banzhaf power - lecture 8 section 2

Banzhaf PowerLecture 8

Section 2.2

Robb T. Koether

Hampden-Sydney College

Fri, Sep 11, 2015

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 1 / 29

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 2 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 3 / 29

Coalitions

Definition (Coalition)A coalition is a group of players who agree to vote as a block.A winning coalition is a coalition whose votes add up to at leastthe quota.A losing coalition is a coalition whose votes add up to less thanthe quota.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 4 / 29

Coalitions

If there are 3 players, how many possible coalitions are there?

How about 4 players?5 players?In general, if there are n players, then there are 2n − 1 possiblecoalitions.So, if there are 10 players, then there are 210 − 1 = 1023 possiblecoalitions.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 5 / 29

Coalitions

If there are 3 players, how many possible coalitions are there?How about 4 players?

5 players?In general, if there are n players, then there are 2n − 1 possiblecoalitions.So, if there are 10 players, then there are 210 − 1 = 1023 possiblecoalitions.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 5 / 29

Coalitions

If there are 3 players, how many possible coalitions are there?How about 4 players?5 players?

In general, if there are n players, then there are 2n − 1 possiblecoalitions.So, if there are 10 players, then there are 210 − 1 = 1023 possiblecoalitions.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 5 / 29

Coalitions

If there are 3 players, how many possible coalitions are there?How about 4 players?5 players?In general, if there are n players, then there are 2n − 1 possiblecoalitions.

So, if there are 10 players, then there are 210 − 1 = 1023 possiblecoalitions.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 5 / 29

Coalitions

If there are 3 players, how many possible coalitions are there?How about 4 players?5 players?In general, if there are n players, then there are 2n − 1 possiblecoalitions.So, if there are 10 players, then there are 210 − 1 = 1023 possiblecoalitions.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 5 / 29

Listing Coalitions

There are two good systematic ways to list all possible coalitions.By sizeBy “choice”

I’ll show you both, but I prefer the first method.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 6 / 29

Listing Coalitions

There are two good systematic ways to list all possible coalitions.By sizeBy “choice”

I’ll show you both, but I prefer the first method.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 6 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 7 / 29

Listing Coalitions By Size

There is coalition of size 0, namely, the empty set ∅, for which wewill have no use.List the coalitions of size 1: A, B, C, . . .List the coalitions of size 2: AB, AC, BC, . . .Etc.

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Listing Coalitions By Size

By Size

∅

The null set

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 9 / 29

Listing Coalitions By Size

By Size

A

B

C

∅

Coalitions of size 1

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 9 / 29

Listing Coalitions By Size

By Size

A

B

C

AB

AC

BC

∅

Coalitions of size 2

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 9 / 29

Listing Coalitions By Size

By Size

A

B

C

AB

AC

BC

ABC∅

Coalition of size 3

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 9 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 10 / 29

Listing Coalitions By “Choice”

Half of the coalitions include A and half do not include A.Half of those that include A also include B and half do not. Thesame for the half that do not include A.Etc.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 11 / 29

Listing Coalitions By “Choice”

By “Choice”

Start here

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 12 / 29

Listing Coalitions By “Choice”

By “Choice”

A A

Choose whether to include A

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 12 / 29

Listing Coalitions By “Choice”

By “Choice”

B B B B

A A

Choose whether to include B

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 12 / 29

Listing Coalitions By “Choice”

By “Choice”

C C C C C C C C

B B B B

A A

Choose whether to include C

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 12 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 13 / 29

Critical Players

Definition (Critical Player)A critical player of a coalition is a player whose membership in thatcoalition takes it from a losing coalition to a winning coalition. That is, ifhe left the winning coalition, it would then be a losing coalition.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 14 / 29

An Example

Example (Coalitions)Consider the situation [4 : 3, 2, 1]. Make a table of all possiblecoalitions and their critical players.

Coalition Weight Critical Players

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 15 / 29

An Example

Example (Coalitions)Consider the situation [4 : 3, 2, 1]. Make a table of all possiblecoalitions and their critical players.

Coalition Weight Critical Players{A} 3{B} 2{C} 1{A, B} 5{A, C} 4{B, C} 3{A, B, C} 6

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 16 / 29

An Example

Example (Coalitions)Consider the situation [4 : 3, 2, 1]. Make a table of all possiblecoalitions and their critical players.

Coalition Weight Critical Players{A} 3{B} 2{C} 1{A, B} 5 A, B{A, C} 4 A, C{B, C} 3{A, B, C} 6 A

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 17 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 18 / 29

Definitions

Definition (Critical Count)The critical count of a player is the number of possible coalitions inwhich he is a critical player.

Definition (Banzhaf Power Index)The Banzhaf power index (BPI) of a player is that player’s critical countdivided by the total of all players’ critical counts.

Definition (Banzhaf Power Distribution)The Banzhaf power distribution is the set of BPI’s for all the players.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 19 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 20 / 29

Example

ExampleFind the power distribution in our original example [14 : 9, 8, 3, 1].

Does this sound right?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 21 / 29

Example

ExampleFind the power distribution in our original example [14 : 9, 8, 3, 1].Does this sound right?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 21 / 29

Example

ExampleFind the power distribution in our modified example [11 : 9, 8, 3, 1].

Does this sound right?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 22 / 29

Example

ExampleFind the power distribution in our modified example [11 : 9, 8, 3, 1].Does this sound right?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 22 / 29

Example

ExampleUse the Javascript program to find the Banzhaf Power Indexes inthe following situations.[14 : 6, 5, 5, 4].[15 : 6, 5, 5, 4].[16 : 6, 5, 5, 4].[17 : 6, 5, 5, 4].

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 23 / 29

Example

Example (Stolen from Wikipedia)California has 55 electoral votes, Texas as 34, and New York as31.Total = 120, majority ≥ 61.If those were the only three states, then we would have[61 : 55, 34, 31].Find the power distribution.

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 24 / 29

Example

Example (Also stolen from Wikipedia)Replace New York with Ohio, with 20 electoral votes.Total = 109, majority ≥ 55.The situation now is [55 : 55, 34, 20].How has the power distribution changed?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 25 / 29

Example

ExampleFind the power distribution in [9 : 5, 4, 3, 2, 1].You are E and you would like to buy one vote from another player.From which player should you buy it?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 26 / 29

Example

ExampleConsider the situation [q : 5, 3, 2, 1].What quota q makes the power distribution most balanced?What quota q makes the power distribution most unbalanced?

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 27 / 29

Outline

1 CoalitionsBy SizeBy Choice

2 Critical Players

3 The Banzhaf Power Index

4 Examples

5 Assignment

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 28 / 29

Assignment

AssignmentCh. 2: Exercises 11, 13, 14, 15, 19, 69, 71. (Use the Javascriptprogram for 69 and 71.)

Robb T. Koether (Hampden-Sydney College) Banzhaf Power Fri, Sep 11, 2015 29 / 29