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

2. A Weighted Voting Game (WVG)

3. 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

4. Annexation and Merging Annexation Merging

5. 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?

6. 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

7. 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]

8. A B C Quota How important is each voter? A A A A A B B B B B C C C C C A claims 2/3 of the power, but look at what happens when the quota changes.

9. A B Quota Banzhaf Power Index A A B C C There are three winning coalitions : {4,2}, {4,2,3},{4,3} -A is critical three times -B is critical once -C is critical once 5 total swing votes A = 3/(3 + 1 + 1) = 3/5; B = C = 1/(3 + 1 + 1) = 1/5 [4,2,3:6] Banzhaf Power Distribution A B C

10. 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.

11. Consider Shapley Shubik 1 2 3 4 5 6 Yellow234432 Blue231132 White201102

12. 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

13. Remove permutations that are redundant 1 x 2 3 x 4 x 5 6 x Merge1/2 11 Orig2/31/25/6 1/22/3 Annex1/2 11 Orig1/31/22/3 1/21/3 Merging can be harmful. Annexing cannot.

14. [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

15. 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/(33+31+5) =.47826

16. 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

17. numden 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

18. 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)

19. 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/(17+15+4) =.47222

20. Suppose my original ratio is 1/3

21. Suppose my decreasing ratio is ½. I lose

22. Suppose my decreasing ratio is 0/2. I improve

23. 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.

24. 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

25. Pseudo-polynomial Manipulation Algorithms… Merging

26. Is that all? NO!

27. Experimental Results-Merging

28. Experimental Results- Annexation

29. Conclusions

30. Thanks

31. Experimental Results-Merging (a) (b) (c) (d)

32. Experimental Results-Annexation (a) (b) (c) (d)