Arrow's paradox

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Kenneth Arrow

In decision theory, Arrow's paradox or Arrow's impossibility theorem states that when voters have three or more alternatives, it is not possible to design a system of voting that allows reflecting the preferences of individuals in a global preference of the community so that at the same time certain "rational" criteria are met:

Absence of a "dictator", that is, of a person who has the power to change the group's preferences.
Partner efficiency
Independence of irrelevant alternatives.

This theorem was disclosed and demonstrated for the first time by the Nobel Prize Winner in Economics Kenneth Arrow in his doctoral thesis Social choice and individual values, and popularized in his book of the same name published in 1951 The original article, A Difficulty in the Concept of Social Welfare, was published in The Journal of Political Economy, in August 1950.

Description of theorem

In the microeconomic field, the behavior of individual economic agents is studied assuming that they are rational. By rationality it is meant that the preferences that the agents have are transitive, complete and reflexive.

We can say that preferences are transient when, if the situation $A$ is preferred to the situation $B$ and the situation $B$ is preferred to the situation $C$ , then the situation $A$ is preferred to the situation $C$ ; this characteristic of the relationship of preference allows to establish a preferential order in the different alternatives presented to us.

The problem arises when we go from the level of individual preferences to social preferences or decisions, that is, when we try to build a rule that allows us to establish an order between the different alternatives, not at the individual level, but at the social level (group). In this case, circular relations can be given where the transitivity of the preference relation disappears (intransitivity).

A case of intransitivity is given, for example, when a set of three voters chooses between three alternatives, using the choice by a simple majority as a method of voting. The voter $A$ , prefer the option $X$ on the $Y$ e $Y$ on $Z$ The voter $B$ prefer $Y$ on $Z$ and $Z$ on $X$ The voter $C$ prefer $Z$ on $X$ and $X$ on $Y$ . In this situation, what is the preferred scale of the whole? It is an example of what is known as Condorcet's paradox.

In this case, the individual preference orders are:

A) $Y;Y>Z;X>Z}" xmlns="http://www.w3.org/1998/Math/MathML">X▪And;And▪Z;X▪Z{displaystyle X rigidY;Y engagedZ;XpurZ}Y;Y>Z;X>Z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ff1e4be7f9a4d5dacaf00006b01f57e3aeced8" style="vertical-align: -0.671ex; width:22.231ex; height:2.509ex;"/>$ (by transitivity)

B) $Z;Z>X;Y>X}" xmlns="http://www.w3.org/1998/Math/MathML">And▪Z;Z▪X;And▪X{displaystyle Y rigidZ;Z purx;Y rigidX}Z;Z>X;Y>X}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f82678586bd3e9a5b2afb2ee26ccaffe61c6b23" style="vertical-align: -0.671ex; width:22.231ex; height:2.509ex;"/>$ (by transitivity)

C) $X;X>Y;Z>Y}" xmlns="http://www.w3.org/1998/Math/MathML">Z▪X;X▪And;Z▪And{displaystyle Z PHPX;X 2005;Z/2005}X;X>Y;Z>Y}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f4892e51b05623cf3dc018d7103eea96b48694" style="vertical-align: -0.671ex; width:22.231ex; height:2.509ex;"/>$ (by transitivity)

Thus, by majority rule, we would have the following set preferences:

(1) $Y}" xmlns="http://www.w3.org/1998/Math/MathML">X▪And{displaystyle X 2005}Y}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75599305ea7a2fe2d50fbef23f7f7516f8dc01de" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;"/>$ (voting) $A$ and $C$ )

(2) $Z}" xmlns="http://www.w3.org/1998/Math/MathML">And▪Z{displaystyle Y/2003/Z}Z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7425765167b69cbb6b8ed05793f20938a6234c36" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;"/>$ (voting) $A$ and $B$ )

(3) $X}" xmlns="http://www.w3.org/1998/Math/MathML">Z▪X{displaystyle Z/2003/X}X}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b4d75b89adef1e0afecdb3b56e1e7ca0f0e943" style="vertical-align: -0.338ex; width:6.759ex; height:2.176ex;"/>$ (voting) $B$ and $C$ )

Now, as a rule of transitivity, we also have $Z}" xmlns="http://www.w3.org/1998/Math/MathML">X▪Z{displaystyle X 2005}Z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfebb824a423eb28c9ebb44dbe68c91eee8b3f9e" style="vertical-align: -0.338ex; width:6.759ex; height:2.176ex;"/>$ Which leads us to a contradictory situation.

The question posed by 'social choice' is: under what conditions is it possible for the aggregate preferences of a set of individuals to be rational (reflexive, transitive and complete), while satisfying certain axiological conditions?

Is it possible a function that aggregates all individual preferences and fulfills a minimum of conditions that we can consider democratic? Arrow conditions the aggregation rule not only to rational criteria (transitivity, completeness, reflexivity), but also to two criteria that we can call "democratic": the principle of non-dictatorship (there are no individuals who determine the ordering of social preferences independently of the preferences of the rest) and the principle of non-imposition (the ordering of social preferences depends on individual orders and is not imposed by other criteria, such as tradition or chance).

The result of Arrow's Theorem concludes that there is no preference aggregation rule that has such desirable normative properties (that the aggregation results in rational preferences, that the rule and the results are valid for any configuration of preferences, that do not go against unanimity and that the social preference between two alternatives is independent of the existence or not of third alternatives), unless the preferences are the faithful reflection of the preferences of some individual, called & #34;dictator".

Simplified statement of theorem

Arrow's Impossibility Theorem is based on establishing that a society needs to agree on an order of preference between different options or social situations. Each individual in society has his own personal preference order and the problem is to find a general mechanism (a social choice rule) that transforms the set of individual preference orders into a preference order for the whole society, which must satisfy several desirable properties:

Unrestricted domain or universality: the rule of social choice should create a full order for every possible set of individual preference orders (the result of the vote should be able to order each other all preferences and the voting mechanism should be able to process all possible sets of voter preferences)
No imposition or criterion of weak couple: if A is socially preferred to B, there must be at least one individual for whom A is preferred to B. This implies that the rule does not go against the criterion of unanimity.
Absence of dictatorship: the rule of social choice It should not be limited to following the order of preference of a single individual ignoring others.
Positive association of individual and social values or monotony: if an individual modifies his or her order of preference by promoting a certain option, the order of preference of society must respond by promoting that same option or, at most, without changing it, but never degrading it.
Independence of irrelevant alternatives: if we restrict our attention to a subset of options and apply rule of social choice to them alone, then the result should be compatible with the corresponding for the whole set of options. The changes in the way that an individual orders the "irrelevant" alternatives (i.e. those that do not belong to the subset) should not have an impact on the system that the "relevant" subset society does.

Arrow's theorem says that if the decision-making body has at least two members and at least three options from which it must decide, then it is impossible to design a social choice rule that simultaneously satisfies all of these conditions. Formally, the set of decision rules that satisfy the required criteria is empty.

Monge (2021) proves an alternative version of the theorem using utility functions, instead of decision rules. Specifically, he shows that no social welfare function can incorporate all the monotonically increasing transformations of each individual's utility functions; consequently, no ordinal aggregate utility criterion authentically represents the preferences of individuals.

Demo

To prove it we will take the axioms as true and we will see that there is a decisive voter who is a dictator (contradiction with axiom 3). Let's start with a definition.

A set $V$ of voters decisive for the alternative $x$ against $y$ Yeah. $x$ is chosen whenever every voter of $V$ prefer $x$ a $y$

Demonstration: Step I {There is a decisive voter} For each pair of alternatives $x$ , $y$ There is at least one non-vailing set, the whole of all voters. Among all these sets we take the minimum set, call it $V$ . If this set has a single voter then it is already, this is our decisive voter. Let's see the case that he has at least two voters. Sea ${displaystyle V^{*}}$ the set contained in $V$ and formed by a single voter, and ${displaystyle {hat {V}}=V-V^{*}}$ . Sea ${displaystyle V'=V^{c}}$ . Let's see what ${displaystyle V*}$ is decisive for some choice, thus coming to contradiction with which $V$ It was minimal. Sea $V$ critical $x$ or $y$ and be $z$ any other alternative, suppose ${displaystyle V^{*}}$ choose ( ${displaystyle xyz}$ ), ${displaystyle {hat {V}}}$ vote ( ${displaystyle zxy}$ ) and all in ${displaystyle V'}$ vote ( ${displaystyle yzx}$ ). We must notice that everyone in $V$ prefer $x$ a $y$ and all in ${displaystyle V'}$ $y$ a $x$ So, like $V$ was decisive in the choice. $x$ instead of a $y$ . Now, ${displaystyle {hat {V}}}$ is less than $V$ , then it is not decisive at all, in particular it is not decisive in the election $y$ or $z$ , then society prefers $y$ a $z$ . We use transientness, society prefers $x$ a $y$ but also $y$ a $z$ , then prefer $x$ a $z$ . But if we see the votes the only one that voted $x$ above $z$ That's it. ${displaystyle V^{*}}$ Then ${displaystyle V^{*}}$ is decisive for $x$ or $z$ and here we have the contradiction that $V$ It was minimal.

Demonstration: Step II (this decisive voter is a dictator)Sea $J$ a member of society, we say that ${displaystyle a{bar {D}}b}$ Yeah. $a$ is preferred by society whenever $J$ prefer to $a$ and no matter the rest of the votes. And we say ${displaystyle aDb}$ Yeah. $a$ is preferred by society if $J$ prefer $a$ and the rest of society to $b$ . We see that ${displaystyle a{bar {D}}b}$ is the condition of dictatorship, while ${displaystyle aDb}$ It is to be decisive.

At this point we must demonstrate a lemma that will be useful to us.

Lema: Suppose we have three alternatives, $a,b,c$ , then:

${displaystyle aDbRightarrow a{bar {D}}c}$

and

${displaystyle aDbRightarrow c{bar {D}}b}$

Demonstration (from the motto): Here $J$ this priority, ${displaystyle abc}$ And suppose the rest prefers $b$ before $a$ or $c$ . Like ${displaystyle aDb}$ , then society prefers $a$ a $b$ . As all individuals prefer $b$ a $c$ society, then, by transitivity, prefers $a$ a $c$ . Axiom 5 assures us that whenever $J$ prefer $a$ a $c$ society will, too. This is it. ${displaystyle a{bar {D}}c}$ . To prove that ${displaystyle aDbRightarrow c{bar {D}}b}$ Suppose $J$ order the alternatives in order ${displaystyle cab}$ and all other voters order them ${displaystyle cba}$ or ${displaystyle bca}$ . How we do ${displaystyle aDb}$ society prefers $a$ instead of a $b$ . unanimously society prefers $c$ a $a$ . Transitivity gives us that society prefers $c$ on $b$ . And for axiom 5 we have to ${displaystyle c{bar {D}}b.quad blacksquare }$

We can follow the test now. We have to see that ${displaystyle a{bar {D}}b}$ for all couple of alternatives. ${displaystyle 1.x{bar {D}}zquad 2.z{bar {D}}yquad 3.x{bar {D}}yquad 4.y{bar {D}}zquad 5.z{bar {D}}xquad 6.y{bar {D}}x}$ Test 1 comes directly from the motto with ${displaystyle a=x,b=y}$ and ${displaystyle c=z}$ . Similarly we have 2. Now we have to ${displaystyle a=x,b=z}$ and ${displaystyle c=y}$ They give us 3 and 4. The 5 and 6 tests are similar. ${displaystyle blacksquare }$

Interpretations of Arrow's Theorem

Arrow's Theorem is often expressed in non-mathematical language by the phrase "No voting system is fair". However, this sentence is incorrect or, at best, imprecise, since it would be necessary to clarify what is meant by a fair voting mechanism. Although Arrow himself uses the term "fair" to refer to his criteria, it is not at all evident that this is the case.

The most discussed criterion is independence from irrelevant alternatives since it seems excessively "strong". And so, with a narrower definition of "irrelevant alternatives" that excludes those candidates from the Smith set, some Condorcet methods satisfy the Arrow properties.

Arrow's impossibility theorem starts from a very curious situation and it is the following: if we face the three alternatives simultaneously to the social vote we would have a triple tie to one vote, since agent A would vote for option X, B for Y and C for Z. Furthermore, if we look closely, each of the three votes in which one has to vote between two alternatives, we obtain a triple tie with two votes, that is: as explained above, between X and Y we have that X obtained two votes from A and C against Y with the vote of B (section 1), in section (2) Y won against Z and in 3) Z won over X with another two votes. Triple tie to two votes. When alternatives are faced two by two, it is shown again that there is no logical possibility of choosing, in fact, the contradiction obtained is indicative of that impossibility (that X is indirectly preferred to Z and that Z was preferred to X in direct voting). This leads us to the next question, is it not that the starting situation is the cause of the impossibility? We are looking for a social decision system starting from a situation in which an impossibility of choosing in the conditions formulated a posteriori already exists, given the tie at three initially and subsequent ties at two votes. But we can go further, given that individual preferences and by extension the ordinal utility functions of each individual are incomparable with each other and therefore cannot be added objectively, Arrow's theorem is a logical consequence of the starting model.

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