# 2.8: Borda Count - Mathematics

Borda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.

Borda Count

In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.

Example 8

A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown to the right.

The votes for where to hold the conference were:

(egin{array}{|l|l|l|l|l|}
hline & 51 & 25 & 10 & 14
hline 1^{ ext {st }} ext { choice } & ext { Seattle } & ext { Tacoma } & ext { Puyallup } & ext { Olympia }
hline 2^{ ext {nd }} ext { choice } & ext { Tacoma } & ext { Puyallup } & ext { Tacoma } & ext { Tacoma }
hline 3^{ ext {rd }} ext { choice } & ext { Olympia } & ext { Olympia } & ext { Olympia } & ext { Puyallup }
hline 4^{ ext {th }} ext { choice } & ext { Puyallup } & ext { Seattle } & ext { Seattle } & ext { Seattle }
hline
end{array})

Solution

In each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 points. Multiplying the points per vote times the number of votes allows us to calculate points awarded:

(egin{array}{|l|l|l|l|l|}
hline & 51 & 25 & 10 & 14
hline 1^{ ext {st choice }} & ext { Seattle } & ext { Tacoma } & ext { Puyallup } & ext { Olympia }
4 ext { points } & 4 cdot 51=204 & 4 cdot 25= 100 & 4 cdot 10=40 & 4 cdot 14=56
hline 2^{ ext {nd choice }} & ext { Tacoma } & ext { Puyallup } & ext { Tacoma } & ext { Tacoma }
3 ext { points } & 3 cdot 51=153 & 3 cdot 25=75 & 3 cdot 10=30 & 3 cdot 14=42
hline 3^{ ext {rd }} ext { choice } & ext { Olympia } & ext { Olympia } & ext { Olympia } & ext { Puyallup }
2 ext { points } & 2 cdot 51=102 & 2 cdot 25=50 & 2 cdot 10=20 & 2 cdot 14=28
hline 4^{ ext {th }} ext { choice } & ext { Puyallup } & ext { Seattle } & ext { Seattle } & ext { Seattle }
1 ext { point } & 1 cdot 51=51 & 1 cdot 25=25 & 1 cdot 10=10 & 1 cdot 14=14
hline
end{array})

• Seattle: (204 + 25 + 10 + 14 = 253) points
• Tacoma: (153 + 100 + 30 + 42 = 325) points
• Puyallup: (51 + 75 + 40 + 28 = 194) points
• Olympia: (102 + 50 + 20 + 56 = 228) points

Under the Borda Count method, Tacoma is the winner of this vote.

Try it Now 4

Consider again the election from Try it Now 1. Find the winner using Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place.

(egin{array}{|l|l|l|l|l|l|l|l|}
hline & 44 & 14 & 20 & 70 & 22 & 80 & 39
hline 1^{ ext {st }} ext { choice } & mathrm{G} & mathrm{G} & mathrm{G} & mathrm{M} & mathrm{M} & mathrm{B} & mathrm{B}
hline 2^{ ext {nd }} ext { choice } & mathrm{M} & mathrm{B} & & mathrm{G} & mathrm{B} & mathrm{M} &
hline 3^{ ext {rd }} ext { choice } & mathrm{B} & mathrm{M} & & mathrm{B} & mathrm{G} & mathrm{G} &
hline
end{array})

Using Borda Count:

We give 1 point for 3rd place, 2 points for 2nd place, and 3 points for 1st place.

(egin{array}{|l|l|l|l|l|l|l|l|}
hline & 44 & 14 & 20 & 70 & 22 & 80 & 39
hline 1^{ ext {st }} ext { choice } & mathrm{G} & mathrm{G} & mathrm{G} & mathrm{M} & mathrm{M} & mathrm{B} & mathrm{B}
& 132 mathrm{pt} & 42 mathrm{pt} & 60 mathrm{pt} & 210 mathrm{pt} & 66 mathrm{pt} & 240 mathrm{pt} & 117 mathrm{pt}
hline 2^{ ext {nd }} ext { choice } & mathrm{M} & mathrm{B} & & mathrm{G} & mathrm{B} & mathrm{M} &
& 88 mathrm{pt} & 28 mathrm{pt} & & 140 mathrm{pt} & 44 mathrm{pt} & 160 mathrm{pt} &
hline 3^{ ext {rd }} ext { choice } & mathrm{B} & mathrm{M} & mathrm{M} 20 mathrm{pt} & mathrm{B} & mathrm{G} & mathrm{G} & mathrm{M} 39 mathrm{pt}
& 44 mathrm{pt} & 14 mathrm{pt} & mathrm{B} 20 mathrm{pt} & 70 mathrm{pt} & 22 mathrm{pt} & 80 mathrm{pt} & mathrm{G} 39 mathrm{pt}
hline
end{array})

G: (132+42+60+140+22+80+39 = 515) pts

M: (88+14+20+210+66+160+39 = 597) pts

B: (44+28+20+70+44+240+117 = 563) pts

McCarthy (M) would be the winner using Borda Count.

## A preference aggregation method through the estimation of utility intervals ☆

In this paper, a preference aggregation method is developed for ranking alternative courses of actions by combining preference rankings of alternatives given on individual criteria or by individual decision makers. In the method, preference rankings are viewed as constraints on alternative utilities, which are normalized, and linear programming models are constructed to estimate utility intervals, which are weighted and averaged to generate an aggregated utility interval. A simple yet pragmatic interval ranking method is used to compare and/or rank alternatives. The final ranking is generated as the most likely ranking with certain degrees of belief. Three numerical examples are examined to illustrate the potential applications of the proposed method.

The aggregation of preference rankings has wide applications in group decision making, social choice, committee election and voting systems. The purpose of this paper is to develop a preference aggregation method through the estimation of utility intervals, in which preference rankings are associated with utility intervals that are estimated using linear programming models and aggregated using the simple additive weighting method.

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### Program Details

Unit 1: EVERYONE HAS PROBLEMS

Lesson 1-1: Be Reasonable (Inductive and Deductive Reasoning)

Objective 1: Explain the difference between inductive and deductive reasoning.

Objective 2: Use inductive reasoning to make conjectures.

Objective 3: Use deductive reasoning to prove or disprove a conjecture.

Lesson 1-2: More or Less (Estimation and Interpreting Graphs)

Objective 1: Use rounding and mental arithmetic to estimate the answers to applied problems.

Objective 2: Obtain and interpret information from bar graphs, pie charts, and time series graphs.

Lesson 1-3: You Got a Problem? (Problem-Solving Strategies)

Objective 1: Identify the four steps in Polya’s problem-solving procedure.

Objective 2: Apply Polya’s procedure to solving problems.

Objective 3: Solve problems using different strategies: trial and error, drawing a diagram, using algebra, and comparing different outcomes.

Lesson 2-1: Giving 110 Percent (Review of Percents)

Objective 1: Perform conversions and calculations involving percents.

Objective 2: Find percent increase or decrease.

Objective 3: Solve problems using percents.

Objective 4: Evaluate the legitimacy of claims based on percents.

Lesson 2-2: Building It Is the Easy Part . . . (Budgeting)

Objective 1: Calculate take-home pay and monthly expenses.

Objective 2: Identify necessary expenses and luxuries.

Objective 3: Prepare a monthly budget.

Objective 4: Prorate long-term expenses to save in advance for them.

Lesson 2-3: A Topic of Interest (Simple Interest)

Objective 1: Define interest and understand related terminology.

Objective 2: Develop simple interest formulas.

Objective 3: Use simple interest formulas to analyze financial issues.

Lesson 2-4: Like a Snowball Rolling Downhill (Compound Interest)

Objective 1: Describe how compound interest differs from simple interest.

Objective 2: Develop compound interest formulas.

Objective 3: Use compound interest formulas to analyze financial issues.

Objective 1: Compute payments and charges associated with installment loans.

Objective 2: Identify the true cost of a loan by computing APR.

Objective 3: Evaluate the costs of buying items on credit.

Lesson 2-6: Investing in Yourself (Education and Home Loans)

Objective 1: Understand different student loan options.

Objective 2: Compute interest and monthly payments on a student loan.

Objective 3: Evaluate the effects of capitalizing interest.

Objective 4: Analyze various aspects of a mortgage.

Objective 5: Compare two mortgages of different lengths.

Objective 6: Prepare an amortization schedule.

Lesson 2-7: A Walk on Wall Street (Stocks and Bonds)

Objective 1: Read information from a stock listing.

Objective 2: Calculate costs of buying stock, and profit or loss from selling.

Objective 3: Study the price-to-earnings ratio, and use it to analyze the value of a stock.

Objective 4: Calculate profit from a bond sale.

Lesson 2-8: A Taxing Situation (Income Taxes)

Objective 1: Understand why we pay taxes.

Objective 2: Explain the basic process of paying taxes.

Objective 3: Determine the amount of tax due based on taxable income.

Objective 4: Complete a 1040 form.

Lesson 3-1: So You’re Saying There’s a Chance . . . (Basic Probability)

Objective 1: Understand key terminology in the study of probability.

Objective 2: Compute and interpret theoretical and empirical probabilities.

Objective 3: Compare theoretical and empirical probability.

Lesson 3-2: Make It Count (Sample Spaces and Counting Techniques)

Objective 1: Describe how counting techniques are useful in probability theory.

Objective 2: Use tree diagrams and tables to determine sample spaces and compute probabilities.

Objective 3: Develop and use the fundamental counting principle.

Lesson 3-3: Combining Forces (Combinatorics)

Objective 1: Understand how combinatorics are useful in probability theory.

Objective 2: Distinguish between permutations and combinations.

Objective 3: Find the number of permutations and combinations of n objects.

Objective 4: Find the number of permutations and combinations of n objects chosen r at a time.

Lesson 3-4: Too Good to Be True? (Probability Using Counting Techniques)

Objective 1: Recognize probability problems where permutations are useful and where combinations are useful.

Objective 2: Use permutations to calculate probabilities.

Objective 3: Use combinations to calculate probabilities.

Lesson 3-5: Odds and Ends (Odds and Expected Value)

Objective 1: Distinguish between odds and probability.

Objective 2: Compute and interpret the odds in favor of and odds against an event.

Objective 3: Compute odds from probability and vice versa.

Objective 4: Develop a procedure for finding expected value.

Objective 5: Compute and interpret expected values.

Lesson 3-6: An Exclusive Club (Addition Rules for Probability)

Objective 1: Distinguish between events that are and are not mutually exclusive.

Objective 2: Develop addition rules for finding probabilities of “or” events that are and are not mutually exclusive.

Objective 3: Use the addition rules to calculate probabilities.

Lesson 3-7: Independence Day (Multiplication Rules and Conditional Probability)

Objective 1: Distinguish between events that are and are not independent.

Objective 2: Develop multiplication rules for finding probabilities of “and” events that are and are not independent.

Objective 3: Use the multiplication rules to calculate probabilities.

Objective 4: Define, compute, and interpret conditional probabilities.

Lesson 3-8: Either/Or (Binomial Probabilities)

Objective 1: Identify binomial experiments.

Objective 2: Compute and interpret probabilities of outcomes in a binomial experiment.

Objective 3: Compute cumulative binomial probabilities.

Unit 4: STATISTICALLY SPEAKING

Lesson 4-1: Crunching the Numbers (Gathering and Organizing Data)

Objective 1: Explain the difference between a population and a sample.

Objective 2: Compare and contrast different sampling methods.

Objective 3: Organize data with frequency distributions.

Objective 4: Analyze data with stem and leaf plots.

Lesson 4-2: Picture This (Representing Data Graphically)

Objective 1: Draw and interpret bar graphs from frequency distributions.

Objective 2: Draw and interpret pie charts from frequency distributions.

Objective 3: Draw and interpret histograms and frequency polygons from frequency distributions.

Objective 4: Draw and interpret time series graphs.

Lesson 4-3: An Average Joe (Measures of Average)

Objective 1: Compute measures of average for given data.

Objective 2: Interpret the story told by measures of average.

Objective 3: Compute and interpret the mean for grouped data.

Objective 5: Use technology to compute measures of average.

Lesson 4-4: Your Results May Vary (Measures of Variation)

Objective 1: Compute measures of variation for a given data set.

Objective 2: Interpret standard deviation for a data set.

Objective 3: Make meaningful comparisons of standard deviation for two data sets.

Objective 4: Analyze the procedure for computing standard deviation.

Lesson 4-5: Where Do You Rank? (Measures of Position in a Data Set)

Objective 1: Compute percentile ranks.

Objective 2: Find data corresponding to a given percentile rank.

Objective 3: Use percentiles to compare data from different sets.

Objective 4: Compute quartiles and use them to analyze spread.

Objective 5: Draw and interpret box plots.

Lesson 4-6: Just a Normal Day (Normal Distributions and Z Scores)

Objective 1: Recognize characteristics of data that are normally distributed.

Objective 2: Understand the connection between area under a normal curve, percentage, and probability.

Objective 3: Make an educated guess about the empirical rule, then use the rule to calculate percentages and probabilities.

Objective 4: Compare data values from different sets using Z scores.

Lesson 4-7: The Way the Cookie Crumbles (Applications of the Normal Distribution)

Objective 1: Use normal distributions to find probabilities, percentages, and percentiles.

Objective 2: Learn how normal distributions are used in manufacturing and packaging.

Objective 3: Recognize data that are approximately normally distributed.

Lesson 4-8: Making Connections (Correlation and Regression Analysis)

Objective 1: Draw and analyze scatter plots for two data sets.

Objective 2: Define correlation coefficient, and decide if correlation coefficients are significant.

Objective 3: Find regression lines and use them to make predictions.

Objective 4: Recognize the difference between data sets being related and being linearly related.

Lesson 4-9: Trust No One (Misuses of Statistics)

Objective 1: Identify misuses of sampling and evaluate their effect on statistical results.

Objective 2: Recognize and describe common misuses of compiling and reporting statistics that make them meaningless or deceiving.

Objective 3: Study ways that graphs can be manipulated to tell a desired story.

Lesson 5-1: Keeping Things in Proportion (Ratios and Proportions)

Objective 1: Compare two quantities using ratios.

Objective 2: Describe the value of using ratios to compare quantities rather than differences.

Objective 3: Solve proportions.

Objective 4: Solve problems that involve proportional reasoning.

Lesson 5-2: Making Some Extra Cash (The Basics of Graphing Functions)

Objective 1: Demonstrate an understanding of the significance of a rectangular coordinate system.

Objective 2: Describe what the graph of an equation is.

Objective 3: Use and interpret function notation.

Objective 4: Graph and interpret linear functions.

Objective 5: Graph and interpret quadratic functions.

Lesson 5-3: A Slippery Slope (Modeling with Linear Functions)

Objective 1: Calculate slope and interpret as rate of change.

Objective 2: Solve problems using linear modeling, both algebraically and using technology.

Objective 1: Identify quantities that are and are not good candidates to be modeled with quadratic functions.

Objective 2: Solve problems using quadratic modeling, both algebraically and using technology.

Lesson 5-5: Progressing Regressively (Linear and Quadratic Regression)

Objective 1: Decide whether a linear or a quadratic model is most appropriate for a data set.

Objective 2: Create a line or parabola of best fit from data.

Lesson 5-6: Phone a Friend (Modeling with Exponential and Log Functions)

Objective 1: Identify quantities that are and are not good candidates to be modeled with exponential equations.

Objective 2: Solve problems by exponential modeling, both algebraically and using technology.

Objective 3: Define logarithms as inverses of exponentials.

Objective 4: Solve problems by logarithmic modeling, both algebraically and using technology.

Lesson 6-1: Setting Up (The Basics of Working with Sets)

Objective 1: Define sets and use different methods to represent them.

Objective 2: Identify when sets are equivalent.

Objective 3: Study cardinality for finite and infinite sets.

Lesson 6-2: Busy Intersections, More Perfect Unions (Operations on Sets)

Objective 1: Find the complement and all subsets for a given set.

Objective 2: Evaluate set statements involving subset notation.

Objective 3: Perform and apply set operations: union, intersection, subtraction.

Lesson 6-3: Worlds Collide (Studying Sets with Two-Circle Venn Diagrams)

Objective 1: Illustrate sets with two-circle Venn diagrams.

Objective 2: Develop and use De Morgan’s laws.

Objective 3: Use Venn diagrams to decide if two sets are equal.

Objective 4: Review how Venn diagrams can be used in probability.

Lesson 6-4: A Dollar for Your Thoughts (Using Sets to Solve Problems)

Objective 1: Illustrate sets with three-circle Venn diagrams.

Objective 2: Decide if two sets are equal using three-circle Venn diagrams.

Objective 3: Solve a variety of applied problems using Venn diagrams.

Lesson 7-1: Opening Statements (Statements and Quantifiers)

Objective 1: Define and identify statements.

Objective 2: Define the logical connectives and identify their use.

Objective 3: Recognize and write negations of statements.

Objective 4: Write statements symbolically, and translate symbolic statements back to verbal.

Lesson 7-2: Finding the Truth (Truth Tables)

Objective 1: Build truth tables for negations, disjunctions, and conjunctions.

Objective 2: Build truth tables for conditional and biconditional statements.

Objective 3: Build truth tables for compound statements.

Objective 4: Use the hierarchy of connectives, and compare it to order of operations.

Lesson 7-3: To Be and Not to Be (Types of Statements in Logic)

Objective 1: Classify a statement as a tautology, a self-contradiction, or neither.

Objective 2: Identify statements that are logically equivalent.

Objective 3: Write and recognize negations of compound statements.

Objective 4: Write and recognize the converse, inverse, and contrapositive of a statement.

Objective 5: Evaluate logical connections between a statement and its converse, inverse, and contrapositive.

Lesson 7-4: Being Argumentative (Evaluating Logical Arguments)

Objective 1: Identify the difference between a valid argument and a fallacy.

Objective 2: Use truth tables to evaluate the validity of arguments.

Objective 3: Determine the validity of common argument forms.

Objective 4: Use common argument forms to decide if arguments are valid.

Unit 8: HOW DO YOU MEASURE UP?

Lesson 8-1: Going to Great Lengths (Unit Conversion, Length, and the Metric System)

Objective 1: Understand the importance of units in measurement.

Objective 2: Understand how dimensional analysis makes converting units easy.

Objective 3: Identify the key components of the metric system.

Objective 4: Convert between U.S. and metric units of length, and describe perspective on the size of these measurements.

Lesson 8-2: New Dimensions (Measuring Area, Volume, and Capacity)

Objective 1: Understand the difference between unit conversions for length, and unit conversions for area and volume.

Objective 2: Convert area and volume measurements within the U.S. system, and describe perspective on sizes of these measurements.

Objective 3: Convert area and volume measurements between the U.S. and metric systems, and describe perspective on sizes of these measurements.

Lesson 8-3: Weighty Matters (Units of Weight and Temperature)

Objective 1: Convert weight and temperature measurements within the U.S. and metric systems.

Objective 2: Convert weight and temperature measurements between the U.S. and metric systems.

Objective 3: Demonstrate an understanding of the sizes of measurements in these systems.

Lesson 8-4: Stocking the Shelves (Evaluating Efficiency in Packaging)

Objective 1: Develop surface area and volume formulas for rectangular solids and cylinders.

Objective 2: Study the volume-to-surface-area ratio for different product packages and identify its significance.

Objective 3: Develop methods for deciding on optimal size and shape given certain goals.

Units 9, 10, and 11 are available online. Though not included in this desk copy, they can be added to custom versions of the text built through Create or accessed in the Instructor Resources area of ALEKS.

Lesson 9-1: State Your Preference (Preference Tables and Plurality Voting)

Objective 1: Interpret the information in a preference table.

Objective 2: Identify the winner of an election using the plurality method.

Objective 3: Identify potential weaknesses of plurality voting.

Lesson 9-2: We’re Number One! (Borda Count and Plurality with Elimination)

Objective 1: Determine the winners of elections using the Borda count method and the plurality-with-elimination method.

Objective 2: Decide if elections violate the majority criterion or the monotonicity criterion.

Lesson 9-3: It’s So Unfair! (Pairwise Comparison and Approval Voting)

Objective 1: Determine the winners of elections using the pairwise comparison method and approval voting.

Objective 2: Decide if an election violates the irrelevant alternatives criterion.

Objective 3: Describe Arrow’s impossibility theorem.

Lesson 9-4: Portion Control (Apportionment)

Objective 1: Describe what apportionment is and why it’s used.

Objective 2: Compute standard divisors and quotas.

Objective 3: Apportion items using a variety of methods.

Unit 10: WARNING: GRAPHIC CONTENT

Lesson 10-1: Color Your World (Basic Concepts of Graph Theory)

Objective 1: Define basic graph theory terms.

Objective 2: Represent relationships with graphs.

Objective 3: Decide if two graphs are equivalent.

Objective 4: Recognize features of graphs.

Objective 5: Apply graph coloring.

Lesson 10-2: Efficiency Experts (Euler’s Theorem)

Objective 1: Define Euler path and Euler circuit.

Objective 2: Use Euler’s theorem to decide if an Euler path or Euler circuit exists.

Objective 3: Use Fleury’s algorithm to find an Euler path or Euler circuit.

Objective 4: Solve practical problems using Euler paths or circuits.

Lesson 10-3: Who Wants to Be a Zillionaire? (Hamilton Paths and Circuits)

Objective 1: Find Hamilton paths and Hamilton circuits on graphs.

Objective 2: Solve a traveling salesperson problem using the brute force method.

Objective 3: Find an approximate optimal solution using the nearest neighbor method.

Objective 4: Draw a complete weighted graph based on provided information.

Lesson 10-4: Tree Hugging (Trees)

Objective 1: Decide if a graph is a tree.

Objective 2: Find a spanning tree for a graph.

Objective 3: Find a minimum spanning tree for a weighted graph.

Objective 4: Apply minimum spanning trees to problems in our world.

Unit 11: DISCOVERING NEW NUMBERS

Lesson 11-1: History Lessons (Early and Modern Numeration Systems)

Objective 1: Use a tally system.

Objective 2: Define and use simple grouping systems.

Objective 3: Define and use multiplicative grouping systems.

Objective 4: Define a positional system and identify place values.

Lesson 11-2: Off Base (Base Number Systems)

Objective 1: Convert between base 10 and other bases.

Objective 2: Convert between binary, octal, and hexadecimal.

Lesson 11-3: Working Out (Operations in Base Number Systems)

Objective 1: Add and subtract in bases other than 10.

Objective 2: Multiply and divide in bases other than 10.

## Contents

After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is [1]

• V is the total number of votes that party received, and
• s is the number of seats that have been allocated so far to that party, initially 0 for all parties.

Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated. The process is repeated until all seats have been allocated.

The Webster/Sainte-Laguë method does not ensure that a party receiving more than half the votes will win at least half the seats nor does its modified form. [11]

### Example Edit

In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, each party's total votes is divided by 1, then by 3, and 5 (and then, if necessary, by 7, 9, 11, 13, and so on). The 8 highest entries, marked with asterisks, range from 100,000 down to 16,000. For each, the corresponding party gets a seat.

For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48.)

The below chart is an easy way to perform the calculation:

Denominator /1 /3 /5 Seats
won (*)
True proportion
Party A 100,000* 33,333* 20,000* 3 3.5
Party B 80,000* 26,667* 16,000* 3 2.8
Party C 30,000* 10,000 6,000 1 1.0
Party D 20,000* 6,667 4,000 1 0.7
Total 8 8

Webster proposed the method in the United States Congress in 1832 for proportional allocation of seats in United States congressional apportionment. In 1842 the method was adopted (Act of June 25, 1842, ch 46, 5 Stat. 491). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced. [3]

According to some observers the method should be treated as two methods with the same result, because the Webster method is used for allocating seats based on states' population, and the Sainte-Laguë based on parties' votes. [12] Webster invented his method for legislative apportionment (allocating legislative seats to regions based on their share of the population) rather than elections (allocating legislative seats to parties based on their share of the votes) but this makes no difference to the calculations in the method.

Webster's method is defined in terms of a quota as in the largest remainder method in this method, the quota is called a "divisor". For a given value of the divisor, the population count for each region is divided by this divisor and then rounded to give the number of legislators to allocate to that region. In order to make the total number of legislators come out equal to the target number, the divisor is adjusted to make the sum of allocated seats after being rounded give the required total.

One way to determine the correct value of the divisor would be to start with a very large divisor, so that no seats are allocated after rounding. Then the divisor may be successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The number of allocated seats for a given region increases from s to s + 1 exactly when the divisor equals the population of the region divided by s + 1/2, so at each step the next region to get a seat will be the one with the largest value of this quotient. That means that this successive adjustment method for implementing Webster's method allocates seats in the same order to the same regions as the Sainte-Laguë method would allocate them.

In 1980 the German physicist Hans Schepers, at the time Head of the Data Processing Group of the German Bundestag, suggested that the distribution of seats according to d'Hondt be modified to avoid putting smaller parties at a disadvantage. [13] German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers. [13]

Some countries, e.g. Nepal, Norway and Sweden, change the quotient formula for parties that have not yet been allocated any seats (s = 0). These countries changed the quotient from V to V/1.4, though from the general 2018 elections onwards, Sweden has been using V/1.2. [14] That is, the modified method changes the sequence of divisors used in this method from (1, 3, 5, 7, . ) to (1.4, 3, 5, 7, . ). This gives slightly greater preference to the larger parties over parties that would earn, by a small margin, a single seat if the unmodified Sainte-Laguë's method were used. With the modified method, such small parties do not get any seats these seats are instead given to a larger party. [1]

Norway further amends this system by utilizing a two-tier proportionality. The number of members to be returned from each of Norway's 19 constituencies (former counties) depends on the population and area of the county: each inhabitant counts one point, while each square kilometer counts 1.8 points. Furthermore, one seat from each constituency is allocated according to the national distribution of votes. [15]

## 2.8: Borda Count - Mathematics

The following exam review will be given to all students in hard copy. The material below does not include certain figures in order to expedite the posting.

Name ________________________ Discrete Mathematics

Exam Review Problems January 2013

1. Compute the value of each of these expressions. Show all work. A calculator generated answer alone will receive no credit on the exam for these however, you should use your graphing calculator to check your work. Use this problem to prove that you know the formulas and how to calculate with them.

For each of the following problems, show your set up ( the symbol for permutation or the symbol for combination with n and r filled in, the numbers being used in the Multiplication/Addition Principle), the specific numbers filled into any factorials being used, and the final answer. Use your calculator efficiently.

2. 10 people want to be on the dance committee. How many 3 person committees are possible?

3. For a short trip you have packed 2 pairs of pants, 5 shirts and 3 sweaters in coordinating colors. How many outfits can you create if an outfit consists of pants, shirt and sweater?

4. 40 people apply for 5 sales positions at the mall.

a) If the 5 positions are identical, in how many ways can the vacancies be filled?

b) If the 5 positions are all different, in how many ways can the vacancies be filled?

5. If the 8 people go out to eat, how many seating arrangements are possible around a table if the chair a person sits in doesn’t matter but only the relative position to everyone else does?

6. How many different looking "words" can be formed using 7 Scrabble tiles all at once if the tiles are: E T E R T E E

7. If a jury has 12 members and all must agree to a verdict, in how many ways can a 2 person dreadlock occur? A 3 person deadlock? A 2 or 3 person deadlock?

8. Seven horses are entered in a race. How many 1st - 2nd - 3rd place finishes are possible?

9. Fifteen players are in a tennis tournament. How many singles matches must be scheduled if each player plays every other player one time?

10. You decide to write every “word” you can create from all of the letters of “Somerville.” If each piece of paper you use is .01 inch thick, how high in feet will your pile of “word” papers be?

NO COMPUTATIONS ARE NECESSARY FOR THESE. NO FINAL ANSWER IS NECESSARY. Just show your set-up. For example, 5 P 2 , 8 C 5 , 5 · 6 · 7 .

11. In a certain state, license plates consist of 3 letters followed by 3 digits (0 to 9) followed by a letter. How many license plates can be formed if numbers and letters may not repeat?

12. There are 25 points marked on a circle. How many line segments can be drawn connecting any two points on the circle?

13. The language of the ancient “Immaculatans” contained the vowels a,e,i,o,u and the consonants c,l,m,n,s,t.

a) How many 3 letter “words” are possible if each contained two different vowels and a consonant between them?

b) How many 5 letter “words” with 2 different vowels at the beginning and the end and a group of 3 different consonants between them?

14. In how many ways can you randomly guess at the answers on a 10 question multiple choice quiz if each question has 5 possible responses from which to choose?

15. At Monmouth Park you bet the 1st place/2 nd place finish in 3 races. If the first race has 8 horses, the second race has 9 horses and the third race has 8 horses, what is the probability that you will choose the winning duo in all 3 races?

Show work and give a solution.

16. Eight people sit in the same row in a theater. Four are men and 4 are women.

a) How many seating arrangement are possible?

b) How many seating arrangements are possible if a man enters the row first and then the sex of the seat occupants alternates?

c) How many seating arrangement are possible if the men and women alternate positions?

17. You have 3 large and 3 small red beads, 5 large and 5 small black beads and 7 white beads,

a) In how many ways can you randomly string them on a cord?

b) How many different looking patterns could you create?

c) How many different looking patterns could you create if you put the black beads on first, then the red ones and then the white ones?

18. a) State the formula for the probability of an event .

b) What does it mean to say that the probability of an event is 1? Give an example of an event whose probability is 1.

19. What is the difference between an empirical and a theoretical probability? Give an example of each using tossing two dice and getting a sum of 5 on the faces. Be specific in your examples and be sure to reference the Law of Large Numbers.

20. One card is chosen at random from a standard deck of 52 playing cards. Find each of these probabilities.

a) Prob(Jack) = b) Prob(black card) =

c) Prob(Ace of Spades) = d) Prob(8,9 or 10)=

e) You start with a standard deck of 52 cards. You randomly choose two cards which happen to be a Queen of Hearts and a Queen of Spades and you then set these two cards aside. Find each of these probabilities based on the remaining cards.

Prob(King of Clubs) = Prob(hearts) =

21. Two girlfriends decide to dress alike during a weekend away. Each one packs the same 3 pairs of pants, the same 4 sweaters and the same 5 blouses. All clothing items are color coordinated so you randomly choose a pair of pants, a sweater and a blouse for the first day of your trip. What is the probability that your friend would randomly choose exactly the same outfit as you?

22. 17 people apply for a 5 person committee as part of the Polish Heritage Society. The committee members will be chosen by a random draw. The moderator of the activity thinks that a committee consisting of Agnieszka, Brunislaw, Cyryl, Dorotka and Eugeniusz would not work well together. What is the probability that this exact committee would be chosen by random draw?

23. There are 15 blue blocks , 17 red blocks and 18 green blocks in a bag. If you pull one out randomly , find each of these :

odds for red _____ : _____ odds against blue _____ : _____

odds for green _____ : _____ probability of green _____

24. If a racing horse has odds 3 : 2 , what does that mean ?

25. The odds in favor of your team's winning a certain game are 7 : 3 as determined by a sportscaster. What is the probability that your team will win ? _____

26. The probability that you will win in a certain game of chance is 2 / 25. What are your odds to win? _____ : _____ What are your odds to lose ? _____ : _____

27. Use the information contained in the chart from a survey of students to answer these questions.

Doesn’t like apples 50 150

a) What is the probability that a person who doesn't like apples is female?

b) What is the probability that a female doesn't like apples?

c) What is the probability that a student likes apples?

28. The Venn diagram represents a group of students and their enrollment in math and psychology.

a) What is the probability that a randomly chosen student takes psychology?

b) What is the probability that a math student takes psychology?

c) What is the probability that a randomly chosen student takes both subjects?

d) What is the probability that a student who doesn’t take math does take psychology?

e) What is the probability that a student who doesn’t take psychology is not taking math?

29. A fair die is rolled 200 times.

a) About how many even numbers would you expect?

b) About how many fours would you expect?

c) About how many twos or threes would you expect?

30. In a restaurant you can choose a complete meal from the regular menu which includes 6 main courses, 5 desserts and 2 beverages OR you can choose a complete meal from the special menu which includes 5 main courses, 2 desserts and 3 beverages. What is the probability that your companion will choose the exact same complete meal that you do?

31. List rows 0 to 5 of Pascal’s Triangle.

32. Use Pascal’s triangle to expand each of these binomials. Show your work.

(Note: Because the quiz on this material was open notes, the material on the exam will be limited.)

33. T rue/False Write the full word in the blank provided. *Correct each false statement.

a. _____ In plurality voting, the declared winner has more than 50% of the votes cast.

b. _____ New Jersey uses the plurality voting procedure to elect its governor.

c. _____ In the United States because of the strength of the two party system, a third party candidate cannot have a major impact on the outcome of a presidential election.

d. _____ A preference schedule is a listing of candidates in order of preference.

e. _____ A disadvantage of majority voting is that it can require election run-offs.

f. _____ Plurality voting requires greater literacy skills than preference schedule voting therefore, it should not be used in areas with high illiteracy.

g. _____ Ballots on which several products can be ranked are often used in consumer surveys.

h. _____ Run-off elections can produce different results from the same data depending on the run-off system used.

i. _____ In a Borda count, each candidate receives “points” for where they fall on a preference schedule. These points are then tallied and compared to determine a winner.

j. _____ You used a paradox each time you did a sequential run-off with data in class.

k. _____ Arrow’s Algorithm states informally that no voting system can exist that will not, at times, yield results that seem to contradict voters’ expectations based on the data.

34. a.) Complete this recurrence formula in words: t n = n + 10t n-1

b.) Complete this table for the recurrence formula t n = n + 10t n-1 . Show work in the space below the table.

## Shapes of Math: It is time to rethink voting methods

Last fall, we witnessed another close presidential election in which the popular vote winner did not succeed in winning the election. The 2016 contest was one of the most unusual and contentious in American history.

As in many prior elections, it has again given impetus toward a re-examination of the very voting method that our nation employs to select its leaders. People of every political stripe would probably agree that it is in keeping with the basic tenets of democracy to use the method that best reflects the aggregate will of the voting populace.

Much research has been done in the last 30 years that shows that replacing our current long-standing plurality method- vote for one, most votes wins- could not only better meet the voters’ will, but might also reduce the vitriolic rhetoric of a hard-fought campaign.

Alternate methods do exist and, in fact, are currently used in this country and elsewhere to decide other types of elections ranging from mayoral races in some of the largest cities to awards for the “Most Valuable Player” or “Rookie of the Year” in most major sports leagues.

Some democracies in the world, like Australia, Ireland, India and Scotland, also have successfully used alternate voting schemes called that most experts believe gauge voter sentiment far more accurately than the plurality method.

The remarks here do not address the pros or cons of the Electoral College that gets scrutinized every four years by the post-election pundits with no effort ever initiated toward changing it. Rather, this discussion is about the way we mark the ballots themselves.

Instead of being limited to voting for just one candidate, consider the possibility of each voter ranking the candidates: #1 for his/her favorite, #2 for the next favorite and so forth.

The example below represents a ballot where voters number the candidates in order of preference.

In the field of social choice, these types of ballots are called preference ballots and a variety of methods exist by which to tabulate them.

For example, suppose the Marywood Ski Club held an election for club president among Alicia, Barzhon and Carlos and that the 15 members used preference ballots. The following table, called a preference table, summarizes the results. Let “A” represent the ranking for Alicia, “B” represent Barzhon and “C” represent Carlos.

This table indicates that six voters voted for Alicia first, Barzhon second and Carlos third. Only one voter ranked Alicia first, Carlos second and Barzhon third.

Alicia would win this election by the plurality method—the same method used to elect our president—because she received seven votes for first place.

However, the Borda Method, which is used in determining the Heisman Trophy and many other sports awards, assigns three points for each first place vote, two points for every second place vote and one point for every third-place vote. You can verify that the Borda count for each candidate is:

Barzhon wins the election instead of Alicia if the Borda Method is used. An examination of the table shows this perhaps corresponds to the fact that Alicia was ranked last by eight voters whereas Barzhon was ranked second by nine voters and last by only one.

By the Borda Method, Barzhon seems to be a more agreeable choice to most of the members of the club even though he ranked at the top of fewer ballots than Alicia. One advantage is that voters whose second choice was the eventual winner would most likely not be wholly unsatisfied with that result.

The Borda method has another distinct advantage for presidential elections in that it encourages voter turnout by fans of third party candidates. They can still rank their underdog first, but place their next favorite candidate in second place.

When using this method, a person’s vote is not “wasted.” I heard from many students who did not vote in this recent election for that very reason. They knew that a vote for Gary Johnson or Jill Stein would not affect the outcome of the election.

The Nobel Prize-Winning Economist Kenneth Arrow devised a number of criteria for judging the fairness of any voting method and concluded that no system is perfectly fair, although some come closer than others. The plurality method, in particular, has many flaws when examined closely and the time to re-examine its effectiveness is long overdue.

## Contemporary Mathematics Contemporary Mathematics at Nebraska

It is often desirable to use a few numbers to summarize a distribution. One important aspect of a distribution is where its center is located. Measures of central tendency are discussed first. A second aspect of a distribution is how spread out it is. In other words, how much the data in the distribution vary from one another. The second section describes measures of variability.

Let's begin by trying to find the most "typical" value of a data set. Note that we just used the word "typical" although in many cases you might think of using the word "average." We need to be careful with the word "average" as it means different things to different people in different contexts. One of the most common uses of the word "average" is what mathematicians and statisticians call the , or just plain old for short. "Arithmetic mean" sounds rather fancy, but you have likely calculated a mean many times without realizing it the mean is what most people think of when they use the word "average".

The of a set of data is the sum of the data values divided by the number of values.

###### Example 2.16

Marci's exam scores for her last math class were: 79, 86, 82, 94. The mean of these values would be:

Typically, we round means to one more decimal place than the orgiinal data had. In this case, we would round 85.25 to 85.3.

###### Example 2.17

The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below:

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

Adding these values, we get 634 total TDs. Dividing by 31, the number of data values, we get 634/31 = 20.4516. It would be appropriate to round this to 20.5.

It would be most correct for us to report that The mean number of touchdown passes thrown in the NFL in the 2000 season was 20.5 passes, but it is not uncommon to see the more casual word average used in place of mean.

###### Exploration 2.5

The price of a jar of peanut butter at 5 stores was: $3.29,$3.59, $3.79,$3.75, and $3.99. Find the mean price Adding the prices and dividing by 5 we get the mean price:$3.682.

###### Example 2.18

The one hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below. Calculating the mean by hand could get trick if we try to type in all 100 values: We could calculate this more easily by noticing that adding 15 to itself six times is the same as = 90. Using this simplification, we get The mean household income of our sample is 33.9 thousand dollars ($33,900).

Note that the calculation you performed above is called a .

###### Example 2.19

Extending off the last example, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand). Adding this to our sample, our mean is now:

While 83.1 thousand dollars ($83,069) is the correct mean household income, it no longer represents a "typical" value. If we graph our household data, the$5 million data value is so far out to the right that the mean has to adjust up to keep things in balance.

For this reason, when working with data that have values far outside the primary grouping it is common to use a different measure of center, the .

###### Median

The of a set of data is the value in the middle when the data is in order.

To find the median, begin by listing the data in order from smallest to largest, or largest to smallest.

If the number of data values, N, is odd, then the median is the middle data value. This value can be found by rounding N/2 up to the next whole number.

If the number of data values is even, there is no one middle value, so we find the mean of the two middle values (values N/2 and N/2 + 1)

###### Example 2.20

Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

Since there are 31 data values, an odd number, the median will be the middle number, the 16th data value (31/2 = 15.5, round up to 16, leaving 15 values below and 15 above). The 16th data value is 20, so the median number of touchdown passes in the 2000 season was 20 passes. Notice that for this data, the median is fairly close to the mean we calculated earlier, 20.5.

###### Example 2.21

Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7

We start by listing the data in order: 2 4 5 5 6 7 7 8 8 10

Since there are 10 data values, an even number, there is no one middle number. So we find the mean of the two middle numbers, 6 and 7, and get (6+7)/2 = 6.5.

The median quiz score was 6.5.

###### Example 2.23

If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is$35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is.

In addition to the mean and the median, there is one other common measurement of the "typical" value of a data set: the .

The In addition to the mean and the median, there is one other common measurement of the "typical" value of a data set: the mode.

The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed.

###### Example 2.24

In our vehicle color survey, we collected the data

For this data, Green is the mode, since it is the data value that occurred the most frequently.

It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once.

## Should you choose Math 161/162 or Math 131/132?

Any questions about placement in calculus or other 100-level courses that remain after reading that section should be directed to John Houlihan, Mathematics Placement Director. Please e-mail him to set up an appointment.

Math 161/162 (Calculus I, Calculus II) is a traditional calculus sequence covering all the basic topics of one-variable calculus. This sequence is a prerequisite for Multivariable Calculus (Math 263) as well as for almost all higher-level math courses. It is required for all students majoring in Chemistry, Engineering Science, Mathematics, Physics and Statistics. It is highly recommended, although not required, for students majoring in Biology, Computer Science and Economics.

Math 131/132 (Applied Calculus I, Applied Calculus II) is more of a survey sequence covering many of the basic topics in one-variable calculus as well as some topics in multivariable calculus and differential equations. It is a terminal sequence in that it does not satisfy the prerequisites of upper-level mathematics and statistics courses. Students who enjoyed mathematics in high school and earned ACT math scores of 28 and higher or SAT math scores of 660 and higher are encouraged to choose the Math 161/162 sequence.

### Installing Mathematica (free!)

Mathematica is a powerful computing environment that is designed for use in engineering, mathematics, finance, physics, chemistry, biology, and a wide range of other fields. Loyola students and faculty can download and install the latest copy of Mathematica for free. You must be logged on to Loyola VPN, and then visit the following ITS webpage, https://digitalmedia.luc.edu/News/NewsItem/View/4/mathematica-version-9-downloads-available.

### Wolfram Demonstrations Project

From the Wolfram Demonstrations Project. ". . . the Wolfram Demonstrations Project is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and a remarkable range of other fields.

Its daily growing collection of interactive illustrations is created by Mathematica users from around the world who participate by contributing innovative Demonstrations."

## A question about voting methodology

I'm working on the constitution for my temple, and we elect 3 board members every 4 years. The way it works now is that we have all the people running in a common ballot, and the voters are allowed to vote for up to 3 people. However, we have a very large number of people turning in only bullet ballots - that is, ballots with only one name. There are complaints that this is hurting the fairness of the voting process. Now, while I can't speak to whether or not that hurts equality, I was wondering if there is in fact a better way to stage this election. Any suggestions?

There are mathematically better voting systems than the one you're using, although convincing the voters that they should trust them can be difficult, because the best ones start to become difficult for a layman to understand. The system you describe is called block voting. Single Transferable Vote is a preferable system that has the advantage of actually being used by some national governments, which might help acceptance. It's also simple enough (if barely) to perform "on paper" without the aid of a computer.

The STV system led me to some more fantastic reading, thank you! It's a bloody complicated method to actually put in place, though, especially if we continue to use paper ballots. We usually have about 3000 voters come through, so counting first preference votes and then second preference and so on may not be very efficient. EDIT: I ate a zero.

Not something I admit to really having much of an understanding of, but I know Arrow's Theorem is somehow related to voting and so that wikipedia page might be a great start.

As far as I remember, Arrow's impossibility theorem basically shows that the only voting system that is perfect (under the 3 stated axioms) is dictatorship. But I believe that's about voting systems where each person gives only one name. I don't think this applies to OP's question where you are allowed to vote for up to 3 people.

As expected, Arrow's Theorem has come up in this comment thread, as it is relatively well-known. However, it seems worth emphasizing a refinement of the result that seems to not be so "popularly" well-known: namely, that the Borda count is the unique positional voting method that satisfies a very reasonable relaxation of IIA.

That relaxation is the following: IIA is usually stated as requiring that the voting method must use only voters' preference between A and B (and not any third candidate C) to determine the relative ranking of A and B in the election outcome. Arrow's Theorem says that if we want this property and unanimity, then the only voting method that satisfies both is a dictatorship.

The situation improves if instead we demand that a voting method use only voter's strength of preference betwen A and B to determine the relative ranking of A and B in the outcome. By "strength of preference" is meant the number of candidates ranked between A and B by each voter.

With this relaxed constraint, there is exactly one positional voting method that answers the mail: the Borda count.

## 2.8: Borda Count - Mathematics

Game Theory: Practice for the Final Examination (2018)

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

1. Given the election below:

i. Compute the pairwise vote matrix for the election above.

ii. Decide the winner (if there is one) of the election using:

e. Condorcet (draw a digraph to show the results of the 2-way races)

f. Sequential run-off based on the Borda Count (Baldwin)

(Should a tie occur, explain how the tie might be broken.)

iii. Be prepared to give a brief account of Arrow's Theorem and the Gibbard-Saitterthwaite Theorems and why they are important important.

iv. Does it ever benefit a voter to "lie" about his/her preferences?

2. i. Use Adams, Jefferson, and Webster to apportion a legislature of 15 seats. Each of three regions is entitled to a minimum of at least one seat. The populations of these regions is shown below. The apportionment should be done "twice:"

a. Using a divisor and rounding (state what divisor you used so that when you rounded as the method requires that 15 seats were distributed). (If a tie occurs when 15 seats are given out, say so.)

b. Using the table method check if you get the same answer as you did for above for Adams, Jefferson, and Webster. For each of these methods indicate the order in which the seats are given out via the table method when there is a tie, indicate this by having the states share these seats. Thus, here for example, seats 1-3 will be shared.

 A B C Population 12300 9500 8200

ii. Use Hamilton's method to apportion 15 seats using the data above.

iii. What is required of an apportionment method which obeys "house monotonicity?"

iv. What is required of an apportionment method which obeys "population monotonicity?"

v. What is required of an apportionment method which obeys "quota?"

vi. What does the Balinksi-Young Theorem state?

vii. What is the method currently used for distributing the seats in the House of Representatives? Give a brief description of this method.

3. Consider the zero-sum games, with payoffs shown from Row's point of view below:

a.

 Row/Column I II III 1 3 -1 7 2 -2 0 5

b.

 Row/Column I II III 1 1 -3 3 2 -4 7 -2

i. What is the best-worst strategy for Row and Column in each game?

ii. Does either game have dominating strategies?

iii. Does either game have a saddle point?

iv. Find the optimal way to play these games for each player. (This means considering issues of dominating strategies or finding a saddle point to find a value for the game, and/or find a small matrix for which one can design optimal spinners.) As part of your solution what is the payoff to each player.

4. Find optimal spinners and the value of the game for the following zero-sum game:
Payoffs below are from Row's point of view.

 Column I Column II Row 1 9 -4 Row 2 -2 1

5. (a) Find any pure and/or mixed Nash equilibrium (equilibria) for the non-zero-sum game below:

 Column I Column II Row 1 (7,7) (-9, 6) Row 2 (6, -9) (-1, -1)

Draw a motion diagram for the game. For each outcome cell of the game, are there any Pareto improvements over this outcome?

(b) Find any pure and/or mixed Nash equilibrium (equilibria) for the non-zero-sum game below:

 Column I Column II Row 1 (0,0) (7, 2) Row 2 (2, 7) (6, 6)

Draw a motion diagram for this game. For each outcome cell of the game, are there any Pareto improvements over this outcome?

c. What makes Chicken and Prisoner's Dilemma "special" as games?

6. For the bankruptcy situations below, find what amount from E is given to each player using:

b. Equality of loss (with possible subsidization)

c. Maimonides gain

d. Maimonides loss

f. Proportionality

g. Contested garment rule ("Talmudic method") (Only use this when there are two claimants)

i. E = 240 A claims 100 B claims 200

ii. E = 240 A claims 180 B claims 220

iii. E = 240 A claims 40 B claims 300

iv. E = 240 A claims 60 B claims 140 C claims 200

Give several real world examples where bankruptcy problems might/do arise.

7. Find the male optimal and female optimal stable matchings (they may be the same) for the preferences shown below. For the stable matchings you find, give the "rank" of the "mate" each man and woman gets. For example if man 4 is paired with woman 2 she may be his 4th highest choice and he may be her 1st choice. For i and ii. if man i is matched with woman i is the matching stable? If not find a "blocking" pair.
i.
Men: (Example: m2 likes w3, 2nd best m4 likes w4 best.)

 m 1 2 3 1 5 4 m 2 1 3 2 5 4 m 3 1 5 2 3 4 m 4 4 3 1 2 5 m 5 5 4 3 2 1

Women: (Example: w2 likes m2 3rd best w5 ranks man 5 (m5) first.)

 w 1 2 3 1 5 4 w 2 1 3 2 4 5 w 3 2 4 1 3 5 w 4 3 4 1 2 5 w 5 5 3 4 1 2

ii.

Men: (Example: m1 likes w4 last m3 likes w4 second best.)

 m 1 2 3 1 5 4 m 2 2 5 1 3 4 m 3 1 4 2 3 5 m 4 2 3 1 4 5 m 5 1 4 5 2 3

Women: (Example: w2 likes m1 best w4 likes m1 third best.)

 w 1 2 3 4 5 1 w 2 1 3 2 4 5 w 3 2 4 3 1 5 w 4 3 4 1 2 5 w 5 3 5 4 1 2

8. Given the weighted voting game: [13 10, 7, 6, 4]

a. List the minimal winning coalitions
b. Determine the Shapley power of each of the players.
c. Determine the Banzhaf power of each of the players.
d. Determine the Coleman power of each of the players.
e. Does this game have any veto players?
f. List all of the winning coalitions for this game.
g. Suppose that secretly the players with weights 6 and 4 agree to always vote together. What are the power relations now as indicated by the Shapley and Banzhaf power indices? (Hint: What is the 3 player game that is now really being played?)