Arrow Impossibility Theorem
“The Arrow impossibility theorem and its implications for voting and elections”
Arrow’s impossibility theorem represents a fascinating problem in the philosophy of economics, widely discussed for insinuating doubt on commonly accepted beliefs towards collective decision making procedures.
This essay will introduce its fundamental assumptions, explain its meaning, explore some of the solutions available to escape its predictions and finally discuss its implications for political voting and elections. I will begin by giving some definitions and presenting the fundamental issue of social choice theory, consisting of the identification of an “ideal” device for preference aggregation, capable of converting individual rankings into collective ones, reflecting each individual’s preference into an optimal societal choice.
Given a finite set of voters having to choose between a finite set of candidates, we call a voting system the function taking as input the voting preferences of each voter and returning as output a collectively valid ranking of the candidates.
Majority voting is the voting system requiring that given two alternative options X and Y, X is preferred to Y by the group if the number of group members prefering X to Y exceeds that of members prefering Y to X.
When group preferences are rational and transitive, for every pairwise comparison between two options, the group-wide valid outcome obtained applying majority voting is a unique winning alternative, which is said to be the “Condorcet winner”.
Sometimes though, group preferences are not rational and for each pairwise comparison a different winner emerges. In such case there is said to be a cycling majority and the situation represents a “Condorcet paradox”.
Named after the distinguished economist and nobel laureate Kenneth Arrow, the “Arrow Impossibility Theorem” was first proposed and demonstrated in his book “Social Choice and Individual Values”, published in 1951.
The
Arrow’s impossibility theorem represents a fascinating problem in the philosophy of economics, widely discussed for insinuating doubt on commonly accepted beliefs towards collective decision making procedures.
This essay will introduce its fundamental assumptions, explain its meaning, explore some of the solutions available to escape its predictions and finally discuss its implications for political voting and elections. I will begin by giving some definitions and presenting the fundamental issue of social choice theory, consisting of the identification of an “ideal” device for preference aggregation, capable of converting individual rankings into collective ones, reflecting each individual’s preference into an optimal societal choice.
Given a finite set of voters having to choose between a finite set of candidates, we call a voting system the function taking as input the voting preferences of each voter and returning as output a collectively valid ranking of the candidates.
Majority voting is the voting system requiring that given two alternative options X and Y, X is preferred to Y by the group if the number of group members prefering X to Y exceeds that of members prefering Y to X.
When group preferences are rational and transitive, for every pairwise comparison between two options, the group-wide valid outcome obtained applying majority voting is a unique winning alternative, which is said to be the “Condorcet winner”.
Sometimes though, group preferences are not rational and for each pairwise comparison a different winner emerges. In such case there is said to be a cycling majority and the situation represents a “Condorcet paradox”.
Named after the distinguished economist and nobel laureate Kenneth Arrow, the “Arrow Impossibility Theorem” was first proposed and demonstrated in his book “Social Choice and Individual Values”, published in 1951.
The
Bibliography: Arrow K. (1951) “Social choice and individual values”, Yale University Press Collins N. (2003) “Arrow’s Theorem Proves No Voting System Is Perfect” , The Tech MacKay A. F. (1980). “Arrow’s Theorem: The paradox of social choice”, Yale University Press Saari D.G. (2001). “Decisions and elections: Expecting the unexpected”, Cambridge University Press Tao T. (1991). “Arrow’s Theorem on voting paradoxes”, University of California Press