In this example, it will be shown how some embodiments of this invention use process 400 as

explained in FIG. 4 to find the spoilers in a given set of Weighted Preference Lists in step 400 in

process 200 FIG. 2a.

Let’s assume, in a hypothetical presidential election with three candidates; Bill, George and Ross

the preferences of voters are as shown in the following lists:

Given WPL:

44,909,806 Bill, Ross, George

29,004,050 George, Ross, Bill

10,100,500 George, Bill, Ross

19,743,821 Ross, George, Bill

If plurality is used Bill will be elected.

In some embodiments of this invention, in process 100, step 200 will be performed. In process

200, in step 300, some embodiments use flowchart in FIG. 3b that uses plurality repeatedly to find

the highest ranked alternative:

After creation of an operational copy in step 302, and decision in step 305, in steps 312 to 318

Bill is selected and removed and then Ross is selected and removed from operational copy. In

Step 320, a ranked list is not required. In step 323, George is erased from provided Weighted

Preference Lists.

Provided copy WPL in process 300 George erased:

44,909,806 Bill, Ross

29,004,050 Ross, Bill

10,100,500 Bill, Ross

19,743,821 Ross, Bill

Repeating the steps with remaining Bill and Ross the highest ranked alternative selected in FIG.

3b is the following:

Highest ranked Alternative: (Bill)

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Strong Alternative=(Bill)

In step 400 of process 200, some embodiments of this invention use process 400 as explained in

FIG. 4 to create a List of Spoilers.

In step 401, the level-indicator is set to 1. In step 405, it is found that the number of alternatives

minus 2 is more than, the level-indicator. In step 414, from the one member combinations of the

three candidates only Ross or George are valid because the Strong Alternative is not considered a

valid spoiler. For the first iteration Ross is selected.

In step 424, performing the same method explained in FIG. 3b, using a copy of given Weighted

Preference Lists without signed alternatives and selected combination, the highest ranked

alternative is found.

Copy of given WPL without Ross used by in process 300:

44,909,806 Bill, George

29,004,050 George, Bill

10,100,500 George, Bill

19,743,821 George, Bill

Highest ranked alternative = (George)

In step 430, Highest ranked alternative (George) and Strong Alternative: (Bill) are compared and

it is found that the result is not the same. Therefore, Ross is added to the List of Spoilers.

List of Spoilers= {Ross}

In step 445, it is found that a valid combination in this level that is George is not tested for spoiler

effect yet. In step 414, George is selected and in step 424, performing the same method explained

in FIG. 3b, using a copy of given Weighted Preference Lists without George, the highest ranked

alternative is found.

Copy of Given WPL without George used by process 300:

44,909,806 Bill, Ross

29,004,050 Ross, Bill

10,100,500 Bill, Ross

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Highest ranked alternative = (Bill)

In step 430, the highest ranked alternative (Bill) and Strong Alternative: (Bill) are compared and it

is found that they are consistent.

In step 445, it is verified that all valid combinations are checked and in step 460, level-indicator is

incremented to 2.

In step 405, it is found that level-indicator (2) is not greater than ‘number of not signed

alternatives minus two’ (4-2), therefore the process 400 is concluded.

Back to FIG.2a, in step 208, it is realized that a spoiler is in List of Spoilers, and in step 214, Ross

who is in List of Spoilers is signed in Weighted Preference Lists.

Given WPL ross signed:

44,909,806 Bill, ross, George

29,004,050 George, ross, Bill

10,100,500 George, Bill, ross

19,743,821 ross, George, Bill

Signed alternative is shown starting with lower case letter.

Step 300, finds the highest ranked alternative using Weighted Preference Lists excluding Ross and

will lead to:

Provided copy of WPL excluding Ross used by process 300:

44,909,806 Bill, George

29,004,050 George, Bill

10,100,500 George, Bill

19,743,821 George, Bill

Highest ranked Alternative: (George)

Which means in step 300 of process 300 the Strong Alternative is determined:

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Step 400 finds no spoiler between two alternatives and step 208 decides that the next step is 206.

In step 206, all the signs added in this process are removed.

And the highest ranking alternative George will be selected as desired preferred collective choice.

In step 110 in FIG. 1, George is selected as the Independent Collective Choice.

FIG. 5a is a flowchart showing the steps used in process 200 by some embodiments of this

invention to find a preferred collective choice using given Weighted Preference Lists while

restricting the impact of spoilers. This embodiment uses a replica of Weighted Preference Lists

and removes detected spoilers from the replica instead of signing the spoilers in the given

Weighted Preference Lists.

In step 201, a replica of the given Weighted Preference Lists is created.

In step 300, a Strong Alternative is found using the given set of Weighted Preference Lists by one

of many possible vote aggregation methods.

To find a Strong Alternative, some embodiments of the invention apply one or more preferential

voting methods selected from the group consisting of but not limited to: a Ranked Pairs method,

an Instant-Runoff Voting, a Schulze method, a Kemeny-Young method, a Condorcet scoring

method, a Borda Count method. Other embodiments apply a collective choice procedure

repeatedly to find a preferred collective choice as shown in FIG. 3a or FIG. 3b using a copy of

said provided Weighted Preference Lists.

In step 600, the Weighted Preference Lists are analyzed and a List of Spoilers (list 204) is created.

Some embodiments of this invention create an ordered List of Spoilers as part of process 600 and

some just create a set of spoilers. The spoilers can be simple (made of one alternative) or

compound (made of a plurality of alternatives).

In Step 208, if no spoiler is detected, the replica is not needed anymore and the process is

concluded. If any spoiler is detected and List of Spoilers is not empty, in step 230, the spoiler

combinations in the List of Spoilers are analyzed and the least favored among them is found and

removed from the replica of Weighted Preference Lists. In some embodiments, if alternatives in a

compound spoiler are the least favored alternatives, all members of the compound are removed.

Some embodiments remove all the members of the compound spoiler if one of the members of a

compound spoiler is the least preferred alternative. Some embodiments remove a plurality or all of

spoilers in said sorted List of Spoilers, out of said replica of Weighted Preference Lists.

Then steps 300, 600 and the rest of steps in process 200 in FIG. 5a are applied to Weighted

Preference Lists.

FIG. 5b is a flowchart showing the steps used in process 200 by some embodiments of this

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restricting the impact of spoilers by recursively using steps in process 100 when deciding which

spoilers should be removed. This embodiment uses a replica of Weighted Preference Lists and

removes detected spoilers from the replica. This embodiment also uses the process 100

recursively which will ensure that the existence of irrelevant alternatives among spoilers when

finding the least favored spoiler will not influence the outcomes.

Steps 201, 300, 600, and 208 are identical to FIG. 5a described before.

If any spoiler is detected and List of Spoilers is not empty, in step 232, process 100 is recursively

employed to produce an Independent Collective Choice Ordered List using a copy of said replica

of Weighted Preference Lists including only the alternatives in said List of Spoilers. If the number

of spoilers detected in List of Spoilers is greater than two then some of them may act as spoilers

for others. The outcome of step 232 is shown as list 231.

In step 233, at least one of the lowest ranked spoilers in Independent Collective Choice Ordered

List of Spoilers is removed from Weighted Preference Lists. In some embodiments, if alternatives

in a compound spoiler are the least favored alternatives, all members of the compound are signed

out. Some embodiments sign all the members of the compound spoiler if one of the members of a

compound spoiler is the lowest ranked alternative. Some embodiments sign a plurality or all of

spoilers in said Independent Collective Choice Ordered List of Spoilers, out of said Weighted

Preference Lists.

Then steps 300 and other steps in process 200 are repeated until no spoilers are detected.

FIG. 5c is a flowchart showing the steps used in process 200 by some embodiments of this

invention to find a preferred collective choice using given Weighted Preference Lists while

restricting the impact of spoilers by making a ranked List of Spoilers when deciding which

spoilers should be removed. This embodiment uses a replica of Weighted Preference Lists and

removes detected spoilers from the replica. It uses a collective choice procedure that creates a

ranked list of spoilers like Borda Count, or uses process 300, as explained in FIG. 3a and FIG. 3b.

Steps 201, 300, 600, and 208 are identical to FIG. 5a described before.

If any spoiler is detected and List of Spoilers is not empty, in step 235, a ranked list of alternatives

is made, some embodiments of the invention generate a sequence of alternatives using said replica

of Weighted Preference Lists including only alternatives in the list of spoilers by applying one or

more preferential voting methods selected from the group consisting of but not limited to: a

Ranked Pairs method, an Instant-Runoff Voting, a Schulze method, a Kemeny-Young method, a

Condorcet scoring method, a Borda Count method. Other embodiments apply a collective choice

procedure repeatedly to make a ranked list of alternatives by applying process 300 shown in FIG.

3a or FIG. 3b using given Weighted Preference Lists including only alternatives in the list of

spoilers. The result of step 235 is a ranked List of Spoilers shown as list 234.

In step 236, at least one of the lowest ranked spoilers in Independent Collective Choice Ordered

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in a compound spoiler are the least favored alternatives, all members of the compound are signed

out. Some embodiments sign all the members of the compound spoiler if one of the members of a

compound spoiler is the lowest ranked alternative. Some embodiments sign a plurality or all of

spoilers in said Independent Collective Choice Ordered List of Spoilers, out of said Weighted

Preference Lists.

Then steps 300 and other steps in process 200 are repeated until no spoilers are detected.

FIG. 6 is a flowchart showing the steps used in process 600 by some embodiments of this

invention to create a new List of Spoilers using a supplied set of Weighted Preference Lists and a

Strong Alternative. Whenever ranking is needed in process 600, it uses the same ranking method

used for the creation of Strong Alternative.

This process is identical to process in FIG. 4, except that it assumes: any alternative that should

not be considered in the current search for spoilers has already been removed from the supplied

Weighted Preference Lists.

In step 601, a level-indicator is set to level 1. The level-indicator is a counter that starts at 1

meaning that the alternatives are tested one by one to check the effect of their removal on the

highest ranked alternative found. And then in next round it changes to 2, which means, in that

iteration the alternatives will be tested two by two to check the effect of their removal.

In step 609, the level-indicator is compared with the number of alternatives in given Weighted

Preference Lists that are not in List of Spoilers. If the level-indicator is more than the number of

alternatives in supplied Weighted Preference Lists excluding those that are in spoilers list minus 2,

the process is concluded.

Some embodiments of this invention also conclude the creation of said List of Spoilers in step 609

if at least one of, but not limited to, the following happens: the level-indicator is more than a

predetermined level, a deadline has reached or a predetermined number of spoilers are found.

In step 615, a new combination of alternatives, in supplied Weighted Preference Lists, is selected

based on a strategy. This combination is a set of alternatives wherein the number of alternatives in

the set is the level-indicator. The selected combination does not include Strong Alternative or

simple or compound spoilers in the List of Spoilers. Some embodiments choose some

optimization strategies to maximize the speed of decision making by choosing from a certain

subset of alternatives when selecting the combinations or by selecting the alternatives with certain

order or according to some algorithms.

In step 626, the highest ranked alternative is found using a copy of said supplied Weighted

Preference Lists without the alternatives in selected combination. The method used for creating

this ranked list must be identical to what has been used for finding the Strong Alternative.

In step 630, the highest ranked alternative and Strong Alternative are compared to see if they are

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to be a spoiler and is added to the anticipated List of Spoilers (list 640).

In step 645, it is verified if all the valid combinations at the current value of level-indicator have

been selected and tested for spoiler effect. The valid combination is a set of alternatives in the

supplied Weighted Preference Lists wherein the number of alternatives in the combination is the

level-indicator and the selected combination does not have the Strong Alternative or the members

of List of Spoilers.

If all of the valid combinations at this level-indicator are not tested, the steps 615 and its next

steps are repeated. If all of the valid combinations at this level-indicator are tested, in step 651,

the level-indicator is Incremented by one and the step 609 and its next steps are repeated.