How to Manipulate Staff Decisions using Voting:

Transcription

1 How to Manipulate Staff Decisions using Voting: Voting is Neither Analysis nor Decision Making, PhD Research Fellow, US Naval War College The opinions contained in this briefing are those of the author and of the world s mathematicians and scientists for the last three centuries who study voting, they do not reflect official policy of the US Naval War College, the US Navy, or the US Government. Introduction Military officers and civilian analysts frequently prioritize or rank alternatives, either as an individual or as part of some form of group (committee, study group, tiger team etc.) for complicated high risk and high pay-off situations. After all the study and assessment of information and data has taken place, it is often the case that the members of the group do not agree on the interpretation of the information and thus do not come to consensus on the ranking of alternatives. A commonly used method of Executive Decision Making to deal with this situation is to vote on the prioritization. However, every possible (not just conceivable) voting method is subject to a number of paradoxes that make them ideal for manipulation to produce many different results. The decision is not what decision alternative is best supported by the facts or best supports the goal but is what voting method gives me an answer closest to the one I want. It is in fact impossible, not just hard(!), to create a fair or objective voting method when there are three or more alternatives AND if the voting method is based on ordinal scoring of preferences. This presentation explores systematic methods for manipulating voting processes to achieve desired results, and how to defend against such behavior. Only by being aware of the opportunities for manipulation can the officer or civilian analyst avoid being manipulated and avoid inadvertently manipulating the process. Only by being aware of the impossibility of creating a voting process free of paradoxes can the intelligent officer or civilian analyst avoid wasting time attempting to do so. This presentation also provides advice on how to proceed when ordered to make a group decision by voting, and how to proceed when faced with an unpalatable decision from a vote. 1

2 Why do you need to know? Manipulate Staff Decisions using Voting 2 Why do you need to know? 1. You will have to get decisions through staffs, committees, study groups, tiger teams, etc. Prioritizing (ranking) alternatives Selecting programs for funding Recommending Courses of Action After all the reading, analysis and discussion Your colleagues and you disagree on how to interpret the information You might not actually come to a consensus One way to deal with this is to vote on the prioritization 2. Staffs, committees etc. often try to do analysis by voting. 3. Voting is also fundamental to our society Everyone has a vote, and every vote counts You swore an oath to uphold our democratic constitution You need to convince your colleagues and boss to accept the voting method that gives the answer you believe is best for National Security. Image: Portrait of Niccolò Machiavelli by Santi di Tito, 2

3 Ballot Box and Terminology Staffer N F D A E C B Staffer 2 B D A C F E Staffer 1 A B C D E F Staff Consensus C E F B A D C > E > F > B > A > D F > N D > N A > N E > N C > N B Manipulate Staff Decisions using Voting 3 Alternatives A, B, C, D,... Individual preferences or choice, for example Ballot box and terminology Each individual has a single preference ranking of alternatives, for example might prefer A to B, B to C, etc. Written as A > B > C etc. Voting Method Staff Staff Choice Numbers Ignore ties, results are not changed but argument is simplified A procedure for combining individual choices into a combined group choice. I.e. replacing the collection of individual choices with a single choice from among all possible rank orders of the alternatives. Any group, committee, staff, tiger team, study team, etc that uses voting as a method to prioritize alternatives A single preference ranking of all alternatives (as though the staff was a single entity). This ranking is one of all possible rankings. For ten items there are 3,628,800 possible rankings without ties, or more than 102 million with ties. 3

4 Is it possible for more than half the voters to prefer someone who lost the election? A Trick Question Manipulate Staff Decisions using Voting 4 A Trick Question 4

5 The Setup Consider three alternatives D, H and J. The best single alternative is desired. Each member of staff or the electorate vote for their top choice. And D wins with 45% of the vote. 45% 44% 11% D H J Manipulate Staff Decisions using Voting 5 The Setup 5

6 In Theory, Yes But when we consider second and third best preference we might get this! More than half the voters (51%) prefer the loser H. Would this ever happen for real? 22% 23% 15% 29% 7% 4% D D H H J J H J D J H D J H J D D H Manipulate Staff Decisions using Voting 6 In Theory, Yes Plurality Voting: each person votes top choice. Alternatives: D, H, J Votes cast by the Staff (or Electorate) D: 45% H: 44% J: 11% So who won? D won with 45% of the vote. But more than half the voters, 51% to be precise, preferred a loser H! So, which was the most preferred alternative? OK, so has this ever happened in the real world? Yes, actually, it has and does! 6

7 1980 U.S. Senate Race NY Alphonse D Amato 45% Elizabeth Holtzman 44% Jacob Javits 11% Manipulate Staff Decisions using Voting US Senate Race New York Plurality Voting: each person votes top choice. Alternatives D: Alphonse D Amato H: Elizabeth Holtzman J: Jacob Javits Votes gained at the election were Alphonse D Amato 45% Elizabeth Holtzman 44% Jacob Javits 11% (Example drawn from chapter 5 of Mathematics and Politics: Strategy, Voting, Power and Proof, Alan D. Taylor, Springer-Verlag 1995.) Images: Alphonse D Amato, Elizabeth Holtzman, Jacob Javits, 7

8 1980 U.S. Senate Race NY Alphonse D Amato 45% Elizabeth Holtzman 22% 23% 15% 29% 7% 4% D D H H J J H J D J H D J H J D D H 44% Jacob Javits 11% Manipulate Staff Decisions using Voting 8 So who won? D Amato won with 45% of the vote US Senate Race in New York But more than half the voters, 51% to be precise, preferred Holtzman (who lost) to D Amato (who won)! So, who was the most preferred candidate? What is causing the problem? Plurality voting does not take into account all the preferences of the voters? How might we fix this? Score the level of preference for a candidate Borda Count Allow voters to vote for more than one candidate modified approval voting (Example drawn from chapter 5 of Mathematics and Politics: Strategy, Voting, Power and Proof, Alan D. Taylor, Springer-Verlag 1995.) 8

9 Borda Count Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A D A B D D B B A B C A A C C B C D B B D D C D A C C A Score Tallies are: A: 12 B: 13 C: 6 D: 11 So ordering is B > A > D > C Use my Borda Count method to take into account strength of preference for each alternative! What if the Staff were never given C in the first place? Manipulate Staff Decisions using Voting 9 Borda count example Jean-Charles, chevalier de Borda (May 4, 1733 February 19, 1799) was a French mathematician, physicist, political scientist, and sailor. In 1770, Borda formulated a ranked preferential voting system that is referred to as the Borda count. Each voter rank orders all the alternatives, and the rank becomes the score for that voter s preference. This is arithmetically identical to COA comparison using rank orders, and we all remember how that turned out In this example, what if the exact same staff was never given C to consider in the first place? Or were told to simply ignore C in further analysis? Do they need to revote? After all C came last... Example taken from chapter 14 of Games of Strategy, Avinash Dixit and Susan Skeath, W W Norton & Company

10 Borda Count Member 1 Member 2 Member 3 Member 4 Member 5 Member 6 Member 7 Score A D A B D D B B A B D A A D D B D A B B A Tallies are: A: B: D: Ordering with C was: B>A>D>C Ordering without C is: D>A>B Completely reversed final list Manipulate Staff Decisions using Voting 10 Borda count example continued 10

11 Introduce Modified Approval Voting Well, that didn t work! But it did open up opportunities for manipulating the mathematically illiterate! OK, let s look at another approach modified approval voting. We ll allow voters to check off their top candidates and count the votes. Manipulate Staff Decisions using Voting 11 Introduce modified approval voting First recorded use of an early form of approval voting is the Papal Enclave of 1294 which enthroned Benedetto Gaetani as Pope Boniface VIII (Bonifacius octavus). Sometimes used by modern professional associations, was used in 19 th Century Britain, sometimes used in local politics. Critically for us however, it is frequently used by uniformed and civilian groups in the US defense industry attempting to do analysis by voting. See Image: Pope Boniface VIII 11

12 Modified Approval Voting Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 Manipulate Staff Decisions using Voting 12 Modified approval voting There are two versions of approval voting. Each voter votes for as many alternatives as he or she approves of, or each voter votes for a given number of approved alternatives (the check your top n approach). If n = 1, then we have plurality voting. If n = 1 less than the total number of alternatives we have anti-plurality, in effect one has voted against one s single least preferred alternative by voting for all the others. If n = the total number of alternatives then one has in effect abstained. Example drawn from chapter 5 of Mathematics and Politics: Strategy, Voting, Power and Proof, Alan D. Taylor, Springer-Verlag

13 Check Top 1 (plurality) Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A D B B E A: 3 B: 2 C: 0 D: 1 E: 1 Top 1: A > B > D = E > C Manipulate Staff Decisions using Voting 13 This is also plurality voting. Check top 1 13

14 Check Top 2 Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A D B B E B D D B D D D A: 3 B: 4 C: 0 D: 6 E: 1 Top 1: A > B > D = E > C Top 2: D > B > A > E > C Manipulate Staff Decisions using Voting 14 Check top 2 14

15 Check Top 3 Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A D B B E B D D B D D D C C C C C C C A: 3 B: 4 C: 7 D: 6 E: 1 Top 1: A > B > D = E > C Top 2: D > B > A > E > C Top 3: C > D > B > A > E Manipulate Staff Decisions using Voting 15 Check top 3 15

16 Check Top 4 (anti-plurality) Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A D B B E B D D B D D D C C C C C C C D E E E A A B A: 5 B: 5 C: 7 D: 7 E: 4 Top 1: A > B > D = E > C Top 2: D > B > A > E > C Top 3: C > D > B > A > E Top 4: C = D > A = B > E Manipulate Staff Decisions using Voting 16 Check top 4 This is also anti-plurality if there are 5 alternatives. 16

17 Check least liked (anti-plurality) Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A D B B E B D D B D D D C C C C C C C D E E E A A B E B B A E E A A: 5 B: 5 C: 7 D: 7 E: 4 Top 1: A > B > D = E > C Top 2: D > B > A > E > C Top 3: C > D > B > A > E Anti-plurality: C = D > A = B > E Manipulate Staff Decisions using Voting 17 Anti-plurality Vote against the single alternative you most dislike by checking it. This is equivalent to selecting the top bar one preferred alternatives. 17

18 What answer do you want? So, what answer do you want? A > B > D = E > C? D > B > A > E > C? C > D > B > A > E? C = D > B = A > E? Manipulate Staff Decisions using Voting 18 What answer do you want? Selecting how many items you check in approval voting can give you very different and contradictory results. We have managed to get an alternative C at the very top and at the very bottom of the list. Item A is also sometimes at the top, and sometimes second to last, and so on. Note that deciding you want a final list of 3 (say) items, does not in any way imply that you should just check off your top 3 items when voting. No matter how many items you check off when voting you will still get a complete list from which you may choose your list of 3 scored winners. 18

19 Introduce five popular methods Let s systematically examine five popular voting methods your colleagues will propose Plurality voting Borda count Hare system Pairwise comparison Dictatorship! (my favorite!) Manipulate Staff Decisions using Voting 19 Introduce five popular methods Plurality voting Borda count Hare system Pairwise comparison Dictatorship 19

20 Example Voting Profile Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A So here s an example voting profile Manipulate Staff Decisions using Voting 20 Example voting profile Example drawn from chapter 5 of Mathematics and Politics: Strategy, Voting, Power and Proof, Alan D. Taylor, Springer-Verlag

21 Plurality Voting Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 Tallies are: A: 3 B: 1 C: 2 D: 0 E: 1 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A And we ll start with Plurality Voting! Declare as the group choice the alternative with the largest number of first place rankings in the individual preference lists Winner is A! Manipulate Staff Decisions using Voting 21 Plurality voting 21

22 Borda Count Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 Score A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Tallies are: A: 14 B: 17 C: 16 D: 16 E: 7 Winner is B Let s try Borda Count! Takes into account individual intensity of preference, i.e. how high up the list is each alternative for each voter Manipulate Staff Decisions using Voting 22 Borda count 22

23 The Hare System Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A How about the Hare System? Successively delete less desirable alternatives. Any alternative at the top of at least half the preference lists is the winner. If none, delete the alternative at the top of the fewest preference lists and start again. Manipulate Staff Decisions using Voting 23 The Hare system Also known as the Single Transferable Vote method (designed to produce proportional representation). Introduced by the Englishman Thomas Hare in

24 The Hare System Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A No alternative at the top of four or more lists D is at the top of the fewest lists (none) so delete Manipulate Staff Decisions using Voting 24 The Hare system continued 24

25 The Hare System Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B B C C C B B B E E E A A B E C C A E E A Still no alternative at the top of four or more lists B and E at the top of fewest lists (one each) so delete Manipulate Staff Decisions using Voting 25 The Hare system continued 25

26 The Hare System Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C C C C A A C appears at the top of four lists A C A C Winner is C! Manipulate Staff Decisions using Voting 26 The Hare system continued 26

27 Sequential Pairwise comparison with Fixed Agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Sequential pairwise comparison with fixed agenda? Successive one-on-one competitions between pairs. The fixed ordering of alternatives is the agenda. First alternative pitted against the second, loser deleted and winner pitted against the third etc. Choose fixed agenda A, B, C, D, E. Manipulate Staff Decisions using Voting 27 Sequential pairwise comparison with fixed agenda 27

28 Sequential pairwise voting with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Note the non-transitive Condorcet Loop! The loop winner will be that candidate that comes after the others in the loop. So choose the fixed agenda A, B, C, D, E B C A E Manipulate Staff Decisions using Voting 28 D Try pairwise comparison with fixed agenda B, C, and D form a non-transitive ring, so D>B>C>D. A is beaten by B, C, and D, so D>B>C>D>A. E is beaten by every other alternative, so D>B>C>D>A>E. The last item in a fixed agenda is guaranteed to win over a non-transitive ring. So any agenda with B and D before C will guarantee C wins. 28

29 Sequential pairwise comparison with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Remember, our agenda is A B C D E A versus B? B wins 4 to 3 So remove A Manipulate Staff Decisions using Voting 29 Sequential pairwise comparison with fixed agenda continued 29

30 Sequential pairwise comparison with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 C C B E B D D B D C C C B B D B D D D E E E B E C C A E E Our agenda is now A B C D E B versus C? B wins 4 to 3 So remove C Manipulate Staff Decisions using Voting 30 Sequential pairwise comparison with fixed agenda continued 30

31 Sequential pairwise comparison with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 B E B D D B D B B D B D D D E E E B E E E Our agenda is now A B C D E B versus D? D wins 4 to 3 So remove B Manipulate Staff Decisions using Voting 31 Sequential pairwise comparison with fixed agenda continued 31

32 Sequential pairwise comparison with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 E D D D D D D D E E E E E E Our agenda is now A B C D E D versus E? D wins 6 to 1 So remove E Manipulate Staff Decisions using Voting 32 Sequential pairwise comparison with fixed agenda continued 32

33 Sequential pairwise comparison with fixed agenda Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 D D D D D D D Our agenda is now A B C D E Winner is D! Manipulate Staff Decisions using Voting 33 Sequential pairwise comparison with fixed agenda continued 33

34 Dictatorship Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Now let s try my favorite: Dictatorship! The dictator s preference list is the group choice. And I, the Dictator, am voter 7 Manipulate Staff Decisions using Voting 34 Dictatorship 34

35 Dictatorship Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 Now let s try my favorite: Dictatorship! The dictator s preference list is the group choice. And I, the Dictator, am voter 7. E C D B A Winner is E! Manipulate Staff Decisions using Voting 35 Dictatorship continued 35

36 Voting method defines the outcome Procedure Plurality Voting: The Borda Count: The Hare System: Sequential pairwise: Dictatorship: Winner A B C D E Manipulate Staff Decisions using Voting 36 Change the voting method, change the outcome Our five reasonable voting methods yielded five different results for our particular voting profile. And yes, Dictatorship is reasonable. The word dictator here is a technical term of art, it is nothing more than the Commander of Boss making the decision after listening or not to advice or opinion from others. There is no moral value attached to any of these procedures... nothing that says this one is the morally correct one to use. So how do you choose a voting method? By voting? If so, how do you choose the voting method to choose the voting method to choose? So what is the decision? Is it to prioritize the list or is it to choose a voting method that gives you the answer you want? The decision is to decide what answer you want, and then to choose the voting method. The purpose of the voting method is to persuade others that their voice has been heard and that the result is fair, reasonable, democratic, add whatever feel-good terms you want. 36

37 How Serious is the Problem? Is this just the result of a bizarre and unlikely example? No For large numbers of voters, more than 2/3 possible voting profiles result in different outcomes as we change the voting method Manipulate Staff Decisions using Voting 37 How serious is the problem? 37

38 Group Voting Methods and outcomes Group Voter N F D A E C B Voter 2 B D A C F E Voter 1 A B C D E F Group Group Group C E F B A D Manipulate Staff Decisions using Voting 38 Voting methods and outcomes A voting method gives you one prioritization, consistent with the voting profile, from a possibly large number of unknown and contradictory prioritizations all of which are also consistent with the voting profile. Consistency here merely means derived from a well defined voting method. One would like voting to yield a many to one mapping. It does not, it yields a many to many mapping. Some questions that might be asked are: Q: OK, so are some procedures better than others? A: What does better mean? Q: Well, can we devise a voting method in which it is impossible for unreasonable results to occur? A: No, that is provably impossible. But you might want to define unreasonable. Q: OK, how about a voting method in which it is never in anyone s best interest to lie? A: No, that is provably impossible. The real decision as I have already said -- is deciding the answer you want and choosing the voting method that gives it. 38

39 Desirable Properties of Voting Methods OK, I simply want to capture the staff consensus! Recommend a method! There exist many proposed desirable properties of voting methods. We will examine three key properties Transitivity Unanimity Independence of irrelevant alternatives Manipulate Staff Decisions using Voting 39 Desirable properties of voting methods 39

40 Transitivity A voting method respects transitivity if the consensus prefers alternative A to C whenever it prefers A to B and B to C. Staff. A. B. C. Manipulate Staff Decisions using Voting 40 Transitivity 40

41 Unanimity A voting method respects unanimity if the consensus prefers alternative A to B whenever every individual on the Staff prefers A to B. Voter 1.. A. B. Voter 2 A.. B.. Voter n. A B... Consensus. A.. B. Manipulate Staff Decisions using Voting 41 Unanimity 41

42 Independence of Irrelevant Alternatives A voting method respects Independence of Irrelevant Alternatives if removing or adding an alternative does not alter the ordering between any of the other pairs in the final outcome. Consensus A E B F D C Consensus B A E F D C Consensus B A E F D C X Manipulate Staff Decisions using Voting 42 Independence of irrelevant alternatives A voting method respects Independence of Irrelevant Alternatives if the following holds for every pair of alternatives A and B if the outcome of the voting method is that A is preferred to B, and an alternative X is subsequently introduced, or removed, and no voter changes their preferred ordering between A and B, then the voting method cannot make B become preferred to A. In the above example, A is preferred to B. Removing the lowest scoring item C from consideration should not allow B to rise above A. Neither should adding another item X cause B to rise above A. Assuming that the removal of C or the introduction of X does not change vioters preferences between the other items. This assumption is reasonable if the items being voed on are independent. If they are not independent then this voting method is completely invalid and you have other serious problems. The requirement must hold for every pair of items. Not just A and B. Many voting schemes fail to satisfy this requirement. 42

43 Properties of some voting methods Transitivity Unanimity Independence Plurality Yes Yes No Borda Yes Yes No Hare Yes Yes No Sequential Pairs No No No Dictator Yes Yes Yes Manipulate Staff Decisions using Voting 43 Properties of some voting methods Note that many other voting methods are possible. If there are 3 or more alternatives and scoring uses rank orders (ordinal numbers), then they fail to guarantee that all the requirements are always satisfied. This does not mean that every vote fails one or more requirement. It means that you cannot guarantee the requirements will be met for your specific voting profile, and you won t know whether they have all been met or not. 43

44 Impossibility Theorem For three or more alternatives and finite number of voters, then the only voting method that satisfies Transitivity and Unanimity and Independence of irrelevant alternatives is a dictatorship OR It is impossible to find a transitive voting method that satisfies unanimity, independence of irrelevant alternatives, and non-dictatorship Manipulate Staff Decisions using Voting 44 Arrow s Impossibility Theorem Discovered and proven by Kenneth Arrow in Joint winner of the Nobel Memorial Prize in Economics with John Hicks in It is as irrational to look for a non-dictatorship satisfying transitivity, unanimity and independence of irrelevant alternatives as it is to look for Euclidean triangles whose internal angles do not add up to 180 o. This does not mean that every specific vote is irrational, it means that every method can result in irrational group choice depending on the individual preference lists, and you won t know. Note that the Impossibility Theorem only applies to ranked voting profiles when there are three or more alternatives, it does not apply when the voter is asked to apply a cardinal score to each alternative or when there are only two alternatives. See and Image: Kenneth Joseph Arrow 44

45 YAPRA Prominent Libertarian Party member Paul Hager announces that he is joining Republicans. For immediate release, 28 April Hager had urged the Libertarian Party to save itself by embracing what he called democratic voting reform. He said the two party system is unassailable as long as we continue to use the current voting method, technically known as plurality voting. Hager sought to use his campaign to promote a very simple alternative called approval voting, which picks the true majority winner in races of three or more candidates while allowing the other candidates to show their true strength... Manipulate Staff Decisions using Voting 45 YAPRA Yet Another Pointless Reform Attempt Note that Approval Voting suffers from all the same problems as all positional voting methods, plus some additional ones as well. And thus are the innumerate confounded... Approval voting is probably the worst possible method to use for serious decision making or analysis. (I have not found a worse one, even looking at chicken entrails would be an improvement in that it would be no worse but faster.) 45

46 Voting Puppet Mastery Voting Puppet Mastery is an Info Op to persuade your colleagues to pick the voting method that gives an outcome you prefer better than the outcome from any other voting method. Plan it like you would any other operation! 1.What is your preferred final outcome? 2.What are other peoples preferred outcomes? 3.Identify alternative voting methods. 4.Calculate results of the different voting methods. 5.Improve the voting method that gives you the best result. 6.Recommend best voting method to the staff. Manipulate Staff Decisions using Voting 46 Voting Puppet Mastery 1. Find out every one s preferences propose a let s check where we stand by voting now without obligation before analysis starts... and then delay real vote until after analysis during analysis check out and negotiate preference lists by one-on-one conversations 2. Calculate the winner under all the different kinds of voting method 3. Before the final vote calculate and privately negotiate preference changes with team members propose the voting method that delivers the alternative highest on your list select the fixed agenda argue for removal of irrelevant alternatives to simplify the vote engage in strategic voting 4. After the vote if you don t like the result point out the unreasonableness of the outcome (there is usually something wrong with any vote), and suggest a different voting method using the same preferences already revealed 46

47 Improve the Result Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Don t forget strategic voting and negotiation! Manipulate Staff Decisions using Voting 47 If you are voter 1, 2 or 3 Improve the result You want A to be the winner, but A comes low on 4 out of seven lists. So try to persuade the other 4 voters to switch A with the candidate immediately above A. That will add 4 points to A in a Borda vote, and likely (except for approval voting) improve A s position. Most people can be persuaded to switch around candidates that are low in their priorities. If you are voter 7 You cannot get E to win, but can you get C to win? You can improve C s chances by voting against your own best interests, i.e. by voting E as the lowest of your preferences (despite the fact it is the highest) and thus improving the position of every other candidate (adding 1 point to each in a Borda vote for example). In addition, B looks strong on the other ballots, so voter seven would then switch around A and B on his list to push B down. The Hare system will make C the winner, but what if you cannot persuade the others to accept this method? Pairwise comparison has a non-transitive loop that includes C and dominates those not in the loop so can make C win using pairwise comparison with judicious selection of the fixed agenda. Borda count with negotiation and strategic voting will also make C the winner. 47

48 Approval Vote Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A I ll try Approval Voting. OK guys, we need to make the hard decision by each of us checking the two items we believe are most important! C (4) first A, B and D (3) second E (1) last Manipulate Staff Decisions using Voting 48 Try approval voting 48

49 Sequential Pairwise Voting Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A I ll try Pairwise comparison. OK guys, let s keep it simple and just compare pairs of items! Manipulate Staff Decisions using Voting 49 Try pairwise comparison 49

50 Sequential Pairwise Voting Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A B C A E D and the fixed agenda should be D, E, B, C, A Manipulate Staff Decisions using Voting 50 Try pairwise comparison B, C, and D form a non-transitive ring, so D>B>C>D. A is beaten by B, C, and D, so D>B>C>D>A. E is beaten by every other alternative, so D>B>C>D>A>E. The candidate from a non-transitive ting that comes after all other candidates from that ring in the fixed agenda will automatically beat the other candidates in the ring. So any agenda with B and D before C will guarantee C wins. 50

51 Borda Count Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Score I ll try Borda Count. OK guys, I propose the well known, simple, commonly used and trusted rank order the priority lists and add them up method! A: 14 B: 17 C: 16 D: 16 E: 7 A: 15 So now I vote strategically, i.e. against my own interests by switching C and E and switching A and B to make C the winner B: 16 C: 17 D: 16 E: 6 Manipulate Staff Decisions using Voting 51 Try Borda count 51

52 Score Plurality Positional Voting Methods Several of the methods we have looked at are examples of positional voting Score Anti-Plurality So lets look at a positional voting example Score Approval Score N Borda Score W 1 W 2.. W n-1 0 General Method Manipulate Staff Decisions using Voting 52 Where 1 = W 1 > W 2 >. > W n-1 > 0 Positional Voting 52

53 Positional Voting Example 2 voters 6 voters 3 voters 4 voters 0 voters 5 voters A A B B C C B C A C A B C B C A B A Score 1 x 0 15 A>B>C B>A>C B>C>A C>B>A Election Tally A = 3x + 8 B = 7x + 7 C = 10x /4 3/7 2/3 0 x 1 Manipulate Staff Decisions using Voting 53 A = B at x = 1/4 A = C at x = 3/7 B = C at x = 2/3 Simple Example 53

54 Which outcome do you want? A > B > C Score What outcome do you want? B > C > A Score B > A > C Score C > B > A Score Manipulate Staff Decisions using Voting 54 Which outcome do you want? For larger numbers of alternatives use Excel real easy to find all possible outcomes given the voting profile 54

55 Our previous voter profile Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 6 Voter 7 A A A C C B E B D D B D C C C B B D B D D D E E E A A B E C C A E E A Score 1 x y z 0 A>B>C>D>E A>C>B>D>E B>A>C>D>E B>C>A>D>E C>A>B>D>E C>B>A>D>E D>A>B>C>E D>B>A>C>E D>A>C>B>E A>C>B>E>D B>A>D>C>E C>A>D>B>E Can get near complete reversals with small score changes C>A>B>D>E with 10,6,4,1,0 D>A>B>C>E with 10,6,5,1,0 Manipulate Staff Decisions using Voting 55 Our previous voter profile Normalize between 1 and 0: 1 >= x >= y >= z >= 0. Can now manipulate the scores as well! For 5 alternatives there are voting profiles that will give you 96 possible outcomes as you vary the x, y, z scores (not including ties). The dramatic changes (even complete reversals) that can occur with minor changes in scores can now be understood, they are simply where candidates tallies intersect each other as the score variables change. 55

56 Which do you want to win? Winner Example Scores A A>B>C>D>E 10, 5, 4, 3, 0 B B>A>C>D>E 10, 6, 5, 4, 0 C C>A>B>D>E 10, 7, 4, 3, 0 D D>A>B>C>E 10, 7, 5, 3, 0 Manipulate Staff Decisions using Voting 56 Which do you want to win? 56

57 How Many Possible Outcomes? For positional methods, the number of possible outcomes depends on the number of alternatives: Number of alternatives n Number of possible outcomes (not counting ties!) n! (n-1)! , , , ,265, ,219,496,076,800 Not all possible outcomes will arise, depends on the specific voter profile Possible outcomes are outcomes of a vote, NOT all the possible rankings. Manipulate Staff Decisions using Voting 57 How many possible outcomes? 57

58 What is Consensus? Staffer N F D A E C B Staffer 2 B D A C F E Staffer 1 A B C D E F What exactly does consensus mean? Staff Consensus C E F B A D Manipulate Staff Decisions using Voting 58 What is Consensus? Distance between two rankings is proportional to the minimum number of contiguous flips necessary to turn one ranking into the other. The more flips necessary, the farther apart or more different the two rankings. One approach to consensus is to find a ranking that minimizes the average distance of that ranking from all staff input rankings. In this example one can turn the staff consensus into Staffer 1 s ranking in nine flips, so the staff consensus is 9 away from Staffer 1. CEFBAD CEBFAD CEBAFD CEBADF CBEADF CBAEDF CBADEF BCADEF BACDEF ABCDEF Compute the distances of all staffers from the proposed consensus and compute the average, and find that consensus ranking that minimizes this average. In real life problems there may be many (100s, even thousands) rankings that are possible consensus solutions. Takes serious compute power. Condorcet method (interpreting intransitive cycles as ties) gives one of the many possible answers. If all that is required is consensus (as opposed to some good answer!) then Condorcet is good enough and can be done by hand or with a simple spreadsheet. 58

59 What is to be done? Manipulate Staff Decisions using Voting 59 What is to be done? Image: Vladimir Ilyich Lenin 59

60 If you are the decision maker If you are the decision maker Do NOT use voting! Demand an advantages / disadvantages matrix of each alternative for each governing factor Make a decision or recommendation based on the advantages / disadvantages matrix Manipulate Staff Decisions using Voting 60 If you are the decision maker 60

61 If you are required to use voting decide what is best for National Security and collect intel on how each person would rank the alternatives If you are required to use voting then construct and recommend a voting method that scores your selection high on the final outcome. Better yet, demand ranking of all alternatives. Then do a Condorcet Vote (pairwise compare every pair of alternatives and count how often each alternative wins, look for a Condorcet Winner. Do sensitivity analysis. Manipulate Staff Decisions using Voting 61 If you are required to use voting 61

62 If you are recipient If you are on of the an receiving unpalatable end decision of an unpalatable decision Request full details of the analysis, to include Voting method used and voting profile. If they refuse, create a feasible analysis that gives the same answer. Identify legitimate interests left out of the analysis. Do sensitivity analysis based on those interests. Show how outcomes can be changed depending on method and on the missing interests. Identify concerns with the soundness of the analysis. And be aware that this method will be used against your decisions. Manipulate Staff Decisions using Voting 62 If you are receiving an unpalatable decision 62

63 The aura of objectivity, fairness and rationality that surrounds voting methods as analysis and decision methods for staffs, committees or society at large is entirely spurious. Conclusions You can usually use peoples ignorance of this to benefit national security and it is your ethical duty to do so. Manipulate Staff Decisions using Voting 63 Conclusions Voting is a great method for reducing violence during transition of political power it is a lousy method for doing analysis, and it s not much better as a method for making decisions Recommend against voting for analysis or decision making. But if your Boss or Colleagues insist on using voting, then learn how to play the game (they won t!) and win (get your COA on or near the top), and remember it s your ethical duty to do so! 63

64 Questions? Manipulate Staff Decisions using Voting 64 Some References Alan D. Taylor, Mathematics and Politics: Strategy, Voting, Power and Proof, Springer- Verlag 1995 Avinash Dixit and Susan Skeath, Games of Strategy, W.W. Norton & Company 2004 Arrow s Theorem: Jerry Kazdan, Notes on Arrow s Impossibility Theorem, John Geanakoplos, Three Brief Proofs of Arrow s Impossibility Theorem, Political Simulation, Alan D. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press 2005 The Center for Range Voting: Steven Brams, Mathematics and Democracy, Princeton University Press 2008 Edward J. Emond, David W. Mason, A New Rank Correlation Coefficient with Application to the Consensus Ranking Problem, Journal of Multi-Criteria Decision Analysis, 2003, 11(1), John G. Kemeny, J. Laurie Snell, Preference Rankings An Axiomatic Approach, in Mathematical Models in the Social Sciences, Cambridge MIT