I need a high quality tutor who can write in proper, native, U.S. English (no broken English, please) and is very familiar with Lucas’ Method of Markers to provide 100% solutions to Problems 1, 2, and 3. Problems 1-3 are related and they all ask for a specific proof. This tutor must be able to provide a 100% correct proof for each problem in proper, native U.S. English. I also need one or more links to a website that verifies that the tutor’s proofs are correct. I want to be able to verify for myself that the solutions to these 3 problems are correct.I have attached the questions to Problems 1 – 3, a PowerPoint on Lucas’ Methods of Markers, and the proof to Lucas’ Methods of Markers. You can use these resources to help answer Problems 1 – 3.Very Important: Please DO NOT BID if you are not familiar with Lucas’ Method of Markers, since it is necessary to provide solutions to problems 1 – 3. Please DO NOT BID if English is not your primary language. These proofs must be written in native U.S. English.Please DO NOT BID if you are not 100% confident that you can provide full, 100% correct solutions to Problems 1 – 3. It is VERY IMPORTANT that the solutions are correct.

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The Method of Markers: Overview

The Method of Markers is a fair-division method for a

multiplayer game with discrete goods (e.g., Halloween candy).

In comparison to the Method of Sealed Bids:

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Everyone gets at least (roughly) a fair share, provided

they bid honestly.

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Disadvantage: Not suitable if the goods have widely

varying values (e.g., an estate)

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Advantage: Doesnt require players to put in cash

The Method of Markers: Example

In Example 3.11 on pp. 98100 of Tannenbaum, four kids

(Alice, Bianca, Carla, Dana) have a pile of Halloween candy

that they need to share fairly.

The Method of Markers: Example

In Example 3.11 on pp. 98100 of Tannenbaum, four kids

(Alice, Bianca, Carla, Dana) have a pile of Halloween candy

that they need to share fairly.

Step 0: Arranging. Line up the booty to be divided in a

single row.

The Method of Markers: Example

In Example 3.11 on pp. 98100 of Tannenbaum, four kids

(Alice, Bianca, Carla, Dana) have a pile of Halloween candy

that they need to share fairly.

Step 0: Arranging. Line up the booty to be divided in a

single row.

The Method of Markers: Step 1

Step 1: Bidding.

The Method of Markers: Step 1

Step 1: Bidding.

Alice places three markers A1 , A2 , A3 , dividing the row into

four segments, each of which she thinks is a fair share.

The Method of Markers: Step 1

The Method of Markers: Step 1

The Method of Markers: Step 1

That is, Alice would consider any one of these shares to be fair:

{1, 2, 3, 4, 5}

{6, 7, 8, 9, 10, 11}

{12, 13, 14, 15, 16}

{17, 18, 19, 20}

(Alices 1st segment)

(2nd segment)

(3rd segment)

(4th segment)

The Method of Markers: Step 1

Similarly, Bianca places markers B1 , B2 , B3 ; Carla places

markers C1 , C2 , C3 ; Dana places markers D1 , D2 , D3 .

The Method of Markers: Step 1

Similarly, Bianca places markers B1 , B2 , B3 ; Carla places

markers C1 , C2 , C3 ; Dana places markers D1 , D2 , D3 .

The Method of Markers: Step 1

Similarly, Bianca places markers B1 , B2 , B3 ; Carla places

markers C1 , C2 , C3 ; Dana places markers D1 , D2 , D3 .

The Method of Markers: Step 1

Similarly, Bianca places markers B1 , B2 , B3 ; Carla places

markers C1 , C2 , C3 ; Dana places markers D1 , D2 , D3 .

The Method of Markers: Step 1

Similarly, Bianca places markers B1 , B2 , B3 ; Carla places

markers C1 , C2 , C3 ; Dana places markers D1 , D2 , D3 .

In order to keep the method fair, the players must all place

their markers at the same time. (For example, they can

submit sealed envelopes with the positions of their markers.)

The Method of Markers: Step 1

There are a total of 12 markers (3 for each of 4 players).

Now what?

The Method of Markers: Step 2

Step 2: Allocations.

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.1: Locate the leftmost 1st marker (A1 , B1 , C1 , or D1 ).

(If there is a tie, choose one randomly.)

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.1: Locate the leftmost 1st marker (A1 , B1 , C1 , or D1 ).

(If there is a tie, choose one randomly.)

The corresponding player (here, Bianca) gets her 1st segment.

In this case, Bianca gets 1 2 3 4

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.1: Locate the leftmost 1st marker (A1 , B1 , C1 , or D1 ).

(If there is a tie, choose one randomly.)

The corresponding player (here, Bianca) gets her 1st segment.

In this case, Bianca gets 1 2 3 4

Then, remove all Biancas markers.

The Method of Markers: Step 2

Biancas share

The Method of Markers: Step 2

Removing Biancas markers

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.2: Locate the leftmost second marker (A2 , C2 , or D2 ).

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.2: Locate the leftmost second marker (A2 , C2 , or D2 ).

The corresponding player (here, Carla) gets her 2nd segment.

Carlas share: 7 8 9

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.2: Locate the leftmost second marker (A2 , C2 , or D2 ).

The corresponding player (here, Carla) gets her 2nd segment.

Carlas share: 7 8 9

Then, remove all Carlas markers.

The Method of Markers: Step 2

Carlas share (focusing on 1st and 2nd markers)

The Method of Markers: Step 2

Removing Carlas markers

The Method of Markers: Step 2

Note that 5 and 6 have not been allocated to anyone yet.

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.3: Locate the leftmost third marker (A3 or D3 ).

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.3: Locate the leftmost third marker (A3 or D3 ).

This time, there is a tie; lets say a coin toss chooses Alice

rather than Dana.

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.3: Locate the leftmost third marker (A3 or D3 ).

This time, there is a tie; lets say a coin toss chooses Alice

rather than Dana.

Alice gets her 3rd segment.

Alices share: 12 13 14 15 16

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.3: Locate the leftmost third marker (A3 or D3 ).

This time, there is a tie; lets say a coin toss chooses Alice

rather than Dana.

Alice gets her 3rd segment.

Alices share: 12 13 14 15 16

Then, remove all Alices markers.

The Method of Markers: Step 2

The Method of Markers: Step 2

Step 2: Allocations.

Step 2.4: The last player left gets her last segment.

Danas share: 17 18 19 20

The Method of Markers: Step 2

The Method of Markers: Step 3

At this point, everyone has been given a fair share, and

usually, there is candy left over!

Step 3: Divide the surplus. (It doesnt matter how for

instance, players get to pick one item at a time in a random

order.)

The Method of Markers: Review

Step 0: Arrange the booty to be divided in a row.

Step 1: Bidding. Each of the N players bids by placing

N – 1 markers to separate the booty into N fair shares (in

that playerss opinion).

(Preserve the Privacy Assumption by having players reveal

their bids simultaneously.)

The Method of Markers: Review

Step 2: Allocation.

I

Locate the leftmost 1st marker.

Give that player his 1st segment (i.e., from the left end to

his 1st marker). Then remove all his markers.

The Method of Markers: Review

Step 2: Allocation.

I

Locate the leftmost 1st marker.

Give that player his 1st segment (i.e., from the left end to

his 1st marker). Then remove all his markers.

I

Locate the leftmost 2nd marker.

Give that player her 2nd segment (i.e., between her 1st

and 2nd markers). Then remove all her markers.

The Method of Markers: Review

Step 2: Allocation.

I

Locate the leftmost 1st marker.

Give that player his 1st segment (i.e., from the left end to

his 1st marker). Then remove all his markers.

I

Locate the leftmost 2nd marker.

Give that player her 2nd segment (i.e., between her 1st

and 2nd markers). Then remove all her markers.

I

Locate the leftmost 3rd marker.

Give that player her 3rd segment (i.e., between his 2nd

and 3rd markers). Then remove all his markers.

…

The Method of Markers: Review

Step 2: Allocation.

I

Eventually, locate the leftmost (N – 1)st marker.

Give that player her (N – 1)st segment

(i.e., between her (N – 2)nd and (N – 1)st markers).

Then remove all her markers.

I

The last player gets his last segment

(i.e., from his (N – 1)st marker to the end of the row).

The Method of Markers: Review

Step 3: Division of Surplus.

If there are items left over, divide them by, e.g., taking turns

choosing one

(or, if there are a lot of items, use the Method of Markers all

over again!)

The Method of Markers: Review

I

Everyone gets at least a fair share: the method is set up

so that the segments allocated never overlap.

The Method of Markers: Review

I

Everyone gets at least a fair share: the method is set up

so that the segments allocated never overlap.

I

Unlike the Method of Sealed Bids, no cash (or

arithmetic!) is required.

The Method of Markers: Review

I

Everyone gets at least a fair share: the method is set up

so that the segments allocated never overlap.

I

Unlike the Method of Sealed Bids, no cash (or

arithmetic!) is required.

I

The players have to be able to divide the booty into

roughly equal shares.

The Method of Markers: Review

I

Everyone gets at least a fair share: the method is set up

so that the segments allocated never overlap.

I

Unlike the Method of Sealed Bids, no cash (or

arithmetic!) is required.

I

The players have to be able to divide the booty into

roughly equal shares.

I

The method works best if the goods are roughly

equivalent in value to each other, and if the players

preferences are fairly close.

(In the example above, what if one of the players is

allergic to peanuts?)

And now for something completely different.

The Seven Bridges of Ko¨nigsberg

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In 1735, the city of Ko¨nigsberg (present-day Kaliningrad)

was divided into four districts by the Pregel River.

I

The four districts were connected by seven bridges.

The Seven Bridges of Ko¨nigsberg

The Seven Bridges of Ko¨nigsberg

The Seven Bridges of Ko¨nigsberg

Is it possible to take a walking tour of Ko¨nigsberg in which

you cross each of the seven bridges exactly once?

…

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