Overview and Analysis of Sen’s Theorem
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The Sen’s theorem has an important impact on the decision and social choice theory. In the Nobel Prize lecture of 1998, Sen said that discovering the societal decision rules will be the main objective of the choice theory. However, Sen found it is difficult to achieve this objective in his paper of “Impossibility of a Paretian Liberal” (Sen, 1970a, b, 1976c). Sen’s theorem states that it is impossible to achieve the minimal aspect of liberalism when it is combined with the Pareto condition. According to the theorem of Sen, it seems that there is a fundamental conflict between the Pareto welfare concepts and the liberalism. After this argument, there is a pooling of studies that focus on the individual rights and there are also studies that find new interpretation for the theorem of Sen.
According to Saari and his collaborators, the cyclic decision outcomes established by Sen's theorem describe a transitional, dysfunctional state of society. This essay will mainly explain how this interpretation arises and discuss its merits relative to the original interpretation offered by Sen. The essay will be structured as follow: in the second session, the essay will briefly introduce the Sen’s Theorem regarding to the conflicting results between Pareto Optimal and Liberalism. Examples will also be used to better illustrate the choice decisions. In the third session, the essay will how and why there will be new interpretations of the theorem of Sen. Particularly, the essay will base its argument on the research results of Saari and his collaborators. In the fourth session, the essay will discuss the merits of the new interpretation compared with the interpretation by Sen himself. In the last session, a brief conclusion will be made based on the analysis in the previous session.
The Sen's impossibility of a paretian libertarian theorem or Sen's paradox Paratian liberal states that weak Pareto criterion and liberalism may have conflicts on the fundamental level, which indicated that the Pareto criterion is flawed. On one hand, it shows that the Pareto criteria only consider aspects of difference in efficiency among relevant states while ignoring the individual utility conflicts arising because of these states. On the other hand, the theorem also shows the in addition to considering some reasonable internal conditions, one also needs to consider issues such as liberalism for social order decision making. According to Sen, there are some choices that are purely of personal nature, such as the state (x) means everything else in the society as Ω, when A is sleeping supine, state (y) means everything else in society as Ω, when A is sleeping prostrate. If A has a preference of y than x, while many others in the society prefer the opposite, then it can be considered that social choice between x and y is a purely personal thing, because A is the only a 'real ' person that is related to the choice while other people are just nosy person. It can also be considered that one can choose one such a collective choice rule where A's preference should be accurately reflected by social preferences in this purely 'personal nature' choices (Sen, 1970). Based on this point, Sen emphasizes that people should pay attention to the study of individual rights and other issues affecting the social choice order and decision making. The new welfare economics and welfare economics research before the new welfare economics are consciously or unconsciously avoid these problems.
Sen (1970) found that the Pareto criterion is incompatible with liberalism. Pareto optimality is used by the economists and other social scientists to measure the efficiency of society, which is also the the most common and widespread, even the only indicator. It refers to such a state that we are good as it gets, no one can make an additional welfare without compromising the welfare of others. And the principles of individual freedom are the relentless pursuit of mankind, both of which are on the intuition that people can fully accept and understand. However, Sen's theorem shows that these two attractive standards are contradictory and cannot be simultaneously true.
There are three assumptions that Sen’s theorem is based on and the realistic of these assumptions will be one reason why there will be new interpretations for the Sen’s theorem. First, the essay will briefly introduce the assumptions related to the Sen’s theorem.
- An unrestricted area principle;
- The Pareto principle;
- The minimum principles of liberalism (ML), which states society should give at least two people the right to choose between at least a pair of social status. If one thinks that A better than B, then society should not interfere and should agree with the preference.
With Sen's words, if you want sleeping supine and did not want to sleep prostrate, the society should agree with it. However, Sen also prove that for two or more people in the society, there exist no social choice functions that simultaneously satisfy these above three conditions because there will be the similar results of cyclical Condorcet voting paradox revealed by Arrow Impossibility Theorem when Pareto optimal is in combination with a minimum principle of liberalism.
Sen’s theorem on freedom of the Pareto impossibility can be accurately described as: there does not exist the collective decision rule that meets the following conditions.
The first condition is a sort of rational conditions. The sort is reflexive, relevance, and the preference relation is not circulating. Reflexive means for any choice x has xRx, which indicates that any choice is at least as good as its own. Relevance means for any two options, there must be xRy or yRx or both. The second condition is weak Pareto criterion. For any choice of x and y, if everyone i think xPiy, then the society also thinks that xPy. The third condition is the minimal liberalism. In this condition, there are two non-empty, disjoint subsets L1 and L2, two different options for the (a, b) and (c, d), if everyone in L1 considers a is better than b, then the society also thinks that a is better than b; if everyone in L1 thinks that b is better than a, then b is also better than a for the society. Similarly, if everyone in L2 believes that c is better than d, then the society also thinks that c is better than d; if everyone in L2 thinks d is better than c, then the society thinks also that d is better than c. The two groups were decisive on the choice between the two groups. Anyone is free to do what he likes to do, which means there are some choices that are entirely personal, personal preferences should not be affected by some other people.
How to walk out of Sen's paradox? Mueller has proposed two solutions in the “public choice theory ". One is to let the Pareto principle in some cases to comply with the right to liberalism. The other is through Pareto transactions. As Mueller noted, the matrix is similar to game theory, for example a state in the famous prisoner's dilemma, and the Pareto inferior results are due to the independent exercise of each person in his own right, regardless of the damage to others such externalities.
The results of Sen’s theorem are established mainly through examples. There are no rigorous proofs on the results why there will be inconsistency of minimal liberalism and Pareto conditions. The results are basically based on the assumption that there are conflicts between the Pareto Conditions and the Minimal Liberalism. There are questions keep asking whether the assumption is true and what if the cyclic societal rankings are not due to these conditions?
In fact, in 1998 and 2001, Saari argues that the real reason of the seminal result of Sen’s theorem is not related to the nature of the Pareto condition and the Minimal Liberalism. The reason is that Pareto conditions and Minimal Liberalism needs the societal rankings to be made over pairs, which dismiss the transitivity of individual preferences. Therefore, it is not the conflicts between societal need and individual rights that undermine the assumption of individual rationality; it is the concentration of pairs that leads to the ignorance of individual rationality. And Saari also made geometric proof on this argument, which provides a new interpretation for the Sen’s theorem.
There are several advantages of the new interpretation compared to the one that Sen present in the original version. First of all, the theorem carried out by Saari and his collaborates are proved using geometric proof, instead of using only examples and assumptions to derive the final results. In addition, the new interpretation can explain all the examples used by Sen in his prior papers and the new interpretation also supports Sen’s own interpretation that the three conditions in Sen’s theorem force the decisions to be made by ignoring the individual rationality. The new interpretation also thinks that the decision rule also wants to meet the demand in the cyclic preference (Saari 2001; Saari and Petron, 2004).
Secondly, the geometric proof of the new interpretations has identified all possible profiles that support any examples of Sen’s model. In addition, Saari and his collaborates have also significantly expanded on earlier observations by providing a new statistical interpretation for Sen’s Theorem. And they also conclude that the cyclic decision outcomes established by Sen's theorem describe a transitional, dysfunctional state of society.
Thirdly, the new interpretation has pointed a new direction for the movements of individual rights. Compared with the interpretation by Sen, the new interpretation focus more on the intensity minimal liberalism, which will leads to social decision procedures without cyclic outcomes and at the same time satisfy weak Pareto conditions. They pointed out the deeper reason of the ignorance of individual rationality instead of concluding that the reason is because of the inconsistency of the Pareto condition and the Minimal Liberalism.
As discussed in the previous session, the Sen’s theorem provides a good direction for the research of individual rights. And there are many researchers working on the topic to find new interpretation for the seminal results of Sen’s theory. Saari and his collaborates find that Minimal Liberalism makes some of the information in the society irrelevant. However, depending on that information, individual preferences may or may not be transitive. Therefore, they conclude that Minimal Liberalism makes transitivity information irrelevant and this happens for any possible example of Sen’s cycles. They find a way to solve this problem and the response to this is to modify Minimal Liberalism in a way that is sensitive to transitivity information. They use the Intensity Minimal Liberalism (IML), which is a decisive that agent can impose his preferences only when the choice does not create a strong negative externality for some other agent. And they finally find that there are social decision procedures without cyclic outcomes that satisfy weak Pareto and IML, which provides a new interpretation for the Sen’s theorem. The new interpretation finds a more appropriate way to proof Sen’s theorem and expends Sen’s theorem in several aspects.
Li, I. and D.G. Saari 2008. ‘Sen’s theorem: geometric proof, new interpretations’, Social Choice and Welfare 31: 393-413. Focus especially on pages 393-401.
Petron, A and D.G. Saari 2006. `Negative externalities and Sen's liberalism theorem',
Economic Theory 28: 265-281. Read Sections 1 to 4.
Saari, D. G. (1995). Basic Geometry of Voting, Springer-Verlag, New York
Saari, D. G. (1998). Connecting and resolving Sen’s and Arrow’s Theorems, Social Choice & Welfare 15, 239-261
Saari, D. G. (2001). Decisions and Elections; Explaining the Unexpected, Cambridge University Press, New York
Saari, D. G., and Petron, A. (2004). (April), Negative Externalities and Sen’s Liberalism Theorem, IMBS working papers, University of California, Irvine, to appear in Economic Theory,June, 2006.
Saari, D. G. and Sieberg, K. (2001). The sum of the parts can violate the whole, American Political Science Review 95, 415-433.
Salles, M. (1997). On Modelling Individual Rights: Some Conceptual Issues: Discussion; p 129-133 in Social Choice Re-examined, Vol. 2; Ed. by K. J. Arrow, A. K. Sen, and K. Suzumura,St Martin’s Press New York.
Sen, A. K. (1966). A Possibility Theorem on Majority Decisions, Econometrica, 34(2), 491-09.
Sen, A. K. (1970a). Collective Choice and Social Welfare, Holden-Day, San Francisco.
Sen, A. K. (1970b). The Impossibility of a Paretian Liberal, The journal of Political Economy, 78(1), 152-57.
Sen, A. K. (1976). Liberty, Unanimity and Rights, Economica 43(171), 217-45.
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Sen, A. K. (1992) Minimal Liberty, Economica, 59 (234), 139-60.
Sen, A. K. (1999) The possibility of social choice. The American Economic Review 89 (3), 349-378
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