Safe Votes, Sincere Votes, and Strategizing

Transcription

1 Safe Votes, Sincere Votes, and Strategizing Rohit Parikh Eric Pacuit April 7, 2005 Abstract: We examine the basic notion of strategizing in the statement of the Gibbard-Satterthwaite theorem and note that to talk about strategizing presumes that we already know what it means to vote sincerely. For a strategic vote is one which differs from a sincere vote, taking into account how others are voting. Now the notion of a sincere vote is clear enough in most commonly used voting systems (which we identify with the aggregation function Ag) but there is no general notion of sincerity. It follows that there cannot be a general notion of stratagizing either. In this paper we define a notion of safe vote and study its properties. Safe votes exist with plurality, approval voting and the Borda count, but not with STV The Gibbard-Satterthwaite theorem 1 says that under certain circumstances, strategizing is unavoidable. In other words, sometimes a voter is better off voting something different from his/her true preference. An example might be the election of 2000 when certain voters whose top preference was Nader voted for their second preference Gore instead. Indeed, if most or all Nader voters had voted strategically in this sense, voting for their second preference, then some of them would have voted for Bush, but most for Gore and Gore would have become president. Talking this way assumes that we know what it means to vote sincerely. For Both authors are at the City University of New York. We thank Eva Antonakos, Steven Brams, John Crossley, Ron van der Meyden and Remzi Sanver for comments. The research of both authors was supported by a grant from the CUNY Research Foundation. The addresses are: rparikh@gc.cuny.edu epacuit@cs.gc.cuny.edu 1 Please see the appendix following the references. 1

2 unless we can define sincerity in some convincing sense, it makes no sense to talk of strategizing either. After all, if we think of an election as a game, then naturally each player will play her best strategy, using whatever information she has, and there need be no charge of insincerity. So how do we define sincerity in voting? Now if we have an aggregation function Ag which takes in the entire preference profile of all voters and returns a winner (breaking ties somehow) then it would seem that sincerity consists in each voter submitting his actual preference order as his vote. But is this really so obvious? Suppose for instance that Ag takes the bottom element of each ordering submitted and then declares as winner the candidate who is at the bottom of most orders submitted. Moreover, this property of Ag is common knowledge. Then surely the voters are likely to submit the reverse of their preference order as their vote and one might argue that the reverse of the preference order is in fact the sincere vote. Strategizing might then consist in not submitting the reverse of one s actual preference order. So there is some room for a discussion of what sincerity might be in a general context. Now in the example we just looked at, we could think of Ag as a function Ag, which first reverses all the orders submitted and then applies Ag. So Ag would count the tops of all orderings submitted and for Ag, it would be best to submit one s actual preference rather than its reverse. Thus in that case a reverse vote for Ag would be a conventionally sincere non-reversed vote for Ag and all would be well. But would it? After all, Ag and Ag are special in that they require actual orders as inputs and most ballots ask for much less. If Ag takes in as its input some sort of a ballot profile and returns a winner, then it is not obvious what the sincere ballot should be given a particular preference that a voter has. We illustrate this by an example. Let us suppose that there are 34 voters and 3 candidates A,B,C, of whom C is the incumbent. Voters may vote for one of A, B, who are new, or vote 2

3 against C if they feel his performance has been less than adequate 2. But only one of A, B, C, can be mentioned on the voter s ballot. Thus each voter names a name, one of A,B,C, and Ag looks at the three numbers, n = the number of times A is mentioned, m = the number of times B is mentioned, and p = 18 the number of times C is mentioned. The winner is A if n is largest, B if m is largest, and C if p is largest, i.e. if C received few negative votes compared to A and B s positive votes. Now what does sincerity mean in this context? Since it is no longer clear what sincerity means in a general context, we shall use the word safe defined precisely below. Safety is a clear, mathematically defined notion which applies generally, and coincides with sincerity when the latter makes sense. Definition 1 Given the aggregation function Ag, and a preference order P, a P -safe ballot by a voter v will be a ballot b such that regardless of how the others are voting, the voter is at least as well off voting b than not voting at all, and is strictly better off at least once. In other words, a safe vote dominates not voting. The aggregation function Ag is safe if it has a safe vote for each preference order P Note that if the aggregation function Ag takes in preference orders themselves as ballots, and is moreover invariant under permutations of candidates, then Ag is safe iff it has a P -safe ballot for some P. A safe vote respects the idea that an election is not a game but an expression of opinion. Since a safe vote allows a voter to do better than not voting, regardless of how others vote, it is not game theoretic, but rather communicative. Now going back to our example, if a voter prefers A (B), surely the safe vote is to vote for A (B). What if the voter prefers C? Clearly the voter cannot be asked to write in C s name in the name of sincerity as this will count against 2 The California election where Gray Davis was recalled and Arnold Schwarzenegger was elected was one where in fact the incumbent was treated differently from the other candidates. 3

4 C. It looks as if the voter should vote for one of A or B, but we will soon show that in fact there is no safe vote in this setting. Note that in some particular profile of other votes, another ballot b can be better than a safe ballot b - that is what strategizing means. But if b is not safe, then at least once, b will be worse than not voting, and b is of course at least as good as not voting. So b can never dominate b - it can only be better than b some of the time. Theorem 2 If Ag is the plurality function then the only safe vote is a vote for one s top candidate. Proof: to see that a vote for one s top candidate is safe, suppose there are k candidates, {1,..., k} and candidate 1 is some voter v s favourite. Suppose that without v s vote the votes for the candidates would be n 1,..., n k. If v votes for 1, the votes would be n 1 + 1, n 2,..., n k, and clearly 1 s chances of winning will improve. To see that this is the only safe vote, let us assume that ties are broken in favour of earlier numbered candidates. Suppose, now, that in the same context, v were to vote for some other candidate, say 2. It is then possible that n 1 = n 2 > n i for i > 2. 1 would win by the tie breaking rule. But now, after v votes for 2, n 1 < n 2 +1 and 2 will win. So in this situation v would have been better off not voting rather than voting for 2. In short, voting for 1 was the only vote which was guaranteed to be as good as or better than not voting. Remark: Naturally when one candidate s score changes by 1 point, then this situation may create a tie where there was none, or destroy one when there was one to start with. Without going into details of tie-breaking rules, we will assume that regardless of the actual rule, for any two candidates A, B, there is some situation where an increase in A s score by 1 point makes A the winner whereas without that increase, B would have been the winner. For instance if the tie breaking rule is that in case of a tie A wins, being alphabetically earlier, then a situation where A has exactly one vote less than B can be converted into one where A and B have the same scores and A wins. Whereas if the tie was supposed to be broken in B s favour, then a situation where A and B have the same score (and B winning) could be converted into 4

5 one where A wins, simply by A getting one more vote. Fishburn and Brams use a married couple (who can change a candidate s score by two) in their example described below to bypass this issue of tie-breaking. But in 1983, when [FB83] was written there still were married couples who voted the same way, so they had access to a device which we shall refrain from using. Theorem 3 If Ag is approval voting, then a safe vote is one where the voter v approves of one candidate x and all higher ranking y in v s preference ordering. Proof: Suppose that there are k candidates and assume for simplicity that v actually prefers them in the order k. Suppose that without v s vote they would have n 1,..., n k approvals respectively. Let m be the candidate who would be elected without v s vote, and suppose that v does vote, approving of candidates 1,..., r, with no gaps. Suppose moreover that v would have been better off not voting. Then it must be that after v s vote some candidate p > m, less preferred by the voter, is now elected. This means that p received an extra vote and m did not, so p r and m > r. But this is impossible given that p > m. Thus all gapless ballots are safe. Conversely, suppose that v votes with a gap, so that v votes for some candidate r but not for candidate numbered r 1. Now it could have been before v s vote that r 1 was doing as well as or better than r, but that after getting one extra point, r does better and gets elected. So v would have been better off not voting. The gap-ful strategy is not safe. In brief, the only strategies which are safe, i.e., dominate not voting are the gapless strategies, where v votes for some set 1,..., r with r < k. Theorem 4 For the Borda count procedure, sincere ballots are exactly the safe ballots. I.e., P is always P -safe. Proof Suppose (again for simplicity) that voter v s preferences are k. Suppose that without v s ballot the scores would have been 5

6 n 1,..., n k and v harms herself by voting honestly. Now with v s vote, the scores would now be n 1 + k 1,..., n i + (k i),..., n k. Since v did worse by voting, it must be that some i was elected before v s vote and some j > i is now elected. There are two kinds of tie-breaking possibilities and we consider just one. Suppose that i was leading before and j is at least even with i after v s vote. So we get, i < j, n i > n j and n i + (k i) n j + (k j). But then n i i n j j and hence n i + j n j + i which is impossible. That (for the typical systems) insincere votes can be worse than not voting, and therefore cannot be safe, is something on which we have already remarked and we skip the (easy) proof for this case. Example: Consider the following situation. There are two candidates and Ag requires as input an order among them. Ag then counts the number of times each candidate occurs at the bottom of the orders. The candidate who occurs more often is the one chosen by Ag. Note that in this case there is always a safe vote, namely to submit the reverse of one s preference order, and if one is so inclined, one could call that an insincere safe vote. So if we are not considering Plurality, Approval voting, or Borda count, then there can be safety without sincerity. Theorem 5 If Ag is the unconventional function we defined above then (i) for a voter who prefers A or B, one safe vote is a mention of that candidate. If C is the voter s last preference, then (and only then) a mention of C is another safe vote. (ii) If the voter prefers C, then there are no safe mentions. (We use the neutral term mention rather than vote as some mentions are positive and some are negative). Proof: (i) Consider the situation before v s vote. Suppose that without v s vote, A, B, C, receive m, n, p mentions respectively, and p = 18 p. If v prefers A and votes for A, then m will increase to m + 1 and there is no way that A can be harmed by this. If v mentions C, then this will decrease C s chance of winning without affecting the relative standings of A,B. This would be fine if C is v s lowest preference. But otherwise, if B, C had been tied, and A fairly low, v could get his last choice B elected. 6

7 (ii) Assume for simplicity that v s preferences are C, B, A in that order. Clearly a mention of C could harm C which v does not want. A mention of A could enable A to win out over B which would not help v. So a mention of B can be the only safe vote. However, if B and C are even, then a mention of B would give B an extra vote and thereby harm C. So in fact there is no safe vote in the case where the top preference is C. It follows from some results of Fishburn and Brams that a safe ballot (in our sense) might not always exist for all aggregation functions Ag, e.g., Hare s STV is an example of this. In our discussion below we use the figures which Fishburn and Brams [FB83] use (page 208). In their example, Mr. and Mrs. Smith s car breaks down on the way to the polls just before closing time and they are unable to vote. Their preferences, in a tight race for mayor in their town, are Mrs. Bitt (first), Mr. Huff (second) and Dr. Wogg (last). In fact the Smiths dislike Dr. Wogg. Now Mrs. Bitt is the first choice of 499 voters, Mr. Huff of 500, and Dr. Wogg of 609. Thus (without the Smiths voting) Mrs. Bitt is eliminated in the first round, and all her second votes go to the other two, 417 to Huff and 82 to Wogg. Huff is then elected. However, if the Smiths had arrived in time and voted for Mrs. Bitt as their first choice, then it would have been Mr. Huff who would have been eliminated in the first round, and his second votes, 324 and 285 respectively, would go to Wogg and Bitt respectively. So Wogg would be elected. The Smiths would have made their situation worse by voting for their favourite candidate, Mrs. Bitt, and they were lucky that their car broke down. We now show why the [FB83] result implies that there are no safe votes in STV. We note the following: 1) When there is a notion of sincerity then every safe vote must be sincere. This observation applies to plurality, approval voting, STV and Borda. Because it is clear that if you vote differently from your own preference, then it must happen at least some time that you will harm your own interest and you would have done better not voting. A non-sincere vote cannot be safe. 2) Therefore if no sincere vote is safe, no vote is safe. 7

8 Since the sincere votes with STV are not safe there are no safe votes with STV but there are with plurality, approval voting and the Borda count. This would seem to indicate at least some advantage to approval voting over STV. Does all this mean that aggregation functions which do allow for safe votes are somehow nice? Unfortunately not always, and we offer the following example. Suppose that each voter is asked to vote by either submitting a card with an A and the name of a single candidate, or else a card with a B, and a ranking of all candidates. Ag works as follows. 1) If all cards are marked with an A then the winner is decided by plurality. 2) If at least one card is marked with a B, then all A cards are ignored, and the winner is determined by STV from all the B cards. It is clear that voting for one s favourite candidate with an A card is safe, it cannot do harm. The only way that card is counted is if all votes are on A cards, in which case plurality is used and voting for one s favourite is safe. But it is extremely unlikely that it will do any good - it takes only one B card to make all A cards useless! This construction also shows easily that all four paradoxes of [FB83] can arise also with safe aggregation functions Ag. I.e., the No-show paradox, the Thwarted-majorities paradox, the Multiple-districts paradox, and the Moreis-less paradox can all arise even with safe aggregation functions. What stronger condition than safety is needed to avoid such examples, we do not know. Conjecture: We conjecture that there is a generalization of the Gibbard- Satterthwaite theorem for general functions Ag, where there is no clear notion of sincere voting. We conjecture that with suitable conditions, there is no dominant strategy. For note that in the cases where there is a notion of sincerity (e.g., where the ballots are entire preference orders, and the preference orders are used in a reasonable way), the necessity of strategizing in some cases is equivalent to the assertion that there is no dominant strategy. If there is a dominant strategy, then when sincerity makes sense, the only dominant strategy can be sincere. If the sincere strategy is not dominant, then clearly no strategy is dominant. 8

9 Now the statement that there is no dominant strategy does not require us to have a notion of sincerity to make it, and thus the non-existence of a dominant strategy ought to be generally provable for a large class of functions Ag. However, some of the pre-conditions imposed for the Arrow and Gibbard- Satterthwaite theorems to hold, only make sense if we are working with preference orders, and it will take ingenuity to come up with their more abstract versions. References [A63] K. J. Arrow, Social choice and individual values (2nd ed.), Wiley, New York, [Ba83] S. Barbera, Strategy-Proofness and Pivotal Voters: A Direct Proof of the Gibbard-Satterthwaite Theorem, International Economic Review, 24:2, , June [Ben02] Jean-Pierre Benoit, Strategic Manipulation in Games when Lotteries and Ties are Permitted, research report, NYU, [B94] S. Brams, Voting Procedures in Handbook of Game Theory, Elsevier, 2: , [BF94] S. Brams and P. Fishburn, Voting Procedures in Handbook of Social Choice and Welfare, North-Holland, [BF83] S. Brams and P. Fishburn, Approval Voting, Birkhauser, Boston, [CPP04] S. Chopra, E. Pacuit and R. Parikh, Knowledge Theoretic Properties of Strategic Voting, Proceedings of JELIA 2004 (ed. Alferes and Leite), Springer 2004, pages [D84] M. Dummett, Voting Procedures, Clarendon Press, Oxford, [FB83] P. Fishburn and S. Brams, Paradoxes of preferential voting, Mathematics Magazine, 56: ,

10 [G73] A. Gibbard, Manipulation of Voting Schemes: Econometrica, 41:4, , A General Result, [S73] M. Satterthwaite, The Existence of a Strategy Proof Voting Procedure: a Topic in Social Choice Theory, University of Wisconsin, [S75] M. Satterthwaite, Strategy-proofness and Arrow s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions, Journal of Economic Theory, 10:2, , [S70] A. Sen, Collective Choice and Social Welfare, Holden Day, San Francisco, Appendix In this appendix we formally state the Gibbard-Satterthwaite Theorem [G73, S73, S75]. The statement of the theorem and the notation follows that found in [Ba83]. For more information and a proof of the following theorem, the reader can consult [Ba83]. Let A = {1,..., n} be a set of agents, or voters, and O = {o 1,..., o k } a set of candidates. Each agent is assumed to have a preference over the set of candidates O. Formally, a preference for agent i A is a complete, reflexive, transitive and antisymmetric binary relation on O. Let P be the set of all such preferences. A preference profile is an element P P n. Let P P be a preference, then T op(p ) (the choice of P ), is the top element of P. Note that T op(p ) will always be a uniquely defined given our assumptions about P. Let P P n be a preference profile. Given a preference P, let P i P denote the preference profile that is just like P except that in the ith position, P i is replaced by P. Formally if P = P 1,..., P i,..., P n, then P i P = P 1,..., P i 1, P, P i+1,..., P n. A voting scheme is a function Ag : P n O. We denote the range of Ag by Ag[P n ]. We shall henceforth assume that this range, and so of course O has at least three elements. A voting scheme is dictatorial iff there is an agent i A such that for each P P n, Ag( P ) = T op(p i ). In other words, a voting scheme Ag is dictatorial 10

11 iff the candidate chosen by Ag is always the top choice of a single agent i (called the dictator). A voting scheme is manipulable iff there is a profile P P n, a preference P P and an agent i A such that Ag( P i P )P i Ag( P ). Here Ag( P ) is what Ag picks if the agent submits her true preference order P and Ag( P i P ) is what Ag picks if the agent submits P instead. And the second is better for the agent according to the agent s own preference ordering P i. In short, a voting scheme is manipulable iff there is an agent i who would do better (i.e., the candidate selected by Ag under P i P is preferred by i to the candidate selected by Ag under P ) by reporting preference P. Theorem 6 (The Gibbard-Satterthwaite Theorem) Let Ag be a voting scheme whose range contains more than two alternatives. Then Ag is either dictatorial or manipulable. 11