![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.
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Power within the team
- 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?
- 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