Vicki Allan 2013

1. Vicki Allan2013

3. Multiagent systems – program computer agents to act for people.If two heads are better than one, how about 2000?

4. 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?

5. 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.

6. 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?

7. 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.

8. 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

9. Suppose we have only three voters Individual Preferences Joe ranks c> d > b > a Sam ranks a > c > d > b Sally ranks b > a > c > d Who should be hired?

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

11. 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!

12. 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

13. reasonable???

14. 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

15. Borda Paradox – remove loser (d), Now: winner changes • a > b > c • b > c >a • c > a > b • a > b > c • b > c > a • c > a > b • a >b >c a=15,b=14, c=13 • 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 When loser is removed, third choice becomes winner!

16. 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?

17. Who wins? (if highest is first choice)

18. Inserted cloneNow who wins?

19. 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.

20. Conclusion • Finding the best mechanism for social choice is not easy

21. 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?

22. 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

23. Optimization Problem Not want a centralized solution • Communication • Privacy • Situation changing • Self-interested

24. Looking for partners for field trip.Arc labels represent goodness of pairing according to agents.

25. 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

26. 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?

27. 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

28. 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.

29. 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?

30. Scenario 6 You consult with local traffic to find a good route home from work But so does everyone else

31. Ramoni Lasisi and Vicki Allan Utah State University A SEARCH-BASED APPROACH TO ANNEXATION AND MERGING IN WEIGHTED VOTING GAMES by

32. 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

33. 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 :

34. WVG Example • Consider a WVG of three agents with quota =5 Weight 3 3 2 • 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?

35. 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)?

36. Annexation and Merging • Annexation Merging C

37. 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?

38. Power Indices • Measure the fraction of the power attributed to each voter • The ability to influence or affect the outcomes of decision-making processes • Voting power is NOT proportional to voting weight Two most popular power indices are Shapley-Shubik index Banzhafindex

39. Shapley-Shubik Power Index Quota A Looks at value added. What do I add to the existing group? Consider the group being formed one at a time. B C [4,2,3: 6]

40. Shapley-Shubik Power Index Quota A B C A C B B A C B C A C A B C B A A = 4/6 B = 1/6 C = 1/6 [4,2,3: 6]

41. Banzhaf Power Index [4,2,3: 6] A = 3/5 B = 1/5 C = 1/5

42. 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.

43. Consider Shapley Shubik

44. Consider merging yellow/white • To understand effect, remove all permutations where yellow and white are not together

45. Remove permutations that are redundant Merging can be harmful. Annexing cannot.

46. [6, 5, 1, 1, 1, 1, 1;11] Consider Banzhaf power index with annexing • 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

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

48. 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

49. In EVERY line you eliminate, SOMETHING was critical! 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 cases you do NOT eliminate, you could have reduced the total number

50. 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)