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A Refutation of Arrow's Theorem
This book attempts to refute a well-known theorem in social choice theory by questioning both the truth of its premises and the validity of its argument. Arrow emphasizes that four 'reasonable' conditions be placed on democratic constitutions and from these assumptions derives his theorem, which states that it is impossible to satisfy these four conditions simultaneously. This book not only refutes the theorem, but, in addition, shows that its philosophical basis is antidemocratic and should be abandoned. It is important to show that there is no justification for the claim that democracy is flawed, even if that claim be made only in a purely logical sense.
102 pages, Hardcover
First published July 5, 1991
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August 2, 2019
Broken Arrow
I have fond memories of learning Richard Arrow's paradox in university. Briefly stated, given a choice between candidates A, B and C, the voters as a group may prefer A to B and B to C but, and this is the apparent paradox, they may prefer C to A.
The reason for this is subtle but it boils down to the fact that preference is based on multiple dimensions. A's experience may be preferred to B. B's fiscal policy may be preferred to C and C's social policy may be preferred to A. If A and B are paired first, then C and A, C wins. If B and C are paired first, then B and A, A wins. If A and C .... then B will win. While we usually don't vote for candidates this way, we do often vote policy resolutions in this fashion. By suggesting a voting order one can rig the result.
Arrow concluded that true democracy is impossible!
DeLong, who is a logician and mathematician, disagrees. Arrow's full proof was based on 4 axioms which DeLong does a good job of taking apart in chapters 3 to5. They are:
1. Preference implies Choice
2. The Pareto Principle: If all members of a society prefer X, then X is the preference of society
3. Independence of Irrelevant Alternatives AKA Partial determination.
4. There are No Dictators Allowed
A variety of arguments are used to dismiss each of these. DeLong argues that choice and preference are not the same when the voters have false or insufficient information or are coerced. To the 2nd point he gives an example that while individually members of a society might prefer to lynch the perpetrator of a gruesome murder, collectively they will hold out for a fair trial. Democracies, DeLong argues, value cool sobriety over momentary impulse. Collective preference is not the same as the sum of individual preferences, and Arrow, in using the Pareto Principle, assumes that it is. The 3rd axiom isn't used in Arrow's proof, so technically it is irrelevant, but DeLong dismisses that too, giving examples of where the presence of alternate choices, even if they are not voted on, can alter the results of an election. Lastly, Athenian democracy permitted temporary dictators. In effect they temporarily amended the rules of democracy, either by electing a tyrant or, as in Plato's Republic, giving everyone a chance at the leadership by randomizing selection along with frequent rotation. Randomization, such flipping a coin, is also an accepted practice in breaking deadlock in America. Key properties of democracy are moral fairness, and randomization, which Arrow fails to consider.
As for the proof itself, DeLong finds the following problems: For the first he refers to Peter Suber's paper on the paradox of self amendment. What if a voter were to repeal one of Arrow's axioms? Would that still be democratic? My own POV is that this is an exceptional case, and, like division by zero, it should be excluded. DeLong however sees this as a problem of the non-empirical nature of Arrow's work. Contrasting Arrow's approach to that of Alonzo Church who proved that there was no decisive procedure for arithmetic, he argues that Arrow did not base his axioms on historic examples of actual democracies. Arrow may be proving something about the hypothetical system he is describing, but he isn't describing democracy. Real democracies have managed to avoid this problem and that capability is not included in his model.
The second problem for DeLong is that Preference, like Love, is not a transitive relationship. If Paul loves Mary and Mary loves Bill, then the odds are probably against Paul loving Bill. The basis of Arrow's paradox dissolves because Arrow treats Preference as if it was transitive, and acts surprised when he finds it is not. Next, the principle of non-dictatorship is concrete applying to individual members. Collective Rationality is a group property and Arrow commits the fallacy of reification when he mixes them in his proof. The final property of Arrow's theorem that DeLong takes to task is its silence on issues of social justice. Arrow's democracy, by elevating Preference, is hedonistic, not moral, but DeLong argues that the basis of democracy is fairness and social justice, whereas Arrow's version lends itself to a tyranny of the majority.
Revisiting this paradox 3 decades later was something I found profoundly satisfying. DeLong's refutation is clearly written and IMHO only a small section of chapter 6 might be considered hard to follow by the average reader.
The audience for this book is likely small as readers should have prior acquaintance with Arrow's Theorem and its conclusions. Nevertheless, De Long has provided a devastating critique that deserves wider attention. Recommended!
I have fond memories of learning Richard Arrow's paradox in university. Briefly stated, given a choice between candidates A, B and C, the voters as a group may prefer A to B and B to C but, and this is the apparent paradox, they may prefer C to A.
The reason for this is subtle but it boils down to the fact that preference is based on multiple dimensions. A's experience may be preferred to B. B's fiscal policy may be preferred to C and C's social policy may be preferred to A. If A and B are paired first, then C and A, C wins. If B and C are paired first, then B and A, A wins. If A and C .... then B will win. While we usually don't vote for candidates this way, we do often vote policy resolutions in this fashion. By suggesting a voting order one can rig the result.
Arrow concluded that true democracy is impossible!
DeLong, who is a logician and mathematician, disagrees. Arrow's full proof was based on 4 axioms which DeLong does a good job of taking apart in chapters 3 to5. They are:
1. Preference implies Choice
2. The Pareto Principle: If all members of a society prefer X, then X is the preference of society
3. Independence of Irrelevant Alternatives AKA Partial determination.
4. There are No Dictators Allowed
A variety of arguments are used to dismiss each of these. DeLong argues that choice and preference are not the same when the voters have false or insufficient information or are coerced. To the 2nd point he gives an example that while individually members of a society might prefer to lynch the perpetrator of a gruesome murder, collectively they will hold out for a fair trial. Democracies, DeLong argues, value cool sobriety over momentary impulse. Collective preference is not the same as the sum of individual preferences, and Arrow, in using the Pareto Principle, assumes that it is. The 3rd axiom isn't used in Arrow's proof, so technically it is irrelevant, but DeLong dismisses that too, giving examples of where the presence of alternate choices, even if they are not voted on, can alter the results of an election. Lastly, Athenian democracy permitted temporary dictators. In effect they temporarily amended the rules of democracy, either by electing a tyrant or, as in Plato's Republic, giving everyone a chance at the leadership by randomizing selection along with frequent rotation. Randomization, such flipping a coin, is also an accepted practice in breaking deadlock in America. Key properties of democracy are moral fairness, and randomization, which Arrow fails to consider.
As for the proof itself, DeLong finds the following problems: For the first he refers to Peter Suber's paper on the paradox of self amendment. What if a voter were to repeal one of Arrow's axioms? Would that still be democratic? My own POV is that this is an exceptional case, and, like division by zero, it should be excluded. DeLong however sees this as a problem of the non-empirical nature of Arrow's work. Contrasting Arrow's approach to that of Alonzo Church who proved that there was no decisive procedure for arithmetic, he argues that Arrow did not base his axioms on historic examples of actual democracies. Arrow may be proving something about the hypothetical system he is describing, but he isn't describing democracy. Real democracies have managed to avoid this problem and that capability is not included in his model.
The second problem for DeLong is that Preference, like Love, is not a transitive relationship. If Paul loves Mary and Mary loves Bill, then the odds are probably against Paul loving Bill. The basis of Arrow's paradox dissolves because Arrow treats Preference as if it was transitive, and acts surprised when he finds it is not. Next, the principle of non-dictatorship is concrete applying to individual members. Collective Rationality is a group property and Arrow commits the fallacy of reification when he mixes them in his proof. The final property of Arrow's theorem that DeLong takes to task is its silence on issues of social justice. Arrow's democracy, by elevating Preference, is hedonistic, not moral, but DeLong argues that the basis of democracy is fairness and social justice, whereas Arrow's version lends itself to a tyranny of the majority.
Revisiting this paradox 3 decades later was something I found profoundly satisfying. DeLong's refutation is clearly written and IMHO only a small section of chapter 6 might be considered hard to follow by the average reader.
The audience for this book is likely small as readers should have prior acquaintance with Arrow's Theorem and its conclusions. Nevertheless, De Long has provided a devastating critique that deserves wider attention. Recommended!
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