80,000 Hours Podcast
#139 Classic episode – Alan Hájek on puzzles and paradoxes in probability and expected value
Episode guests
Podcast summary created with Snipd AI
Quick takeaways
- The St. Petersburg paradox illustrates the absurdity of traditional expected value reasoning, suggesting an infinite expected value despite finite outcomes.
- Philosophical thought experiments challenge common assumptions, prompting deeper inquiry into complex concepts like knowledge and decision-making.
- Maximizing expected value becomes problematic in extreme scenarios, as seen in high-stakes gambling and its irrational implications for rational players.
- Counterfactuals play a crucial role in moral reasoning, revealing complexities in causation and the validity of objective consequentialism.
- Modifications to expected value calculations, such as incorporating diminishing marginal utility, enable more rational decision-making regarding risky propositions.
Deep dives
Thought Experiments in Philosophy
Philosophical thought experiments, such as Mary in the room or Searle's Chinese room, are often used to explore key concepts. These thought experiments challenge assumptions and prompt discussions around complex philosophical points. For example, Mary knows all the physical facts but has never seen red, only to discover something new when she finally does, highlighting potential gaps in knowledge. This usage of fictional scenarios emphasizes the need for philosophical inquiry beyond empirical facts.
Expected Value in Decision-Making
The concept of maximizing expected value plays a significant role in decision-making, as illustrated by the extreme example of Sam Bergman-Fried wanting to gamble with the universe's existence. His choice to take high-stakes risks demonstrates the potential absurdity of traditional expected value reasoning, especially when it comes to extreme outcomes. This leads to the St. Petersburg paradox, which shows how maximizing expected value can produce counterintuitive conclusions. Consequently, discussions arise around modifications to expected value theory and the limitations of applying it to all cases.
St. Petersburg Paradox Explained
The St. Petersburg paradox presents a gambling scenario where one tosses a fair coin until it lands on heads, with escalating payouts. This paradox reveals the contradiction in valuing this gamble, as it suggests an infinite expected value based on high payouts, making it seem irrational for most players. When using expected value calculations, the paradox highlights the illogical conclusion that one should always be willing to play, regardless of the investment needed. This raises questions about the interpretations of values and probabilities within expected value theory.
Responses to the St. Petersburg Paradox
Various responses to the St. Petersburg paradox involve addressing the core components of expected value calculations, such as modifying the probabilities, adjusting the values, or changing the decision-making framework. One notable solution is to account for diminishing marginal utility, suggesting that the value of additional winnings decreases as wealth increases. This adjustment shifts the expected value from being infinite to a finite amount, allowing for rational decision-making. These suggestions lead to a considerable reevaluation of how individuals approach expected value in unfamiliar or extreme scenarios.
Challenges in Objective Consequentialism
Objective consequentialism evaluates actions based solely on their outcomes, but this framework struggles when incorporating counterfactuals into moral judgments. Given the potential ripple effects and vast number of possible outcomes stemming from a single action, assessing what could have happened becomes complex and uncertain. The implications of such uncertainty challenge the validity of objective consequentialism, leading to intense debates about how reality and potential consequences interact. Particularly, long-term impacts of actions may distort moral assessments based on finite outcomes in the present.
Counterfactuals and Conditional Reasoning
Counterfactual statements explore what could happen under hypothetical scenarios, playing crucial roles in moral reasoning, causal relationships, and decision-making. The challenge lies in accurately attributing truth conditions to counterfactuals, as traditional definitions often fail to differentiate between true and false claims effectively. By examining the nuances of possible worlds and considering various factors impacting similarity and causal history, philosophers strive to construct a coherent understanding of these thought experiments. The assertability condition highlights the importance of context in stating counterfactuals, allowing for a clearer representation of how assertions are communicated.
Importance of Context in Counterfactuals
Asserting counterfactuals requires attention to the context in which they are made, as different situations may lead to various interpretations of truth. The notion that most counterfactuals are false emphasizes that strict conditions must be analytically defined to ensure consistent reasoning. Examples highlight the implications of overly specific conclusions resulting from vague antecedents, showcasing the need for a more flexible interpretation of counterfactual statements. By recognizing that context-dependent factors influence how we evaluate counterfactuals, philosophers can better grasp their implications and address flaws in traditional analyses.
Challenges with Most Similar Worlds
Philosophers have criticized the approach of relying solely on the closest possible worlds to assess counterfactuals, arguing that this method can yield implausibly specific outcomes. Examples, such as the nuances of well-known historical events and specific hypothetical situations, emphasize how this approach can lead to inaccuracies in reasoning. An overreliance on closest possible worlds risks oversimplifying the complexities inherent in real-world scenarios and can overlook essential details that may prove relevant to the analysis. As discussions evolve, the quest for a more comprehensive understanding of counterfactuals continues.
Implications for Moral Theories
The discussions around counterfactuals and their relevance to consequentialist moral theories highlight the tensions that arise when prioritizing outcomes. The struggle of assigning value to choices based on their potential consequences becomes more complex as the implications of actions ripple through time. In evaluating moral behavior, whether through actual consequences or counterfactual scenarios, philosophers must grapple with the inherent uncertainties in causation and historical context. Ultimately, the complexities of counterfactual reasoning necessitate a reconsideration of conventional moral frameworks and their foundations.
The Role of Probability in Ethical Decisions
The intersections between probability theory and ethical decision-making reveal a host of philosophical challenges and insights. Counterfactuals inevitably implicate probabilistic reasoning, as the likelihood of potential outcomes influences the assessment of moral choices. This relationship indicates that ethical theories cannot be fully disentangled from questions of probability, raising concerns over the foundations of consequentialist frameworks. As philosophers confront these challenges, exploring how probability affects moral conclusions becomes increasingly crucial for refining ethical reasoning.
A casino offers you a game. A coin will be tossed. If it comes up heads on the first flip you win $2. If it comes up on the second flip you win $4. If it comes up on the third you win $8, the fourth you win $16, and so on. How much should you be willing to pay to play?
The standard way of analysing gambling problems, ‘expected value’ — in which you multiply probabilities by the value of each outcome and then sum them up — says your expected earnings are infinite. You have a 50% chance of winning $2, for '0.5 * $2 = $1' in expected earnings. A 25% chance of winning $4, for '0.25 * $4 = $1' in expected earnings, and on and on. A never-ending series of $1s added together comes to infinity. And that's despite the fact that you know with certainty you can only ever win a finite amount!
Today's guest — philosopher Alan Hájek of the Australian National University — thinks of much of philosophy as “the demolition of common sense followed by damage control” and is an expert on paradoxes related to probability and decision-making rules like “maximise expected value.”
Rebroadcast: this episode was originally released in October 2022.
Links to learn more, highlights, and full transcript.
The problem described above, known as the St. Petersburg paradox, has been a staple of the field since the 18th century, with many proposed solutions. In the interview, Alan explains how very natural attempts to resolve the paradox — such as factoring in the low likelihood that the casino can pay out very large sums, or the fact that money becomes less and less valuable the more of it you already have — fail to work as hoped.
We might reject the setup as a hypothetical that could never exist in the real world, and therefore of mere intellectual curiosity. But Alan doesn't find that objection persuasive. If expected value fails in extreme cases, that should make us worry that something could be rotten at the heart of the standard procedure we use to make decisions in government, business, and nonprofits.
These issues regularly show up in 80,000 Hours' efforts to try to find the best ways to improve the world, as the best approach will arguably involve long-shot attempts to do very large amounts of good.
Consider which is better: saving one life for sure, or three lives with 50% probability? Expected value says the second, which will probably strike you as reasonable enough. But what if we repeat this process and evaluate the chance to save nine lives with 25% probability, or 27 lives with 12.5% probability, or after 17 more iterations, 3,486,784,401 lives with a 0.00000009% chance. Expected value says this final offer is better than the others — 1,000 times better, in fact.
Ultimately Alan leans towards the view that our best choice is to “bite the bullet” and stick with expected value, even with its sometimes counterintuitive implications. Where we want to do damage control, we're better off looking for ways our probability estimates might be wrong.
In this conversation, originally released in October 2022, Alan and Rob explore these issues and many others:
- Simple rules of thumb for having philosophical insights
- A key flaw that hid in Pascal's wager from the very beginning
- Whether we have to simply ignore infinities because they mess everything up
- What fundamentally is 'probability'?
- Some of the many reasons 'frequentism' doesn't work as an account of probability
- Why the standard account of counterfactuals in philosophy is deeply flawed
- And why counterfactuals present a fatal problem for one sort of consequentialism
Chapters:
- Cold open {00:00:00}
- Rob's intro {00:01:05}
- The interview begins {00:05:28}
- Philosophical methodology {00:06:35}
- Theories of probability {00:40:58}
- Everyday Bayesianism {00:49:42}
- Frequentism {01:08:37}
- Ranges of probabilities {01:20:05}
- Implications for how to live {01:25:05}
- Expected value {01:30:39}
- The St. Petersburg paradox {01:35:21}
- Pascal’s wager {01:53:25}
- Using expected value in everyday life {02:07:34}
- Counterfactuals {02:20:19}
- Most counterfactuals are false {02:56:06}
- Relevance to objective consequentialism {03:13:28}
- Alan’s best conference story {03:37:18}
- Rob's outro {03:40:22}
Producer: Keiran Harris
Audio mastering: Ben Cordell and Ryan Kessler
Transcriptions: Katy Moore
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