Fair Division

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Fair Division

• 1. Fair Division Problem A problem that involves
  the dividing up of an object or set of objects
  among several individuals (players) so that each
  individual considers the part he or she receives
  to be a fair portion.
• a. Adjusted Winner Procedure (2 person)
• b. Knaster Inheritance Procedure (n3 or more
  players, 1 asset)
• c. Taking Turns (2 players, 1 player selects)
• d. Cake Division Problem-Proportionality     

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Adjusted Winner Procedure (2 person)

• i. Give each party 100 points to independently
  and simultaneously distribute over the items in
  dispute in a way that reflects how much they
  value each item.
• ii. Tabulate the data
• iii. Award each item to the party that weighted
  it more heavily. (Ties are not awarded at this
  time)
• iv. Sum points for each party. Let party A be
  the party with the smallest point total and
  party B be the party with the largest point
  total.

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Adjusted Winner Procedure (2 person)

• v.  Distribute ties to party A unless by doing
  so it makes As point total larger than point
  Bs point total. In that case, award the tie to
  B.
• vi.  Now all items have been distributed and B
  still has the largest point total. We must begin
  to transfer items from B to A as follows
• 1.  For each item that now belongs to B,
  calculate the quotient
• 2. Arrange the quotients in
  increasing order.

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Adjusted Winner Procedure (2 person)

• 3. beginning with the smallest quotient, compute
  the portion of that item that is transferred to A
  as follows
•        A. solve the equation formed below for
  x point total
  of A (As point value of item)x
  point total of B (Bs
  point value of item)x
•    B. x represents the portion of the asset
  that is transferred to A, while 1-x represents
  the portion of the asset that is left to B.
• C. if the sum totals of A and B are now equal,
  then transfer is complete, otherwise, continue
  with the transfer using the item with the 2nd
  smallest quotient.

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Adjusted Winner Procedure (2 person)

• Ties go to loser unless losers total becomes gt
  winner, then give to winner. Get new total.

• 2. Compute winner quotients list in
• increasing order 15/15,50/25,6/3,10/2.
• Begin with item of smallest quotient.
• Solve equation
• 81-15x 55 15x (-55 to both sides)
• 26-15x 15x (15x to both sides)
• 26 30x (divide both sides by 30)
• 26/30 x goes to loser 4/30 to winner
• Check work
• Mike 66(15)( 4/30) 68
• Phil 55(15)(26/30) 68

New Total 81 55
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Properties of Adjusted Winner Procedure

•  
• 1. Equitable both parties receive the same
  amount.
• 2. Envy-free neither party would be happier
  with what the other party received.
• 3. Pareto-optimal no other allocation of items
  by any other means can make one party better
  off without making the other party worse off.

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   Knaster Inheritance Procedure (n3 or more
players, 1 asset)

• i.    Each player independently places a value on
  the asset.
• ii.  Asset is awarded to the highest bidder, say
  player A.
• iii.  Now subtract from A an amount equal to
  (value A placed on the asset).
  
• iv. Place this amount in a kitty.

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Knaster Inheritance Procedure

• v.  Add to each of the remaining players an
  amount (value that player placed on the
  asset)/n and subtract that amount from the
  kitty.
• vi.  Add to each player (including A) an amount
  equal to (remainder in kitty)/n
• vii.  For more than one asset, repeat this
  procedure for each asset.
• viii. Drawback players need to have substantial
  front money available.

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Knaster Inheritance Procedure

• Award item to highest bidder subtract from that
  player an amt
• (her amt)(n-1)/n
• Place that amt into kitty.
• Do 1-5 for each item.
• Add to each remaining
• player an amt (players amt)/n
• subtract that amt from
• kitty.
• 5. Award to each player an amt Remainder in
  kitty/n.

Kitty 1 (3)(90,000)/4 67,500 80,000/4
-20,000 75,000/4 -18,750 60,000/4
-15,000 13,750 13,750/4
3,437.50
Kitty 2 (3)(15,000)/4 11,250 10,000/4 -
2,500 12,000/4 - 3,000 13,000/4 -
3,250
2,500 2,500/4 625
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Taking Turns 2 players

• i.  One party selects an item then other party
  selects an item. This procedure is continued
  until all items have been distributed.
• ii.  Bottom Up Strategy how to get the best
  advantage.
• 1. Players are aware of each others preference
  lists for the items in question.
• 2. Assume A selects 1st and B selects 2nd. Fill
  in the following chart beginning from the right
  (assume 5 items)
• A Bs last Bs 2nd
  last Bs last
• B As 2nd last
  As last

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Taking Turns 2 players
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Cake Division Problem - Proportionality

• Origins go back 5,000 years. Modern era of fair
  division in math began in Poland during WWII.
• Proportional a scheme is proportional if each
  players strategy guarantees him a piece of size
  at least 1/n
• Envy-free a scheme is envy-free if each players
  strategy guarantees him a piece he considers tied
  for the largest.

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Cake Division Problem

• Divide and Choose (2 players).
• Steinhaus Proportional Procedure or Lone Divider
  Scheme (3 players).
• Banach-Knaster Procedure or Last Diminisher
  Scheme (4 or more players).
• Selfridge-Conway Envy-free Procedure (3 players).
• 1992 Envy-free Procedure (4 or more players).

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Cake Division Problem Divide Choose

• Player A divides cake into 2 parts in any way he
  or she desires.
• Player B chooses the piece he or she wants.
• Envy-free scheme.

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Cake Division Steinhaus Proportional Procedure

• Player A divided cake into what he thinks
  represents 3 equal pieces, X, Y, and Z.
• If players B and C approve of a different piece
  then they receive the pieces they approved of and
  player A gets the remaining piece.
• If players B and C approve of the same piece X
  and disapprove of the same piece Z then player A
  gets piece Z and pieces X and Y are put together
  for B and C to split by the divide and choose
  scheme.

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Cake Division Steinhaus Proportional Procedure
A gets leftover
B approves
X
Y
Z
C approves
X Y put together for B C to divide
choose
B C approve of X
X
Y
Y
Z
B C disapprove of Z Then A gets Z
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Credits

• COMAP, For All Practical Purposes, 5th ed