Condorcet Paradox
The Condorcet Paradox refers to a situation in which majority preferences are inconsistent when individuals vote on multiple options. This paradox arises when there is no clear winner because different groups of voters prefer different options. It highlights the challenges of aggregating individual preferences into a collective decision, and it has implications for voting systems and decision-making processes.
5 Key excerpts on "Condorcet Paradox"
eBook - ePubGame Theory in the Social Sciences
A Reader-friendly Guide
- Luca Lambertini(Author)
- 2011(Publication Date)
- Routledge(Publisher)
...In summary, the method consists in pitting every candidate against every other candidate in a series of imaginary one-on-one contests, with pairwise counting. I will not delve into the details of the different procedures that can be adopted to solve the paradox, but this way out is precisely what is being adopted in the US presidential elections and elsewhere – yielding a winner, but not necessarily selecting as such the candidate who would have the largest consensus across the population of voters. A direct, although much later, follow-up to the Condorcet Paradox is Arrow's impossibility theorem, or Arrow's paradox (Arrow, 1951). Arrow's theorem proves that, when voters face three or more options, there exists no voting system capable of converting individually ranked preferences into a community-wide ranking while also meeting a certain set of criteria. An equivalent formulation of the theorem is that, in the absence of restrictions on either individual preferences or neutrality of the constitution as to feasible alternatives, there exists no social choice satisfying a set of seemingly plausible requirements. The result generalizes the Condorcet voting paradox. To make things as simple as possible, one has to think that the subject of Arrow's theorem is the problem of aggregating in an acceptable and satisfactory way the preferences of a number of individuals. This aggregation procedure is supposed to produce a system of what this literature calls social preferences, through some generally accepted mechanism that is labelled as a social welfare function, which has to meet a set of sensible conditions or rules...
eBook - ePub- Yasusuke Murakami(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...Arrow’s Condition 3 may be regarded as an attempt to secure workability for any issues. The paradox of social decision is, therefore, due to a society’s actual inability to make an overall comparison in any cases. If a society were supposed to be omniscient in this sense, the paradox would disappear. However, such dissolution is not practicable at all. 4. Intensity of Preference. We have generally assumed in this book that an individual’s decision is nothing more nor less than his preference ordering on alternatives. Few would deny that ordinality of preference is an essential element of consistent individual decisions. Particularly, it has been an accepted fact in modern economic analysis that ordinal preference theory is sufficient, as well as necessary, for rationalizing economic choice behaviours in static situations. As such theory is sufficient, any additional assumptions are to be regarded as superfluous. In other disciplines such as political science, however, individual decisions are often considered as something more than preference orderings. In this section, we shall try to introduce an additional attribute of individual decisions, and so modify Condition 3-b and the related arguments. As we suggested in the last section, an individual might be capable not only of ordering the alternatives according to his preference, but also of comparing preference intensities with respect to alternatives. For example, Mr. Jones might feel that he passionately prefers a Conservative candidate to a Liberal candidate, whereas he barely prefers the Liberal to a Labour candidate. Like Mr. Jones, we sometimes feel that we can differentiate among preference intensities. Moreover, we may conceive a society where the minority prefers an alternative much more ardently than the majority prefers the contrary alternative. We may well doubt that the majority principle still makes sense...
eBook - ePub- Bill J Gerrard, Bill J Gerrard(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...In the case of simply 20 commodities, this involves a very large reduction from 419 parameters to 39. Without special restrictions on individual preferences, such as homotheticity and resultant linear Engel curves, the existence of transitive community indifference curves and the associated symmetry of the Slutsky matrix for aggregate demand behaviour will not be guaranteed. We then have no a priori justification for making the above major reduction in the number of parameters facing the econometrician in the empirical estimation of market demand, thereby substantially increasing the difficulties of empirical estimation. THE PARETO PRINCIPLE Even aside from problems related to intransitivity of social preferences, there arises the question of whether society is ‘rational’ enough to choose an alternative x rather than alternative y, when all individuals in the society would prefer x to y. If not, then we might infer that society is not even rational enough to make a choice that is in the best interest of all its members. In line with both traditional welfare economics and the ethics of democracy, we can define the Pareto principle as being that a unanimous preference in favour of x rather than y by all members of the society (i.e. xP i y for all i, where P i denotes individual strict preference) implies that the society prefers x rather than y, that is, Condition P : If for any pair (x, y) of alternatives in S, xP i y for all individuals i in the society, then xPy. In some circumstances, condition P might be regarded of dubious desirability. Thus, if all members of the society are addicted to an activity, such as smoking or taking heroin, that is not conducive to their welfare, it might be considered desirable socially to disregard their preferences. A further possible objection to the Pareto principle occurs under conditions of uncertainty, where the alternatives x and y are risky prospects...
...Notice this is quite different from saying that there is no decision, or even from saying that they should be counted equally because there is no decision. Here there is the positive judgment, derived from the individual judgments, that they should count equally. If this instinct about a possible solution is correct, thenwe should not try and handle this kind of case, in which there aremore than two alternatives, by simple pairwise comparison. Instead it would seem to be preferable to use a more sophisticated method, which collects more information, such as the so-called Borda count. (The paradox we started with is sometimes named after Condorcet; both Condorcet and Borda were late eighteenth-century Frenchman interested in the theory of voting.) In the Borda count, everyone assigns points to their preferences, giving most points for their first preference, one less for the second and so on. It can easily be seen that if the three people (or groups) in the original example voted in this way, then the three outcomes would come out equally, each having the same number of points. Something like the Borda count, then, would seem to accord better with the intuitions lying behind democratic decision theory. However, we are not out of the minefield yet. As will be seen, there are problems about theBorda count. Merely adopting it does notmean thatwe need think nomore;we could still run into an explosion. First, however, let us examine why the Borda countmight satisfy our intuitions better. One way it could be considered is as a sort of satisfaction score with a particular outcome: the most favoured outcome would give three units of satisfaction, the next two units, and so on. So what we would be trying to discover by voting ishow various outcomeswould satisfy people...
eBook - ePub- Antonio Merlo(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...So, we took away a choice that was completely dominated, removed an irrelevant alternative, and we have seen a complete reversal in the ordering of group preferences. Do not think that this example has never occurred in the real world where an alternative was removed that could not be mentioned for political reasons, and the outcome changed completely. Economically that is an important decision. Figure 3.7 Preferences and Borda count scores after option D has been removed. These examples have all been very close races; there were no heavily favored alternatives. If some alternative has landslide support, it will win regardless, but you do not need a slim majority of support to observe these pathologies in voting outcomes. What is particularly alarming is that under these voting rules monotonicity fails. Increasing the number of people who support a particular alternative does not necessarily improve the probability that alternative will win. So, is there a light at the end of the tunnel? Are there situations where the pathologies illustrated here are unlikely to happen? We will see next that these situations do indeed exist. 3.2 Median voter theorems Arrow’s impossibility theorem is a negative result, which is important because it places bounds on what voting rules can and cannot do. In the remainder of this chapter, we will learn about three other important theorems for modern political economy, which provide positive results about when we can expect a Condorcet winner to exist. The three median voter theorems we will introduce here provide sufficient conditions for the existence of a CW and ways to characterize the CW that have wide applicability in political economy. Before we state the theorems though, we need to introduce some additional notation. Recall in our student unionization example that under the preferences in Figure 3.2 we had no CW, but under the preferences in Figure 3.3 we did have a CW...
