Unique Winner Research Articles

Parameterized complexity of candidate nomination for elections based on positional scoring rules

Consider elections where the set of candidates is partitioned into parties, and each party must nominate exactly one candidate. The Possible President problem asks whether some candidate of a given party can become the unique winner of the election for some nominations from other parties. We perform a multivariate computational complexity analysis of Possible President for several classes of elections based on positional scoring rules. We consider the following parameters: the size of the largest party, the number of parties, the number of voters and the number of voter types. We provide a complete computational map of Possible President in the sense that for each choice of the four possible parameters as (i) constant, (ii) parameter, or (iii) unbounded, we classify the computational complexity of the resulting problem as either polynomial-time solvable or NP-complete, and for parameterized versions as either fixed-parameter tractable or W[1]-hard with respect to the parameters considered.

Hardness of candidate nomination

We consider elections where the set of candidates is split into parties and each party can nominate just one candidate. We study the computational complexity of two problems. The Possible President problem asks whether a given party candidate can become the unique winner of the election for some nominations from other parties. The Necessary President is the problem to decide whether a given candidate will be the unique winner of the election for any possible nominations from other parties. We consider several different voting rules and show that for all of them the Possible President problem is NP-complete, even if the size of each party is at most two; for some voting rules we prove that the Necessary President is coNP-complete. Further, we formulate integer programs to solve the Possible President and Necessary President problems and test them on real and artificial data.

Preserving Condorcet Winners under Strategic Manipulation

Condorcet extensions have long held a prominent place in social choice theory. A Condorcet extension will return the Condorcet winner as the unique winner whenever such an alternative exists. However, the definition of a Condorcet extension does not take into account possible manipulation by the voters. A profile where all agents vote truthfully may have a Condorcet winner, but this alternative may not end up in the set of winners if agents are acting strategically. Focusing on the class of tournament solutions, we show that many natural social choice functions in this class, such as the well-known Copeland and Slater rules, cannot guarantee the preservation of Condorcet winners when agents behave strategically. Our main result in this respect is an impossibility theorem that establishes that no tournament solution satisfying a very weak decisiveness requirement can provide such a guarantee. On the bright side, we identify several indecisive but otherwise attractive tournament solutions that do guarantee the preservation of Condorcet winners under strategic manipulation for a large class of preference extensions.

On the Likelihood of the Borda Effect: The Overall Probabilities for General Weighted Scoring Rules and Scoring Runoff Rules

The Borda Effect, first introduced by Colman and Poutney (Behav Sci 23:15–20, 1978), occurs in a preference aggregation process using the Plurality rule if given the (unique) winner there is at least one loser that is preferred to the winner by a majority of the electorate. Colman and Poutney (1978) distinguished two forms of the Borda Effect: the Weak Borda Effect, describing a situation under which the unique winner of the Plurality rule is majority dominated by only one loser; and the Strong Borda Effect, under which the Plurality winner is majority dominated by each of the losers. The Strong Borda Effect is well documented in the literature as the Strong Borda Paradox. Colman and Poutney (1978) showed that the probability of the Weak Borda Effect is not negligible; but they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing, for three-candidate elections, representations of the limiting probabilities of the (Weak) Borda Effect for the whole family of scoring rules and scoring runoff rules. Our analysis leads us to highlight that there is a relation between the (Weak) Borda Effect and Condorcet efficiency. We perform our analysis under the assumptions of Impartial Culture and Impartial Anonymous Culture, which are two well-known assumptions often used for such a study.

The truncated geometric election algorithm: Duration of the election

The present paper makes three distinct improvements over an earlier investigation of Kalpathy and Ward. We analyze the length of the entire election process (not just one participant’s duration), for a randomized election algorithm, with a truncated geometric number of survivors in each round. We not only analyze the mean and variance; we analyze the asymptotic distribution of the entire election process. We also introduce a new variant of the election that guarantees a unique winner will be chosen; this methodology should be more useful in practice than the previous methodology. The method of analysis includes a precise analytic (complex-valued) approach, relying on singularity analysis of probability generating functions.

Ties Matter: Complexity of Manipulation when Tie-Breaking with a Random Vote

We study the impact on strategic voting of tie-breaking by means of considering the order of tied candidates within a random vote. We compare this to another non deterministic tie-breaking rule where we simply choose candidate uniformly at random. In general, we demonstrate that there is no connection between the computational complexity of computing a manipulating vote with the two different types of tie-breaking. However, we prove that for some scoring rules, the computational complexity of computing a manipulation can increase from polynomial to NP-hard. We also discuss the relationship with the computational complexity of computing a manipulating vote when we ask for a candidate to be the unique winner, or to be among the set of co-winners.

Determining Possible and Necessary Winners Given Partial Orders

Usually a voting rule requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a voting rule, a profile of partial orders, and an alternative (candidate) c, two important questions arise: first, is it still possible for c to win, and second, is c guaranteed to win? These are the possible winner and necessary winner problems, respectively. Each of these two problems is further divided into two sub-problems: determining whether c is a unique winner (that is, c is the only winner), or determining whether c is a co-winner (that is, c is in the set of winners). We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We completely characterize the complexity of possible/necessary winner problems for the following common voting rules: a class of positional scoring rules (including Borda), Copeland, maximin, Bucklin, ranked pairs, voting trees, and plurality with runoff.

A Voting System for Internet Based Democracy

Many voting systems are focused on resolving tight or inconclusive group preferences and produce a unique winner within a limited time frame. Such approaches are not feasible for typical web based interest groups with decentralized organization structure. These groups are often loosely organized. If at all, they show low entry and exit barriers which make it hard to define clear rules about who is in and out. Associated security issues make it hard to verify such a distinction even if it did exist.This paper suggests a voting system that is suitable for decision making in a decentralized and loosely organized group of activists. The voting mechanics employs an iterative voting procedures where participants have to solve a sequence of social dilemma games in order to identify themselves as group members. In theory this mechanism could work even without any central registration service through transitive approvals from universally accepted representatives.

Redistributive policies with heterogeneous social preferences of voters

There is growing evidence on the roles of fairness and other-regarding preferences as fundamental human motives. Call voters with fair preferences, as in Fehr and Schmidt (1999), fair-voters. By contrast, traditional political economy models are based on selfish-voters who derive utility solely from “own” payoff. In a general equilibrium model with endogenous labor supply, a mixture of fair and selfish voters choose optimal policy through majority voting. First, we show that majority voting produces a unique winner in pairwise contests over feasible policies (the Condorcet winner). Second, we show that a preference for greater fairness leads to greater redistribution. An increase in the number of fair voters can also lead to greater redistribution. Third, we show that in economies where the majority are selfish-voters, the decisive policy could be chosen by fair-voters, and vice versa. Fourth, while choosing labor supply, even fair voters behave exactly like selfish voters. We show how this apparently inconsistent behavior in different domains (voting and labor supply) can be rationalized within the model.

Powerball, Expected Value, and the Law of (very) Large Numbers

In this paper, we consider some combinatorial and statistical aspects of the popular “Powerball” lottery game. It is not difficult for students in an introductory statistics course to compute the probabilities of winning various prizes, including the “jackpot” in the Powerball game. Assuming a unique jackpot winner, it is not difficult to find the expected value and the variance of the probability distribution for the dollar prize amount. In certain circumstances, the expected value is positive, which might suggest that it would be desirable to buy Powerball tickets. However, due to the extremely high coefficient of variation in this problem, we use the law of large numbers to show that we would need to buy an untenable number of tickets to be reasonably confident of making a profit. We also consider the impact of sharing the jackpot with other winners.

A high-precision high-resolution WTA-MAX circuit of O(N) complexity

The design and implementation of a high-precision, high-resolution winner-takes-all-maximum (WTA-MAX) circuit of O(N) complexity are presented. The proposed technique also allows straightforward implementation of loser-takes-all-minimum (LTA-MIN) circuit. To satisfy a given precision specification, conventional techniques rely on matching of an N-size transistor array, where N is the input count of the circuit. Thanks to the algorithmic structure of the proposed technique, the circuit performs the computation locally, which makes the precision of the overall system independent of the input count. Some circuit topologies use positive feedback to assure selection of a unique winner. Instead proposed technique relies on priority assignment to the inputs, making it well-suitable for continuous-time operation. Finally, a WTA-MAX circuit is designed in AMS 0.8-/spl mu/m double-poly double-metal 5-V CMOS process, and the simulation results are presented for several input counts.

64% Majority rule in Ducal Venice: Voting for the Doge*

A recent result of Caplin and Nalebuff (1988) demonstrates that, under certain conditions on individual preferences and their distribution across society, super-majority rule performs well as a social decision rule. If the required super-majority is chosen appropriately, the rule yields a unique winner and voter cycles cannot occur. The voting procedure for electing a Doge in medieval Venice, developed in 1268, employed a super-majority requirement agreeing with the Caplin and Nalebuff formula. We present a brief history of the Venetian political institutions, show how the rule was employed, and argue that it contributed to the remarkable centuries-long political stability of Venice.

On Lotteries with Unique Winners

Lotteries with the unique maximum property and the unique winner property are considered. Tight lower bounds are proven on the domain size of such lotteries.

Joint concept formation

Many concept formation systems construct disjoint-concept trees. However, a priori imposed tree structures may restrict the application of these systems in some domains. A joint concept formation scheme is thus proposed, which learns from observation, and constructs acyclic directed concept graphs (trees are a special case). We show that the joint concept formation system can avoid or alleviate some problems the disjoint concept formation system would face, such as the unique winner and oscillation problems. We also demonstrate that a joint concept formation system is able to generate a concept tree if such a regularity is found among the data. The experimental results are consistent with the expectations that the joint system is a generalized version of the disjoint system and improves the learning performance. Joint concept formation extends the classic works, such as COBWEB and ARACHNE.

On the sum-of-ranks winner when losers are removed

Suppose that m alternatives are linearly ranked from best to worst by each of a number of judges, and that alternative x is the unique winner on the sum-of-ranks basis. It is shown that it is possible to construct a situation (with an appropriate number of judges) such that the initial winner x will be a sum-of-ranks loser within every proper subset of the original set of alternatives that contains x and at least one other alternative, except that x is the winner in exactly one subset that contains x and one other alternative.

Some inconsistencies in judging problems

Suppose that I individuals are ordered on the basis of the sums of ranks assigned independently by J judges, and that there is a unique winner. If the winner is deleted and the ranks assigned to the remaining individuals are adjusted, then an ordering of the reduced set is obtained. This ordering is said to be consistent with the original ordering if the relative positions of all remaining individuals are unchanged. An examination is made of conditions under which an individual, who was beaten by the winner and at least one other person in the original ranking, emerges as the winner in the reduced ranking. Such an inconsistency is called an interchange.