Sleeping Beauty problem

Last updated

The Sleeping Beauty problem is a puzzle in decision theory in which whenever an ideally rational epistemic agent is awoken from sleep, she has no memory of whether she has been awoken before. Upon being told that she has been woken once or twice according to the toss of a coin, once if heads and twice if tails, she is asked her degree of belief for the coin having come up heads.

History

The problem was originally formulated in unpublished work in the mid 1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience") [1] followed by a paper by Adam Elga. [2] A formal analysis of the problem of belief formation in decision problems with imperfect recall was provided first by Michele Piccione and Ariel Rubinstein in their paper: "On the Interpretation of Decision Problems with Imperfect Recall" where the "paradox of the absent minded driver" was first introduced and the Sleeping Beauty problem discussed as Example 5. [3] [4] The name "Sleeping Beauty" was given to the problem by Robert Stalnaker and was first used in extensive discussion in the Usenet newsgroup rec.puzzles in 1999. [5]

The problem

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

• If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
• If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

Solutions

This problem continues to produce ongoing debate.

Thirder position

The thirder position argues that the probability of heads is 1/3. Adam Elga argued for this position originally [2] as follows: Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, given that the coin lands tails, her credence that it is Monday should equal her credence that it is Tuesday, since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

P(Tails and Tuesday) = P(Tails and Monday).

Suppose now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should hold that P(Tails | Monday) = P(Heads | Monday), and thus

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).

Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third by the previous two steps in the argument.

An alternative argument is as follows. Credence can be viewed as the amount a rational risk-neutral bettor would wager if the payoff for being correct is 1 unit (the wager itself being lost either way). In the heads scenario, Sleeping Beauty would spend her wager amount one time, and receive 1 money for being correct. In the tails scenario, she would spend her wager amount twice, and receive nothing. Her expected value is therefore to gain 0.5 but also lose 1.5 times her wager, thus she should break even if her wager is 1/3.

Halfer position

David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2. [6] Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment. This directly contradicts one of the thirder's premises, since it means P(Tails | Monday) = 1/3 and P(Heads | Monday) = 2/3.

Nick Bostrom argues that Sleeping Beauty does have new evidence about her future from Sunday: "that she is now in it," but does not know whether it is Monday or Tuesday, so the halfer argument fails. [7] In particular, she gains the information that it is not both Tuesday and the case that Heads was flipped.

Double halfer position

The double halfer position [8] argues that both P(Heads) and P(Heads | Monday) equal 1/2. Mikaël Cozic, [9] in particular, argues that context-sensitive propositions like "it is Monday" are in general problematic for conditionalization and proposes the use of an imaging rule instead, which supports the double halfer position.

Connections to other problems

Nick Bostrom argues that the thirder position is implied by the Self-Indication Assumption.

Credence about what precedes awakenings is a core question in connection with the anthropic principle.

Variations

Extreme Sleeping Beauty

This differs from the original in that there are one million and one wakings if tails comes up. It was formulated by Nick Bostrom, and is used to argue for the thirder position.

Sailor's Child problem

The Sailor's Child problem, introduced by Radford M. Neal, is somewhat similar. It involves a sailor who regularly sails between ports. In one port there is a woman who wants to have a child with him, across the sea there is another woman who also wants to have a child with him. The sailor cannot decide if he will have one or two children, so he will leave it up to a coin toss. If Heads, he will have one child, and if Tails, two children. But if the coin lands on Heads, which woman would have his child? He would decide this by looking at The Sailor's Guide to Ports and the woman in the port that appears first would be the woman that he has a child with. You are his child. You do not have a copy of The Sailor's Guide to Ports. What is the probability that you are his only child, thus the coin landed on Heads (assume a fair coin)? [10]

Related Research Articles

The anthropic principle is the principle that there is a restrictive lower bound on how statistically probable our observations of the universe are, given that we could only exist in the particular type of universe capable of developing and sustaining sentient life. Proponents of the anthropic principle argue that it explains why this universe has the age and the fundamental physical constants necessary to accommodate conscious life, since if either had been different, we would not have been around to make observations. Anthropic reasoning is often used to deal with the notion that the universe seems to be fine tuned.

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the incorrect belief that, if a particular event occurs more frequently than normal during the past, it is less likely to happen in the future, when it has otherwise been established that the probability of such events does not depend on what has happened in the past. Such events, having the quality of historical independence, are referred to as statistically independent. The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the usual number of sixes.

The unexpected hanging paradox or surprise test paradox is a paradox about a person's expectations about the timing of a future event which they are told will occur at an unexpected time. The paradox is variously applied to a prisoner's hanging, or a surprise school test. It was first introduced to the public in Martin Gardner's March 1963 Mathematical Games column in Scientific American magazine.

In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future.

Statistical bias is a systematic tendency which causes differences between results and facts. The bias exists in numbers of the process of data analysis, including the source of the data, the estimator chosen, and the ways the data was analyzed. Bias may have a serious impact on results, for example, to investigate people's buying habits. If the sample size is not large enough, the results may not be representative of the buying habits of all the people. That is, there may be discrepancies between the survey results and the actual results. Therefore, understanding the source of statistical bias can help to assess whether the observed results are close to the real results.

In probability theory, an event is said to happen almost surely if it happens with probability 1. In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory.

Selection bias is the bias introduced by the selection of individuals, groups, or data for analysis in such a way that proper randomization is not achieved, thereby failing to ensure that the sample obtained is representative of the population intended to be analyzed. It is sometimes referred to as the selection effect. The phrase "selection bias" most often refers to the distortion of a statistical analysis, resulting from the method of collecting samples. If the selection bias is not taken into account, then some conclusions of the study may be false.

Nick Bostrom is a Swedish-born philosopher at the University of Oxford known for his work on existential risk, the anthropic principle, human enhancement ethics, superintelligence risks, and the reversal test. In 2011, he founded the Oxford Martin Program on the Impacts of Future Technology, and is the founding director of the Future of Humanity Institute at Oxford University. In 2009 and 2015, he was included in Foreign Policy's Top 100 Global Thinkers list.

Counterfactual conditionals are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.

Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

Analysis is a peer-reviewed academic journal of philosophy established in 1933 that is published quarterly by Oxford University Press on behalf of the Analysis Trust. Prior to January 2009, the journal was published by Blackwell Publishing. Electronic access to this journal is available via JSTOR (1933–2013), Wiley InterScience (1996–2008), and Oxford Journals (2009–present). The journal publishes short, concise articles in virtually any field of the analytic tradition.

The simulation hypothesis is a proposal regarding the nature of existence which posits that all of existence is an artificial simulation, such as a computer simulation. Some versions rely on the development of a simulated reality, a proposed technology that would be able to convince its inhabitants that the simulation was "real".

The propensity theory of probability is one interpretation of the concept of probability. Theorists who adopt this interpretation think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome.

In philosophy, Pascal's mugging is a thought-experiment demonstrating a problem in expected utility maximization. A rational agent should choose actions whose outcomes, when weighed by their probability, have higher utility. But some very unlikely outcomes may have very great utilities, and these utilities can grow faster than the probability diminishes. Hence the agent should focus more on vastly improbable cases with implausibly high rewards; this leads first to counter-intuitive choices, and then to incoherence as the utility of every choice becomes unbounded.

Credence is a statistical term that expresses how much a person believes that a proposition is true. As an example, a reasonable person will believe with 50% credence that a fair coin will land on heads the next time it is flipped. If the prize for correctly predicting the coin flip is \$100, then a reasonable risk-neutral person will wager \$49 on heads, but they will not wager \$51 on heads.

Anthropic Bias: Observation Selection Effects in Science and Philosophy (2002) is a book by philosopher Nick Bostrom. Bostrom investigates how to reason when one suspects that evidence is biased by "observation selection effects", in other words, when the evidence presented has been pre-filtered by the condition that there was some appropriately positioned observer to "receive" the evidence. This conundrum is sometimes called the "anthropic principle," "self-locating belief," or "indexical information". Discussed concepts include the self-sampling assumption and the self-indication assumption.

Surveillance capitalism is an economic system centred around the commodification of personal data with the core purpose of profit-making. The concept of surveillance capitalism, as described by Shoshana Zuboff, arose as advertising companies, led by Google's AdWords, saw the possibilities of using personal data to target consumers more precisely.

Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based on the idea that beliefs can be interpreted as subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesians have applied these fundamental principles to various epistemological topics but Bayesianism does not cover all topics of traditional epistemology. The problem of confirmation in the philosophy of science, for example, can be approached through the Bayesian principle of conditionalization by holding that a piece of evidence confirms a theory if it raises the likelihood that this theory is true. Various proposals have been made to define the concept of coherence in terms of probability, usually in the sense that two propositions cohere if the probability of their conjunction is higher than if they were neutrally related to each other. The Bayesian approach has also been fruitful in the field of social epistemology, for example, concerning the problem of testimony or the problem of group belief. Bayesianism still faces various theoretical objections that have not been fully solved.

Economists and marketers use of the Search, Experience, Credence (SEC) classification of goods and services, which is based on the ease or difficulty with which consumers can evaluate or obtain information. These days most economics and marketers treat the three classes of goods as a continuum. Archetypal goods are:

Arnold Stuart Zuboff is an American philosopher who has worked on topics such as personal identity, philosophy of mind, ethics, metaphysics, epistemology and the philosophy of probability. He is the original formulator of the Sleeping Beauty problem and a view analogous to open individualism—the position that there is one subject of experience, who is everyone—which he calls "universalism".

References

1. Arnold Zuboff (1990). "One Self: The Logic of Experience". Inquiry: An Interdisciplinary Journal of Philosophy. 33 (1): 39–68. doi:10.1080/00201749008602210.(subscription required)
2. Elga, A. (2000). "Self-locating Belief and the Sleeping Beauty Problem". Analysis. 60 (2): 143–147. CiteSeerX  . doi:10.1093/analys/60.2.143. JSTOR   3329167.
3. Michele Piccione and Ariel Rubinstein (1997) “On the Interpretation of Decision Problems with Imperfect Recall,” Games and Economic Behavior 20, 3-24.
4. Michele Piccione and Ariel Rubinstein (1997) “The Absent Minded Driver's Paradox: Synthesis and Responses,” Games and Economic Behavior 20, 121-130.
5. Nick Wedd (June 14, 2006). "Some "Sleeping Beauty" postings" . Retrieved November 7, 2014.
6. Lewis, D. (2001). "Sleeping Beauty: reply to Elga" (PDF). Analysis. 61 (3): 171–76. doi:10.1093/analys/61.3.171. JSTOR   3329230.
7. Bostrom, Nick (July 2007). "Sleeping beauty and self-location: A hybrid model" (PDF). Synthese. 157 (1): 59–78. doi:10.1007/s11229-006-9010-7. JSTOR   27653543. S2CID   12215640.
8. Meacham, C. J. (2008). "Sleeping beauty and the dynamics of de se beliefs". Philosophical Studies. 138 (2): 245–269. CiteSeerX  . doi:10.1007/s11098-006-9036-1. JSTOR   40208872. S2CID   26902640.
9. Mikaël Cozic (February 2011). "Imaging and Sleeping Beauty: A case for double-halfers". International Journal of Approximate Reasoning. 52 (2): 137–143. doi:10.1016/j.ijar.2009.06.010.
10. Neal, Radford M. (2006). "Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning". arXiv:.