math 180 - fair divisionchhays/lecture29.pdf · fair division i fair division problems are...
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![Page 1: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/1.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 2: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/2.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 3: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/3.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)
I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 4: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/4.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 5: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/5.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in Tokyo
I House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 6: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/6.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, Kansas
I SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 7: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/7.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI Sailboat
I Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 8: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/8.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 9: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/9.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 10: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/10.jpg)
Fair Division
I Fair division problems are concerned with fairly dividinggoods among parties
I We will consider a division to be fair if every party gets ≥ 1n of
the goods (according to them)
I Today, considering indivisible goods (of varying values)I Albert, Beatrice, Clyde, and Diane are heirs to an estate:
I House in TokyoI House in Manhattan, KansasI SailboatI Painting
I Note that previous methods don’t work well
I Instead, have heirs bid on items
![Page 11: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/11.jpg)
Method of Sealed Bids
1. Bidding: each player determines how much each item isworth, then submits a bid on each item
I Bids are hidden until everyone submits their bids
2. Allocation: each item goes to the highest bidder
3. First Settlement: each player then pays or receives money tomake sure they have their fair share
4. Division of Surplus: remaining money is divided equallyamong the parties
![Page 12: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/12.jpg)
Method of Sealed Bids
1. Bidding: each player determines how much each item isworth, then submits a bid on each item
I Bids are hidden until everyone submits their bids
2. Allocation: each item goes to the highest bidder
3. First Settlement: each player then pays or receives money tomake sure they have their fair share
4. Division of Surplus: remaining money is divided equallyamong the parties
![Page 13: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/13.jpg)
Method of Sealed Bids
1. Bidding: each player determines how much each item isworth, then submits a bid on each item
I Bids are hidden until everyone submits their bids
2. Allocation: each item goes to the highest bidder
3. First Settlement: each player then pays or receives money tomake sure they have their fair share
4. Division of Surplus: remaining money is divided equallyamong the parties
![Page 14: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/14.jpg)
Method of Sealed Bids
1. Bidding: each player determines how much each item isworth, then submits a bid on each item
I Bids are hidden until everyone submits their bids
2. Allocation: each item goes to the highest bidder
3. First Settlement: each player then pays or receives money tomake sure they have their fair share
4. Division of Surplus: remaining money is divided equallyamong the parties
![Page 15: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/15.jpg)
Method of Sealed Bids
1. Bidding: each player determines how much each item isworth, then submits a bid on each item
I Bids are hidden until everyone submits their bids
2. Allocation: each item goes to the highest bidder
3. First Settlement: each player then pays or receives money tomake sure they have their fair share
4. Division of Surplus: remaining money is divided equallyamong the parties
![Page 16: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/16.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
TokyoManhattan
BoatPainting
![Page 17: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/17.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Note: We know how each party values the entire inheritance
![Page 18: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/18.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Note: We know how each party values the entire inheritance
![Page 19: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/19.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510
Note: We know how each party values the entire inheritance
Note: We know what constitutes a fair share for each person
![Page 20: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/20.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510
Note: We know how each party values the entire inheritanceNote: We know what constitutes a fair share for each person
![Page 21: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/21.jpg)
Method of Sealed Bids
Bidding: each player determines how much each item is worth,then submits a bid on each item (in 1000s)
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5
Note: We know how each party values the entire inheritanceNote: We know what constitutes a fair share for each person
![Page 22: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/22.jpg)
Method of Sealed Bids
Allocation: each item goes to the highest bidder
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5
![Page 23: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/23.jpg)
Method of Sealed Bids
Allocation: each item goes to the highest bidder
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5Item Value 50 520 60 0
![Page 24: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/24.jpg)
Method of Sealed BidsFirst Settlement: each player then pays or receives money tomake sure they have their fair share
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5Item Value 50 520 60 0Settlement
![Page 25: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/25.jpg)
Method of Sealed BidsFirst Settlement: each player then pays or receives money tomake sure they have their fair share
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5Item Value 50 520 60 0Settlement 71.5 -370 56 127.5
![Page 26: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/26.jpg)
Method of Sealed Bids
I Summary:
I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 27: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/27.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500
I Beatrice gets the Tokyo house, the painting, and has to paythe estate $370, 000
I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 28: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/28.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000
I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 29: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/29.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000
I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 30: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/30.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 31: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/31.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?
I No - there is $115, 000 in the estateI Split this equally
![Page 32: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/32.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?I No - there is $115, 000 in the estate
I Split this equally
![Page 33: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/33.jpg)
Method of Sealed Bids
I Summary:I Albert gets the boat, and receives $71,500I Beatrice gets the Tokyo house, the painting, and has to pay
the estate $370, 000I Clyde gets the Manhattan house, and receives $56, 000I Diane receives $127, 500
I Are we done?I No - there is $115, 000 in the estateI Split this equally
![Page 34: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/34.jpg)
Method of Sealed BidsDivision of Surplus: remaining money is divided equally amongthe parties
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5Item Value 50 520 60 0Settlement 71.5 -370 56 127.5
Surplus 28.75 28.75 28.75 28.75
![Page 35: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/35.jpg)
Method of Sealed BidsDivision of Surplus: remaining money is divided equally amongthe parties
Albert Beatrice Clyde Diane
Tokyo 400 500 360 420Manhattan 30 50 60 40
Boat 50 30 40 40Painting 6 20 4 10
Total 486 600 464 510Fair Share 121.5 150 116 127.5Item Value 50 520 60 0Settlement 71.5 -370 56 127.5
Surplus 28.75 28.75 28.75 28.75
Final Settlement 100.25 -341.25 84.75 156.25
![Page 36: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/36.jpg)
Method of Sealed Bids
I Do the bids need to be secret?
I Auction theory says expected values would equal (with openbids)
I If bids were open, would want to just undercut highest bidder:
![Page 37: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/37.jpg)
Method of Sealed Bids
I Do the bids need to be secret?I Auction theory says expected values would equal (with open
bids)
I If bids were open, would want to just undercut highest bidder:
![Page 38: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/38.jpg)
Method of Sealed Bids
I Do the bids need to be secret?I Auction theory says expected values would equal (with open
bids)I If bids were open, would want to just undercut highest bidder:
![Page 39: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/39.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626
![Page 40: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/40.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5
![Page 41: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/41.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5Item Value 50 520 60 0Settlement
![Page 42: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/42.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5Item Value 50 520 60 0Settlement 106.75 -363 96.75 156.5
![Page 43: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/43.jpg)
Open BiddingSuppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5Item Value 50 520 60 0Settlement 106.75 -363 96.75 156.75
Surplus .75 .75 .75 .75
![Page 44: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/44.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5Item Value 50 520 60 0Settlement 106.75 -363 96.75 156.75
Surplus .75 .75 .75 .75
Final Settlement 107.5 -362.25 97.5 157.5
Open bidding encourages false bids (can get more $$)
![Page 45: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/45.jpg)
Open Bidding
Suppose bids were open:
Albert Beatrice Clyde Diane
Tokyo 499 500 499 499Manhattan 59 59 60 59
Boat 50 49 49 49Painting 19 20 19 19
Total 627 628 627 626Fair Share 156.75 157 156.75 156.5Item Value 50 520 60 0Settlement 106.75 -363 96.75 156.75
Surplus .75 .75 .75 .75
Final Settlement 107.5 -362.25 97.5 157.5
Open bidding encourages false bids (can get more $$)
![Page 46: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/46.jpg)
Method of Sealed Bids
I Perks:
I Parties receive more than their fair share (by $28, 750 in theexample
I Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 47: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/47.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
example
I Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 48: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/48.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying values
I Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 49: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/49.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 50: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/50.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreases
I If you overbid, you’ll pay a lot of money for something youdon’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 51: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/51.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 52: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/52.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?
I Requires players to have enough cash to cover bids
![Page 53: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/53.jpg)
Method of Sealed Bids
I Perks:I Parties receive more than their fair share (by $28, 750 in the
exampleI Works well when goods consist of a few items of varying valuesI Encourages honesty
I If you underbid, your “fair share” decreasesI If you overbid, you’ll pay a lot of money for something you
don’t want
I Problems?I Requires players to have enough cash to cover bids
![Page 54: Math 180 - Fair Divisionchhays/lecture29.pdf · Fair Division I Fair division problems are concerned with fairly dividing goods among parties I We will consider a division to be fair](https://reader033.vdocument.in/reader033/viewer/2022050323/5f7ccc378ccd537b2318e736/html5/thumbnails/54.jpg)
Method of Sealed Bids
I See Handout #7