Vicki Allan2013

Multiagent systems – program computer agents to act for

people.

If two heads are better than one, how about 2000?

Monetary Auction

• Object for sale: a one dollar bill• Rules

– Highest bidder gets it– Highest bidder and the second highest bidder

pay their bids– New bids must beat old bids by 5¢.– Bidding starts at 5¢. – What would your strategy be?

Give Away

• Bag of candy to give away• Put your name and vote on piece of paper.• If everyone in the class says “share”, the

candy is split equally.• If only one person says “I want it”, he/she

gets the candy to himself.• If more than one person says “I want it”, I

keep the candy.

Regret?

• Seeing how everyone else played, do you wish you would have played differently?

• If you could have talked to others before (collusion), what would you have said? Would it change anything?

The point?

• You are competing against others who are as smart as you are.

• If there is a “weakness” that someone can exploit to their benefit, someone will find it.

• You don’t have a central planner who is making the decision.

• Decisions happen in parallel.

Social Choice

• Hiring several new professor this year.• Committee of five people to make decision• Have narrowed it down to four candidates.• Each person has a different ranking for the

candidates.• How do we make a decision?• Termed a social choice function

Who should be hired?

Individual PreferencesJoe ranks c > d > b > aSam ranks a > c > d > bSally ranks b > a > c > d

Suppose we have only three voters

Who should be hired?

Runoff - Binary ProtocolJoe ranks c > d > b > aSam ranks a > c > d > bSally ranks b > a > c > d

One idea – consider candidates pairwise

winner (c, (winner (a, winner(b,d)))

Runoff - Binary ProtocolOne voter ranks c > d > b > aOne voter ranks a > c > d > bOne voter ranks b > a > c > dwinner (c, (winner (a,

winner(b,d)))=awinner (d, (winner (b, winner(c,a)))=d

winner (c, (winner (b, winner(a,d)))=c

winner (b, (winner (a, winner(c,d)))=bsurprisingly, order of pairing yields different winner!

Borda protocol assigns an alternative |O| points for the

highest preference, |O|-1 points for the second, and so on

The counts are summed across the voters and the alternative with the highest count becomes the social choice

12

reasonable???

Borda Paradox• a > b > c >d • b > c > d >a• c > d > a > b• a > b > c > d• b > c > d> a• c > d > a >b• a > b >c >da=18, b=19, c=20, d=13

Is this a good way?

Clear loser

Borda Paradox – remove loser (d), Now: winner changes

• a > b > c >d • b > c > d >a• c > d > a > b• a > b > c > d• b > c > d> a• c > d > a > b• a > b >c > da=18, b=19, c=20,d=13

a > b > c b > c >a c > a > b a > b > c b > c > a c > a > b a >b >ca=15,b=14, c=13

When loser is removed, third choice becomes winner!

Issues with Borda

• favorite betrayal. How can anyone report different preference to gain advantage?

B wins in this example, but the middle player can change the winner to something he likes better. How?

Who wins? (if highest is first choice)

Inserted cloneNow who wins?

Other issues with Borda• less expressive• voter strategy Ex: 3 candidates each with strong supporters. Many

non-entities that no one really cared about. the strategic votes are: A > nonentities > B > C (cast by about 1/3 of the voters) B > nonentities > C > A (cast by about 1/3 of the voters) C > nonentities > A > B (cast by about 1/3 of the voters) ---------------------------------------------------------------- A,B, and C each get an average score of N/3. Non-entities score about

N/2. So a non-entity always wins and the 3 good candidates always are ranked below average.

Conclusion

• Finding the best mechanism for social choice is not easy

Coalition Formation Overview

• Tasks: Various skills required by team members

• Agents form coalitions• Agent types - Differing policies regarding

which coalition to join• How do policies interact?

Multi-Agent Coalitions

• “A coalition is a set of agents that work together to achieve a mutually beneficial goal” (Klusch and Shehory, 1996)

• Reasons agent would join Coalition– Cannot complete task alone– Complete task more quickly

Optimization Problem

Not want a centralized solution• Communication• Privacy• Situation changing• Self-interested

Looking for partners for field trip.Arc labels represent goodness of

pairing according to agents.

Scenario 1 – Bargain Buy(supply-demand)

• Store “Bargain Buy” advertises a great price

• 300 people show up• 5 in stock• Everyone sees the advertised

price, but it just isn’t possible for all to achieve it

Scenario 2 – selecting a spouse(agency)

• Bob knows all the characteristics of the perfect wife

• Bob seeks out such a wife

• Why would the perfect woman want Bob?

Scenario 3 – hiring a new PhD(strategy)

• Universities ranked 1,2,3• Students ranked a,b,cDilemma for second tier university• offer to “a” student• likely rejected• rejection delayed - see other options• “b” students are gone

Scenario 4 (trust)What if one person talks a good story, but his claims of skills are really inflated?

He isn’t capable of performing. the task.

Scenario 5

The coalition is completed and rewards are earned. How are they fairly divided among agents with various contributions?If organizer is greedy, why wouldn’t others replace him with a cheaper agent?

Scenario 6You consult with local traffic to find a good route home from work

But so does everyone else

A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES

Ramoni Lasisi and Vicki Allan

Utah State University

by

Consider the US electoral college –

A weighted voting game(California 55;Texas 38;Florida 29; New York 29;Illinois 20; Pennsylvania 20;Ohio 18;Georgia 16;Michigan 16;North Carolina 15;New Jersey 14;Virginia 13;Washington 12;Arizona 11;Indiana 11;Massachusetts 11;Tennessee 11;Maryland 10;Minnesota 10;Missouri 10;Wisconsin 10;Alabama 9;Colorado 9;South Carolina 9;Kentucky 8;Louisiana 8;Connecticut 7;Oklahoma 7;Oregon 7;Arkansas 6;

Iowa 6;Kansas 6;Mississippi 6;Nevada 6; Utah 6;Nebraska 5;New Mexico 5;West Virginia 5;Hawaii 4;Idaho 4;Maine 4;New Hampshire 4;Rhode Island 4;Alaska 3;Delaware 3;D.C. 3;Montana 3;North Dakota 3;South Dakota 3;Vermont 3;Wyoming 3; quota = 270) 538 total votes

A Weighted Voting Game (WVG) Consists of a set of agents

Each agent has a weight

A game has a quota

A coalition wins if

In a WVG, the value of a coalition is either (i.e., ) or (i.e., )

Notation for a WVG :

WVG Example Consider a WVG of three agents with quota =5

3 3 2Weight

Any two agents form a winning coalition. We attemptto assign power based on their ability to contribute to a winning

coalition. How would you divide power?

Questions? Would Texas have more power if it split

into more states (splitting)? Would Maryland be better off to grab the

votes of Washington DC (annexation)? Would several of the smaller states be

better off combining into a coalition (merging)?

Annexation and Merging

Annexation Merging

C

Annexation and Merging

Annexation Merging

The focus of this talk:To what extent or by how much can agents improve their

power via annexation or merging?

Power Indices

The ability to influence or affect the outcomes of decision-making processes

Voting power is NOT proportional to voting weight

Measure the fraction of the power attributed to each voter

Two most popular power indices are Shapley-Shubik index Banzhaf index

A

B

C

Quota

Shapley-Shubik Power Index

Looks at value added. What do I add to the existing group?

Consider the group being formed one at a time.

[4,2,3: 6]

A B C

Quota

Shapley-Shubik Power Index

[4,2,3: 6]

A

A

A

A

A

C

C

C

C

C

B

B

B

B

B

A = 4/6 B = 1/6 C = 1/6

Banzhaf Power Index [4,2,3: 6]

A B C

A B C

A B C

A = 3/5 B = 1/5 C = 1/5

Consider annexing and merging

We expect annexing to be better

as you don’t have to split the power With merging, we must gain

more power than is already in the agents individually.

Consider Shapley Shubik1            

2            

3            

4            

5          

6            

Yellow 2 3 4 4 3 2

Blue 2 3 1 1 3 2

White 2 0 1 1 0 2

Consider merging yellow/white To understand effect, remove all

permutations where yellow and white are not together

1             x

2            

3             x

4            

5          

6            

Remove permutations that are redundant

1             x

2            

3             x

4             x

5          

6             x

Merged 1/2 1/2 1 1 1/2 1/2

Original (white and yellow) 2/3 1/2 5/6 5/6 1/2 2/3

Annexed 1/2 1/2 1 1 1/2 1/2

Original  (yellow) 1/3 1/2 2/3 2/3 1/2 1/3

Merging can be harmful. Annexing cannot.

[6, 5, 1, 1, 1, 1, 1;11] Consider player A (=6) as the annexer. We expect annexing to be non-harmful,

as agent gets bigger without having to share the power.

Bloc paradox Example from Aziz, Bachrach, Elkind, &

Paterson

Consider Banzhaf power index with annexing

Original GameShow onlyWinning coalitions

A = critical 33B = critical 31C = critical 1D = critical 1E = critical 1F = critical 1G = critical 1

1 A B C D E F G

2 A B C D E F G

3 A B C D E F G

4 A B C D E F G

5 A B C D E F G

6 A B C D E F G

7 A B C D E F G

8 A B C D E F G

9 A B C D E F G

10 A B C D E F G

11 A B C D E F G

12 A B C D E F G

13 A B C D E F G

14 A B C D E F G

15 A B C D E F G

16 A B C D E F G

17 A B C D E F G

18 A B C D E F G

19 A B C D E F G

20 A B C D E F G

21 A B C D E F G

22 A B C D E F G

23 A B C D E F G

24 A B C D E F G

25 A B C D E F G

26 A B C D E F G

27 A B C D E F G

28 A B C D E F G

29 A B C D E F G

30 A B C D E F G

31 A B C D E F G

32 A B C D E F G

33 A B C D E F G

Power A =33/(33+31+5)= .47826

Paradox Total number of winning coalitions shrinks as

we can’t have cases where the members of bloc are not together.

If agent A was critical before, since A got bigger, it is still critical.

If A was not critical before, it MAY be critical now.

BUT as we delete cases, both numerator and denominator are changing

Surprisingly, bigger is not always better

Eliminate num den

A Org C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G

A B C D E F G x 1 2

A B C D E F G

A B C D E F G

A B C D E F G x 1 2

A B C D E F G x 1 2

A B C D E F G

A B C D E F G x 1 2

A B C D E F G

A B C D E F G

A B C D E F G x 1 2

A B C D E F G

A B C D E F G

A B C D E F G

A B C D E F G

A B C D E F G x 1 2

A B C D E F G

A B C D E F G

A B C D E F G

A B C D E F G

A B C D E F G 1 1

n total agentsd in [1,n-1]1/d0/d

In this example, we only see cases of1/21/1

In EVERY line youeliminate, SOMETHINGwas critical!

In cases you do NOT eliminate, you could have reduced the total number

So what is happening? Let k=1Consider all original winning coalitions.Since all coalitions are considered originally, there are

no additional winning coalitions created.The original set of coalitions to too large. Remove any

winning coalitions that do not include the bloc.Notice:If both of the merged agents were critical, only one is

critical (decreasing numerator/denominator)If only one was in the block, you could remove many

critical agents from the total count of critical agents.If neither of the agents was critical, the bloc could be (increasing numerator/denominator)

Original GameShow onlyWinning coalitions

A = critical 17B = critical 15C = critical 1D = critical 1E = critical 1F = critical 1

1 A B C D E F

2 A B C D E F

3 A B C D E F

4 A B C D E F

5 A B C D E F

6 A B C D E F

7 A B C D E F

8 A B C D E F

9 A B C D E F

10 A B C D E F

11 A B C D E F

12 A B C D E F

13 A B C D E F

14 A B C D E F

15 A B C D E F

16 A B C D E F

17 A B C D E F

Power A =17/(17+15+4)= .47222

Suppose my original ratio is 1/3

Suppose my decreasing ratio is ½.I lose

Suppose my decreasing ratio is 0/2.I improve

Suppose my increasing ratio is 1/1.I improve

Win/Lose depends on the relationship between the original ratio and the new ratioand whether you are increasing or decreasing by that ratio.

Pseudo-polynomial Manipulation Algorithms

Merging

The NAÏVE approach checks all subsets of agents to find the best merge – EXPONENTIAL!

. . . Our idea sacrifices optimality for “good”

merge

1 2 n

Finding a good candidate Determining if there is a beneficial

merge is NP-hard because of the combinatorial numbers of merges to check.

We restrict the size of the merge and look for good candidates within that size.

Idea In computing the Shapley-Shubik and

Banzhaf power indices, the generative technique used by Bilboa computes a variety of terms.

These terms are helpful in estimating the power of merged coalitions.

Manipulation via merging

10 20 30 40 500.80.9

11.11.21.31.41.51.61.71.81.9

2

n=10, k=5

n=20, k=510 20 30 40 50

0.80.9

11.11.21.31.41.51.61.71.81.9

2

SS SearchBanzhaf SearchSS best 3Banzhaf best 3

Manipulation via Annexationn=10, k=5

n=20, k=510 20 30 40 500

102030405060708090

100110120130140

SS SearchBanzhaf SearchSS best 3

10 20 30 40 500

102030405060708090

100110120130140

Conclusions Shapley-Shubik is more vulnerable to

manipulation. Our method for finding a beneficial

merge increased the power from between 28% to 45% on average.

Our method for finding a beneficial annexation increased power by over 300%.

Questions?