Vicki Allan 2013

Computer occupations dominate STEM. Source Georgetown Center on Education and the Workforce, STEM. Used with permission.

Annual Degrees and Job Openings in broad S&E Fields ( ). Data taken from the Computing Research Association (cra.org).

Life After the PhD SIAM: Mathematics in Industry Roughly half of all mathematical scientists hired into industry are statisticians. The second-largest group by academic specialty is applied mathematics. Strongest employers of mathematicians are the finance/insurance pharmaceutical/medical. Almost none of the mathematicians have “mathematics” in their job title. By contrast, the title of statisticians often refers to their specialty. The job satisfaction high, nearly 90 percent reporting satisfaction with their compensation and benefits. Median pay was $100,000 for both men and women. Compared to the 1996 survey, fewer reported “modeling and simulation” as an important academic specialty for their jobs, and more reported “statistics.” Contradictory finding, the most important item evaluated in performance reviews was reported to be mathematical models. Programming and computer skills continue to be the most important technical skill that new hires bring to their jobs. Very few people in this survey were forced into taking industrial jobs because they couldn’t get a job in academia. (19 women, 37 men)

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?

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.

Cooperation 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 Preferences One voter ranks c > d > b > a One voter ranks a > c > d > b One voter ranks b > a > c > d

Who should be hired? Runoff - Binary Protocol One voter ranks c > d > b > a One voter ranks a > c > d > b One voter ranks b > a > c > d One idea – consider candidates pairwise winner (c, (winner (a, winner(b,d)))

Runoff - Binary Protocol One voter ranks c > d > b > a One voter ranks a > c > d > b One voter ranks b > a > c > d winner (c, (winner (a, winner(b,d)))=a winner (d, (winner (b, winner(c,a)))=d winner (c, (winner (b, winner(a,d)))=c winner (b, (winner (a, winner(c,d)))=b surprisingly, order of pairing yields different winner!

Suppose we have seven voters How choose winner? a > b > c >d b > c > d> a Who is really the most preferred candidate?. Are they honest?

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 15

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 >d a=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 > d a=18, b=19, c=20,d=13 n a > b > c n b > c >a n c > a > b n a > b > c n b > c > a n c > a > b n a >b >c a=15,b=14, c=13 When loser is removed, third choice becomes winner!

Conclusion Finding the correct mechanism 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,c Dilemma 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 6 You 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)

WVG Example  Consider a WVG of three agents with quota =5 332Weight Any two agents form a winning coalition. We attempt to 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

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 BC 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] ABC ABC ABC 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 Shubik Yellow Blue White201102

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 Merged1/2 11 Original (white and yellow)2/31/25/6 1/22/3 Annexed1/2 11 Original (yellow)1/31/22/3 1/21/3 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 Game Show only Winning coalitions A = critical 33 B = critical 31 C = critical 1 D = critical 1 E = critical 1 F = critical 1 G = critical 1 1 ABCDEFG 2 ABCDEFG 3 ABCDEFG 4 ABCDEFG 5 ABCDEFG 6 ABCDEFG 7 ABCDEFG 8 ABCDEFG 9 ABCDEFG 10 ABCDEFG 11 ABCDEFG 12 ABCDEFG 13 ABCDEFG 14 ABCDEFG 15 ABCDEFG 16 ABCDEFG 17 ABCDEFG 18 ABCDEFG 19 ABCDEFG 20 ABCDEFG 21 ABCDEFG 22 ABCDEFG 23 ABCDEFG 24 ABCDEFG 25 ABCDEFG 26 ABCDEFG 27 ABCDEFG 28 ABCDEFG 29 ABCDEFG 30 ABCDEFG 31 ABCDEFG 32 ABCDEFG 33 ABCDEFG Power A = 33/( ) =.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

Eliminatenumden AOrgCDEFG x12 ABCDEFG x12 ABCDEFG x12 ABCDEFG x12 ABCDEFG x12 ABCDEFG ABCDEFG x12 ABCDEFG x12 ABCDEFG x12 ABCDEFG ABCDEFG x12 ABCDEFG x12 ABCDEFG ABCDEFG x12 ABCDEFG ABCDEFG ABCDEFG x12 ABCDEFG x12 ABCDEFG ABCDEFG x12 ABCDEFG ABCDEFG ABCDEFG x12 ABCDEFG ABCDEFG ABCDEFG ABCDEFG ABCDEFG x12 ABCDEFG ABCDEFG ABCDEFG ABCDEFG ABCDEFG 11 n total agents d in [1,n-1] 1/d 0/d In this example, we only see cases of 1/2 1/1 In EVERY line you eliminate, SOMETHING was critical! In cases you do NOT eliminate, you could have reduced the total number

So what is happening? Let k=1 Consider 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 Game Show only Winning coalitions A = critical 17 B = critical 15 C = critical 1 D = critical 1 E = critical 1 F = critical 1 1ABCDEF 2ABCDEF 3ABCDEF 4ABCDEF 5ABCDEF 6ABCDEF 7ABCDEF 8ABCDEF 9ABCDEF 10ABCDEF 11ABCDEF 12ABCDEF 13ABCDEF 14ABCDEF 15ABCDEF 16ABCDEF 17ABCDEF Power A = 17/( ) =.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 ratio and 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 n=10, k=5 n=20, k=5

Manipulation via Annexation n=10, k=5 n=20, k=5

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?