Arrow's Impossibility Theorem
Arrow's Impossibility Theorem, formulated by economist Kenneth Arrow, states that it is impossible to create a voting system that satisfies a set of desirable criteria. These criteria include unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem has significant implications for understanding the limitations of democratic decision-making processes.
7 Key excerpts on "Arrow's Impossibility Theorem"
eBook - ePubPhilosophy, Politics, and Economics
An Introduction
- Gerald Gaus, John Thrasher(Authors)
- 2021(Publication Date)
- Princeton University Press(Publisher)
...One can easily see why Arrow’s theorem is seen as a challenge to the rationality of democracy. If the aim of democracy is to generate a social decision that (1) represents the preferences of the citizens no matter what their preferences, and yet (2) is coherent, it seems that democracy aims at the impossible. Not all the conditions can be met; the set is contradictory. The Importance of Arrow’s Theorem Does Arrow’s Theorem Challenge Democracy? Arrow’s theorem shows that there is no way to construct a Social Welfare Function that is guaranteed to meet his conditions. Democracy can be seen as a way of aggregative preferences (notions of betterness) into a social decision. So then does Arrow’s theorem undermine the rationality of democracy? Interestingly, some insist that it must cause us to question whether democracy can be said to be a way to generate a reasonable social choice, while others dismiss the theorem as interesting, but not crucial. There are four important ways to challenge it. First, we might dispute whether Arrow’s conditions are really intuitively compelling; to the extent that we do not mind dropping one of the conditions, the proof should not cause concern. The pairwise independence and unrestricted domain conditions both have been subject to considerable debate. Second, it is sometimes argued that Arrow’s theorem is concerned with mere “preferences,” but democratic decision-making pertains to rational judgments about what is in the common good; so, it is said, Arrow’s problem of how to aggregate individual preferences into a social preference is irrelevant to democratic decision- making. This challenge is, we think, misguided, for at least two reasons. (a) As we have stressed throughout, a “preference” is simply a ranking of one option over another—it does not necessarily involve a liking, any sort of selfishness, etc...
eBook - ePubCollective Decision Making
Applications from Public Choice Theory
- Clifford S. Russell(Author)
- 2013(Publication Date)
- RFF Press(Publisher)
...It seems to me useful to think of public choice theory as having two major distinct, yet occasionally intertwined threads. The first of these is the study of rules (institutions) for arriving at a collective choice or ranking of alternatives (which may be individual issues or comprehensive combinations of such issues) on the basis of the choices or preferences of the individuals making up the collective unit (committee, neighborhood, state, or nation). Such choices are made all the time by actual governments as, for example, when by one means or another we decide how much pollution to tolerate, how much national defense to provide, and how much to tax ourselves to pay for the desired "outputs." Arrow's contribution was a very powerful and mildly depressing impossibility theorem in this area. He showed that there is no mechanism for aggregating individual preferences that satisfies certain quite plausible and even innocuous-sounding conditions. It would not be unfair to say that a large fraction of the energy of public choice theorists since Arrow has been devoted to qualifying, expanding, and interpreting this fundamental theorem, an enterprise which has involved the use of set theory, symbolic logic, and other mathematical tools that have given the field a special flavor of abstraction. This part of public choice theory begins with the assumption that the individuals making up the collective have well-defined (and, almost always, rational) preferences among the alternatives to be ranked or chosen from...
...This is the viability of the key assumption that it is possible to derive a single judgment about social utility, or welfare, given people’s individual utilities. Distinguishing this key assumptionmeans thatwe are, once again, in well-trodden territory and so can benefit from the considerable amount of precise and careful work which has been done in this area. The central idea here is of a social welfare function, a function which would take sets of functions relating the welfare of individuals to possible outcomes and produce from them a combined single function. The central result is the famous Arrowimpossibility theoremwhich shows that, if very modest assumptions are made, then it is not possible to have such a function. It is Arrow’s results which lie behind some of the conclusions presented episodically so far, and this is recognised when the impossibility theorem is called the Condorcet-Arrow theorem. Given Arrow’s assumptions, it can be presumed that his result follows. So, for our purposes, the most important thing is to see what these assumptions are, andwhat they and the conclusionmean. Ifwe understand the assumptions, we then know what we have to avoid if we wish to avoid his conclusions. One of Arrow’s assumptions is what he calls non-dictatorship. This is the assumption that the group choice should not be dictated by the choice of one member. Obviously if we were allowed dictatorship, we could always reach social decisions:we would just nominate one person and say that their preferences were to count as the social choice. We could, for example, reach a social decision in our original example by saying that onlyX’s preferenceswere to count. However the interesting question is not whether social choice is possible or impossible but, rather, the conditions of its possibility. That is, the plausibility of the assumptions from which impossibility follows...
eBook - ePubExperts and Democratic Legitimacy
Tracing the Social Ties of Expert Bodies in Europe
- Eva Krick, Cathrine Holst, Eva Krick, Cathrine Holst(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...A very large part of the discussion in this approach has been focused on the alleged inability of representative democracy to handle the ‘aggregation of preferences’ problem in a way that will result in also representing the will of the voters (List & Goodin, 2001). The main problem held forth in this literature is the well-known Arrow impossibility theorem, stating that when voters have three or more distinct alternatives, no ranked order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a number of criteria that are the basis for a democratic voting procedure. Following Arrow, rational choice theorists inspired by William Riker’s Rochester school, denied value to voting. Based largely on Arrow’s theory, Riker’s Liberalism against Populism (1982), declared democratic voting impossible, arbitrary and therefore meaningless. Cohen’s (1986) article, which coined the term ‘epistemic democracy’, was largely written as a response to Riker and the Rochester School (Schwartzberg, 2015). Largely accepting Riker’s critique of populist democracy ‘in the abstract’, Cohen argued that an epistemic conception of democracy required, inter alia, ‘an independent standard of correct decisions – that is, an account of justice or the common good that is independent of current consensus and the outcomes of votes’. However, as argued by Mackie (2003), Arrow’s independence condition, which is central to his approach, is not theoretically justified (see also Regenwetter, 2006; Wittman, 1995). Moreover, it is rejected by almost all human subjects as shown in behavioural social choice experiments (Mackie, 2014). Empirically, in ‘real politics’, voting cycles are almost completely absent, or trivial amongst the preferences of mass voters (Mackie, 2003, pp. 86–92; cf. Regenwetter, 2006). Mackie also shows that they are empirically undemonstrated (Mackie, 2003, pp...
eBook - ePub- Yasusuke Murakami(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...A voting paradox occurs not only in simple majority voting, but also in any democratic decision, so far as it is based on pairwise comparison. Generalized voting paradox can now be established, if Arrow’s theorem does hold in the case of exactly three alternatives or—which is equivalent—if any one of the above three theorems holds. In the following sections, we shall investigate several logical properties of Arrow’s ‘democratic’ social decision function, and finally prove Arrow’s original contention in the form of Theorem 5–3. 3. Monotonicity and Unanimity Rule. In deriving several important properties of Arrow’s social decision function, we intend to prove that Arrow’s five conditions defining the function are inconsistent. From inconsistent premises, we can derive any statement whatsoever. Any property of Arrow’s social decision function is meaningful only if it is derived from those of the five conditions which are mutually consistent. It is imperative to specify from what conditions a property in question is deduced. Let us first examine Arrow’s Condition 2 and the related concepts. As we mentioned before, Condition 2 is slightly weaker than our monotonicity, in the sense that the former concerns only a social preference while the latter makes a statement not only for social preferences but also for social indifferences. The difference is due to the fact that Arrow was interested in proving the inconsistency, rather than in presenting a complete formulation. Theorem 5–5: Monotonicity implies Arrow’s Condition 2. If Condition 2 is coupled with Condition 3, the condition of pairwise comparison, we may consider a simpler formulation introduced by Blau: Condition 2’ (Blau): Then, obviously follows: Theorem 5–6: Under Condition 3, Condition 2′ implies Condition 2...
eBook - ePub- Raghbendra Jha(Author)
- 2009(Publication Date)
- Routledge(Publisher)
...Unfortunately, it is not possible to get around this difficulty in a simple way. A whole new area of welfare economics (social choice theory) has developed around Arrow’s result. Economists have tried every conceivable method to get around Arrow’s difficulty. But so long as they stuck to the basic Arrowian framework, they found that all they managed to do was to come up with some new absurdity. To give readers an idea of such results, we present two important theorems, Gibbard’s oligarchy theorem and Sen’s impossibility of a Paretian liberal. Gibbard's oligarchy theorem Gibbard relaxed RC(1) to RC(2) and came up with a powerful oligarchy instead of a dictator. Before we prove this theorem, we need a lemma. Lemma 3.3 Let S have at least three distinct alternatives and let the GDR satisfy RC(2), WP, and binariness. If for some a, b, a′, b′ ∈ S and for some subsets L 1 and L 2 we have aD L 1 b and a ′ D L 2 b ′ then for all distinct x, y ε S it must be the case that x D ¯ L 1 ∩ L 2 y. Thus if one set of people is nearly decisive over two alternatives and another set of people is nearly decisive over two other alternatives then the intersection of these two sets is fully decisive over all alternatives. Proof Let L 1, L 2, a, b, a′, b′ exist. Partition the set of individuals into four subsets: L 1 ⋂ L 2, L 1 − L 2, L 2 − L 1, L − L 1 ⋃ L 2 as in Figure 3.5. Consider three distinct alternatives x, y, z in S. Assign preference orderings as: Figure 3.5 Illustration of Gibbard's oligarchy theorem xP i yP i z for all i ∈ L 1 ⋂ L 2 yP i zP i x for all i ∈ L 2 − L 1 zP i xP i y for all i ∈ L 1 − L 2 zP i yP i x for all i ∈ {L − (L 1 ⋃ L 2)}. Now, everyone in L 1 prefers x to y and everyone else opposes this choice. By near decisiveness we have xPy. Everyone in L 2 prefers y to z and everyone else opposes it. By near decisiveness we have yPz. By RC(2) we must have xPz. But only few people in L 1 ⋂ L 2 prefer x to z, to z, everyone else opposes this choice...
eBook - ePub- Bill J Gerrard, Bill J Gerrard(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...As we have seen, single-peakedness guarantees stability of the outcome under majority rule so long as there are two competing parties without ideological constraints that inhibit their spatial mobility across the one-dimensional policy spectrum. However, once there is more than one policy dimension, and individuals differ in their marginal trade-offs between these different dimensions, preferences are no longer single-peaked and the required extremal restrictedness condition, that guarantees stability under majority rule, will be broken. THE RATIONALITY OF ARROW’S IIA CONDITION Relaxing the unrestricted domain condition, therefore, does not provide a particularly attractive escape route under majority rule once one moves to a multidimensional policy space. If one wishes to maintain the unrestricted domain condition, together with the Pareto and non-dictatorship conditions, a corollary of Arrow’s theorem is that one must relax the independence of irrelevant alternatives (IIA) condition if the ‘rationality’ properties of a social ordering are to be achieved. It should be noted that the achievement of a social ordering itself implies a form of ‘independence of irrelevant alternatives condition’, which can be defined by two conditions involving the social choice set C(S). As noted above, in the ‘irrational’ case of socially intransitive preferences resulting from Table 8.1, C(S) may be empty with no ‘majority winner’ capable of beating all other alternatives in S = { x,y,z } under majority rule. We then have the following conditions for C (.) itself to behave ‘rationally’: Condition I : The social choice rule over S generates a non-empty choice set C(T) when faced with any non-empty subset T of S. Condition I thus excludes cases of cyclical preferences, such as may occur under majority rule...
