Arrow's Theorems

In order to consider a social welfare function over only two alternatives it is necessary to modify Condition 1. As stated earlier, Condition 1 requires three distinct alternatives, and that is obviously not going to be the case when there are only two alternatives. Instead of Condition 1, Arrow requires, "every set of individual orderings of the two alternatives in question give rise to a social ordering satisfying Axioms I and II."

This next section is an extended quote of the remainder of Arrow ([1951] 2012, vol. 12, chap. V section 1).

Definition 9: By the method of majority decision is meant the social welfare function in which \(x R y\) holds if and only if the number of individuals such that \(x R_{i} y\) is at least as great as the number of individuals such that \(y R_{i} x\).

It is not hard to see that the method of majority decision satisfies Conditions 2-5 when there are only two alternatives. To show that it satisfies Condition 1 we must show that \(R\), as defined, is a weak ordering, i.e., is connected and transitive. For convenience, let \(N(x,y)\) be the number of individuals such that \(x R_{i} y\). Then:

\begin{align} \label{orgecade9c} x R y \mathit{\ if\ and\ only\ if}\ N(x,y)\ge{N}(y,x). \end{align}

Clearly, always either \(N(x,y)\ge{N}(y,x)\) or \(N(y,x)\ge{N}(x,y)\), so that,

\begin{align} \label{org326aeac} \mathit{for\ all\ x\ and\ y,}\ x R y\ \mathit{or}\ y R x, \end{align}

by \eqref{orgecade9c}, and \(R\) is connected. To show transitivity, suppose \(x R y\) and \(y R z\). Since there are only two alternatives, two of \(x\), \(y\), and \(z\) are equal. As already shown in the case of the Scitovsky compensation principle⁷, the conclusion \(x R z\) is trivial if \(x=y\) or \(y=z\). To show \(x R z\) in the case \(x=z\) is equivalent to showing \(x R x\). But, by \eqref{orgecade9c}, \(x R x\) is equivalent to the proposition \(N(x,x)\ge{N}(x,x)\), and is certainly true. Hence, transitivity holds. In conjunction with \eqref{org326aeac}, this proves \(R\) is a weak ordering, so that

\begin{align} \label{org384dc02} \mathit{the\,method\,of\,majority\,decision\,satisfies\,Condition\,1.} \end{align}

Now consider Condition 2. Let \(R_{1},\dotsc,R_{n}\) be such that \(x P y\), i.e. \(x R y\) and not \(y R x\). By \eqref{orgecade9c}, this means \(N(x,y)\ge{N}(y,x)\) but not \(N(y,x)\ge{}N(x,y)\) i.e.,

\begin{align} \label{org7c0a15f} N(x,y)\ge{}N(y,x) \end{align}

Let \(R_{i}^{\prime},\dotsc,R_{n}^{\prime}\)⁸ be a new set of individual orderings satisfying the hypothesis of Condition 2, i.e., for \(x^{\prime}\ne{x}\), \(y^{\prime}\ne{x}\), [a] \(x^{\prime} R_{i}^{\prime} y^{\prime}\) if and only if \(x^{\prime} R_{i} y^{\prime}\); [b] \(x R_{i} y^{\prime}\) implies \(x R_{i}^{\prime} y^{\prime}\); and [c] \(x P_{i} y^{\prime}\) implies \(x P_{i}^{\prime} y^{\prime}\). Consider, in particular, the last two conditions with \(y^{\prime}=y\).

\begin{align} \label{org927ada5} x R_{i} y\ \mathit{implies}\ x R_{i}^{\prime} y; \end{align} \begin{align} \label{orgd4f02b4} x P_{i} y\ \mathit{implies}\ x P_{i}^{\prime} y. \end{align}

Suppose, for some \(i\), \(y R_{i}^{\prime} x\). By Definition 1, not \(x P_{i}^{\prime} y\), and therefore, by \eqref{orgd4f02b4}, not \(x P_{i} y\). Hence, by Lemma 1(e)⁹, \(y R_{i} x\). That is,

\begin{align} \label{orga87c13c} y R_{i}^{\prime} x\ \mathit{implies}\ y R_{i} x; \end{align}

Let \(N^{\prime}(x,y)\) be the number of individuals for whom \(x R_{i}^{\prime} y\); similarly, \(N^{\prime}(y,x)\) is the number of individuals for whom \(y R_{i}^{\prime} x\). By \eqref{org927ada5}, every individual for whom \(x R_{i} y\) has the property \(x R_{i}^{\prime} y\);¹⁰ hence,\(N^{\prime}(x,y)\gt{N}(x,y)\). Similarly, from \eqref{orga87c13c}, \(N(y,x)\ge{N}^{\prime}(y,x)\). From \eqref{org7c0a15f}, \(N^{\prime}(x,y)\ge{N}^{\prime}(y,x)\) or \(N^{\prime}(x,y)\ge{N}^{\prime}(y,x)\) and not \(N^{\prime}(y,x)\ge{N}^{\prime}(x,y)\). By \eqref{orgecade9c}, this means that \(x R^{\prime} y\) but not \(y R^{\prime} x\),¹¹ where \(R^{\prime}\) is the social ordering corresponding to the set of individual orderings \(R_{1}^{\prime},\dotsc,R_{n}^{\prime}\), or \(x P^{\prime} y\), by Definition 1. Therefore, Condition 2 is satisfied.

Condition 3 (independence of irrelevant alternatives) is trivial in this case because the only set \(S\) that contains more than one member contains the entire universe, which consists of two members. If \(S\) contains one element, \(C(S)\) is that one element independent of tastes about alternatives not in \(S\); if \(S\) contains two elements, \(C(S)\) is certainly determined by individual orderings for elements in \(S\) since there are no others.

As for Condition 4 for any \(x\) and \(y\), suppose that individual orderings were such that \(y P_{i} x\) for all \(i\). Then, for everybody, \(y R_{i} x\), while, for nobody, \(x R_{i} y\). Hence, \(N(y,x)\ge{N}(x,y)\) but not \(N(x,y)\ge{N}(y,x)\), so, by \eqref{orgecade9c}, \(y P x\), and therefore not \(x R y\), by Definition 1. Hence, we do not have \(x R y\) independent of the individual orderings \(R_{1},\dotsc,R_{n}\).

Finally, as for Condition 5 (nondictatorship), suppose that there were an individual \(i\) satisfying the conditions of Definition 6. Call him \(1\). Suppose \(x P_{1} y\), while \(y P_{i} x\) for all \(i \ne 1\).¹² Then, \(x R_{1} y\), not \(x R_{i}^{\prime} y\) for \(i\ne{1}\), by Definition 1, so that \(N(x,y)=1\). Also, \(y R_{i} x\) for \(1\), so that \(N(y,x)\ge{1}=N(x,y)\). By \eqref{orgecade9c}, \(y R x\), and therefore, by Definition 1, not \(x P y\). By Definition 6, however, \(x P_{1} y\) implies \(x P y\). Hence, there cannot be any dictator, so that Condition 5 is satisfied.

Theorem 1 (Possibility Theorem for Two Alternatives): If the total number of alternatives is two, the method of majority decision is a social welfare function which satisfies Conditions 2-5 and yields a social ordering of the two alternatives for every set of individual orderings.

Theorem 1 is, in a sense the logical foundation of the Anglo-American two-party system.

For later reference, observe that the proof given above that the method of majority decision satisfies Conditions 2, 4, and 5 was independent of the assumption that there were only two alternatives. It is also true that the method of majority decision satisfies Condition 3 regardless of the total number of alternatives. From Definition 9, it is obvious that the truth or falsity of the statement \(x R y\) is invariant under any change of individual orderings which leaves invariant, for each individual, the relative position of \(x\) and \(y\). By Definition 3, \(C(S)\) is completely determined by knowing the truth or falsity of the statement \(x R y\) for every pair \(x,y\) of elements in \(S\); hence, \(C(S)\) is certainly invariant under any change of the individual orderings which leaves the orderings within \(S\) invariant.

Lemma 3: For any space of alternatives, the method of majority decision is a social welfare function satisfying Conditions 2-5.

The example of the paradox of voting given in Chapter I, Section 1, shows that the method of majority decision does not satisfy Condition 1 when there are more than two alternatives. We are now prepared to examine the construction of social welfare functions in this last case.

We shall hereafter assume that Condition 1 holds in its original form.

The method of majority decision is a satisfactory social welfare function when there are only two alternatives. Moreover, May (1952) shows that any social welfare function satisfying Arrow's condition for two alternatives is equivalent to Arrow's definition of the method of majority decision.

On the way to his general possibility theorem, Arrow looks at the special case of two individuals and three alternatives. Chapter IV, section 2 is also quoted from his book:

To illustrate the methods of analysis and serve as an introduction to the more general case, we shall consider first the formation of a social welfare function for two individuals expressing their preferences for three alternatives. Some consequences will be drawn from Conditions 1-5. It will be shown that the supposition that there is a social welfare function satisfying those conditions leads to a contradiction.

Let \(x,y,z\) be three alternatives among which choice is to be made, e.g., three possible distributions of commodities. Let \(x^{\prime}\) and \(y^{\prime}\) be variable symbols which represent possible alternatives, i.e., which range over the values \(x, y, z\). Let the individuals be designated as \(1\) and \(2\), and let \(R_{1}\) and \(R_{2}\) be the orderings by \(1\) and \(2\), respectively, of the alternatives \(x, y, z\). Let \(P_{1}\) and \(P_{2}\) be the corresponding preference relations; e.g., \(x^{\prime} P_{1} y^{\prime}\) means that individual \(1\) strictly prefers \(x^{\prime}\) to \(y^{\prime}\). It is assumed that there is no a priori reason to suppose that the individuals will not order the alternatives in any given way. For example, if it is supposed that each individual values each distribution of commodities in accordance with his preference for his personal share alone (individualistic behavior), if there is more than one commodity, and if no alternative gives any individual more of all commodities than any other alternative, then, by suitable variation of tastes, each individual may order the alternatives in any logically possible manner (see Chapter VI, Section 4, for an example).

Consequence V.2.1: if \(x^{\prime} P_{1} y^{\prime}\) and \(x^{\prime} P_{2} y^{\prime}\) then \(x^{\prime} P y^{\prime}\).

That is, if both prefer \(x^{\prime}\) to \(y^{\prime}\), then the society must prefer \(x^{\prime}\) to \(y^{\prime}\).

Proof: By Condition 4, there are orderings \(R_{1}^{\prime}\) and \(R_{2}^{\prime}\) for individuals \(1\) and \(2\), respectively, such that, in the corresponding social preference, \(x^{\prime} P^{\prime} y^{\prime}\). Form \(R_{1}^{\prime\prime}\) from \(R_{1}^{\prime}\) by raising \(x^{\prime}\), if need be, to the top while leaving the other two alternatives alone. Form \(R_{2}^{\prime\prime}\) from \(R_{2}^{\prime}\) in the same way.¹³ Since all we have done is raise alternative \(x^{\prime}\) in everyone's esteem while leaving the others alone, \(x^{\prime}\) should still be preferred to \(y^{\prime}\) by society in accordance with Condition 2, so that \(x^{\prime} P^{\prime\prime} y^{\prime}\). But, by construction, both individuals prefer \(x^{\prime}\) to \(y^{\prime}\), and society prefers \(x^{\prime}\) to \(y^{\prime}\). Since, by Condition 3, the social choice between \(x^{\prime}\) ando \(y^{\prime}\) depends only on the individual orderings of those two alternatives, it follows that whenever both individuals prefer \(x^{\prime}\) to \(y^{\prime}\), regardless of the rank of the third alternative, society will prefer \(x^{\prime}\) to \(y^{\prime}\), which is the statement to be proved.

Consequence V.2.2: Suppose that for some \(x^{\prime}\) and \(y^{\prime}\), whenever \(x^{\prime} P_{1} y^{\prime}\) and \(y^{\prime} P_{2} x^{\prime}\), \(x^{\prime} P y^{\prime}\).¹⁴ Then, whenever \(x^{\prime} P_{1} y^{\prime}\), \(x^{\prime} P y^{\prime}\).

That is, if in a given choice the will of individual \(1\) prevails against the opposition of \(2\), then individual \(1\)'s views will certainly prevail if \(2\) is indifferent or if he agrees with \(1\).

Proof: Let \(R_{1}\) be an ordering in which \(x^{\prime} P_{1} y^{\prime}\), and let \(R_{2}\) be any ordering. Let \(R_{1}^{\prime}\) be the same ordering as \(R_{1}\), while \(R_{2}^{\prime}\) is derived from \(R_{2}\) by depressing \(x^{\prime}\) to the bottom while leaving the relative positions of the other two alternatives unchanged. By construction, \(x^{\prime} P_{1}^{\prime} y^{\prime}\), \(y^{\prime} P_{2}^{\prime} x^{\prime}\). By hypothesis, \(x^{\prime} P^{\prime} y^{\prime}\), where \(P^{\prime}\) is the social preference relation derived from the individual orderings \(R_{1}^{\prime},R_{2}^{\prime}\).¹⁵ Now the only difference between \(R_{1}^{\prime},R_{2}^{\prime}\) and \(R_{1},R_{2}\) is that \(x^{\prime}\) is¹⁶ raised in the scale of individual \(2\) in the latter as compared with the former. Hence, by Condition 2, (interchanging the \(R_{i}\)'s and the \(R_{i}^{\prime}\)'s), it follows from \(x^{\prime} P^{\prime} y^{\prime}\) that \(x^{\prime} P y^{\prime}\). That is, whenever \(R_{1},R_{2}\) are such that \(x^{\prime} P_{1} y^{\prime}\) then \(x^{\prime} P y^{\prime}\).

Consequence V.2.3: If \(x^{\prime} P_{1} y^{\prime}\) and \(y^{\prime} P_{2} x^{\prime}\), then \(x^{\prime} I y^{\prime}\).

That is, if the two individuals have exactly opposing interests in the choice between two given alternatives, then society will be indifferent between the alternatives.

Proof: Suppose the consequence is false. Then, for some orderings \(R_{1}\) and \(R_{2}\) and¹⁷ and for some pair of alternatives \(x^{\prime}\) and \(y^{\prime}\) we would have \(x^{\prime} P_{1} y^{\prime}\), \(y^{\prime} P_{2} x^{\prime}\), but not \(x^{\prime} I y^{\prime}\). In that case, either \(x^{\prime} P y^{\prime}\) or \(y^{\prime} P x^{\prime}\). We will suppose \(x^{\prime} P y^{\prime}\) and show that this supposition leads to a contradiction; the same reasoning would show that the assumption \(y^{\prime} P x^{\prime}\) also leads to a contradiction.

Without loss of generality, it can be assumed that \(x^{\prime}\) is the alternative \(x\) and \(y^{\prime} = y\).¹⁸ Then we have, for the particular orderings in question, \(x P_{1} y\), \(y P_{2} x\) and \(x P y\). Since the social choice between \(x\) and \(y\) depends, by Condition 3, only on the individual choices between \(x\) and \(y\), we must have

\begin{align} \label{org92d93e1} \mathit{whenever}\ x P_{1} y\ \mathit{and}\ y P_{2} x,\ x P y. \end{align}

It will be shown that \eqref{org92d93e1} leads to a contradiction.

Suppose than individual \(1\) prefers \(x\) to \(y\) and \(y\) to \(z\), while individual \(2\) prefers \(y\) to \(z\) and \(z\) to \(x\). By \eqref{org92d93e1}, society prefers \(x\) to \(y\). Also, both prefer \(y\) to \(z\); by Consequence V.2.1, society prefers \(y\) to \(z\). Since society prefers \(x\) to \(y\) and \(y\) to \(z\), it must prefer \(x\) to \(z\). Therefore we have exhibited orderings \(R_{1},R_{2}\) such that \(x P_{1} z\), \(z P_{2} x\), but \(x P z\). Since the social choices between \(x\) and \(z\) depends only on the individual preferences for \(x\) and \(z\),

\begin{align} \label{org7e3ccd2} \mathit{whenever}\ x P_{1} z\ \mathit{and}\ z P_{2} x,\ x P z. \end{align}

Now suppose that \(R_{1}\) is the ordering \(y,x,z\) and \(R_{2}\) is the ordering \(z,y,x\). By Consequence V.2.1, \(y P x\); by \eqref{org7e3ccd2}, \(x P z\), so that \(y P z\). By the same reasoning as before,

\begin{align} \label{orgd472f38} \mathit{whenever}\ y P_{1} z\ \mathit{and}\ z P_{2} x,\ y P z. \end{align}

If \(R_{1}\) is the ordering \(y,z,x\), and \(R_{2}\) is the ordering \(z,x,y\), it follows from Consequence V.2.1 and \eqref{orgd472f38} that \(z P x\) and \(y P z\), so that \(y P x\). Hence,

\begin{align} \label{orgba9120d} \mathit{whenever}\ y P_{1} x\ \mathit{and}\ x P_{2} y,\ y P x. \end{align}

If \(R_{1}\) is the ordering \(z,y,x\), and \(R_{2}\) is the ordering \(x,z,y\), then from Consequence V.2.1 and \eqref{orgba9120d} that \(z P y\) and \(y P x\), so that \(z P x\).

\begin{align} \label{org2dd4a8a} \mathit{Whenever}\ z P_{1} x\ \mathit{and}\ x P_{2} z,\ z P x. \end{align}

If \(R_{1}\) is the ordering \(z,x,y\), and \(R_{2}\) is the ordering \(x,y,z\), then using \eqref{org2dd4a8a}, \(z P x\) and \(x P y\), so that \(z P y\).

\begin{align} \label{org87e6e98} \mathit{Whenever}\ z P_{1} y\ \mathit{and}\ y P_{2} z,\ z P y. \end{align}

From \eqref{org92d93e1} it follows from Consequence V.2.2 that, whenever \(x P_{1} y\), \(x P y\). Similarly, from \eqref{org92d93e1}-\eqref{org87e6e98}, it follows that for any pair of alternatives \(x^{\prime},y^{\prime}\), whenever \(x^{\prime} P_{1} y^{\prime}\), then \(x^{\prime} P y^{\prime}\). That is, by Definition 6, individual \(1\) would be a dictator. This is prohibited by Condition 5, so that \eqref{org92d93e1} must be false. Therefore Consequence V.2.3 is proved.

Now suppose that individual \(1\) has the ordering \(x,y,z\), while individual \(2\) has the ordering \(z,x,y\). By Consequence V.2.1,

\begin{align} \label{org425b0d0} x P y. \end{align}

Since \(y P_{1} z\), \(z P_{2} y\), it follows from Consequence V.2.3 that

\begin{align} \label{org4124c9b} y I z. \end{align}

From \eqref{org425b0d0} and \eqref{org4124c9b}, \(x P z\). but also \(x P_{1} z\), \(z P_{2} x\),¹⁹ which implies \(x I y\) by Consequence V.2.3. It cannot be that \(x\) is both preferred and indifferent to \(z\). Hence the assumption that there is a social welfare function compatible with Conditions 1-5 has led to a contradiction.

Next, Arrow provides a proof of his general possibility theorem, now generally known as Arrow's impossibility theorem.

Note: The below quote is Arrow ([1951] 2012, vol. 12, chap. IV, section3) in its entirety. It is here because it is very detailed, but Arrow's notation was apparently difficult for the typesetter to decipher, and there are many mistakes where the notation is not consistent with the descriptions. Since this is a key section of Arrow's work, I figured my corrections are worth including in as close to their intended context as I could manage.

In the following proof we assume a given social welfare function satisfying Conditions 1-5 and show that the assumption leads to a contradiction. Without loss of generality we may suppose that the entire universe is the set of three alternatives mentioned in the statement of Condition 1. In this set, all sets of individual orderings are admissible, so that we need not discuss in each case whether or not a given set is admissible. That is, the orderings which appear in the argument will be orderings only of the three alternatives in question. If we wish to be formally correct and consider the ordering of all alternatives, we can replace each set of orderings of the three given alternatives by a correspondingly admissible set of individual orderings which orders the three given alternatives in the same way.

In what follows, \(V\) will stand for a set of individuals. In particular, \(V^{\prime}\) will be a set containing a single individual and \(V^{\prime\prime}\) will be the set of all individuals.

Definition 10: The set \(V\) is said to be decisive for \(x\) against \(y\) if \(x\ne{y}\) and \(x P y\) for all sets of admissible individual ordering relations such that \(x P_{i} y\) for all \(i\) in \(V\).

This definition can be explained as follows: Let \(\overline{R}\) stand for the set of individual ordering relations \(R_{1},\dotsc,R_{n}\). The condition \(x P y\) for all \(i\) in \(V\) restricts \(\overline{R}\)'s under consideration by restricting the range of variation of those components of \(\overline{R}\) whose subscripts are in \(V\) to ordering relations having the given property with respect to \(x\) and \(y\). To each \(\overline{R}\), a given social welfare function assigns a social ordering \(R\); according to this scale we may have, in general, \(x P y\) or \(x I y\) or \(y P x\). Suppose that it so happens that, for all \(\overline{R}\) consistent with the condition that \(x P_{i} y\) for all \(i\) in \(V\), the resultant \(R\) is such that \(x P y\); then we can say that \(V\) is decisive for \(x\) against \(y\). Intuitively, the concept of decisive set can be explained as follows: A set of individuals is decisive if, whenever they all prefer \(x\) to \(y\), society prefers \(x\) to \(y\) regardless of what preferences any individuals may have concerning any alternatives other than \(x\) or \(y\). Note that a set may be decisive for \(x\) against \(y\) without being decisive for \(y\) against \(x\). For example, in the process of ratification of treaties by the Senate, any set of \(64\) senators is decisive for acceptance against rejection, any set of \(33\) senators is decisive for rejection against acceptance.

It should be emphasized that the question of whether or not a given set of individuals is decisive with respect to a given pair of alternatives, \(x\) and \(y\), is determined by the society welfare function and does not vary with the actual orderings of individuals at any given time.

Consequence V.3.1: Let \(R_{1},\dotsc,R_{n}\) and \(R_{1}^{\prime},\dotsc,R_{n}^{\prime}\) be²⁰ two sets of individual orderings such that for a given distinct \(x\) and \(y\), \(x P_i^{\prime} y\) for all \(i\) for which \(x R_{i} y\). Then, if \(x P y\), \(x P^{\prime} y\), where \(P\) and \(P^{\prime}\) are the social preference relations corresponding to \(R_{1},\dotsc,R_{n}\) and \(R_{1}^{\prime},\dotsc,R_{n}^{\prime}\), respectively.

This consequence extends Condition \(2\). If \(x\) rises or does not fall relative to \(y\) for each individual and actually rises if \(x\) and \(y\) were indifferent, and if \(x\) was socially preferred to \(y\) to begin with, then \(x\) is still preferred to \(y\), regardless of changes in preferences for alternatives other than \(y\).

Proof: In accordance with the preceding remarks, we assume there are only three alternatives altogether. Let \(z\) be the alternative which is distinct from \(x\) and \(y\). For each \(i\), define the ordering \(R_{i}^{\prime\prime}\),²¹ as follows:

\begin{align} \label{orgbdad68c} x^{\prime} R_{i}^{\prime\prime} y^{\prime} \mathit{if\ and\ only\ if\ either}\ x^{\prime} R_{i} y^{\prime} \mathit{and}\ x^{\prime}\ne{z}\ \mathit{or}\ y^{\prime}=z. \end{align}

This amounts to moving \(z\) from its position in \(R_{i}\) to the bottom but otherwise leaving \(R_{i}\) unchanged. It is easy to verify that \(R_{i}^{\prime\prime}\) is an ordering, i.e., satisfies Axioms I and II. Also, for each \(i\), \(R_{i}^{\prime\prime}\) orders the elements \(x,y\) in the same way as \(R_{i}\); i.e.,

\begin{align} \label{orgde54c95} x^{\prime} R_{i}^{\prime\prime} y^{\prime}\ \mathit{if\ and\ only\ if} \ x^{\prime} R_{i} y^{\prime}\ \mathit{for}\ x^{\prime}, y^{\prime} \ \mathit{in}\ [x,y]. \end{align}

From \eqref{orgde54c95}²² and Condition \(3\), \(C([x,y])=C^{\prime\prime}([x,y])\), where \(C(S)\) and \(C^{\prime\prime}(S)\) are the social choices made from an environment \(S\) when \(R_{1},\dotsc,R_{n}\) and \(R_{1}^{\prime\prime},\dotsc,R_{n}^{\prime\prime}\) are the sets of individual orderings, respectively. By hypothesis, \(x P y\); from Lemma 2, \(C([x,y])\) contains the single element \(x\). Hence, \(C^{\prime\prime}([x,y])\) contains the single element \(x\), or, by Lemma 2,

\begin{align} \label{org675a075} x P^{\prime\prime} y. \end{align}

Define the individual orderings \(R_{1}^{\ast},\dotsc,R_{n}^{\ast}\), as follows:

\begin{align} \label{org800e78d} x^{\prime} R_{i}^{\ast} y^{\prime}\ \mathit{if\ and\ only\ if\ either} \ x^{\prime} R_{i}^{\prime} y^{\prime}\ \mathit{and}\ x^{\prime}\!\ne\!z \ \mathit{or}\ y^{\prime}\!=\!z. \end{align}

\eqref{org800e78d} is exactly parallel to \eqref{orgbdad68c}. From \eqref{orgbdad68c}, \eqref{org800e78d}, and Definition 1, \(y P_{i}^{\prime\prime} z\), \(y P_{i}^{\ast} z\), for all \(i\). Hence,²³

\begin{align} \label{org80564e4} \mathit{if}\ x^{\prime}\ne{z}, y^{\prime}\ne{z}, x^{\prime} R_{i}^{\prime\prime} y^{\prime} \ \mathit{if\ and\ only\ if}\ x^{\prime} R_{i}^{\ast}y^{\prime}. \end{align}

Also, \(x P_{i}^{\prime\prime} z\), \(x P_{i}^{\ast} z\) for all \(i\). By \eqref{orgbdad68c}, for all \(i\) such that \(x R_{i}^{\prime\prime} y\), \(x R_{i} y\);²⁴ by hypothesis, \(x P_{i}^{\prime\prime} y\) for such \(i\), and therefore, by \eqref{org800e78d}, \(x P_{i}^{\ast} y\). Hence,

\begin{align} \label{orgb5ed9dc} \mathit{for\ all}\ y^{\prime}, x R_{i}^{\prime\prime} y^{\prime} \ \mathit{implies}\ x R_{i}^{\ast} y^{\prime}; \end{align} \begin{align} \label{org7b15a6b} \mathit{for\ all}\ y^{\prime}, x P_{i}^{\prime\prime} y^{\prime} \ \mathit{implies}\ x P_{i}^{\ast} y^{\prime}; \end{align}

By \eqref{org80564e4}-\eqref{org7b15a6b}²⁵ and \eqref{org675a075}, the hypotheses of Condition \(2\) are satisfied; hence, \(x P^{\ast} y\), From \eqref{org800e78d}, it follows, in the same manner as above, that \(C^{\ast}([x,y])=C^{\prime}([x,y])\), so that \(x P^{\prime} y\). Q.E.D.

This proof is really simple in principle. The purpose in introducing the auxiliary ordering relations \(R_{i}^{\prime\prime}\) and \(R_{i}^{\ast}\) was to permit a comparison between the two sets which would satisfy the hypotheses of Condition 2. At the same time, as far as the choice between alternatives \(x\) and \(y\) is concerned, the relations \(R_{i}^{\prime\prime}\) are essentially equivalent to the relations \(R_{i}\),²⁶ and the relations \(R_{i}^{\ast}\) are equivalent to the relations \(R_{i}^{\prime}\); this is shown by the latter part of the proof.²⁷

Consequence V.3.2: If there is some set of individual ordering relations \(R_{1},\dotsc,R_{n}\) such that \(x P_{i} y\) for all \(i\) in \(V\) and \(y P_{i} x\)²⁸ for all \(i\) not in \(V\), for some particular \(x\) and \(y\), and such that the corresponding social preference relation yields the outcome \(x P y\), then \(V\) is decisive for \(x\) against \(y\).

Proof: Let \(R_{i}^{\prime},\dotsc,R_{n}^{\prime}\) be any set of individual orderings subject only to the condition that

\begin{align} \label{orgf6ebf01} x P_{i}^{\prime} y\ \mathit{for\ all}\ i\ \mathit{in}\ V. \end{align}

To show that \(V\) is decisive, it is necessary according to Definition \(10\) to show that, for every such set \(R_{1}^{\prime},\dotsc,R_{n}^{\prime}\),²⁹ the corresponding social ordering \(R^{\prime}\) is such that \(x P^{\prime} y\). But from \eqref{orgf6ebf01} and the hypothesis that \(x P_{i} y\)³⁰ for \(i\) in \(V\), \(y P_{i} x\) for \(i\) not in \(V\), it follows that \(x P_{i} y\) whenever \(x R_{i} y\). By Consequence V.3.1, \(x P^{\prime} y\). Q.E.D. The meaning of this consequence may be formulated somewhat as follows: Imagine an observer seeing individuals write down their individual orderings and hand them to the central authorities who then form a social ordering based on the individual orderings in accordance with the social welfare function. Suppose further that this observer notices that, for a specific pair of alternatives \(x\) and \(y\), every individual in a certain set \(V\) of individuals prefers \(x\) to \(y\), while everybody not in \(V\) prefers \(y\) to \(x\), and that the resultant social ordering ranks \(x\) higher than \(y\). Then, the observer is entitled to say, without looking at any other aspects of the individual and social orderings, that \(V\) is a decisive set for \(x\) against \(y\), i.e., that, if tastes change, but in such a way that all the individuals in \(V\) still prefer \(x\) to \(y\) (though they might have changed their ranking for all other alternatives and though the individuals not in \(V\) might have changed their scale completely), then the social ordering will still rank \(x\) higher than \(y\).

Consequence V.3.3: For every \(x\) and \(y\) such that \(x\ne{y}\), \(V^{\prime\prime}\) is a decisive set for \(x\) against \(y\).

That is, if every individual prefers \(x\) to \(y\), then society prefers \(x\) to \(y\).

Proof: If we interchange \(x\) and \(y\) in Definition \(5\), then Condition \(4\) says that there exists a set of individual orderings \(R_{1},\dotsc,R_{n}\) such that not \(y R x\), where \(R\) is the social ordering corresponding to the set of individual ordering relations \(R_{1},\dots,R_{n}\). That is to say, by Lemma \(1\) (e),

\begin{align} \label{orgd20706e} x P y. \end{align}

Let \(R_{1}^{\prime},\dotsc,R_{n}^{\prime}\) be any set of individual orderings such that

\begin{align} \label{org3a098d0} x P_{i}^{\prime} y\ \mathit{for\ all}\ i. \end{align}

From \eqref{org3a098d0},³¹ certainly \(x P_{i}^{\prime} y\) for all \(i\) such that \(x R_{i}^{\prime} y\).³² Then from \eqref{orgd20706e} and Consequence V.3.1, \(x P^{\prime} y\). Since this holds for any set of orderings satisfying \eqref{org3a098d0}, it follows from the definition of \(V^{\prime}\)³³ that \(x P^{\prime} y\) for any set of orderings such that \(x P_{i} y\) for \(i\) in \(V^{\prime\prime}\), \(y P_{i} x\) for \(i\) not in \(V^{\prime\prime}\) (i.e. for no \(i\)). By Consequence V.3.2, \(V^{\prime\prime}\)³⁴ is decisive for \(x\) against \(y\).

Consequence V.3.4: If \(V^{\prime}\) is decisive for either \(x\) against \(y\) or \(y\) against \(z\), \(V^{\prime}\)³⁵ is decisive for \(x\) against \(z\), where \(x\), \(y\), and \(z\) are distinct alternatives.

Recall that \(V^{\prime}\)³⁶ is a set consisting of a single individual. The consequence asserts that, if a single individual is decisive for a given \(x\) against any other alternative, he is decisive for \(x\) against any alternative, and that, if he is decisive for any alternative against a given alternative \(z\) , he is decisive for any alternative against \(z\). This is the first consequence in which some paradoxes begin to appear.

Proof: (a) Assume than \(V^{\prime}\) is decisive for \(x\) against \(y\). We seek to prove that \(V^{\prime}\) is decisive for \(x\) against any \(z\ne{x}\).

Let the individual in \(V^{\prime}\)³⁷ be given the number \(1\). Let \(R_{1},\dotsc,R_{n}\) be a set of individual ordering relations satisfying the conditions

\begin{align} \label{org94c9e4e} x P_{1} y, \end{align} \begin{align} \label{org634be31} y P_{i} z\ \mathit{for\ all}\ i, \end{align} \begin{align} \label{org917d5b5} z P_{i} x\ \mathit{for}\ i\ne{1}. \end{align}

From \eqref{org94c9e4e}, \(x P_{i} y\) for all \(i\) in \(V^{\prime}\) therefore, by Definition \(10\),

\begin{align} \label{org4de4b4a} x P y, \end{align}

where \(P\) is the social preference relation corresponding to the set of individual orderings \(R_{1},\dotsc,R_{n}\). From \eqref{org634be31}, \(y P_{i} z\) for all \(i\) in \(V^{\prime\prime}\) so that, from Consequence V.3.3 and the definition of a decisive set,

\begin{align} \label{org39daca7} y P z. \end{align}

By Condition \(1\), the social ordering relation satisfies Axioms I and II and hence Lemma \(1\)(c). Therefore, from \eqref{org4de4b4a} and \eqref{org39daca7},

\begin{align} \label{org484cbd7} x P z. \end{align}

But, from \eqref{org94c9e4e} and \eqref{org634be31}, \(x P_{1} y\) and \(y P_{1} z\), so that \(x P_{1} z\), or

\begin{align} \label{orgba989e2} x P_{i} z\ \mathit{for\ all}\ i\ \mathit{in}\ V^{\prime}. \end{align}

\eqref{org917d5b5} may be written

\begin{align} \label{org039e23e} z P_{i} x\ \mathit{for\ all}\ i\ \mathit{not\ in}\ V^{\prime}. \end{align}

By \eqref{org484cbd7}-\eqref{org039e23e}, the hypotheses of Consequence V.3.2 are satisfied, so that \(V^{\prime}\) must be decisive for \(x\) against \(z\). That is, there is one set of individual ordering relations in which all the individuals in \(V^{\prime}\) (in this case, one individual) prefer \(x\) to \(z\) while all other individuals prefer \(z\) to \(x\), and the social welfare function is such as to yield a social preference for \(x\) against \(z\). This suffices, by Consequence V.3.2, to establish than \(V^{\prime}\) is decisive for \(x\) against \(z\).

(b) Now assume that \(V^{\prime}\) is decisive for \(y\) against \(z\). Let the individual in \(V^{\prime}\)³⁸ have the number \(1\), and let \(R_{1},\dotsc,R_{n}\) be a set of individual ordering relations such that³⁹

\begin{align} \label{orgfd106f7} x P_{1} y, \end{align} \begin{align} \label{orge85262b} y P_{i} z\ \mathit{for\ all}\ i, \end{align} \begin{align} \label{org9c64fb7} z P_{i} x\ \mathit{for}\ i\ne{1}. \end{align}

Then, as in part (a) of the proof, \eqref{orgfd106f7} implies that \(x P y\), while \eqref{orge85262b} implies that \(y P z\), so that \(x P z\). But, from \eqref{orgfd106f7} and \eqref{orge85262b}, \(x P_{1} z\), which, in conjunction with \eqref{org9c64fb7}, shows that the hypotheses of Consequence V.3.2 are satisfied, and therefore \(V^{\prime}\) is decisive for \(x\)⁴⁰ against \(z\) again.

Consequence V.3.5: For every pair of alternatives \(x\), \(y\) and every one-member set of individuals \(V^{\prime}\),⁴¹ it is not true that \(V^{\prime}\) is decisive for \(x\) against \(y\).

This consequence states that no individual can be a dictator for even one pair of alternatives; i.e., there is no individual such that, with the given social welfare function, the community automatically prefers a certain \(x\)⁴² to a certain \(y\) whenever the individual in question does so.

Proof: Suppose the consequence is false. Let the one member of \(V^{\prime}\) be designated by \(1\).

Let \(y^{\prime}\) be any alternative distinct from \(x\) and \(y\). Then, from the hypothesis and Consequence V.3.4, \(V^{\prime}\) is decisive for \(x\) against \(y^{\prime}\). Since this statement is still true for \(y^{\prime}=y\), we may say

\begin{align} \label{orgd12d1a0} V^{\prime}\ \mathit{is\ decisive\ for}\ x\ \mathit{against\ any}\ y^{\prime}\ne{x}. \end{align}

For a fixed \(y^{\prime}\ne{x}\),⁴³ let \(x^{\prime}\) be an alternative distinct from \(x\) and \(y^{\prime}\). This choice is possible by Condition \(1\) (there are three alternatives). Then, from \eqref{orgd12d1a0} and Consequence V.3.4, \(V^{\prime}\) is decisive for \(x^{\prime}\) against \(y^{\prime}\). By \eqref{orgd12d1a0}, this statement still holds if \(x^{\prime}=x\).⁴⁴

\begin{align} \label{orgd6bec59} V^{\prime}\!\mathit{is\,decisive\,for}\,x^{\prime}\!\mathit{against}\,y^{\prime}\!\mathit{ provided}\ x^{\prime}\ne{y}^{\prime}, y^{\prime}\ne{x}. \end{align}

Choose any \(x^{\prime}\ne{x}\), and a particular \(y^{\prime\prime}\) distinct from both \(x\) and \(x^{\prime}\). This choice is possible by Condition 1. Then \eqref{orgd6bec59}⁴⁵ holds; since \(x^{\prime}, y^{\prime\prime}, x\) are distinct, it follows from Consequence V.3.4, if we substitute \(x^{\prime}\) for \(x\), \(y^{\prime\prime}\) for \(y\), and \(x\) for \(z\), that

\begin{align} \label{orged636ac} V^{\prime}\ \mathit{is\ decisive\ for}\ x^{\prime}\ \mathit{against}\ x\ \mathit{ provided}\ x^{\prime}\ne{x}. \end{align}

\eqref{orgd6bec59} and \eqref{orged636ac} together can be written

\begin{align} \label{orgbda6670} V^{\prime}\ \mathit{is\ decisive\ for}\ x^{\prime}\ \mathit{against\ any}\ y^{\prime} \ \mathit{provided}\ x^{\prime}\ne{y}^{\prime}. \end{align}

But, by Definition 10, \eqref{orgbda6670} says that, for all \(x^{\prime}\) and \(y^{\prime}\) (distinct), \(x^{\prime} P y^{\prime}\) whenever \(x^{\prime} P_{1} y^{\prime}\). By Definition 6, this means that the social welfare function is dictatorial, which, however, is excluded by Condition 5. Hence, the supposition that the consequence is false leads to a contradiction with one of the conditions. Q.E.D.

It will now be shown that Conditions 1-5 lead to a contradiction. Use will be made of the preceding five consequences of the conditions. Let \(S\) be the set composed of three distinct alternatives which occurs in the statement of Condition 1. For each possible ordered pair \(x^{\prime},y^{\prime}\) such that \(x^{\prime}\) and \(y^{\prime}\) both belong to \(S\) and \(x^{\prime}\ne{y}^{\prime}\) (there are six such ordered pairs), there is at least one set of individuals which is decisive for \(x^{\prime}\) against \(y^{\prime}\) by Consequence V.3.3. Consider all sets of individuals who are decisive for some \(x^{\prime}\) in \(S\) against some \(y^{\prime}\), distinct from \(x^{\prime}\) in \(S\). Among these sets, choose the one with the fewest number of individuals; if this condition does not uniquely specify the set, choose any of those decisive sets which does not have more members in it than some other decisive set. For example, if, among all the sets which are decisive for some \(x^{\prime}\) in \(S\) against some (distinct) \(y^{\prime}\) in \(S\), there is one with two members and all others have more than two members, choose that one; on the other hand, if there are two sets decisive for some \(x^{\prime}\) in \(S\) against some \(y^{\prime}\) in \(S\) which have three members each while all other decisive sets have more than three members, choose any one of the three member sets. Designate the chosen set by \(V_{1}\). It is decisive for some alternative in \(S\) against some other one in \(S\); by suitable labeling, we may say that \(V_{1}\) is decisive for \(x\) against \(y\). \(S\) contains just one other alternative other than \(x\) and \(y\); call that alternative \(z\). Let the number of members in \(V_{1}\) be \(k\); designate the members of \(V_{1}\) by the numbers \(1,\dotsc,k\), and number the remaining individuals \(k+1,\dotsc,n\). Let \(V^{\prime}\) contain the single individual \(1\), \(V_{2}\) the individuals \(2,\dotsc,k\), and \(V_{3}\) individuals \(k+1,...,n\). Note that \(V_{3}\) may contain no members. From the construction of \(V_{1}\) we may conclude that

\begin{align} \label{orgee2a6be} V_{1}\ \mathit{is\ decisive\ for}\ x\ \mathit{against}\ y, \end{align} \begin{align} \label{orga141d02} & \mathit{any\ set\ which\ is\ decisive\ for\ some\ alternative} \end{align} \begin{align*} & \mathit{in}\ S\ \mathit{against\ some\ other\ alternative\ in}\ S\\ & \mathit{contains\ at\ least}\ k\ \mathit{members.} \end{align*}

By construction, \(V_{2}\) contains \(k-1\) members. Hence, from \eqref{orga141d02},

\begin{align} \label{orgb9ee821} V_{2}\ \mathit{is\ not\ decisive\ for\ any\ alternative\ in}\ S \end{align} \begin{align*} \ \mathit{against\ any\ other\ alternative\ in}\ S. \end{align*}

Consequence V.3.5 is equivalent to stating that, if \(V^{\prime}\) contains exactly one member, then

\begin{align} \label{orgaf72385} V^{\prime}\ \mathit{is\ not\ decisive\ for\ any\ alternative\ in} \end{align} \begin{align*} \ S\ \mathit{against\ any\ other\ alternative\ in}\ S. \end{align*}

Let \(R_{i},/dotsc,R_{n}\) be a set of individual ordering relations such that,

\begin{align} \label{org0dda594} \mathit{for}\ i\ \mathit{in}\ V^{\prime}, x P_{i} y\ \mathit{and}\ y P_{i} z, \end{align} \begin{align} \label{org0796a42} \mathit{for}\ i\ \mathit{in}\ V_{2}, z P_{i} x\ \mathit{and}\ x P_{i} y, \end{align} \begin{align} \label{orge10665f} \mathit{for}\ i\ \mathit{in}\ V_{3}, y P_{i} z\ \mathit{and}\ z P_{1} x. \end{align}

From \eqref{org0dda594},⁴⁶ \eqref{org0796a42},⁴⁷ and the definitions of \(V_{1}\), \(V_{2}\) and \(V^{\prime}\), \(x P_{i} y\) for all \(i\) in \(V_{1}\). From \eqref{orgee2a6be}

\begin{align} \label{orgc2abdda} x P y, \end{align}

Where \(P\) is the social preference relation corresponding to \(R_{1},\dotsc,R_{n}\).⁴⁸ From \eqref{org0796a42}, and the fact that \(R_{1}\) is a weak ordering and hence transitive,

\begin{align} \label{org1a91c2d} z P_{i} y\ \mathit{for\ all}\ i\ \mathit{in}\ V_{2}. \end{align}

From \eqref{org0dda594} and \eqref{orge10665f},⁴⁹

\begin{align} \label{org7e27b0f} y P_{i} z\ \mathit{for\ all}\ i\ \mathit{not\ in}\ V_{2}. \end{align}

Suppose \(z P y\). Then from \eqref{org1a91c2d}, \eqref{org7e27b0f}, and Consequence V.3.2, it would follow that \(V_{2}\) was decisive for \(y\) against \(z\); but this contradicts \eqref{orgb9ee821}. Hence, we must say not \(z P y\), or

\begin{align} \label{org3813ad1} y R z. \end{align}

where \(R\) is the social ordering relation corresponding to \(R_{1},\dotsc,R_{n}\), the relation from which the preference relation \(P\) was derived. By Condition 1, the relation \(R\) is a weak ordering relation, having all the usual properties assigned to preference scales, including that of transitivity. Hence, from \eqref{orgc2abdda} and \eqref{org3813ad1},

\begin{align} \label{org7565ab2} x P z. \end{align}

From \eqref{org0dda594}, it follows from the transitivity of \(R_{1}\) that

\begin{align} \label{org395b98d} x P_{i} z\ \mathit{for}\ i\ \mathit{in} V^{\prime}, \end{align}

while, from \eqref{org0796a42} and \eqref{orge10665f},

\begin{align} \label{orga834d95} z P_{i} x\ \mathit{for}\ i\ \mathit{not\ in}\ V^{\prime}. \end{align}

From \eqref{org7565ab2}-\eqref{orga834d95}⁵⁰ and Consequence V.3.2, it follows that \(V^{\prime}\) is decisive for \(x\) against \(Z\). But this contradicts \eqref{orgaf72385}. Thus, we have shown that Conditions 1-5 taken together lead to a contradiction. Put another way, if we assume that our social welfare function satisfies Conditions 2 and 3, and further suppose that Condition 1 holds, i.e., that there are at least three alternatives which the individuals can order in any way and still get a social order, then either Condition 4 or Condition 5 must be violated. Condition 4 states that the social welfare function is not imposed; Condition 5 states that it is not dictatorial.

Theorem 2 (General Possibility Theorem): If there are at least three alternatives which the members of the society are free to order in any /way, then every social welfare function satisfying Conditions 2 and 8 and yielding a social ordering satisfying Axioms I and II must be either imposed or dictatorial.

Theorem 2 shows that, if no prior assumptions are made about the nature of individual orderings, there is no method of voting which will remove the paradox of voting discussed in Chapter I, Section 1, neither plurality voting nor any scheme of proportional representation, no matter how complicated. Similarly, the market mechanism does not create a rational social choice.