Articles

8.4: Discrete Methods - Sealed Bids and Markers


Example (PageIndex{1}): Method of Sealed Bids, #1

Three heirs, Alice, Betty and Charles inherit an estate consisting of a house, a painting and a tractor. They decide to use the method of sealed bids to divide the estate among themselves.

  1. The players each submit a list of bids for the items. The bid is the value that a player would assign to the item. The bids are done privately and independently. The bids are usually listed in a table.
Table (PageIndex{1}): Initial Bids

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

  1. For each item, the player with the highest bid wins the item. The winning bids are highlighted in the table.
Table (PageIndex{2}): Winning Bids

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

  1. For each player find the sum of his/her bids. This amount is what the player thinks the whole estate is worth. For three players, each player is entitled to one third of the estate. Divide each sum by three to get a fair share for each player. Remember that each player sees the values differently so the fair shares will not be the same.
Table (PageIndex{3}): Total Bids and Fair Shares

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

Total Bids

$411,000

$405,000

$435,000

Fair Share

$137,000

$135,000

$145,000

  1. Each player either gets more than his/her fair or less than his/her fair share when the items are awarded. Find the difference between the fair share and the items awarded for each player. If a player was awarded more than his/her fair share, the player owes the difference to the estate. If a player was awarded less than his/her fair share, the estate owes the player the difference.

Alice: $137,000 - $75,000 = $62,000. The estate owes Alice $62,000.

Betty: $135,000 - $0 = $135,000. The estate owes Betty $135,000.

Charles: $145,000 – ($300,000 + $63,000) = -$218,000. Charles owes the estate $218,000.

Table (PageIndex{4}): Owed to Estate and Estate Owes

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

Total Bids

$411,000

$405,000

$435,000

Fair Share

$137,000

$135,000

$145,000

Owed to Estate

$218,000

Estate Owes

$62,000

$135,000

  1. At this point in the game, there is always some extra money in the estate called the surplus. To find the surplus, we find the difference between all the money owed to the estate and all the money the estate owes.

$218,000 – ($62,000 + $135,000) = $21,000.

Divide this surplus evenly between the three players.

Table (PageIndex{5}): Share of Surplus

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

Total Bids

$411,000

$405,000

$435,000

Fair Share

$137,000

$135,000

$145,000

Owed to Estate

$218,000

Estate Owes

$62,000

$135,000

Share of Surplus

$7,000

$7,000

$7,000

  1. Finish the problem by combining the share of the surplus to either the amount owed to the estate or the amount the estate owes. Include the items awarded in the final share as well as any money.

Alice: $62,000 + $7,000 = $69,000

Betty: $135,000 + $7,000 = $142,000

Charles: -$218,000 + $7,000 = -$211,000

Table (PageIndex{6}): Final Share

Alice

Betty

Charles

House

$280,000

$275,000

$300,000

Painting

$75,000

$70,000

$72,000

Tractor

$56,000

$60,000

$63,000

Total Bids

$411,000

$405,000

$435,000

Fair Share

$137,000

$135,000

$145,000

Owed to Estate

$218,000

Estate Owes

$62,000

$135,000

Share of Surplus

$7,000

$7,000

$7,000

Final Share

Gets painting and

$69,000 cash

Gets $142,000 cash

Gets house and tractor and pays $211,000

Alice gets the painting and $69,000 cash. Betty gets $142,000 cash. Charles gets the house and the tractor and pays $211,000 to the estate.

Now, we find the value of the final settlement for each of the three heirs in this example. Remember that each player has their own value system in this game so fair shares are not the same amount.

Alice: Painting worth $75,000 and $69,000 cash for a total of $144,000. This is $7,000 more than her fair share of $137,000.

Betty: $142,000 cash. This is $7,000 more than her fair share of $135,000.

Charles: House worth $300,000, tractor worth $63,000 and pays $211,000 for a total share of $152,000. This is $7,000 more that his fair share of $145,000.

At the end of the game, each player ends up with more than a fair share. It always works out this way as long as the assumptions are satisfied.

Summary of the Method of Sealed Bids:

  1. Each player privately and independently bids on each item. A bid is the amount the player thinks the item is worth.
  2. For each item, the player with the highest bid wins the item.
  3. For each player find the sum of the bids and divide this sum by the number of players to find the fair share for that player.
  4. Find the difference (fair share) – (total of items awarded) for each player. If the difference is negative, the player owes the estate that amount of money. If the difference is positive, the estate owes the player that amount of money.
  5. Find the surplus by finding the difference (sum of money owed to the estate) – (sum of money the estate owes). Divide the surplus by the number of players to find the fair share of surplus.
  6. Find the final settlement by adding the share of surplus to either the amount owed to the estate or the amount the estate owes. Include any items awarded and any cash owed in the final settlement. The sum of all the cash in the final settlement should be $0.

Example (PageIndex{2}): Method of Sealed Bids, #2

Doug, Edward, Frank and George have inherited some furniture from their great-grandmother’s estate and wish to divide the furniture equally among themselves. Use the method of sealed bids to find a fair division of the furniture.

Note: We start with one table and add lines to the bottom as we go through the steps.

  1. List the bids in table form.
Table (PageIndex{7}): Initial Bids

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

  1. Award each item to the highest bidder.
Table (PageIndex{8}): Winning Bids

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

  1. Find the fair share for each player.

Doug:

Edward:

Calculate similarly for Frank and George.

Table (PageIndex{9}): Total Bids and Fair Shares

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

Total Bids

$3,035.00

$3,025.00

$3,050.00

$3,050.00

Fair Share

$758.75

$756.25

$762.50

$762.50

  1. Find the amount owed to the estate or the amount the estate owes.

Doug:

Doug owes the estate $241.25.

Calculate similarly for Edward and Frank.

George:

The estate owes George $462.50.

Table (PageIndex{10}): Owes to Estate and Estate Owes

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

Total Bids

$3,035.00

$3,025.00

$3,050.00

$3,050.00

Fair Share

$758.75

$756.25

$762.50

$762.50

Owes to Estate

$241.25

$393.75

$87.50

Estate Owes

$462.50

  1. Find the share of surplus for each player.
Table (PageIndex{11}): Share of Surplus

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

Total Bids

$3,035.00

$3,025.00

$3,050.00

$3,050.00

Fair Share

$758.75

$756.25

$762.50

$762.50

Owes to Estate

$241.25

$393.75

$87.50

Estate Owes

$462.50

Share of Surplus

$65.00

$65.00

$65.00

$65.00

  1. Find the final share for each player.

Doug:

Calculate similarly for Edward and Frank.

George:

Table (PageIndex{12}): Final Shares

Doug

Edward

Frank

George

Dresser

$280.00

$275.00

$250.00

$300.00

Desk

$480.00

$500.00

$450.00

$475.00

Wardrobe

$775.00

$800.00

$850.00

$800.00

Dining Set

$1,000.00

$800.00

$900.00

$950.00

Poster Bed

$500.00

$650.00

$600.00

$525.00

Total Bids

$3,035.00

$3,025.00

$3,050.00

$3,050.00

Fair Share

$758.75

$756.25

$762.50

$762.50

Owes to Estate

$241.25

$393.75

$87.50

Estate Owes

$462.50

Share of Surplus

$65.00

$65.00

$65.00

$65.00

Final Share

Dining set and pays $176.25

Desk and poster bed and pays $328.75

Wardrobe and pays $22.50

Dresser and gets $527.50

Doug gets the dining set and pays $176.25. Edward gets the desk and poster bed and pays $328.75. Frank gets the wardrobe and pays $22.50. George gets the dresser and $527.50 in cash.

Note that the sum of all the money in the final shares is $0 as it should be. Also note that each player’s final share is worth $65.00 more than the fair share in his eyes.

Example (PageIndex{3}): Method of Sealed Bids in Dissolving a Partnership

Jack, Kelly and Lisa are partners in a local coffee shop. The partners wish to dissolve the partnership to pursue other interests. Use the method of sealed bids to find a fair division of the business. Jack bids $450,000, Kelly bids $420,000 and Lisa bids $480,000 for the business.

Make a table similar to the table for dividing up an estate and follow the same set of steps to solve this problem.

Table (PageIndex{13}): Method of Sealed Bids for Dissolving a Partnership

Jack

Kelly

Lisa

Business

$450,000

$420,000

$480,000

Total Bids

$450,000

$420,000

$480,000

Fair Share

$150,000

$140,000

$160,000

Owes to Business

$320,000

Business Owes

$150,000

$140,000

Share of Surplus

$10,000

$10,000

$10,000

Final Share

$160,000 cash

$150,000 cash

Business and pays $310,000

Lisa gets the business and pays Jack $160,000 and Kelly $150,000.

Method of Markers

The method of markers is used to divide up a collection of many objects of roughly the same value. The heirs could use the method of markers to divide up their grandmother’s jewelry collection. The basic idea of the method is to arrange the objects in a line. Then, each player puts markers between the objects dividing the line of objects into distinct parts. Each part is a fair share to that particular player. Based on the placement of the markers, the objects are allotted to the players. If there are players, each player places markers among the objects. We will use notation A1 to represent the first marker for player A, A2 to represent the second marker for player A, and so on.

Many times when the players have done all the steps in the method of markers there are some objects left over. If many objects remain, the players can line them up and do the method of markers again. If only a few objects remain, a common approach is to randomly choose an order for the players, then let each player pick an object until all the objects are gone.

It is interesting to see that one player may only receive one or two objects while another player may receive four or five objects. The number of objects allotted depends on each player’s value system. First we will look at the allocation of the pieces after the markers have been placed. Once we understand that, we will look at placing the markers in the correct places for each player.

Example (PageIndex{4}): Method of Markers, #1

Three players Albert (A), Bertrand (B), and Charles (C), wish to divide a collection of 15 objects using the method of markers. Determine the final allocation of objects to each player. Since there are three players, each player uses two markers.

Let’s start by looking at the line of objects and Albert’s markers.

The markers divide the line of objects into three pieces. Each piece of the line is a fair share in Albert’s value system. He would be satisfied with any of the three pieces in the final allocation. For now, do not worry about how Albert determined where to place the markers. We will look at that in Example (PageIndex{6}).

Now let’s add the markers for Bertrand and Charles.

Step 1: As you examine the objects from left to right, find the first marker, B1. Give Bertrand all the objects from the beginning of the line to the marker B1. Bertrand removes the rest of his markers and leaves the game for now.

Step 2: Now, continuing from left to right, find the first marker out of the second group of markers (A2 and C2). The first marker from this group we come across is A2. Give Albert all the objects from his first marker A1 to his second marker A2. Remember that a fair share is from one marker to the next. Object #4 is not part of Albert’s fair share since it is before his first marker. Albert removes the rest of his markers and leaves the game for now.

Step 3: Charles is the only player left in the game. He considers everything from his second marker to the end of the line to be a fair share so give it to him. Any objects not allocated at this point are left over.

Step 4: Typically some objects are left over at this point. Objects numbered 4, 9, 10 and 11 are left over in this game. The three players could draw straws to determine an order. Then each player in order would choose an object until all the object are allocated.

Note: Normally when we do the method of markers, we only draw the figure once.

Summary of the Method of Markers for n players:

  1. Arrange the objects in a line. Each of the n players places n-1 markers among the objects.
  2. Find the 1st first marker, say A1. Give player A all the objects from the beginning of the line to the 1st first marker. Player A removes his/her remaining markers and leaves the game for now.
  3. Find the 1st second marker, say B2. Give player B all the objects from the 1st second marker back to B’s previous marker B1. In other words, all the objects from B1 to B2. Player B removes his/her remaining markers and leaves the game for now.
  4. Find the 1st third marker, say C3. Give player C all the objects from the 1st third marker back to C’s previous marker C2. In other words, all the objects from C2 to C3. Player C removes his/her remaining markers and leaves the game for now.
  5. Continue this pattern until one player remains. Give the last player all the objects from his/her last marker to the end of the line of objects.
  6. Divide up the remaining objects. If many objects remain, do the method of markers again. If only a few objects remain, randomly choose an order then let each player choose an object in order until all the objects are gone.

Example (PageIndex{5}): Method of Markers, #2

Four cousins, Amy, Becky, Connie and Debbie wish to use the method of markers to divide a collection of jewels. The jewels are lined up and the cousins place their markers as shown below in Figure (PageIndex{23}). What is the final allocation of the jewels?

The 1st first marker is C1 so give Connie all the jewels from the beginning of the line to her first marker. Connie removes her remaining markers and leaves the game for now.

The 1st second marker is D2 so give Debbie all the jewels between markers D1 and D2. Debbie removes her remaining markers and leaves the game for now.

The 1st third marker is a tie between A3 and B3 so randomly choose one. One possibility is to have Amy and Becky toss a coin and the winner gets the next fair share. Assume Becky wins the coin toss. Give Becky all the jewels between markers B2 and B3. Becky removes her remaining markers and leaves the game for now.

Amy is the last player in the game. Give Amy all the jewels from her last marker to the end of the line.

Jewels numbered 4, 8, and 9 are left over. The players can draw straws to determine an order. Each player, in order, chooses a jewel until all the jewels have been allocated.

Example (PageIndex{6}): Determining Where to Place the Markers

Four roommates want to split up a collection of fruit consisting of 8 oranges (O), 8 bananas (B), 4 pears (P), and 4 apples (A). The fruit are lined up as shown below in Figure (PageIndex{28}).

To determine where to place the markers, each player assigns a value to each type of fruit. Jack loves oranges, likes apples and pears equally, but dislikes bananas. He assigns a value of $1 to each apple and each pear, a value of $2 to each orange, and a value of $0 to each banana. In Jack’s value system, the collection of fruit is worth $24. Jack’s fair share is $6. He needs to place his markers so that the fruit is divided into groups worth $6. It can be helpful to work from both ends of the line. Jack has no choice about the placement of his first and third markers. Because he sees the bananas as worth $0 he has three possible places for his second marker. These possibilities are shown below in Figure (PageIndex{29}) as dotted lines.

Kent dislikes apples and oranges, like bananas and really loves pears. He assigns a value of $0 to each apple or orange, a value of $1 to each banana, and a value of $3 to each pear. In Kent’s value system, the collection of fruit is worth $20. Since there are four players, Kent’s fair share is $5. He needs to place his markers so that the fruit is divided into groups worth $5. Jack has no choice about the placement of his third marker. He has a few possibilities for his first two markers. The possibilities are shown below in Figure (PageIndex{30}) as dotted lines.

The other two roommates would follow the same process to place their markers. Once all the markers are placed, the allocation by the method of markers begins.

Imagine if the order of the fruit in Example (PageIndex{6}) was rearranged. It might not be possible for Kent to divide up the line of fruits into groups worth $5. He might have to use a group worth $6 next to a group worth $4. This is a good time to remember that none of our fair division methods are perfect. They work well most of the time but sometimes we just have to make do. If Kent was allocated a group of fruit worth only $4 he might get some of the missing value back when the left over fruits are allocated.


Part 14 - Sealed Bidding

Sealed bidding is a method of contracting that employs competitive bids, public opening of bids, and awards. The following steps are involved:

(a) Preparation of invitations for bids. Invitations must describe the requirements of the Government clearly, accurately, and completely. Unnecessarily restrictive specifications or requirements that might unduly limit the number of bidders are prohibited. The invitation includes all documents (whether attached or incorporated by reference) furnished prospective bidders for the purpose of bidding.

(b) Publicizing the invitation for bids. Invitations must be publicized through distribution to prospective bidders, posting in public places, and such other means as may be appropriate. Publicizing must occur a sufficient time before public opening of bids to enable prospective bidders to prepare and submit bids.

(c) Submission of bids. Bidders must submit sealed bids to be opened at the time and place stated in the solicitation for the public opening of bids.

(d) Evaluation of bids. Bids shall be evaluated without discussions.

(e) Contract award. After bids are publicly opened, an award will be made with reasonable promptness to that responsible bidder whose bid, conforming to the invitation for bids, will be most advantageous to the Government, considering only price and the price-related factors included in the invitation.

14.102 [Reserved]

14.103 Policy.

14.103-1 General.

(a) Sealed bidding shall be used whenever the conditions in 6.401(a) are met. This requirement applies to any proposed contract action under part 6.

(b) Sealed bidding may be used for classified acquisitions if its use does not violate agency security requirements.

(c) The policy for pricing modifications of sealed bid contract appears in 15.403-4(a)(1)(iii).

14.103-2 Limitations.

No awards shall be made as a result of sealed bidding unless-

(a) Bids have been solicited as required by subpart 14.2

(b) Bids have been submitted as required by subpart 14.3

(c) The requirements of 1.602-1(b) and part 6 have been met and

(d) An award is made to the responsible bidder (see 9.1) whose bid is responsive to the terms of the invitation for bids and is most advantageous to the Government, considering only price and the price related factors included in the invitation, as provided in subpart 14.4.

14.104 Types of contracts.

Firm-fixed-price contracts shall be used when the method of contracting is sealed bidding, except that fixed-price contracts with economic price adjustment clauses may be used if authorized in accordance with 16.203 when some flexibility is necessary and feasible. Such clauses must afford all bidders an equal opportunity to bid.

14.105 Solicitations for informational or planning purposes.


Contemporary Mathematics Contemporary Mathematics at Nebraska

The Sealed Bids method provides a method for discrete fair division, allowing for the division of items that cannot be split into smaller pieces, like a house or a car. Because of this, the method requires that all parties have a large amount of cash at their disposal to balance out the difference in item values.

The method begins by compiling a list of items to be divided. Then:

  1. Each party involved lists, in secret, a dollar amount they value each item to be worth. This is their sealed bid.
  2. The bids are collected. For each party, the value of all the items is totaled, and divided by the number of parties. This defines their fair share.
  3. Each item is awarded to the highest bidder.
  4. For each party, the value of all items received is totaled. If the value is more than that party's fair share, they pay the difference into a holding pile. If the value is less than that party's fair share, they receive the difference from the holding pile. This ends the initial allocation.
  5. In most cases, there will be a surplus, or leftover money, in the holding pile. The surplus is divided evenly between all the players. This produces the final allocation.

While the assumptions we made for fair division methods specified that an arbitrator should not be necessary, it is common for an independent third party to collect the bids and announce the outcome. While not technically necessary, since the method can be executed without a third party involved, this protects the secrecy of the bids, which can sometimes help avoid resentment or bad feelings between the players.

Example 7.1

Sam and Omar have lived together for the last 3 years, during which time they shared the expense of purchasing several items for their home. Sam has accepted a job in another city, and now they find themselves needing to divide their shared assets.

Each records their value of each item, as shown below.

Use the sealed bids procedure to allocate the items between Sam and Omar.

Sam's total valuation of the items is $150+$200+$250+$50 = $650, making a fair share for Sam $650 2 = $325.

Omar's total valuation of the items is $100+$250+$150+$100 = $600, making a fair share for Omar $600 2 = $300.

Each item is now awarded to the highest bidder. Sam will receive the couch and video game system, providing $150+$250 = $400 of value to Sam. Since this exceeds his fair share, he has to pay the difference, $75, into a holding pile.

Omar will receive the TV and surround sound system, providing $250+$100=$350 in value. This is more than his fair share, so he has to pay the difference, $50, into the holding pile.

Thus, in the initial allocation, Sam receives the couch and video game system and pays $75 into the holding pile. Omar receives the TV and surround sound system and pays $50 into the holding pile. At this point, both players would feel they have received a fair share.

There is now $125 remaining in the holding pile. This is the surplus from the division. This is now split evenly, and both Sam and Omar are given back $62.50. Since Sam had paid in $75, the net effect is that he paid $12.50. Since Omar had originally paid in $50, the net effect is that he receives $12.50.

Thus, in the final allocation, Sam receives the couch and video game system and pays $12.50 to Omar. Omar receives the TV and surround sound system and receives $12.50. At this point, both players feel they have received more than a fair share.

Example 7.2

Four small graphic design companies are merging operations to become one larger corporation. In this merger, there are a number of issues that need to be settled. Each company is asked to place a monetary value (in thousands of dollars) on each issue:

Super Designs DesignByMe LayoutPros Graphix
Company name $5 $3 $3 $6
Company location $8 $9 $7 $6
CEO $10 $5 $6 $7
Chair of the board $7 $6 $6 $8

We can use the method of sealed bids to resolve these issues. We can allocate the issues as if they were items, using the usual sealed bids rules. Then, whichever company wins each issue can make the decision on that issue.

The items would then be allocated to the company that bid the most for each:

  • Company name would be awarded to Graphix
  • Company location would be awarded to DesignByMe
  • CEO would be awarded to Super Designs
  • Chair of the board would be awarded to Graphix

In the table below, we calculate the first and final settlements. Recall that all amounts are in thousands of dollars. Note that after the first settlemebt, the total surplus is (2.50+3.25-5.50+7.25 = 7.50 ext<,>) so each company gets (7.50/4=1.875) as their extra share.

Super Designs DesignByMe LayoutPros Graphix
Total value of all issues $30 $23 $22 $27
Fair share $7.50 $5.75 $5.50 $6.75
Total value of issues awarded $10 $9 $14
Cash portion of the (7.50-10=-2.50) (5.75-9=-3.25) (5.50-0=5.50) (6.75-14=-7.25)
first settlement Pay $2.50 Pay $3.25 Get $5.50 Pay $7.25
Extra share $1.875 $1.875 $1.875 $1.875
Cash portion of the (-2.50+1.875=-0.625) (-3.25+1.875=-1.375) (5.50+1.875=7.375) (-7.25+1.875=-5.375)
final settlement Pay .625 Pay $1.375 Get $7.375 Pay $5.375
Table 7.3

In summary, the final settlement is:

  • Super Designs wins the CEO, and pays $625 (.625 thousand)
  • DesignByMe wins the company location and pays $1,375 ($1.375 thousand)
  • LayoutPros wins no issues, but receives $7,375 in compensation
  • Graphix wins the company name and chair of the board, and pays $5,375.
Exploration 7.1

Jamal, Maggie, and Kendra are dividing an estate consisting of a house, a vacation home, and a small business. Their valuations (in thousands) are shown below. Determine the final settlement.

  • Jamal's total value is $250 + $170 + $300 = $720. His fair share is $240 thousand.
  • Maggie's total value is $300 + $180 + $255 = $735. Her fair share is $245 thousand.
  • Kendra's total value is$280 + $200 + $270 = $750. Her fair share is $250 thousand.
  • Jamal receives the business, and pays $300 - $240 = $60 thousand into holding.
  • Maggie receives the house, and pays $300 - $245 = $55 thousand into holding.
  • Kendra receives the vacation home, and gets $250 - $200 = $50 thousand from holding.

There is a surplus of $60 + $55 - $50 = $65 thousand in holding, so each person will receive $21,667 from surplus. In the final settlement,

  • Jamal receives the business, and pays $38,333.
  • Maggie receives the house, and pays $33,333.
  • Kendra receives the vacation home, and gets $71,667.

Fair division does not always have to be used for items of value. It can also be used to divide undesirable items. To do so, we simply follow the same procedure but assign the items negative values. We illustrate this in the following example.

Example 7.4

Suppose Chelsea and Mariah are sharing an apartment, and need to split the chores for the household. They list the chores, assigning a negative dollar value to each item in other words, the amount they would pay for someone else to do the chore (a per week amount). We will assume, however, that they are committed to doing all the chores themselves and not hiring someone else.

Now we award each item to the player who bid the highestbut remember that the values are negative. So, for example, $8 is greater than $10, and thus Mariah is awarded the vacuuming. This makes sense, since her value of $8 indicates that she dislikes it less than Chelsea, who valued it at $10. Similarly, we give cleaning the bathroom and doing the dishes to Chelsea, while Mariah also gets dusting. Now we can find the first and final settlements, given in the table below.

Chelsea Mariah
Total value of all chores $34 $38
Fair share $17 $19
Total value of chores assigned $18 $12
Cash portion of the (-17-(-18)=-17+18=1) (-19-(-12)=-19+12=-7)
first settlement Get $1 Pay $7
Extra share $3 $3
Cash portion of the (1+3=4) (-7+3=-4)
final settlement Get $4 Pay $4
Table 7.5

The overall surplus is 71=6, so each player receives a surplus of 6 2=3, giving the final settlement values shown in the table above. Notice that the cash in the final settlement comes out even: Mariah pays Chelsea $4, with nothing left over. This is a good way to check we've done our math right.


2 CFR § 200.320 - Methods of procurement to be followed.

The non-Federal entity must have and use documented procurement procedures, consistent with the standards of this section and §§ 200.317, 200.318, and 200.319 for any of the following methods of procurement used for the acquisition of property or services required under a Federal award or sub-award.

(a) Informal procurement methods. When the value of the procurement for property or services under a Federal award does not exceed the simplified acquisition threshold (SAT), as defined in § 200.1, or a lower threshold established by a non-Federal entity, formal procurement methods are not required. The non-Federal entity may use informal procurement methods to expedite the completion of its transactions and minimize the associated administrative burden and cost. The informal methods used for procurement of property or services at or below the SAT include:

(i) Distribution. The acquisition of supplies or services, the aggregate dollar amount of which does not exceed the micro-purchase threshold (See the definition of micro-purchase in § 200.1). To the maximum extent practicable, the non-Federal entity should distribute micro-purchases equitably among qualified suppliers.

(ii) Micro-purchase awards. Micro-purchases may be awarded without soliciting competitive price or rate quotations if the non-Federal entity considers the price to be reasonable based on research, experience, purchase history or other information and documents it files accordingly. Purchase cards can be used for micro-purchases if procedures are documented and approved by the non-Federal entity.

(iii) Micro-purchase thresholds. The non-Federal entity is responsible for determining and documenting an appropriate micro-purchase threshold based on internal controls, an evaluation of risk, and its documented procurement procedures. The micro-purchase threshold used by the non-Federal entity must be authorized or not prohibited under State, local, or tribal laws or regulations. Non-Federal entities may establish a threshold higher than the Federal threshold established in the Federal Acquisition Regulations (FAR) in accordance with paragraphs (a)(1)(iv) and (v) of this section.

(iv) Non-Federal entity increase to the micro-purchase threshold up to $50,000. Non-Federal entities may establish a threshold higher than the micro-purchase threshold identified in the FAR in accordance with the requirements of this section. The non-Federal entity may self-certify a threshold up to $50,000 on an annual basis and must maintain documentation to be made available to the Federal awarding agency and auditors in accordance with § 200.334. The self-certification must include a justification, clear identification of the threshold, and supporting documentation of any of the following:

(A) A qualification as a low-risk auditee, in accordance with the criteria in § 200.520 for the most recent audit

(B) An annual internal institutional risk assessment to identify, mitigate, and manage financial risks or,

(C) For public institutions, a higher threshold consistent with State law.

(v) Non-Federal entity increase to the micro-purchase threshold over $50,000. Micro-purchase thresholds higher than $50,000 must be approved by the cognizant agency for indirect costs. The non-federal entity must submit a request with the requirements included in paragraph (a)(1)(iv) of this section. The increased threshold is valid until there is a change in status in which the justification was approved.

(i) Small purchase procedures. The acquisition of property or services, the aggregate dollar amount of which is higher than the micro-purchase threshold but does not exceed the simplified acquisition threshold. If small purchase procedures are used, price or rate quotations must be obtained from an adequate number of qualified sources as determined appropriate by the non-Federal entity.

(ii) Simplified acquisition thresholds. The non-Federal entity is responsible for determining an appropriate simplified acquisition threshold based on internal controls, an evaluation of risk and its documented procurement procedures which must not exceed the threshold established in the FAR. When applicable, a lower simplified acquisition threshold used by the non-Federal entity must be authorized or not prohibited under State, local, or tribal laws or regulations.

(b) Formal procurement methods. When the value of the procurement for property or services under a Federal financial assistance award exceeds the SAT, or a lower threshold established by a non-Federal entity, formal procurement methods are required. Formal procurement methods require following documented procedures. Formal procurement methods also require public advertising unless a non-competitive procurement can be used in accordance with § 200.319 or paragraph (c) of this section. The following formal methods of procurement are used for procurement of property or services above the simplified acquisition threshold or a value below the simplified acquisition threshold the non-Federal entity determines to be appropriate:

(1) Sealed bids. A procurement method in which bids are publicly solicited and a firm fixed-price contract (lump sum or unit price) is awarded to the responsible bidder whose bid, conforming with all the material terms and conditions of the invitation for bids, is the lowest in price. The sealed bids method is the preferred method for procuring construction, if the conditions.

(i) In order for sealed bidding to be feasible, the following conditions should be present:

(A) A complete, adequate, and realistic specification or purchase description is available

(B) Two or more responsible bidders are willing and able to compete effectively for the business and

(C) The procurement lends itself to a firm fixed price contract and the selection of the successful bidder can be made principally on the basis of price.

(ii) If sealed bids are used, the following requirements apply:

(A) Bids must be solicited from an adequate number of qualified sources, providing them sufficient response time prior to the date set for opening the bids, for local, and tribal governments, the invitation for bids must be publicly advertised

(B) The invitation for bids, which will include any specifications and pertinent attachments, must define the items or services in order for the bidder to properly respond

(C) All bids will be opened at the time and place prescribed in the invitation for bids, and for local and tribal governments, the bids must be opened publicly

(D) A firm fixed price contract award will be made in writing to the lowest responsive and responsible bidder. Where specified in bidding documents, factors such as discounts, transportation cost, and life cycle costs must be considered in determining which bid is lowest. Payment discounts will only be used to determine the low bid when prior experience indicates that such discounts are usually taken advantage of and

(E) Any or all bids may be rejected if there is a sound documented reason.

(2) Proposals. A procurement method in which either a fixed price or cost-reimbursement type contract is awarded. Proposals are generally used when conditions are not appropriate for the use of sealed bids. They are awarded in accordance with the following requirements:

(i) Requests for proposals must be publicized and identify all evaluation factors and their relative importance. Proposals must be solicited from an adequate number of qualified offerors. Any response to publicized requests for proposals must be considered to the maximum extent practical

(ii) The non-Federal entity must have a written method for conducting technical evaluations of the proposals received and making selections

(iii) Contracts must be awarded to the responsible offeror whose proposal is most advantageous to the non-Federal entity, with price and other factors considered and

(iv) The non-Federal entity may use competitive proposal procedures for qualifications-based procurement of architectural/engineering (A/E) professional services whereby offeror's qualifications are evaluated and the most qualified offeror is selected, subject to negotiation of fair and reasonable compensation. The method, where price is not used as a selection factor, can only be used in procurement of A/E professional services. It cannot be used to purchase other types of services though A/E firms that are a potential source to perform the proposed effort.

(c) Noncompetitive procurement. There are specific circumstances in which noncompetitive procurement can be used. Noncompetitive procurement can only be awarded if one or more of the following circumstances apply:

(1) The acquisition of property or services, the aggregate dollar amount of which does not exceed the micro-purchase threshold (see paragraph (a)(1) of this section)

(2) The item is available only from a single source

(3) The public exigency or emergency for the requirement will not permit a delay resulting from publicizing a competitive solicitation

(4) The Federal awarding agency or pass-through entity expressly authorizes a noncompetitive procurement in response to a written request from the non-Federal entity or

(5) After solicitation of a number of sources, competition is determined inadequate.


8.4: Discrete Methods - Sealed Bids and Markers

Each player assigns a (subjective) value to each of the items at hand and submits the bids in a sealed envelope. (The average of the bids by a single player provides that player's estimate for the fair share of the goods.) Players are not aware about each other's estimates. Envelopes are open simultaneously, and items are assigned to the highest bidder. The goods received by a player may exceed or not add up to that player's subjective value of the fair share. Depending on the circumstances the player than agrees to pay the excess over or receive a cash payment to make up to the fair share.

The total of the payments and the receipts is never negative. If it's positive, it may be evenly divided between the participants.

(Bold numbers could be clicked upon. To increase the number, click to the right of its vertical center line. To decrease it click to the left of the line. Dragging the mouse near the center line will accomplish the same task, but faster.)


Title 2

ECFR Content

Editorial codification of the general and permanent rules published in the Federal Register.

The non-Federal entity must have and use documented procurement procedures, consistent with the standards of this section and §§ 200.317, 200.318, and 200.319 for any of the following methods of procurement used for the acquisition of property or services required under a Federal award or sub-award.

( a ) Informal procurement methods. When the value of the procurement for property or services under a Federal award does not exceed the simplified acquisition threshold (SAT), as defined in § 200.1, or a lower threshold established by a non-Federal entity, formal procurement methods are not required. The non-Federal entity may use informal procurement methods to expedite the completion of its transactions and minimize the associated administrative burden and cost. The informal methods used for procurement of property or services at or below the SAT include:

( i ) Distribution. The acquisition of supplies or services, the aggregate dollar amount of which does not exceed the micro-purchase threshold (See the definition of micro-purchase in § 200.1). To the maximum extent practicable, the non-Federal entity should distribute micro-purchases equitably among qualified suppliers.

( ii ) Micro-purchase awards. Micro-purchases may be awarded without soliciting competitive price or rate quotations if the non-Federal entity considers the price to be reasonable based on research, experience, purchase history or other information and documents it files accordingly. Purchase cards can be used for micro-purchases if procedures are documented and approved by the non-Federal entity.

( iii ) Micro-purchase thresholds. The non-Federal entity is responsible for determining and documenting an appropriate micro-purchase threshold based on internal controls, an evaluation of risk, and its documented procurement procedures. The micro-purchase threshold used by the non-Federal entity must be authorized or not prohibited under State, local, or tribal laws or regulations. Non-Federal entities may establish a threshold higher than the Federal threshold established in the Federal Acquisition Regulations (FAR) in accordance with paragraphs (a)(1)(iv) and (v) of this section.

( iv ) Non-Federal entity increase to the micro-purchase threshold up to $50,000. Non-Federal entities may establish a threshold higher than the micro-purchase threshold identified in the FAR in accordance with the requirements of this section. The non-Federal entity may self-certify a threshold up to $50,000 on an annual basis and must maintain documentation to be made available to the Federal awarding agency and auditors in accordance with § 200.334. The self-certification must include a justification, clear identification of the threshold, and supporting documentation of any of the following:

( A ) A qualification as a low-risk auditee, in accordance with the criteria in § 200.520 for the most recent audit

( B ) An annual internal institutional risk assessment to identify, mitigate, and manage financial risks or,

( C ) For public institutions, a higher threshold consistent with State law.

( v ) Non-Federal entity increase to the micro-purchase threshold over $50,000. Micro-purchase thresholds higher than $50,000 must be approved by the cognizant agency for indirect costs. The non-federal entity must submit a request with the requirements included in paragraph (a)(1)(iv) of this section. The increased threshold is valid until there is a change in status in which the justification was approved.

( i ) Small purchase procedures. The acquisition of property or services, the aggregate dollar amount of which is higher than the micro-purchase threshold but does not exceed the simplified acquisition threshold. If small purchase procedures are used, price or rate quotations must be obtained from an adequate number of qualified sources as determined appropriate by the non-Federal entity.

( ii ) Simplified acquisition thresholds. The non-Federal entity is responsible for determining an appropriate simplified acquisition threshold based on internal controls, an evaluation of risk and its documented procurement procedures which must not exceed the threshold established in the FAR. When applicable, a lower simplified acquisition threshold used by the non-Federal entity must be authorized or not prohibited under State, local, or tribal laws or regulations.

( b ) Formal procurement methods. When the value of the procurement for property or services under a Federal financial assistance award exceeds the SAT, or a lower threshold established by a non-Federal entity, formal procurement methods are required. Formal procurement methods require following documented procedures. Formal procurement methods also require public advertising unless a non-competitive procurement can be used in accordance with § 200.319 or paragraph (c) of this section. The following formal methods of procurement are used for procurement of property or services above the simplified acquisition threshold or a value below the simplified acquisition threshold the non-Federal entity determines to be appropriate:

( 1 ) Sealed bids. A procurement method in which bids are publicly solicited and a firm fixed-price contract (lump sum or unit price) is awarded to the responsible bidder whose bid, conforming with all the material terms and conditions of the invitation for bids, is the lowest in price. The sealed bids method is the preferred method for procuring construction, if the conditions.

( i ) In order for sealed bidding to be feasible, the following conditions should be present:

( A ) A complete, adequate, and realistic specification or purchase description is available

( B ) Two or more responsible bidders are willing and able to compete effectively for the business and

( C ) The procurement lends itself to a firm fixed price contract and the selection of the successful bidder can be made principally on the basis of price.

( ii ) If sealed bids are used, the following requirements apply:

( A ) Bids must be solicited from an adequate number of qualified sources, providing them sufficient response time prior to the date set for opening the bids, for local, and tribal governments, the invitation for bids must be publicly advertised

( B ) The invitation for bids, which will include any specifications and pertinent attachments, must define the items or services in order for the bidder to properly respond

( C ) All bids will be opened at the time and place prescribed in the invitation for bids, and for local and tribal governments, the bids must be opened publicly

( D ) A firm fixed price contract award will be made in writing to the lowest responsive and responsible bidder. Where specified in bidding documents, factors such as discounts, transportation cost, and life cycle costs must be considered in determining which bid is lowest. Payment discounts will only be used to determine the low bid when prior experience indicates that such discounts are usually taken advantage of and

( E ) Any or all bids may be rejected if there is a sound documented reason.

( 2 ) Proposals. A procurement method in which either a fixed price or cost-reimbursement type contract is awarded. Proposals are generally used when conditions are not appropriate for the use of sealed bids. They are awarded in accordance with the following requirements:

( i ) Requests for proposals must be publicized and identify all evaluation factors and their relative importance. Proposals must be solicited from an adequate number of qualified offerors. Any response to publicized requests for proposals must be considered to the maximum extent practical

( ii ) The non-Federal entity must have a written method for conducting technical evaluations of the proposals received and making selections

( iii ) Contracts must be awarded to the responsible offeror whose proposal is most advantageous to the non-Federal entity, with price and other factors considered and

( iv ) The non-Federal entity may use competitive proposal procedures for qualifications-based procurement of architectural/engineering (A/E) professional services whereby offeror's qualifications are evaluated and the most qualified offeror is selected, subject to negotiation of fair and reasonable compensation. The method, where price is not used as a selection factor, can only be used in procurement of A/E professional services. It cannot be used to purchase other types of services though A/E firms that are a potential source to perform the proposed effort.

( c ) Noncompetitive procurement. There are specific circumstances in which noncompetitive procurement can be used. Noncompetitive procurement can only be awarded if one or more of the following circumstances apply:

( 1 ) The acquisition of property or services, the aggregate dollar amount of which does not exceed the micro-purchase threshold (see paragraph (a)(1) of this section)

( 2 ) The item is available only from a single source

( 3 ) The public exigency or emergency for the requirement will not permit a delay resulting from publicizing a competitive solicitation

( 4 ) The Federal awarding agency or pass-through entity expressly authorizes a noncompetitive procurement in response to a written request from the non-Federal entity or

( 5 ) After solicitation of a number of sources, competition is determined inadequate.


'Best and final' offers

When more than one person wants to make an offer, some estate agents will ask buyers to submit their 'best and final' offer by a set date.

The best and final offer process works in a very similar way to sealed bids the main difference is that your offer doesn't have to be formally submitted in an envelope.

You might initially submit your best and final offer over the phone, but we'd advise doing it in writing too (email or letter) so that all of the details of your offer and situation are laid out for the vendor to see.

Along with the amount you're offering, including all of the information that you would in a sealed bid (see above).


8.4: Discrete Methods - Sealed Bids and Markers

Day One. Read all of Chapter 1 quickly to get an overview, and then go back and read pages 3-9 carefully. Pay special attention to the distinction between a method and a criterion .

Day Two. Chapter 1, pages 9-20.

Day Three. Chapter 1, pages 20-26.

Note the distinction between extended methods and recursive methods. The word ``recursive'' indicates a repetition of a process. (To ``recur'' is to ``occur again,'' sometimes over and over.)

C HAPTER 2: W EIGHTED V OTING S YSTEMS

Day One. Read all of Chapter 2 quickly to get an overview, and then go back and read pages 51-54 carefully.

Problems: 6-9, 49, 50. In 50, assume that the players are listed in descending order of weight.

Day Two. Pages 54-63. The Banzhaff power index.

Day Three. Pages 63-70. The Shapley-Shubik power index.

Problems: 24, 25, 27, 29, 42, 56, 58. If these take you a long time, then you're doing them the hard way. Try to find the easier way.

Test No. 1 (Chapters 1 and 2)

C HAPTER 3: F AIR D IVISION

Day One. Read all of Chapter 3 quickly to get an overview. Then go back and read pages 87-97 carefully. These cover the preliminaries and the lone divider method.

Problems: 3, 5, 7, 9, 15, 20, 21, 23, 54.

Day Two. Chapter 3, pages 100-107. On this day we will work on the last diminisher method and start on the method of sealed bids. We will omit the lone chooser method, but you should at least read about it quickly.

Day Three. Chapter 3, pages 107-110. Methods for discrete problems (in contrast to continuous problems): The method of sealed bids and the method of markers.

Problems: 39, 41, 42, 47-51, 62, 64.

C HAPTER 4: A PPORTIONMENT

Day One. Read all of Chapter 4 quickly to get an overview, and then go back and read up through page 143 carefully. This covers the preliminary ideas, and then Hamilton's method, without yet starting on the paradoxes that plague it.

Problems: 1, 3, 6, 7, 9, 12. Keep copies of your answers these will be helpful on the next assignment.

Day Two. Chapter 4, pages 144-152. This covers the paradoxes that plague Hamilton's method, and it covers the methods of Jefferson and John Quincy Adams (which suffer from problems of their own).

Problems: 13, 14, 17, 19, 23, 25, 30.

Day Three. Chapter 4, pages 152-159. This covers the method of Daniel Webster, the impossibility theorem of Balinski and Young, and the history of apportionment in the United States. There are also two appendices. You should certainly read Appendix 2 (page 174). Read Appendix 1 (pages 171-173) if you want to--it explains the method (slightly different from Webster's method) that is now actually used to apportion the United States House of Representatives.

Test No. 2 (Chapters 3 and 4)

C HAPTER 5: E ULER C IRCUITS

Day One. Read all of Chapter 5 quickly to get an overview, and then go back and read pages 179-186 carefully. Starting with the famous problem of the seven bridges of Königsberg, this covers the basic ideas of graph theory.

Day Two. Chapter 5, pages 186-192. This covers the concepts and vocabulary of graph theory. It also covers Euler's theorems on the existence of Euler circuits and Euler paths.

Problems: 11, 13, 15, 23, 25, 63.

Day Three. Chapter 5, pages 192-201. This covers Fleury's algorithm, and also the process of ``Eulerizing'' a graph.

C HAPTER 6: T RAVELING S ALESMAN P ROBLEM

Day One. Read all of Chapter 6 quickly to get an overview, and then go back and read pages 221-230 carefully. This covers Hamilton circuits and Hamilton paths, weighted graphs, and complete graphs (weighted and unweighted). It also describes the famous traveling-salesman problem . If you find an efficient, optimal solution to this problem, then you have essentially solved the famous `` P versus NP '' Problem, and you will win $1,000,000 . For more details, see

Problems: 2, 4, 6, 7, 9, 11, 17, 19.

Day Two. Chapter 6, pages 231-239. This covers three algorithms for the traveling-salesman problem on a complete weighted graph, namely, the brute-force algorithm, the nearest-neighbor algorithm, and the repetitive nearest-neighbor algorithm.

Problems: 23, 25, 26, 29a, 32, 61.

Day Three. Chapter 6, pages 239-245. This covers the cheapest-link algorithm.

Test No. 3 (Chapters 5 and 6)

Day One. First read pages 393-411 of Chapter 10 quickly to get an overview. (You may skip Example 1 if you want to.) We will not study the last part of the chapter, on logistic growth. Then go back and read pages 393-395 (preliminaries) and pages 397-402 (linear growth) carefully. Especially note the recursive description (page 398 this is our transition rule), the explicit description (page 398), the graphing (page 400), and the summation formula (page 402).

Problems: 1, 3, 5, 9, 11, 14, 17. These are all due on Day Three , not Day Two. But you must attempt them all for Day Two.

Day Two. We will continue our study of linear growth (arithMETic sequences). In particular, we will examine difficulties you faced when you tried to do the problems assigned last time.

Day Three. Chapter 10, pages 403-411 (exponential growth). Especially note the recursive description (this is our transition rule), the explicit description , the graphing (all on page 405), the compounding rules (involving money ) (pages 407-408), and the summation formula (page 409).

Problems: 19, 23, 25, 33, 35, 53 (alas!), 59, 61.

Day Four. We will continue our study of exponential growth (geometric sequences).

I NFINITY AND I MPOSSIBILITY

We have seen statements (specifically, the theorems of Arrow and of Balinski and Young) alleging that it was impossible to do something or other. Should you believe such statements just because they're made by someone in authority? Of course not! The reason to believe a theorem is that someone has proved it. However, you probably haven't seen proofs before, and the proofs of the theorems above are too complicated for this course.

I want you to see an example of a historically important impossibility theorem whose proof is easy enough for you to understand. Once you have understood it, you will have a better idea of what it means for something to be proved, and what it means for something to be impossible. This is worth writing home about.

Problems: To be handed out. This subject is not covered in our textbook.

Test No. 4 (Chapter 10 and beyond)

C HAPTER 13: C OLLECTING S TATISTICAL D ATA

Day One. Read How to Lie With Statistics , which is one of the best books ever. And it's short! Read all of Chapter 13 in Tannenbaum quickly to get an overview, and then go back and read pages 517-526 carefully. Pay special attention to the terms in boldface type and to the three case studies. (You might find the overview on page 535 helpful.)

Day Two. Chapter 13, pages 526-530.

Day Three. Read pages 530-535. As you are reading, try to understand clearly how this part of the chapter differs from the earlier parts.

C HAPTER 14: D ESCRIPTIVE S TATISTICS

Day One. Chapter 14 is a bit long, and contains many concepts. Even so, it is probably best to read all of the chapter quickly to get an overview of it. Then go back and carefully study this day's specific reading assignment, pages 549-558. This deals with graphical representations of data.

Problems: 1-4, 13, 14, 17, 19-22.

Day Two. Read pages 558-567. Here we study a few ``numerical summaries of data.'' The idea is to find just a few numbers to summarize a large data set so that we can understand the data without being overwhelmed by too many numbers. In particular, we will see the ``three M's'': mean, median, and mode. (The textbook mentions mode only in the exercises.)

Problems: 24, 34, 38, 45, 47, 51, 67, 69. Start looking at 71, 73, and 77, even though they won't be assigned until next time.

Day Three. Chapter 14, pages 567-571. We reserve this whole day for one concept (and, if necessary, for a little catch-up). That one concept is ``measures of spread.'' The reason for devoting a whole class session to it is so that we can make sure that everyone understands its meaning (which is important).

Problems: 56 (note that these data sets all have the same average how do they differ?), 71, 73, 74, 77.

Test No. 5 on Chapters 13 and 14.

This document was generated using the LaTeX 2HTML translator Version 2002 (1.62)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.


Smith, Matthew J

Hon Discrete Syllabus Spring 2016 :This syllabus contains most necessary information regarding the curriculum, policies, and expectations of the class. Included is a list of topics covered, contact information for Mr. Smith, classroom expectations, tutoring hours, make up policies, and discipline policy.

Hon Discrete: Parent/Guardian Letter :This letter provides an introduction of Mr. Smith to parents and guardians, an overview of methods for contacting Mr. Smith and Mount Tabor regarding your student, information regarding this website, a supply list and wish list of the class, and an explanation of weekly email contact you can expect.

Parent/Guardian Contact Information Form :This contact form provides Mr. Smith with detailed contact information to help increase communication and support for your student to be successful this year.

Student Information Sheet : This survey gives students an opportunity to introduce themselves to Mr. Smith. This is the first opportunity for students to let Mr. Smith know about any specific concerns or challenges that they may have toward being successful this year.


Tumor Markers

A tumor marker is anything present in or produced by cancer cells or other cells of the body in response to cancer or certain benign (noncancerous) conditions that provides information about a cancer, such as how aggressive it is, what kind of treatment it may respond to, or whether it is responding to treatment.

Tumor markers have traditionally been proteins or other substances that are made at higher amounts by cancer cells than normal cells. These can be found in the blood, urine, stool, tumors, or other tissues or bodily fluids of some patients with cancer. Increasingly, however, genomic markers (such as tumor gene mutations, patterns of tumor gene expression, and nongenetic changes in tumor DNA) that are found in tumors themselves and in tumor fragments shed into bodily fluids are being used.

Many different tumor markers have been characterized and are in clinical use. Some are associated with only one type of cancer, whereas others are associated with multiple different cancer types

How are tumor markers used in cancer care?

There are two main types of tumor markers: circulating tumor markers and tumor tissue markers.

Circulating tumor markers can be found in the blood, urine, stool, or other bodily fluids of some patients with cancer. Circulating tumor markers are used to:

  • estimate prognosis
  • determine the stage of cancer
  • detect cancer that remains after treatment (residual disease) or that has returned after treatment
  • assess how well a treatment is working
  • monitor whether the treatment has stopped working

Although an elevated level of a circulating tumor marker may suggest the presence of cancer and can sometimes help to diagnose cancer, this alone is not enough to diagnose cancer. For example, noncancerous conditions can sometimes cause the levels of certain tumor markers to increase. In addition, not everyone with a particular type of cancer will have a higher level of a tumor marker associated with that cancer. Therefore, measurements of circulating tumor markers are usually combined with the results of other tests, such as biopsies or imaging, to diagnose cancer.

Tumor markers may also be measured periodically during cancer therapy. Such “serial measurements,” which show how the level of a marker is changing over time, are usually more meaningful than a single measurement. For example, a decrease in the level of a circulating tumor marker may indicate that the cancer is responding to treatment, whereas an increasing or unchanged level may indicate that the cancer is not responding.

Circulating tumor markers may also be measured periodically after treatment has ended to check for recurrence (the return of cancer).

Examples of commonly used circulating tumor markers include calcitonin (measured in blood), which is used to assess treatment response, screen for recurrence, and estimate prognosis in medullary thyroid cancer CA-125 (measured in blood), to monitor how well cancer treatments are working and if cancer has come back in ovarian cancer and beta-2-microglobulin (measured in blood, urine, or cerebrospinal fluid), to estimate prognosis and follow response to treatment for multiple myeloma, chronic lymphocytic leukemia, and some lymphomas. Additional circulating tumor markers are described in the list of tumor markers in common use.

Tumor tissue (or cell) markersare found in the actual tumors themselves, typically in a sample of the tumor that is removed during a biopsy. Tumor tissue markers are used to:

  • diagnose, stage, and/or classify cancer
  • estimate prognosis
  • select an appropriate treatment (e.g., treatment with a targeted therapy)

Tumor tissue markers that indicate whether someone is a candidate for a particular targeted therapy are sometimes referred to as biomarkers for cancer treatment. Tests for these biomarkers are usually genetic tests that look for changes in genes that affect cancer growth. More information is available on the Biomarker Testing for Cancer Treatment page and in the Targeted Cancer Therapies fact sheet.

Examples of tumor tissue markers that are used as biomarkers for cancer treatment include estrogen receptor and progesterone receptor, which are tested for to determine whether someone with breast cancer should get treatment with hormone therapy FGFR3 gene mutation analysis, to help determine treatment for patients with bladder cancer and PD-L1, to see if people with any of a number of cancer types are candidates for treatment with an immune checkpoint inhibitor.

Because some tumors shed cells and genetic material into blood, it is sometimes possible to examine biomarkers in blood samples. Although these "liquid biopsies" are not yet routinely used, they have several potential advantages. Because they don’t involve surgery, they can be done more frequently than standard biopsies. They can also be performed when surgical biopsies cannot, such as when tumors are difficult to reach or patients can’t tolerate surgery.

Liquid biopsy tests can often detect multiple cancer-associated biomarkers. For example, the Foundation One Liquid CDx test is approved for the detection of genetic mutations in 324 genes and two genomic signatures in any solid tumor type. The test can also identify which patients with non-small cell lung cancer, melanoma, breast cancer, colorectal cancer, or ovarian cancer may benefit from 15 different FDA-approved targeted treatment options.

Does NCI have guidelines for the use of tumor markers?

NCI does not have guidelines for the use of tumor markers. However, some national and international organizations have guidelines for the use of tumor markers for some types of cancer:

  • The American Society of Clinical Oncology (ASCO) has developed and published clinical practice guidelines on a variety of topics, including tumor markers for breast cancer, colorectal cancer, lung cancer, and others.
  • The National Academy of Clinical Biochemistry publishes laboratory medicine practice guidelines, including Use of Tumor Markers in Clinical Practice: Quality Requirements , which focuses on the appropriate use of tumor markers for specific cancers.

Because tumor markers are detected using biospecimens (tissue and blood samples), the National Cancer Institute publishes best practices for biospecimen collection, processing, and storage.

What tumor markers are currently being used, and for which cancer types?

A number of tumor markers are currently being used for a wide range of cancer types. See the list of tumor markers in common use for more information. Although most of these can be tested in laboratories that meet standards set by the Clinical Laboratory Improvement Amendments , some cannot be and may therefore be considered experimental.

Can tumor markers be used in cancer screening?

Because tumor markers can be used to predict the response of a tumor to treatment and for prognosis, researchers have hoped that they might also be useful in screening tests that aim to detect cancer early, before there are any symptoms.

However, studies to see whether circulating tumor markers can be used to screen for cancer have generally found that these markers don’t identify everyone with the disease (they are not sensitive enough) or that they indicate the possible presence of cancer in people who don’t have it (they are not specific enough). When a test has low specificity, people have to have further testing to determine whether cancer is present. And some screening tests based on tumor markers have been shown to lead to overdiagnosis, which happens when people are diagnosed with cancers that would never have affected them during their lifetimes.

For example, the prostate-specific antigen (PSA) test was used routinely in the past to screen men for prostate cancer. However, as more was learned about the limitations of the test (including relatively low specificity), medical groups began to recommend against using it for routine population screening.

Several liquid biopsy–based assays that test for multiple tumor markers to detect cancer early, in people without symptoms, are in development:

  • PapSEEK identifies ovarian and endometrial cancer–related alterations in DNA obtained from fluids collected during a routine Pap test. In a study that included women already diagnosed with cancer, the test was able to detect some endometrial and ovarian cancers at early, more treatable stages.
  • CancerSEEK is a blood test that detects DNA mutations and protein biomarkers linked to multiple types of cancer. In a large trial of women with no history of cancer that combined the blood test with whole-body PET imaging, 65% of cancers that were detected were at an early stage.
  • UroSEEK is a urine-based test that detects the most common alterations in 11 genes linked to bladder and upper tract urothelial cancers. In a study that included patients not yet diagnosed with bladder cancer but at high risk of the disease because of symptoms, UroSEEK identified 83% of those who developed bladder cancer.

Although these tests are able to detect early cancers, it is not yet known whether treating those cancers would actually reduce deaths from these cancers.

What research is under way to develop additional tumor markers?

NCI’s Early Detection Research Network (EDRN), a collaborative consortium of academic and private-sector investigators, has focused on the systematic discovery, development, and validation of biomarkers and imaging methods to detect early-stage cancers and to assess risk for developing cancer. One goal of EDRN is to develop biomarkers that can distinguish aggressive early-stage cancers from slow-growing cancers that would never cause symptoms to reduce overtreatment.

Cancer researchers are turning to proteomics (the study of protein structure, function, and patterns of expression) and proteogenomics (the integration of proteomics with genomics and gene expression analysis, or transcriptomics) with the hope of developing novel biomarkers that can be used to identify cancer in its early stages, to predict the effectiveness of treatment, and to predict the chance of cancer recurrence.

NCI’s Clinical Proteomic Tumor Analysis Consortium (CPTAC) is using a proteogenomic approach for tumor marker discovery for a growing number of cancers, including colorectal, breast, and ovarian cancers. By systematically identifying proteins (and associated biological processes) that originate from alterations in cancer genomes, CPTAC researchers have discovered new tumor subtypes, tumor microenvironment variations, and new potential proteins for targeted drug therapy. Recent innovations have suggested that these analyses could be done on a microscale using very small amounts of tumor tissue obtain from a biopsy.

NCI’s Molecular Applications for Therapy Choice (NCI-MATCH) and NCI-COG Pediatric MATCH clinical trials are using a precision medicine approach to assign patients to treatment by gene mutations in their tumors rather than by the type of cancer they have. By analyzing the response of patients to these targeted agents, and the underlying genomic alterations associated with these responses, researchers are identifying potentially new molecular targets for cancer therapy. Companion studies associated with these trials will also allow researchers to identify new biomarkers for determining response to therapies and for predicting treatment resistance.

The NCI Cancer Moonshot SM Biobank is working with patient participants at community hospitals around the country to encourage them to donate tissue and blood samples over the course of their cancer treatment. The samples are sent to researchers who use them to better understand cancer and potentially identify tumor markers.

More information on NCI’s role in supporting research on novel tools and methods for diagnosing cancer is available on the Cancer Diagnosis Research page.

Selected References

Bettegowda C, Sausen M, Leary RJ, et al. Detection of circulating tumor DNA in early- and late-stage human malignancies. Science Translational Medicine 2014 6(224):224ra24.

Duffy MJ. Clinical uses of tumor markers: a critical review. Critical Reviews in Clinical Laboratory Sciences 2001 38(3):225–262.

Duffy MJ. Tumor markers in clinical practice: A review focusing on common solid cancers. Medical Principles and Practice 2013 22(1):4–11.

Lennon AM, Buchanan AH, Kinde I, et al. Feasibility of blood testing combined with PET-CT to screen for cancer and guide intervention. Science 2020 369(6499):eabb9601.

Springer SU, Chen CH, Rodriguez Pena MDC, et al. Non-invasive detection of urothelial cancer through the analysis of driver gene mutations and aneuploidy. Elife 2018 7:e32143.

Wang Y, Li L, Douville C, et al. Evaluation of liquid from the Papanicolaou test and other liquid biopsies for the detection of endometrial and ovarian cancers. Science Translational Medicine 2018 10(433):eaap8793.


Watch the video: Survey: Pairwise Comparison Method of Voting (December 2021).