Vicki Allan 2013
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Transcript of Vicki Allan 2013
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?