skating team event
Event Gold Silver Bronze
Men's
singles
Evan Lysacek
United States
(USA)
Evgeni
Plushenko
Russia (RUS)
Daisuke
Takahashi
Japan (JPN)
Ladies'
singles
Kim Yuna
South Korea
(KOR)
Mao Asada
Japan (JPN)
Joannie
Rochette
Canada (CAN)
Pair
skating
Shen Xue
and Zhao
Hongbo
China (CHN)
Pang Qing
and Tong Jian
China (CHN)
Aliona
Savchenko
and Robin
Szolkowy
Germany
(GER)
Ice
dancing
Tessa Virtue
and Scott Moir
Canada (CAN)
Meryl Davis
and Charlie
White
United States
(USA)
Oksana
Domnina
and Maxim
Shabalin
Russia (RUS)
 Medal counts
 5 countries
each earned
two of the 12
available
medals
(China, US,
Canada,
Russia,
Japan)
 Gold medals
 A different
country won
each of the 4
events

8. Youth Olympic Games 2012
 Ties decided by sum of International Judging System
scores
 3 disciplines taken into account for team score;
insufficient pairs entries
 Skaters on teams didn’t represent countries
 Separate skates were not held from the individual

9. World Team Trophy 2009
 All four disciplines were contested (only 3 here
considered for this exercise)
 The overall countries’ placements are the same if we
remove the ice dance entries

10. Banzhaf & Shapley-Shubik comparisons
For the case where n = 3:
 The rankings for the Banzhaf and Shapley-Shubik
power indices are identical (Saari & Sieberg, 2000)
 Even though the two power indices assign different
numbers to a player (differ on the amount of power
that a player has), the two indices always agree on
the relative ranking of the amounts of power that the
players have

11. Banzhaf & Shapley Shubik for n = 3
 What are the distinct Shapley-Shubik power index
profiles?
 What are the distinct Banzhaf power index profiles?
 What are the pairs of SSI and BI power index
profiles?

12. Cases for weights

13. Consider the case
 Consider the following quantities:
 Where could the quantities lie in relationship to
one another?

14. N = 4 example: 2010 Winter Olympic Games
 Four countries which qualified entries in all 4
disciplines – Canada, Italy, Russia, US
 In this analysis – only the highest placing skater
representing each of these four countries will be
considered even though several of these countries
had multiple entries per event

15. N = 4 example: 2014 Winter Olympic Games
 Ten countries were eligible
 There were two rounds; after the first round (short
program), five countries were eliminated
 Five countries competed in the second round (free
skate)

16. Uses of power indices in figure skating
 Rewarding Most Valuable Player or providing financial
stipends
 Answering hypothetical questions (eg, 2010 Winter
Olympic Games analysis) based on individuals’ relative
rankings
 Selection of athletes to teams (countries do not need to
select athletes to fill entry spots in hopes of earning the
greatest International Judging System scores; they only
need to select athletes with equivalent power within each
discipline)
 If a country’s team has chances of earning both gold and silver
medals in the mens event, the country could choose to send
the second-place finishing man to the team event
 Intercollegiate team figure skating – skaters placing in the
top five of their events earn 5, 4, 3, 2, 1 points towards
their team totals by placing first, second, third, fourth, fifth