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

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

2. 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 :

3. 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

4. Annexation and Merging • Annexation Merging C

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 • 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

7. 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]

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

9. Banzhaf Power Index Quota [4,2,3:6] B A B A C A 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 Banzhaf Power Distribution C A = 3/(3 + 1 + 1) = 3/5; B = C = 1/(3 + 1 + 1) = 1/5 A B

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

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

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

14. [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

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 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. 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

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 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 We limit the size of the coalition to constant using the following assumptions: • Manipulators prefer smaller-sized coalitions – easier to form and manage • Intra-coalition coordination, communication, other overheads increase with coalition size . . . 1 2 n • The NAÏVE approach checks all subsets of agents to find the best merge – EXPONENTIAL! • Our idea sacrifices optimality for “good” merge

25. Pseudo-polynomial Manipulation Algorithms… Merging • Note that computing Shapley-Shubik and Banzhaf Index is NP-Hard • We need to search only coalitions for good merge • By considering the possibilities in a reasonable order, we can often prune less likely candidates.

26. Is that all? NO! • The problem remains NP-hard even with limitation on coalition size • Also, coalitions may be large to search in practice • So, we employ informed heuristic search strategy to improve the search.

27. Experimental Results-Merging

28. Experimental Results-Annexation

29. Conclusions • We present two search-based Pseudo-polynomial manipulation algorithms • We complement the algorithms with informed heuristic search strategies to improve performance • Our manipulation algorithm for annexation improves annexer’s benefit by more than • Our manipulation algorithm for merging improves manipulators’ benefits between to

30. Thanks

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

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