LECTURE 8 Coalitions Voting Power and Computational Social

LECTURE 8: Coalitions, Voting Power, and Computational Social Choice An Introduction to Multi. Agent Systems http: //www. csc. liv. ac. uk/~mjw/pubs/imas 7 -1

Forming Coalitions: Coalitional Games n n Coalitional games model scenarios where agents can benefit by cooperating Issues in coalitional games (Sandholm et al, 1999): q q q Coalition Structure Generation Teamwork Dividing the benefits of cooperation 7 -2

Coalition Structure Generation n n Deciding in principle who will work together The basic question: q n n Which coalition should I join? The result: partitions agents into disjoint coalitions The overall partition is a coalition structure 7 -3

Solving the optimization problem of each coalition n n Deciding how to work together Solving the “joint problem” of a coalition Finding how to maximize the utility of the coalition itself Typically involves joint planning, etc. 7 -4

Dividing the Benefits n n n Deciding “who gets what” in the payoff Coalition members cannot ignore each other’s preferences, because members can defect: if you try to give me a bad payoff, I can always walk away We might want to consider issues such as fairness of the distribution 7 -5

Formalizing Cooperative Scenarios n n n A coalitional game: where Ag = {1, …, n} is a set of agents v = 2 Ag -> R is the characteristic function of the game Usual interpretation: if ν(C) = k, then coalition C can cooperate in such a way they will obtain utility k, which may then be distributed among the team members Characteristic functions are often assumed to be superadditive (the value of a union of disjoint coalitions is greater than or equal to the sum of the coalitions’ separate values) 7 -6

Which Coalition Should I Join? n n n Most important question in coalitional games: is a coalition stable? that is, is it rational for all members of coalition to stay with the coalition, or could they benefit by defecting from it? (There is no point in me trying to join a coalition with you unless you want to form one with me, and vice versa. ) Stability is a necessary but not sufficient condition for coalitions to form 7 -7

The Core n n The core of a coalitional game is the set of feasible distributions of payoff to members of a coalition that no sub-coalition can reasonably object to An outcome for a coalition C in game ν is a vector of payoffs to members of C, 1, . . . , xk which represents a feasible distribution of payoff to members of Ag. Feasible means: 7 -8

Example n If ν({1, 2}) = 20, then possible outcomes are <20, 0 1> 18, 2 . . . , 20 >, <19, , < >, <0, >. (Actually there will be infinitely many!) 7 -9

Objections n n n Intuitively, a coalition C objects to an outcome if there is some outcome for them that makes all of them strictly better off Formally, C ⊆ Ag objects to an outcome 1, …, xn for the grand coalition if there is some outcome for C such that x’i xi for all i ∈ C > The idea is that an outcome is not going to happen if somebody objects to it! 7 -10

The Core n n The core is the set of outcomes for the grand coalition to which no coalition objects (The grand coalition is particularly interesting if the characteristic function v is superadditive) If the core is non-empty then the grand coalition is stable, since nobody can benefit from defection Thus, asking is the grand coalition stable? is the same as asking: is the core non-empty? 7 -11

Problems with the Core n n n Sometimes, the core is empty; what happens then? Sometimes it is non-empty but isn’t “fair”. Suppose: Ag = {1, 2}, ν({1}) = 5, ν({2}) = 5, ν({1, 2}) = 20. Then outcome 0 (i. e. , agent 1 gets <20, > everything) is not in the core, since the coalition {2} can object. (He can work on his own and do better. ) However, outcome 5 is in the core: <15, > even though this seems unfair to agent 2, this agent has no objection. Why unfair? Because the agents are identical! 7 -12

What if the core is empty? n n Example from “Subsidies, Stability, and Restricted Cooperation in Coalitional Games”, Meir, Rosenschein and Malizia, IJCAI 2011: “Consider three companies, A, B, and C, interested in a cooperative advertising campaign. Expected profit increases as more companies cooperate (e. g. , due to exposure in multiple media). A joint effort by all three companies will result in a total profit of $12 M (the value of the coalition {A, B, C}). ” 7 -13

What if the core is empty? n n n “Alternatively, the campaign can be carried out by just A and B (with profit of $10 M), or each company can choose to advertise alone (with profit of $4 M). ” v{A} = v{B} = v{C} = 4 v{A, B} = 10 v{A, B, C} = 12 The core is empty – the grand coalition is unstable 7 -14

Cost of Stability n n n Some outside source injects a subsidy into the coalition, in order to create stability In the above example, a subsidy of $2 M enables a payment of $5 M to A, $5 M to B, and $4 M to C, creating grand coalition stability Another example: 3 hospitals who could cooperate to buy an expensive piece of equipment (and a government subsidy would make it worthwhile for them to cooperate) 7 -15

An area of ongoing research n n n Bounding the Cost of Stability in Games over Interaction Networks, Reshef Meir, Yair Zick, Edith Elkind, and Jeffrey S. Rosenschein. The Twenty-Seventh National Conference on Artificial Intelligence (AAAI 2013), Bellevue, Washington, July 2013. To appear. Subsidies, Stability, and Restricted Cooperation in Coalitional Games, Reshef Meir, Jeffrey S. Rosenschein, and Enrico Malizia. The Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI 2011), Barcelona, July 2011, pages 301 -306. The Cost of Stability in Weighted Voting Games (Extended Abstract), Yoram Bachrach, Reshef Meir, Michael Zuckerman, Joerg Rothe, and Jeffrey S. Rosenschein. The Eighth International Joint Conference on Autonomous Agents and Multiagent Systems, Budapest, Hungary, May 2009, pages 1289 -1290. The Cost of Stability in Network Flow Games, Ezra Resnick, Yoram Bachrach, Reshef Meir, and Jeffrey S. Rosenschein. Mathematical Foundations of Computer Science 2009: The Thirty-Fourth International Symposium on Mathematical Foundations of Computer Science, Novy Smokovec, High Tatras, Slovakia, August 2009. Lecture Notes in Computer Science Volume 5734, R. Kralovic and D. Niwinski (editors), Springer, 2009, pages 636650. Power and Stability in Connectivity Games, Yoram Bachrach, Jeffrey S. Rosenschein, and Ely Porat. The Seventh International Joint Conference on Autonomous Agents and Multiagent Systems, Estoril, Portugal, May 2008, pages 999 -1006. 7 -16

How To Share Benefits of Cooperation? n n n The Shapley value is best known attempt to define how to divide benefits of cooperation fairly. It does this by taking into account how much an agent contributes. The Shapley value of agent i is the average amount that i is expected to contribute to a coalition. Axiomatically: the unique value which satisfies axioms: efficient, symmetric, additive, and gives zero to dummy players. 7 -17

Shapley Value n n Efficient: The payoff vector exactly splits the total value Symmetric: The payoff vector treats two equivalent players identically (equivalent in their contributions to all coalitions) Additive: The payoff vector gives a player the same allocation in a sum of two games as the sum of what it gives to the player in each individual game Dummy: Zero Allocation to Null Players 7 -18

Imputations n n n A pre-imputation is an efficient payoff vector An individually rational pre-imputation (i. e. , one that gives each player at least as much as it could get alone) is called an imputation Most solution concepts (and in particular, the Shapley Value for superadditive games) are imputations 7 -19

Shapley Defined n n Let δi(S) be the amount that i adds by joining S ⊆ Ag: …the marginal contribution of i to S Then the Shapley value for i, denoted φi, is: where R is the set of all orderings of Ag and Si(r) is the set of agents preceding i in ordering r. 7 -20

Representing Coalitional Games n n n It is important for an agent to know (for example) whether the core of a coalition is non-empty…so, how hard is it to decide this? Problem: naive, obvious representation of coalitional game is exponential in the size of Ag Now such a representation is: – utterly infeasible in practice; and – so large that it renders comparisons to this input size meaningless: stating that we have an algorithm that runs in (say) time linear in the size of such a representation means it runs in time exponential in the size of Ag 7 -21

How to Represent Characteristic Functions? n Two approaches to this problem: q q q try to find a complete representation that is succinct in “most” cases try to find a representation that is not complete but is always succinct A common approach: interpret characteristic function over combinatorial structure 7 -22

Representation 1: Induced Subgraph n n Represent ν as an undirected graph on Ag, with integer weights wi, j between nodes i, j ∈ Ag Value of coalition C then: i. e. , the value of a coalition C ⊆ Ag is the weight of the subgraph induced by C 7 -23

Representation 1: Induced Subgraph 7 -24

Representation 1: Induced Subgraph (Deng & Papadimitriou, 94) n Computing Shapley: in polynomial time n Determining emptiness of the core: NP-complete n Checking whether a specific distribution is in the core: co-NP-complete n But this representation is not complete 7 -25

Representation 2: Weighted Voting Games n For each agent i ∈ Ag, assign a weight wi, and define an overall quota, q. n Shapley value: #P-complete, and “hard to approximate” (Deng & Papadimitriou, 94) Core non-emptiness: in polynomial time Not a complete representation n n 7 -26

Representation 3: Marginal Contribution Nets (Ieong & Shoham, 2005) n Characteristic function represented as rules: pattern → value 7 -27

Representation 3: Marginal Contribution Nets n n Pattern is conjunction of agents, a rule applies to a group of agents C if C is a superset of the agents in the pattern Value of a coalition is then sum over the values of all the rules that apply to the coalition Example: We have: ν({a}) = 0, ν({b}) = 2, and ν({a, b}) = 7 We can also allow negations in rules (agent not present) 7 -28

Representation 3: Marginal Contribution Nets n n Shapley value: in polynomial time Checking whether distribution is in the core: co-NP-complete Checking whether the core is non-empty: co-NP-hard. A complete representation, but not necessarily succinct 7 -29

Voting Power Assume that there is a body voting “yes” or “no” on a law. How much voting power does each member have? Example: Four member body, A, B, C, D. If there is a tie, A (the chairman) gets to break the tie. How much extra voting power does A have? 30

Shapley-Shubik Index Shapley-Shubik index: Assume all orderings are possible, see when each member is pivotal (i. e. , the losing coalition becomes the winning coalition when that member joins it). The pivotal member holds the power. Arrange the members in order of their enthusiasm for the bill. We don’t know what members’ preferences are, so assume a priori that any order is equally likely (big assumption in the real world). 31

Possible orderings (pivotal) 4 members, 4!=24 orders, A breaks ties: ABCD ADBC BCAD CABD CDAB DBAC ABDC ADCB BCDA CADB CDBA DBCA ACBD BACD BDAC CBAD DABC DCAB ACDB BADC BDCA CBDA DACB DCBA A is pivotal 12/24 = 1/2 of the time, B, C, D each 1/6 of the time. A’s voting power is three times greater than other members. 32

Weighted Voting Bodies Sometimes there are bodies where different members’ votes are weighted: [q ; w 1, w 2, w 3, … , wn] – q is required to pass a law, each member i gets wi weight for his vote. Example: Council of Ministers of the European Community in 1958 (France, Great Britain, Italy, Belgium, Netherlands, Luxembourg): [12 ; 4, 4, 4, 2, 2, 1] – 12 to pass resolution F G I B N L 33

60 different ways to arrange 4, 4, 4, 2, 2, 1 [6!/(3!2!1!)]. A “ 4” occupies the pivotal position in 42 orders, a “ 2” occupies the pivotal position in 18 orders, and a “ 1” never occupies the pivotal position. Shapley-Shubik index: ( 14/60, 9/60, 0) F G I B N L % votes % power France, Germany, Italy 23. 5% 23. 3% Belgium, Netherlands 11. 8% 15. 0% Luxembourg 5. 9% 0. 0. % Luxembourg is a “dummy. ” 34

Another example (no dummy) [5 ; 2, 2, 1, 1] (A, B, C, D) Six possible orders: 2211 2121 2112 1221 1212 1122 A “ 2” pivots in five of them, so power indices are: (5/12, 1/12, 1/12) [Voting bodies with same winning coalitions are “structurally equivalent”; e. g. , body with dummy equivalent to same body without the dummy. ] 35

Shapley-Shubik Indices of Structurally Distinct Weighted Voting Bodies with Four or Fewer Voters Minimal Winning Coalitions A [1; 1] AB AB, AC, BC AB, AC ABC, ABD, ACD, BCD AB, AC, AD, BCD ABC, ABD, ACD AB, AC, AD AB, ACD, BCD ABC, ABD AB, AC, BCD Sample Weighted Shapley-Shubik Voting Body Indices (1) [2; 1, 1] (1/2, 1/2) [2; 1, 1, 1] (1/3, 1/3) [3; 2, 1, 1] (4/6, 1/6) [3; 1, 1, 1, 1] (1/4, 1/4) [4; 1, 1, 1, 1] (1/4, 1/4) [3; 2, 1, 1, 1] (3/6, 1/6, 1/6) [4; 3, 1, 1, 1] (9/12, 1/12, 1/12) [4; 2, 2, 1, 1] (2/6, 1/6, 1/6) [5; 2, 2, 1, 1] (5/12, 1/12, 1/12) [5; 3, 2, 1, 1] (7/12, 3/12, 1/12) [5; 3, 2, 2, 1] (5/12, 3/12, 1/12) 36

Power of Voting Blocs Voting blocs can be thought of as a single member with a weighted vote: [4; 3, 1, 1] has power indices (6/10, 1/10, 1/10) Single bloc among heterogeneous voters has disproportionate power (another example): [19; 13, 1, 1, … , 1] (23 “ 1”’s) – 24 distinct orders (“ 13” listed 1 st, … , 24 th), “ 13” is pivotal from 7 th through 19 th, i. e. , 13 out of 24, so has 13/24 = 54% of the voting power. 37

Things change, however, when there are two blocs in the voting body (bloc E, bloc I): Power of Two Voting Blocs in 11 -member Voting Body: Number of Bloc members % of vote of E power of E of I % of power of I |E|=2, |I|=2 18. 2 19. 4 |E|=4, |I|=2 36. 4 47. 6 18. 2 14. 3 |E|=4, |I|=4 36. 4 30. 0 Two small blocs gain power at the expense of others. One large bloc gains power at the expense of small bloc and others. Two large blocs lose power to other voters (since when they are opposed, non-bloc members hold balance). 38

Committees Example: 3 -member committee (A’s) in 9 -member body (6 other B’s). Laws must pass committee, then main body. To be approved, law must have support of 5 members, including at least 2 A’s. Orders: 39

For a B to be pivotal, it must be preceded by exactly 4 others (including either 2 or 3 A’s): (AABB) B (ABBB) (AAAB) B (BBBB) B’s pivot 28 ways, so A’s pivot 56 ways. Power index of each A is (1/3 * 56/84) = 2/9, each B is (1/6 * 28/84) = 1/18. A’s are four times more powerful than B’s. 40

Decisions made by Two Voting Bodies Two voting bodies will share power equally: Example: 3 -person body (A’s) and 5 -person body (B’s), 8 choose 3 = 56 orders. For an A to pivot, must be second A preceded by 3 or 4 or 5 B’s: (ABBB) A (ABB) (ABBBB) A (AB) (ABBBBB) A A = 28 ways Individual A’s more powerful than B’s (A’s 1/6 each of power, B’s 1/10 each of power) 41

If the voting rules change, however, and A’s must agree unanimously, their power increases dramatically: (AABBB) A BB (AABBBB) A B (AABBBBB) A = 46 ways The A’s together now have 46/56 = 82% of the power; each A has one third of that, i. e. , 0. 274; each B has 1/5 * 10/56 = 0. 036. An A’s power goes from 1. 67 to 7. 67 of a B. 42

Fill Out the Seker Hora’a! n Don’t forget to fill out the “seker hora’a” for Intro to Multi-Agent Systems, Course 67715: http: //www. huji. ac. il/seker 7 -43

In my Daughter Elianna’s 7 th Grade Class… n Class has to decide about class shirt: 44

Preferences n n n n 10 students like pink, green, blue, red 8 students like blue, green, red, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink What should be chosen? The teacher uses plurality voting, and declares pink the winner My daughter, who knows about voting and doesn’t like pink, protests 45

Preferences n n n 10 students like pink, green, blue, red 8 students like blue, green, red, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink My daughter wants a run-off election (plurality with run-off), and blue will win 46

Preferences n n n 10 students like pink, green, blue, red 8 students like blue, green, red, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink But why not use all the information, and assign 3 points for a first place vote, 2 points for second place, 1 point for third place… add up the points (Borda count) – winner is one with most points 47

Preferences n n n n 10 students like pink, green, blue, red 8 students like blue, green, red, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink– 30 points (30 + 0 + 0) blue – 44 points (10 +24 + 6 + 4) green – 47 points (20 + 16 + 9 + 2) red – 17 points (0 + 8 + 3 + 6) Winner! 48

Preferences n n n n 10 students like pink, green, blue, red 8 students like blue, red, green, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink Lying! pink – 30 points (30 + 0 + 0) Winner! blue – 44 points (10 +24 + 6 + 4) green – 39 points (20 + 8 + 9 + 2) (was 47) red – 25 points (0 + 16 + 3 + 6) (was 17) 49

Preferences n n n n n 10 students like pink, green, blue, red 8 students like blue, red, green, pink 3 students like green, blue, red, pink 2 students like red, blue, green, pink Lying! Do beneficial manipulations (lies) exist for every reasonable voting rule? (Basically, yes. ) How hard computationally is it to figure out these beneficial manipulations? In the worst case? In the “average” case? For other voting rules? Other kinds of manipulations? 50

Palestinian Authority Legislative Election, 2006 Fatah Hamas Popular Vote 410, 554 440, 409 % Popular Vote 41. 43% 44. 45% Seats (out of 132) 45 74 % Seats 56% 34% By popular vote, Hamas has about 7% more support than Fatah, but gets 64% more seats 51

Palestinian Authority Legislative Election, 2006 n n This could be a result of district division of votes, but more likely was a result of the voting rule used (mixed proportional and district voting) Simplified Example: Multiple candidates from same party split vote; A and B are from same party q q q n n 32% of voters prefer A > B > C 30% of voters prefer B > A > C 36% of voters prefer C > A > B Plurality vote: C wins Couldn’t we figure out a way to combine the first two groups of voters? We could, if we used Single Transferable Vote, STV (also known as Instant Runoff Voting, IRV) 52

Preference Aggregation: motivation p p Agents have to reach a consensus regarding a preferred alternative in a shared environment Examples: n n p Collaborative filtering (votes over movies, books…) Trust and Reputation Systems Google as a platform for (fast) preference aggregation Creation of Multiagent Plans Social choice theory gives a well-studied framework for preference aggregation 53

Social Choice Theory n n Studies how decisions or rankings are made among a collection of alternatives, when there are voters with separate opinions Group choice should reflect the individual voters’ desires (by some definition, as much as possible…) 54

Ordinal Voting Methods n n Group of voters with ranked ordinal preference over more than two alternatives, decide on an ordering, or on a choice Example (order of preferences over candidates a, b, c, d): 1 voter a c b b a d d b c c d a 55

Voting Protocol as Function p p p A voting protocol is a 1 voter function from all the a c b voters’ (linearly ordered) b a d d b c preferences to an c d a outcome ordered list, or to a winning candidate Not always just an ordering (can be cardinal preferences), not always linearly ordered… To understand social choice, think of all the functions… 56

Voting Protocol as Function n n There are many possible functions… Voting can take many more varied forms than most people think of There is no “best” function that chooses the “right” answer, but different functions can satisfy different criteria that we might find desirable What might those criteria look like? 57

Arrow’s Impossibility Theorem n n n Universality: should create a deterministic, complete social preference order from every possible set of individual preference orders Citizen sovereignty: every possible order should be achievable by some set of individual preference orders Non-dictatorship: the social welfare function should be sensitive to more than the wishes of a single voter Monotonicity: change favorable to candidate x does not hurt x Independence of irrelevant alternatives: if we restrict attention to a subset of options and apply the social welfare function only to those, then the result should be compatible with the outcome for the whole set of options No system meets all these criteria when there are two or more voters, and three or more choices 58

Gibbard–Satterthwaite Theorem (Regarding systems that choose a single winner) n For three or more candidates, one of the following three things must hold for every voting rule: q q q The rule is dictatorial; or There is some candidate who cannot win, under the rule, in any circumstances; or The rule is manipulable 59

Manipulations n n n Voters may prefer to reveal their intentions untruthfully Can happen in the full knowledge case (where a manipulating voter knows others’ votes), or 1 3 strategically 2 (heuristically) without full knowledge of others’ votes This is presumably undesirable, since the outcome may be one that does not maximize 11 10 10 social welfare 4 3 60

Many Kinds of Election Tampering p Manipulation (strategic voting) n p p p p p Coalitional manipulation Bribery Micro-bribery Adding candidates Removing candidates Partition of candidates (cascading elections) Run-off partition of candidates Adding voters Removing voters Partition of voters (hierarchical 2 -stage election) 61

A Few Examples, A Few Criteria n n n Sequential Pairwise Voting Pareto Criterion Plurality Voting Condorcet Winner and Condorcet Loser Criteria Plurality with Run-off Copeland Monotonicity Criterion The Borda Count Scoring Protocols Black’s Procedure Smith’s Generalized Condorcet Criterion Single Transferable Vote 62

Ordinal Voting Methods n n Group of voters with ranked ordinal preference over more than two alternatives, decide on an ordering, or on a choice Example (order of preferences over candidates a, b, c, d): 1 voter a c b b a d d b c c d a 63

Sequential Pairwise Voting 1 c a d c d Agenda i b c a b d a a b c d b b d a c a d d b c c d a Agenda iii a b b Agenda ii a c c b a b 1 a Different possible agendas 1 Agenda iv In this example, anyone can be a winner! Rule of thumb: bring up your favorite as late as possible 64

Manipulation n n The vote, of course, is also susceptible to insincere, manipulative voting Example: third voter votes insincerely for c instead of for b (just in first election): 1 b c a c d 1 a Agenda ii (insincere third voter) 1 c b b a d d b c c d a Third voter gets second choice (d) instead of last choice (a), by lying 65

c a b b c a a b More General Form for Sequential Pairwise Voting: Voting trees c ? ? a a c b 66

Pareto Criterion n n “If every voter prefers an alternative x to an alternative y, a voting rule should not produce y as a winner” Sequential pairwise voting violates this criterion; for example, in Agenda i, d wins, even though everyone prefers b to d a b c a d c 1 d Agenda i 1 1 a c b b a d d b c c d a 67

Plurality Voting n n Each voter votes for one alternative; the one with the most votes wins Example (9 voters): 3 voters a b c n 2 voters b a c 4 voters c b a Plurality voting has c winning, even though 5 -4 majority rate c last 68

Majority Graphs n n Pairwise elections are easy to illustrate by using a majority graph A directed graph with: q q n vertices = candidates an edge (i, j) if i would beat j is a simple majority election A compact representation of voter preferences 7 -69

Even More Disturbing… n In pairwise decisions, b, which came in last in the plurality vote, would beat both c and a; and c, which won the plurality vote, would have lost all pairwise contests: Majority Graph for previous example a 6 -to-3 5 -to-4 b 3 a b c 2 b a c 4 c b a 5 -to-4 c 70

Another Majority Graph n Give preference orderings for each candidate to win with the following majority graph: 7 -71

Condorcet Winners n A Condorcet winner is a candidate that would beat every other candidate in a pairwise election. Here, ω1 is a Condorcet winner. 7 -72

Condorcet winner p p p a beats b in a pairwise election if the majority of voters prefers a to b a is a Condorcet winner if a beats any other candidate in a pairwise election Often there is no Condorcet winner, but if there is, it is obviously unique 73

Satisfying the Condorcet Criterion n n Condorcet-consistency: if a Condorcet winner exists, it must be elected Plurality voting (as we have seen) is not Condorcet-consistent q n In fact, it doesn’t even satisfy the Condorcet-loser criterion (it might choose a candidate that loses to every other candidate in pairwise elections) Sequential pairwise voting, despite its other faults, is Condorcet-consistent 74

The Condorcet Paradox c a b Voter 1 a Voter 2 b b a c b c Voter 3 75

Another Condorcet voting rule n n n Copeland: a’s score is the number of other candidates that a beats in pairwise elections The winner is the candidate with the highest score If a is a Condorcet winner, of course, its score = m-1 (where m is the number of candidates), and for any b≠a, score < m-1 76

Plurality with Run-off p Example (17 voters): 6 voters 5 voters 4 voters 2 voters a c b b b a c p p p (last 2 voters, strategic) a b c Plurality voting, a and b are top two, and a beats b in run-off by 11 -to-6 But, if last two voters changed their minds in favor of a, i. e. , a b c instead of b a c, then a and c are top two, and c beats a by 9 -to-8 a gets more first place votes, and loses an election it would have won! 77

Monotonicity Criterion “If x is a winner under a voting rule, and one or more voters change their preferences in a way favorable to x (without the changing the order in which they prefer any other alternatives), then x should still be the winner” n Straight plurality voting satisfies monotonicity n Plurality with Run-off violates it n 78

The Borda Count n n Each voter submits preferences over the m alternatives Each alternative receives q q n no points for being ranked last 1 point for being ranked second-to-last … up to m-1 points for being ranked first Points for each alternative are summed across all voters, and the alternative with the highest total is the winner 79

Borda Count Example 1 voter a b d n n 1 voter b d c c n 1 voter c a b d a 3 points 2 points 1 point 0 points With Borda count, a gets 3 points from first voter, 2 points from the second, and 0 from the third Final Borda count totals: a: 5, b: 6, c: 4, d: 3 b is the Borda winner 80

Violates Condorcet Winner Criterion 3 voters a n n 2 points c 1 point c n b b n 2 voters a 0 points Borda counts, a: 6, b: 7, c: 2 b wins, but a is the Condorcet winner Even worse: a has an absolute majority of first place votes! The existence of c allows the 2 voters to weight b over a more heavily than the 3 voters choose to weight a over b, enabling b to win the Borda count 81

Borda Count (heuristic) strategic manipulation In general, a good heuristic for manipulating the Borda count is: if a voter favors x and believes y is the most dangerous competitor, he can minimize the risk that y will beat x by putting y at the bottom of his preference list (we’ll look at this idea more in a moment…) 82

Scoring Rules n n = < 1, …, m> where i ≥ i+1. Candidate receives i points for each voter that ranks it in i’th place Examples: q q n Plurality: <1, 0, …, 0> Veto: <1, …, 1, 0> Borda: Sensitive scoring rules have m= 0 Borda is thus one example of the general class of (sensitive) scoring rules 83

A Hybrid Condorcet Voting Method A combined voting method that satisfies the Condorcet Winner criterion: Black’s Procedure If there’s a Condorcet winner, choose it. Otherwise, use the Borda count. Satisfies Pareto, Condorcet winner, and Monotonicity Criteria. 84

Smith’s Generalized Condorcet Criterion If the alternatives can be partitioned into two sets A and B such that every alternative in A beats every alternative in B in pairwise contests, then a voting rule should not select an alternative in B. Smith’s Criterion implies both the Condorcet Winner and Condorcet Loser Criteria, but not vice versa. Black’s voting procedure does not satisfy Smith’s generalized Condorcet Criterion. 85

Example (3 voters): 1 voter a b c a x x x y y y z z z w w w c a b If we partition so that A=[a, b, c] and B=[w, x, y, z], then every alternative in A beats every one in B by a 2 -to-1 vote. There is no Condorcet winner (a, b, and c beat each other cyclically). Borda counts are a: 11, b: 11, c: 11, w: 3, x: 12, y: 9, z: 6, so Black’s method chooses x. 86

Single Transferable Vote p p Some voting procedures try to eliminate ‘undesirable’ alternatives, one after another, until a winner is chosen STV is one such example Election proceeds in rounds In each round, each voter awards one point to candidate he ranks highest out of surviving candidates – candidate with least points is eliminated 87

STV: example a b c b b b d c a a d d d c a 2 voters b 1 voter 88

Axiomatic Evaluation of Some Voting Rules criteria voting rules Sequential Plurality w/Runoff STV Borda Copeland pareto N Y Y Y monotonicity Y Y N N Y Y Condorcet Loser Y N Y Y Majority Y Y N Y Condorcet Winner Y N N Y Smith Y N N Y 89

A sobering example a b c d e b d c e a c d b a e Condorcet STV Borda Plurality with Runoff 33 voters c e b d a 8 voters 16 voters d e c b a 18 voters 3 voters e c b d a 22 voters

An Aside – Dodgson’s rule n n Distance of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner Elect candidate with minimum dist=score 91

Complexity of Dodgson p p DODGSON-SCORE: given candidate x, a preference profile, and a threshold k, is the Dodgson score of x at most k ? [BTT 89] DODGSON-SCORE is NP-complete, DODGSON-WINNER is NP-hard [HHR 97] DODGSON-WINNER is complete for Parallel access to NP [Procaccia, Feldman, Rosenschein]: O(logm) LP-based randomized rounding algorithm (matches lower bound) 92

Manipulation n n As we saw, it is often in the voters’ interest to reveal false preferences May lead to the election of a socially bad candidate p 0 1 2 3 4 a 0 1 2 3 4 b 0 1 2 3 4 93

Other Kinds of Manipulation n n Center (running the election) removes candidates Center adds voters Center bribes voters … 94

Gibbard-Satterthwaite Theorem (reminder) n n If m=2, Plurality is nonmanipulable Let m 3. The following properties are incompatible: 1. 2. 3. Onto: any candidate can be elected Nondictatorship: there is no single voter who dictates the outcome of the election Nonmanipulability 95

Single Peaked Preferences p p A grocery store is being built. Each voter (resident of the street) wants it as close as possible to his own house. Need to choose a spot. Nonmanipulable solution: choose the median voter 96

Various “solutions” n Circumvent Gibbard-Satterthwaite by: q q n n n Single-peaked preferences Mechanism design [Bartholdi et al. SC&W 89]: Computational hardness to the rescue! [Bartholdi and Orlin SC&W 91]: STV is NPhard to manipulate A lot of recent work

Coalitional Manipulation

Coalitional manipulation p p p A coalition of manipulators cooperates in order to make p C win the election Votes are weighted Formulation as decision problem (CCWM): n n p p p Instance: a set of weighted votes which have been cast, the weights of the manipulators, p C Question: Can p win the election? Manipulation is (presumably) undesirable Bounded rationality comes to the rescue! Conitzer et al. [JACM 07]: NP-hard for a variety of voting rules, even when m is constant

Complexity as Scourge or Savior n n Computational complexity can be an obstacle to desirable computations Computational complexity can be an obstacle to undesirable computations Example: RSA encryption Manipulation is (basically) always possible, but if it’s too hard to calculate, perhaps a voting process can be manipulation-resistant 100

Some Previous Results n n n [Bartholdi and Orlin 1991] There are voting protocols that are NP-hard for a single voter to manipulate [Conitzer and Sandholm 2002, 2003 a] Some manipulations of common voting protocols are NP-hard, even for a small number of candidates [Conitzer and Sandholm 2003 b] Adding a preround to some voting protocols can make manipulation hard (even PSPACE-hard in some cases) 101

NP-hard manipulations p p Individual manipulation of some protocols is NP-hard when the number of candidates m is large Coalitional manipulation of sensitive scoring protocols is NP-hard, even when m=3 But…this may be a weak guarantee of resistance to manipulation Given a reasonable distribution, how hard is it to manipulate? (ongoing research) 102

Junta Distributions p p p Procaccia and Rosenschein [JAIR 07]: Junta distributions are hard. Susceptibility to manipulation if one can manipulate with high prob. w. r. t. a Junta distribution. Scoring rules are susceptible; very loose bound on the error window of a greedy algorithm Only scoring rules and constant m

The greedy algorithm n n Reminder: in Borda, each voter awards m-k points to candidate ranked k Greedy algorithm for coalitional manipulation [Procaccia and Rosenschein, JAIR 07]: each manipulator ranks p first, and the other candidates by inverse score

A heuristic polynomial time alg n Greedy algorithm: each voter in T ranks p first, and the other candidates in an order inversely proportional to their current score. a b p p 3. 1 2. 6 2. 1 2 1 T = 0. 5 b 3 4 3. 9 2. 9 a 0. 5 Case 1 1 2. 1 1. 6 T= 2. 1 1 0. 5 Case 2 1 105

A heuristic polynomial time alg n Greedy algorithm: each voter in T ranks p first, and the other candidates in an order inversely proportional to their current score. a b p 4 3. 9 2. 9 p 3. 1 2. 6 2. 1 2 1 T= 2. 6 1. 6 T= Case 1 3 2. 6 2. 1 1 0. 5 Case 2 1 106

Example: Algorithm is correct p 5 5 10 0 10 20 30 40 a 0 10 20 30 40 b 0 10 20 30 40

Example: Algorithm is wrong p 5 10 0 10 20 30 40 a 0 10 20 30 40 b 0 10 20 30 40

Theorem: Borda n Theorem: Given an instance of CWM with weights W for the manipulators 1. 2. If there is no manipulation, the greedy algorithm will return false If there is a manipulation with weights W max W, then the algorithm will find the manipulation with W

Example for theorem p 5 10 10 20 30 40 a 0 10 20 30 40 b 0 10 20 30 40

Window of Error Algorithm fails Manipulation exists All instances

Some Papers p p Algorithms for the Coalitional Manipulation Problem, Michael Zuckerman, Ariel D. Procaccia, and Jeffrey S. Rosenschein. SODA 2008, San Francisco, January 2008, pages 277 -286 On the Robustness of Preference Aggregation in Noisy Environments, Ariel D. Procaccia, Jeffrey S. Rosenschein, and Gal A. Kaminka. AAMAS’ 07, Honolulu, May 2007, pages 416 -422 Automated Design of Scoring Rules by Learning from Examples, Ariel D. Procaccia, Aviv Zohar, and Jeffrey S. Rosenschein. AAMAS’ 08, Estoril, Portugal, May 2008, pages 951 -958 Learning Voting Trees, Ariel D. Procaccia, Aviv Zohar, Yoni Peleg, and Jeffrey S. Rosenschein. AAAI’ 07, Vancouver, British Columbia, July 2007, pages 110 -115 http: //www. cs. huji. ac. il/~jeff/papers. html 112