DOCSLIB.ORG

Decision Theory with a Human Face

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Decision Theory with a Human Face Richard Bradley April 2016 ii For Shura Contents Preface xi Introduction xiii I Rationality, Uncertainty and Choice 1 1 Decision Problems 3 1.1 Maximisation of Expected Bene…t . 3 1.2 Framing Decisions . 6 1.3 Savage’sTheory . 9 2 Rationality 13 2.1 Moderate Humeanism . 13 2.2 The Choice Principle . 16 2.3 Subjectivism . 19 3 Uncertainty 23 3.1 Evaluative Uncertainty . 24 3.2 Option uncertainty . 27 3.3 Modal Uncertainty . 30 4 Justifying Bayesianism 33 4.1 Pragmatism . 34 4.2 Interpretations of Preference . 36 4.3 Representation Theorems . 38 4.4 Savage’sRepresentation Theorem . 40 4.5 Evaluation of Savage’saxioms . 44 4.5.1 The Sure-thing Principle . 45 4.5.2 State Independence / Rectangular Field . 46 4.5.3 Probability Principle . 47 4.6 Evaluation of Savage’sargument . 49 II Prospective Rationality 53 5 Rational Belief and Desire 55 5.1 Agents . 55 5.2 Prospects . 56 iii iv CONTENTS 5.2.1 Vocabulary . 58 5.2.2 Boolean Algebras* . 58 5.2.3 Conditional Algebras* . 59 5.3 Attitudes . 60 5.3.1 Comparative Relations . 63 5.3.2 Numerical Representation of Comparative Relations* . 65 5.4 Rational Belief and Desire . 66 5.4.1 Numerical Degrees of Belief and Desire . 66 5.4.2 Probabilism . 68 5.4.3 Desirabilism . 69 5.5 Aspirations of Objectivity . 73 6 Conditional Attitudes 77 6.1 Suppositions . 78 6.2 Suppositional Probability and Desirability . 80 6.3 Evidential Supposition . 82 6.3.1 Evidential Conditional Belief . 82 6.3.2 Evidential Conditional Desire . 84 6.4 Independence . 85 6.5 Foundations* . 86 6.5.1 Representation of Conditional Attitudes . 87 6.5.2 Evidential Supposition . 88 6.5.3 Independence and Separability . 89 6.6 Hypothetical Reasoning . 90 6.6.1 Dominance Arguments . 90 6.6.2 Averaging Arguments . 92 6.6.3 Summary . 94 7 Conditionals and the Ramsey Test 95 7.1 The Ramsey Test for Belief . 95 7.2 The Ramsey Test for Desire . 97 7.3 Chances and Counterfactuals . 98 7.4 Foundations . 101 7.5 Facing the Triviality Results . 103 7.6 Partitioning Conditionals . 106 7.7 Summary . 109 III Facing the World 111 8 Multidimensional Possible World Semantics 113 8.1 Prospects and Possible Worlds . 113 8.1.1 Possible Worlds Models . 113 8.1.2 Triviality Strikes Again . 117 8.2 Multidimensional Semantics . 119 8.2.1 Centering . 121 8.2.2 Multidimensional Possible World Semantics . 123 8.3 Probability . 124 8.3.1 Independence . 126 8.4 Desirability and Counterfactual Value . 128 CONTENTS v 8.5 Value Actualism . 130 8.5.1 World Actualism . 130 8.5.2 Prospect Actualism . 131 8.5.3 Restricted Actualism and Evidential Desirability . 132 8.6 Partitioning Conditionals . 133 8.7 Concluding Remarks . 135 9 Taking Action 139 9.1 Decision Principles . 139 9.2 Preference-based Choice . 141 9.3 Decisions Under Risk . 144 9.4 Decisions Making Under Uncertainty . 146 9.4.1 Savage Acts . 148 9.5 Bayesian Decision Theory . 150 9.5.1 Allais’Paradox . 152 9.5.2 Ellsberg’sParadox . 155 9.6 Causal Decision Theory . 157 10 The Learning Agent 163 10.1 Introduction . 163 10.2 Experience and Learning . 165 10.3 Classical Bayesian Conditioning . 168 10.4 Je¤rey Conditioning . 172 10.5 Adams Conditioning . 175 10.6 Preference Change . 178 10.6.1 Generalised Conditioning . 180 10.6.2 Belief-Driven Preference Change . 184 10.6.3 Desire-Driven Preference Change . 187 IV Rationality Within Bounds 189 11 Imprecise Bayesianism 191 11.1 Introduction . 191 11.2 Bounded Minds . 192 11.2.1 Unawareness and Incompleteness . 192 11.2.2 Types of Inconsistency . 194 11.2.3 Coherent Extendability . 196 11.3 Imprecise Bayesianism . 197 11.4 Severe Uncertainty . 202 11.5 Rational Incomplete Preference . 206 11.5.1 Coherently Extendable Preferences . 207 11.5.2 Representation Theorems . 208 11.6 Challenges to Imprecise Bayesianism . 210 12 Changing Your Mind 215 12.1 Attitude Change . 215 12.2 Changing An Incomplete State of Mind . 217 12.2.1 Attitude Revision . 220 12.2.2 Attitude Formation . 221 vi CONTENTS 12.2.3 Attitude Withdrawal . 222 12.2.4 Connections* . 223 12.3 Unawareness . 224 12.3.1 Rational Unawareness . 225 12.3.2 Representations of Unawareness* . 227 12.4 Changes in States of Awareness . 228 12.4.1 Domain Extension* . 229 12.4.2 Domain Restriction* . 230 12.4.3 The Problem of New Hypotheses . 231 13 Decision Making under Ambiguity 233 13.1 Introduction . 233 13.2 Reaching a Judgement . 234 13.2.1 Picking the ‘Best’ . 236 13.2.2 Aggregating . 237 13.3 Alternative Decision Rules . 240 13.3.1 Caution . 241 13.3.2 Con…dence . 243 13.3.3 Robustness . 244 13.3.4 Flexibility . 245 13.4 Cautious Decision Making . 246 13.5 Models of Ambiguity Aversion* . 248 13.6 The Rationality of Ambiguity Aversion . 251 13.7 Concluding Remarks . 254 14 Con…dence 257 14.1 The Paradox of Ideal Evidence . 257 14.2 Con…dence-Based Learning . 260 14.3 A Model of Con…dence . 263 14.3.1 Con…dence Relations* . 265 14.3.2 Deciding with Su¢ cient Con…dence . 267 14.4 Concluding Remarks . 269 15 Appendix: Proofs 271 List of Figures 1 Flow of Chapters . xvi 6.1 Causal Graph for Breathing Di¢ culties . 79 8.1 A Small-World Model . 118 10.1 Evidence-based Constraints on Belief . 164 10.2 Je¤rey Conditioning . ..
Recommended publications
  • Extracting Money from Causal Decision Theorists

    Extracting Money from Causal Decision Theorists

    Extracting Money from Causal Decision Theorists Caspar Oesterheld∗ , Vincent Conitzer Duke University, Department of Computer Science focaspar, [email protected] Abstract that Newcomb’s problem closely resembles playing a Pris- oner’s Dilemma against a similar opponent [Brams, 1975; Newcomb’s problem has spawned a debate about Lewis, 1979]. Whereas CDT recommends defecting even which variant of expected utility maximization (if in a Prisoner’s Dilemma against an exact copy, EDT rec- any) should guide rational choice. In this paper, ommends cooperating when facing sufficiently similar op- we provide a new argument against what is proba- ponents. What degree of similarity is required and whether bly the most popular variant: causal decision the- EDT and CDT ever come apart in this way in practice has ory (CDT). In particular, we provide two scenarios been the subject of much discussion [Ahmed, 2014, Sec- in which CDT voluntarily loses money. In the first, tion 4.6]. A common view is that CDT and EDT come an agent faces a single choice and following CDT’s apart only under fairly specific circumstances that do not recommendation yields a loss of money in expec- include most human interactions. Still, Hofstadter [1983], tation. The second scenario extends the first to a for example, argues that “superrational” humans should co- diachronic Dutch book against CDT. operate with each other in a real-world one-shot Prisoner’s Dilemma, reasoning in a way that resembles the reasoning done under EDT. Economists have usually been somewhat 1 Introduction dismissive of such ideas, sometimes referring to them as In Newcomb’s problem [Nozick, 1969; Ahmed, 2014], a “be- “(quasi-)magical thinking” when trying to explain observed ing” offers two boxes, A and B.
  • Contingent Free Choice: on Extending Quantum Theory to a Contextual, Deterministic Theory with Improved Predictive Power

    Contingent Free Choice: on Extending Quantum Theory to a Contextual, Deterministic Theory with Improved Predictive Power

    Research Collection Working Paper Contingent Free Choice: On Extending Quantum Theory to a Contextual, Deterministic Theory With Improved Predictive Power Author(s): Fourny, Ghislain Publication Date: 2019-07-03 Permanent Link: https://doi.org/10.3929/ethz-b-000351179 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Noname manuscript No. (will be inserted by the editor) Contingent Free Choice: On Extending Quantum Theory to a Contextual, Deterministic Theory With Improved Predictive Power Ghislain Fourny Originally March 2019, Updated July 2019. Abstract The non-extensibility of quantum theory into a theory with im- proved predictive power is based on a strong assumption of independent free choice, in which the physicists pick a measurement axis independently of any- thing that couldn't have been caused by their decision. Independent free choice is also at the core of the Nash equilibrium and classical game theory. A more recent line of game-theoretical research based on weakening free choice leads to non-trivial solution concepts with desirable properties such as at-most uniqueness, Pareto optimality, and contextuality. We show how introducing contingent free choice in the foundations of quan- tum theory yields a class of deterministic and contextual theories with an improved predictive power, and contrast them with the pilot-wave theory. Specifically, we suggest that quantum experiments, such as the EPR exper- iment, involving measurements located in spacetime, can be recast as dynamic games with imperfect information involving human agents and the universe.
  • Arxiv:1712.05723V4 [Cs.GT] 18 Mar 2020 Department of Computer Science E-Mail: Ghislain.Fourny@Inf.Ethz.Ch 2 Ghislain Fourny

    Arxiv:1712.05723V4 [Cs.GT] 18 Mar 2020 Department of Computer Science E-Mail: [email protected] 2 Ghislain Fourny

    Noname manuscript No. (will be inserted by the editor) Perfect Prediction in Normal Form: Superrational Thinking Extended to Non-Symmetric Games Ghislain Fourny Originally December 15, 2017. Updated February 19, 2020. Abstract This paper introduces a new solution concept for non-cooperative games in normal form with no ties and pure strategies: the Perfectly Transpar- ent Equilibrium. The players are rational in all possible worlds and know each other's strategies in all possible worlds { which, together, we refer to as Perfect Prediction. The anticipation of a player's decision by their opponents is coun- terfactually dependent on the decision, unlike in Nash Equilibria where the decisions are made independently. The equilibrium, when it exists, is unique and is Pareto optimal. This equilibrium is the normal-form counterpart of the Perfect Prediction Equilibrium; the prediction happens \in another room" rather than in the past. The equilibrium can also be seen as a natural extension of Hofstadter's superrationality to non-symmetric games. Algorithmically, an iterated elimi- nation of non-individually-rational strategy profiles is performed until at most one remains. An equilibrium is a strategy profile that is immune against knowl- edge of strategies in all possible worlds and rationality in all possible worlds, a stronger concept than common knowledge of rationality but also stronger than common counterfactual belief of rationality. We formalize and contrast the Non-Nashian Decision Theory paradigm, common to this and several other papers, with Causal Decision Theory and Evidential Decision Theory. We define the Perfectly Transparent Equilibrium algorithmically and prove (when it exists) that it is unique, that it is Pareto- optimal, and that it coincides with Hofstadter's Superrationality on symmetric games.
  • Multiverse-Wide Cooperation Via Correlated Decision Making

    Multiverse-Wide Cooperation Via Correlated Decision Making

    Multiverse-wide Cooperation via Correlated Decision Making Caspar Oesterheld Abstract Some decision theorists argue that when playing a prisoner’s dilemma-type game against a sufficiently similar opponent, we should cooperate to make it more likely that our opponent also cooperates. This idea, which Hofstadter calls superrationality, has strong implications when combined with the insight from modern physics that we probably live in a large universe or multiverse of some sort. If we care about what happens in civilizations located elsewhere in the multiverse, we can superrationally cooperate with some of their inhabitants. That is, if we take their values into account, this makes it more likely that they do the same for us. In this paper, I attempt to assess the practical implications of this idea. I argue that to reap the full gains from trade, everyone should maximize the same impartially weighted sum of the utility functions of all collaborators. I also argue that we can obtain at least weak evidence about the content of these utility functions. In practice, the application of superrationality implies that we should promote causal cooperation, moral pluralism, moral reflection, and ensure that our descendants, who will be smarter and thus better at finding out how to benefit other superrationalists in the universe, engage in superrational cooperation. Contents 1 Introduction – the basic idea 3 1.1 An overview of the paper . 5 2 Superrationality 7 2.1 Lack of knowledge is evidential power, part I: the other agents . 9 2.2 A short survey of decision theories and their relation to superrationality . 11 2.3 CDT would self-modify to behave like a non-causal decision theory in some Newcomb-like problems .
  • Newcomb's Problem, Prisoners' Dilemma, and Collective Action*

    Newcomb's Problem, Prisoners' Dilemma, and Collective Action*

    S. L, HURLEY NEWCOMB'S PROBLEM, PRISONERS' DILEMMA, AND COLLECTIVE ACTION* ABSTRACT. Among various cases that equally admit of evidentialist reasoning, the supposedly evidentialist solution has varying degrees of intuitive attractiveness. I suggest that cooperative reasoning may account for the appeal of apparently evidentialist behavior in the cases in which it is intuitively attractive, while the inapplicability of cooperative reasoning may account for the unattractiveness of evidentialist behaviour in other cases. A collective causal power with respect to agreed outcomes, not evidentialist reasoning, makes cooperation attractive in the Prisoners' Dilemma. And a natural though unwar- ranted assumption of such a power may account for the intuitive appeal of the one-box response in Newcomb's Problem. 1. NEWCOMB~S PROBLEM AND COMMON CAUSES In Newcomb's Problem, I am offered the choice of taking either just the opaque box in front of me, or both it and another transparent box, in which I can see $1,000. I am told, and believe, that a predictor of human behaviour has already put either nothing or $1,000,000 into the opaque box. He has done so on the basis of whether he predicts that I will take both boxes, or only one. That is, he has put $1,000,000 in the opaque box if and only if he has predicted that I will take just it, and not the other as well. Concerning 99% (or some very high per- centage) of the many other cases of choices made by persons in the same circumstances as myself, the predictor has predicted correctly.