Введение
Очевидное нарушение предсказаний теории ожидаемой полезности
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory. Rather than adhering to rationality, the Allais paradox proves that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox. involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B. The inconsistency stems from the fact that in expected utility theory, equal outcomes (e. g. $1 million for all gambles) added to each of the two choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out". In each experiment the two gambles give the same outcome 89% of the time (starting from the top row and moving down, both 1A and 1B give an outcome of $1 million with 89% probability, and both 2A and 2B give an outcome of nothing with 89% probability). If this 89% ‘common consequence’ is disregarded, then in each experiment the choice between gambles will be the same – 11% chance of $1 million versus 10% chance of $5 million. After re writing the payoffs, and disregarding the 89% chance of winning — equalising the outcome — then 1B is left offering a 1% chance of winning nothing and a 10% chance of winning $5 million, while 2B is also left offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be seen as the same choice. In the same manner, 1A and 2A can also be seen as the same choice, i. e.:
Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 89% $1 million 89% Nothing 89% Nothing 89%$1 million11% Nothing 1%$1 million11% Nothing 1% $5 million 10% $5 million 10%
Allais presented his paradox as a counterexample to the independence axiom. Independence means that if an agent is indifferent between simple lotteries and , the agent is also indifferent between mixed with an arbitrary simple lottery with probability and mixed with with the same probability Violating this principle is known as the "common consequence" problem (or "common consequence" effect). The idea of the common consequence problem is that as the prize offered by increases, and become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk and disappointment in case they do not win the higher prize offered by
Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew), rank dependent expected utility by John Quiggin, and regret theory. The point of these models was to allow a wider range of behavior than was consistent with expected utility theory. Michael Birnbaum performed experimental dissections of the paradox and showed that the results violated the theories of Quiggin, Kahneman, Tversky, and others, but could be explained by his configural weight theory that violates the property of coalescing. The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you had chosen 1A. This feeling of disappointment, however, is contingent on the outcome in the other portion of the gamble (i. e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge of our rational action (1B cannot be valued independently of 1A as the independence or sure thing principle requires of us). We don't act irrationally when choosing 1A and 2B; rather expected utility theory is not robust enough to capture such "bounded rationality" choices that in this case arise because of complementarities.
Парадокс Аллеса — это проблема выбора, разработанная для демонстрации несоответствия фактических наблюдаемых выборов предсказаниям теории ожидаемой полезности. Вместо следования рациональности, парадокс Аллеса доказывает, что люди редко принимают рациональные решения последовательно, когда требуется сделать это немедленно. Аксиома независимости теории ожидаемой полезности, требующая, чтобы предпочтения индивида не менялись при изменении двух лотерей в равных пропорциях, была опровергнута этим парадоксом. Исследования, связанные с гипотетическими и небольшими денежными выигрышами, а также недавние исследования, касающиеся результатов в области здравоохранения, подтвердили утверждение, что при выборе между 1А и 1В большинство людей выбирают 1А. Аналогично, при выборе между 2А и 2В большинство людей выбирают 2В. Алле также утверждал, что разумно выбрать только 1А или только 2В. Однако, тот факт, что тот же человек (выбравший 1А или 2В) выберет и 1А, и 2В вместе, противоречит теории ожидаемой полезности. Согласно теории ожидаемой полезности, человек должен выбрать либо 1А и 2А, либо 1В и 2В. Несоответствие возникает из-за того, что в теории ожидаемой полезности равные исходы (например, выигрыш в 1 миллион долларов для всех вариантов) добавленные к каждому из двух выборов не должны влиять на относительную привлекательность одного варианта по сравнению с другим; равные исходы должны "взаимно уничтожаться". В каждом эксперименте два варианта дают один и тот же исход в 89% случаев (начиная с верхней строки и двигаясь вниз, и 1А, и 1В дают выигрыш в 1 миллион долларов с вероятностью 89%, а 2А и 2В дают нулевой выигрыш с вероятностью 89%). Если не учитывать эти 89% "общего исхода", то выбор между вариантами в каждом эксперименте будет одинаковым — 11% шанс выиграть 1 миллион долларов против 10% шанса выиграть 5 миллионов долларов. После переформулировки выигрышей и игнорирования 89% шанса на выигрыш — уравнивания исходов — 1B остается с 1% шансом не выиграть ничего и 10% шансом выиграть 5 миллионов долларов, а 2B также остается с 1% шансом не выиграть ничего и 10% шансом выиграть 5 миллионов долларов. Следовательно, выбор 1B и 2B можно рассматривать как один и тот же выбор. Аналогично, 1A и 2A также можно рассматривать как один и тот же выбор, то есть:
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory. Rather than adhering to rationality, the Allais paradox proves that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox. involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B. The inconsistency stems from the fact that in expected utility theory, equal outcomes (e. g. $1 million for all gambles) added to each of the two choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out". In each experiment the two gambles give the same outcome 89% of the time (starting from the top row and moving down, both 1A and 1B give an outcome of $1 million with 89% probability, and both 2A and 2B give an outcome of nothing with 89% probability). If this 89% ‘common consequence’ is disregarded, then in each experiment the choice between gambles will be the same – 11% chance of $1 million versus 10% chance of $5 million. After re writing the payoffs, and disregarding the 89% chance of winning — equalising the outcome — then 1B is left offering a 1% chance of winning nothing and a 10% chance of winning $5 million, while 2B is also left offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be seen as the same choice. In the same manner, 1A and 2A can also be seen as the same choice, i. e.:
Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 89% $1 million 89% Nothing 89% Nothing 89%$1 million11% Nothing 1%$1 million11% Nothing 1% $5 million 10% $5 million 10%
Allais presented his paradox as a counterexample to the independence axiom. Independence means that if an agent is indifferent between simple lotteries and , the agent is also indifferent between mixed with an arbitrary simple lottery with probability and mixed with with the same probability Violating this principle is known as the "common consequence" problem (or "common consequence" effect). The idea of the common consequence problem is that as the prize offered by increases, and become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk and disappointment in case they do not win the higher prize offered by
Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew), rank dependent expected utility by John Quiggin, and regret theory. The point of these models was to allow a wider range of behavior than was consistent with expected utility theory. Michael Birnbaum performed experimental dissections of the paradox and showed that the results violated the theories of Quiggin, Kahneman, Tversky, and others, but could be explained by his configural weight theory that violates the property of coalescing. The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you had chosen 1A. This feeling of disappointment, however, is contingent on the outcome in the other portion of the gamble (i. e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge of our rational action (1B cannot be valued independently of 1A as the independence or sure thing principle requires of us). We don't act irrationally when choosing 1A and 2B; rather expected utility theory is not robust enough to capture such "bounded rationality" choices that in this case arise because of complementarities.
Эксперимент 1 Эксперимент 2 Вариант 1А Вариант 1Б Вариант 2А Вариант 2Б Выигрыш Шанс Выигрыш Шанс Выигрыш Шанс Выигрыш Шанс 1 миллион 89% 1 миллион 89% Ничего 89% Ничего 89% 1 миллион 11% Ничего 1% 1 миллион 11% Ничего 1% 5 миллионов 10% 5 миллионов 10%
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory. Rather than adhering to rationality, the Allais paradox proves that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox. involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B. The inconsistency stems from the fact that in expected utility theory, equal outcomes (e. g. $1 million for all gambles) added to each of the two choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out". In each experiment the two gambles give the same outcome 89% of the time (starting from the top row and moving down, both 1A and 1B give an outcome of $1 million with 89% probability, and both 2A and 2B give an outcome of nothing with 89% probability). If this 89% ‘common consequence’ is disregarded, then in each experiment the choice between gambles will be the same – 11% chance of $1 million versus 10% chance of $5 million. After re writing the payoffs, and disregarding the 89% chance of winning — equalising the outcome — then 1B is left offering a 1% chance of winning nothing and a 10% chance of winning $5 million, while 2B is also left offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be seen as the same choice. In the same manner, 1A and 2A can also be seen as the same choice, i. e.:
Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 89% $1 million 89% Nothing 89% Nothing 89%$1 million11% Nothing 1%$1 million11% Nothing 1% $5 million 10% $5 million 10%
Allais presented his paradox as a counterexample to the independence axiom. Independence means that if an agent is indifferent between simple lotteries and , the agent is also indifferent between mixed with an arbitrary simple lottery with probability and mixed with with the same probability Violating this principle is known as the "common consequence" problem (or "common consequence" effect). The idea of the common consequence problem is that as the prize offered by increases, and become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk and disappointment in case they do not win the higher prize offered by
Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew), rank dependent expected utility by John Quiggin, and regret theory. The point of these models was to allow a wider range of behavior than was consistent with expected utility theory. Michael Birnbaum performed experimental dissections of the paradox and showed that the results violated the theories of Quiggin, Kahneman, Tversky, and others, but could be explained by his configural weight theory that violates the property of coalescing. The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you had chosen 1A. This feeling of disappointment, however, is contingent on the outcome in the other portion of the gamble (i. e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge of our rational action (1B cannot be valued independently of 1A as the independence or sure thing principle requires of us). We don't act irrationally when choosing 1A and 2B; rather expected utility theory is not robust enough to capture such "bounded rationality" choices that in this case arise because of complementarities.
Алле представил свой парадокс как контрпример аксиоме независимости. Независимость означает, что если агент безразличен между простыми лотереями и , то он также безразличен между лотереей смешанной с произвольной простой лотереей с вероятностью и лотереей смешанной с той же вероятностью . Нарушение этого принципа известно как проблема "общего исхода" (или эффект "общего исхода"). Идея проблемы общего исхода заключается в том, что по мере увеличения предлагаемого приза и превращения других исходов в утешительные, агент будет корректировать свои предпочтения между двумя лотереями, чтобы минимизировать риск и разочарование в случае, если он не выиграет более высокий приз. Трудности, подобные этим, привели к появлению ряда альтернатив и обобщений теории, в частности, теории перспективы, разработанной Даниэлем Канеманом и Амосом Тверски, взвешенной полезности (Чью), ожидаемой полезности, зависящей от ранга, разработанной Джоном Квиггином, и теории сожаления. Целью этих моделей было позволить более широкий спектр поведения, чем это было совместимо с теорией ожидаемой полезности. Майкл Бирнбаум провел экспериментальную деконструкцию парадокса и показал, что результаты противоречат теориям Квиггина, Канемана, Тверски и других, но могут быть объяснены его теорией конфигуративного веса, которая нарушает свойство коалесценции. Главный тезис Алле состоял в том, что аксиома независимости теории ожидаемой полезности может быть недействительной аксиомой. Аксиома независимости утверждает, что два идентичных исхода в рамках лотереи должны рассматриваться как несущественные для анализа лотереи в целом. Однако это упускает из виду понятие взаимодополняемости, а именно тот факт, что ваш выбор в одной части лотереи может зависеть от возможного исхода в другой ее части. В приведенном выше выборе 1B есть 1% шанс не получить ничего. Однако этот 1% шанс не получить ничего также несет в себе сильное чувство разочарования, если вы выберете этот вариант и проиграете, зная, что могли бы выиграть со 100% уверенностью, если бы выбрали 1A. Это чувство разочарования, однако, зависит от исхода в другой части лотереи (то есть от чувства уверенности). Следовательно, Алле утверждает, что невозможно оценивать части лотереи или выборы независимо друг от друга, как того требует аксиома независимости, и, следовательно, она является плохим показателем нашей рациональности (1B нельзя оценивать независимо от 1A, как того требует принцип независимости или достоверности). Мы не действуем иррационально, выбирая 1A и 2B; скорее, теория ожидаемой полезности недостаточно надежна, чтобы охватить такие выборы "ограниченной рациональности", которые в данном случае возникают из-за взаимодополняемости.
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory. Rather than adhering to rationality, the Allais paradox proves that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox. involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B. The inconsistency stems from the fact that in expected utility theory, equal outcomes (e. g. $1 million for all gambles) added to each of the two choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out". In each experiment the two gambles give the same outcome 89% of the time (starting from the top row and moving down, both 1A and 1B give an outcome of $1 million with 89% probability, and both 2A and 2B give an outcome of nothing with 89% probability). If this 89% ‘common consequence’ is disregarded, then in each experiment the choice between gambles will be the same – 11% chance of $1 million versus 10% chance of $5 million. After re writing the payoffs, and disregarding the 89% chance of winning — equalising the outcome — then 1B is left offering a 1% chance of winning nothing and a 10% chance of winning $5 million, while 2B is also left offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be seen as the same choice. In the same manner, 1A and 2A can also be seen as the same choice, i. e.:
Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 89% $1 million 89% Nothing 89% Nothing 89%$1 million11% Nothing 1%$1 million11% Nothing 1% $5 million 10% $5 million 10%
Allais presented his paradox as a counterexample to the independence axiom. Independence means that if an agent is indifferent between simple lotteries and , the agent is also indifferent between mixed with an arbitrary simple lottery with probability and mixed with with the same probability Violating this principle is known as the "common consequence" problem (or "common consequence" effect). The idea of the common consequence problem is that as the prize offered by increases, and become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk and disappointment in case they do not win the higher prize offered by
Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew), rank dependent expected utility by John Quiggin, and regret theory. The point of these models was to allow a wider range of behavior than was consistent with expected utility theory. Michael Birnbaum performed experimental dissections of the paradox and showed that the results violated the theories of Quiggin, Kahneman, Tversky, and others, but could be explained by his configural weight theory that violates the property of coalescing. The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you had chosen 1A. This feeling of disappointment, however, is contingent on the outcome in the other portion of the gamble (i. e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge of our rational action (1B cannot be valued independently of 1A as the independence or sure thing principle requires of us). We don't act irrationally when choosing 1A and 2B; rather expected utility theory is not robust enough to capture such "bounded rationality" choices that in this case arise because of complementarities.
Критика
В то время как парадокс Алле рассматривается как контрпример теории ожидаемой полезности, Люк Ватье, профессор маркетинга в Джорджтаунском университете, утверждал, что парадокс Алле демонстрирует необходимость в модифицированной функции полезности и не является парадоксальным по своей сути. В работе "Критика парадокса Алле" (1993) Ватье утверждает, что парадокс "не представляет собой обоснованного теста аксиомы независимости", необходимой в теории ожидаемой полезности. Это связано с тем, что парадокс предполагает сравнение предпочтений между двумя различными ситуациями, а не предпочтений в рамках одного набора вариантов. Результаты данного эксперимента показали, что изменение предпочтений, наблюдаемое в парадоксе Алле, обусловлено состоянием индивида, включающим возможность банкротства и уровень благосостояния.