Discrete Math
3.1
Fair Division: To divide S into shares (one for
each player) in such a way that each player
gets a fair share.
 Fair Share: Given a share s of S and a player P,
we will say that s, in the opinion of P, is worth
at least (1/N)th of the total value of S. (N) is
the number of players.
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To be continued…
Discrete Math
3.1 (Continued...)
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Continuous case: the set S is divisible in
infinitely many ways. (Physically)
Discrete Case: Set S is made up of objects that
are indivisible like houses/cars.
Discrete Math
3.2

Divider Chooser Method: Made for two players,
one player divides the object into two pieces,
and the second player (chooser) picks the piece
he/she wants, leaving the other piece to the
divider.
Discrete Math
3.3

The Lone Divider Method: Steinhaus’ method
for N=3 players.
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Step 1:Division: The divider (chosen at random)
divides the cake into three pieces.
Step 2: Bidding: Player2 declares anonymously which
of the three pieces are fair. Player3 does the same.
Step 3: Distribution:
Case 1: Distribute each player a piece they think is fair
if possible. They may swap at the end if it makes them
happier.
 Case 2: If both P2 and P3 want the same piece, give P1
(divider) one of the unwanted pieces and then place the
other two pieces back together in the divider-chooser
method.

Discrete Math
3.3

The Lone Divider for more than three players:
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Case 1: Same Steps and distribute a fair share to each
player.
Case 2: A Standoff occurs: We first set aside the
shares and players involved in the standoff. The
remaining players can be assigned a fair share. The
method is repeated for those players in the standoff
by recombining the standoff shares.
To be continued…
Discrete Math
3.4 The Lone Chooser Method (Three
Players): One Chooser and Two Dividers:
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Step 1: Division: P1 and P2 divide S between
themselves into two fair shares (divider-chooser
method)
Step 2: Subdivision: P1 and P2 divide their shares
into N subshares.
Step 3: Selection: The chooser [C] now selects one
share from P1 and P2. These two subshares make up
C’s final share.
To be continued…
Discrete Math
3.4 (Continued...)
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The Lone Chooser for N players:
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Step 1: P1… PN-1 divide fairly the set S among
themselves as if C does not exist.
Step2: Each divider subdivides his or her share into N
subshares.
Step 3: The chooser [C] finally gets to choose one
subshare from each divider to get their fair share.
Discrete Math
3.5

The Last Diminisher (For any number of
players)
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Round 1: P1 starts by cutting what they believe to be
exact (1/N) piece of S. P2 now has the opportunity to
pass the piece or diminish it by trimming it to what
they think it is (1/N)th of S. This process is continued
by passing to each player. The player to last trim the
piece will get the piece and be removed from the
game.
Round 2: Repeat the process. If p1 is out (no one
trimmed), then p2 resumes the cutting. Trimmings
are added back to the remains.
The final two players will use the divider-chooser
method.
Discrete Math
3.6

Method of Sealed Bids: Estate division.
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Step 1: Bidding: Each player makes a secret bid on
each item in the estate.
Step 2: Allocation: Each item goes to the highest
bidder.
Step 3: Payments: Calculate fair share (% of total
bids). Subtract the money value of allocated items to
get remaining claims.
Step 4: Dividing the Surplus: Add all remaining claims
to acquire the money left over and divide this among
players (%). Each player is then given their final
settlement (list items won and money received or
paid).
To be continued…
Discrete Math
3.6 (Continued...)
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My Seven Steps:
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1: Sum of Bids- add bids
2: Fair Share - % of them
3: Allocator - value of items awarded
4: Remaining Claims- (Fairshare - Allocations)
5: Total Surplus: sum of all remaining claims (make
positive)
6: Share of Surplus: % of total surplus.
7: Final Settlement: (Remaining claims and share of
surplus) & List items Awarded.
Discrete Math
3.7

Method of Markers: Does not require players to
put up any of their own money. Can’t be used
effectively unless there are many more items
to be divided than there are players, and items
are close in value.

Step 1: Bidding: Line up items in an array. Each player
will privately cut up the array string into N segments
that they can see as an acceptable share.
To be continued…
Discrete Math
3.7 (Continued...)
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Step 2: Allocation: Scanning the array from left to
right, the first segment is given to the player whose
marker is found first. These markers are now
removed and the next segment goes to the player
whose second marker is found first (only the segment
between their two markers). This continues until each
player has a segment. In case of a tie, break it
randomly.
Step 3: Dividing the leftovers: Randomly draw lots
and let the players go in order picking one piece at a
time or if enough pieces are left, do the method
again.
Discrete Math