Some notes on Decision Theory

Once upon a time, I took a course in Decision Theory at ETH. Here are some notes:

normative theory vs descriptive theory (maths vs psychology)

prospect theory

options vs choices

money pump

bond vs shares

Main notions:

1. Slide
   1. Normative: Perfect rationality, optimality, payoff maximality
   2. Descriptive: How real people think and make decisions.
      1. bias: persistent and systematic deviation from rationality
   3. Prescriptive: How can we help people to make better decisions. How can we structure the choice environment to minimize decision errors and biases.
   4. People use limited heuristics in making decisions
   5. Decision maker
      1. individual, couple, family, firm, country
      2. any monolithic cognitive agent
   6. Goals, preferences, wants, desires
   7. options, alternatives, courses of action
   8. beliefs
   9. Decision under
      1. certainty
         1. all options and outcomes are completely known to the DM (military diet, ups example)
      2. risk
         1. When all the choices and options are known but there are well defined stochastic nodes (gambling, insurance, investing)
      3. uncertainty
         1. options are not well defined, or probabilities are not defined (or both) (examples most of life)
   10. Preferences
       1. Revealed, Complete, Transitive, Independent, Stable
   11. People often satisfice instead of optimize
   12. People guess, estimate and try to do “well enough’’ rather than optimally.
   13. Rationality is bounded in two ways.
       1. Information from the environment
       2. cognitive capacity
   14. Sometimes close enough is fine, sometimes not
   15. Cognitive illusions: Monty hall problem
   16. Limitations in attention and memory. There are tools which compensate for that.
   17. Multi Attribute Evaluation (MAUT)
       1. identify options
       2. define goals
       3. quantify subjective evaluations
       4. weight and scale different evaluations so they can be aggregated meaningfully.
       5. make reasonable and justifiable decisions
   18. Problem, Objectives, Alternatives, Consequences, Tradeoffs (PrOACT)
   19. Comparative, multiple stakeholders (probably not equally important), multiple goals (maybe not equally important), multiple choices with multiple attibutes, judgements are necessary, quantify!
       1. Identify the decision maker
       2. Identify the alternatives
       3. the relevant attributes
       4. assign values to the attributes of each alternatives
       5. determine a weight for each attribute
       6. for each alternative, compute a weighted mean
       7. make a provisional decision
       8. get feedback and run a sensitivity analysis
   20. compensatory vs noncompensatory
       1. sometimes some alternatives are not acceptable (minimum requirements)
   21. minor changes may cause a change in the resulting decision
   22. “fair’’ method and easy to develop political support
   23. analytic hierarchy process: binary comparison of pairs of attributes in terms of their importance and then assign a weight (how much better is on alternative compared to the other)
   24. built in check for transitivity and consistency
   25. a well defined process to turn subjective judgements into more objective ratings
2. Secretary problem
   1. What makes a good decision (good outcome or good reasons?)
   2. expected value
   3. probability (classical, frequency, subjective)
   4. it can be hard for people to reason about probabilities correctly (monty hall, other pitfalls)
   5. birthday problem
   6. bayes rule
   7. st petersburg gamble
   8. Moral worth: Bernoulli: the worth of something is not the same as its value → Definition of utility (utility = value^a)
   9. Expected utility does not work always
   10. Allais gambles. The result is a contradiction to EUT.
   11. Feelings about risk: People will accept 1000 times greater risks if they are their own volition, rather than involuntary
   12. O.J. SImpson example
   13. People are generally risk averse, but to varying degrees
   14. EVT defines rationality if the goal of the DM is to maximize value in the long run
   15. EUT defines rationality if the goal of the DM is to maximize utility (moral worth) in the long run
   16. Risk seeking in the domain of losses, risk averse in the domain of gains
   17. Decision framing
   18. Money trump
3. Sequential investment problem
   1. Persistent cognitive noise
   2. systematic departures from EUT – biases
   3. Friedman-Savage utility function
   4. Prospect theory
      1. descriptive account of decision theory under risk
      2. aims to account for systematic deviations from eut and evt
      3. Coding — outcomes relative to the current state in terms of gains and losses
      4. combination — joining similar prospects
      5. segregation — isolation of sure things
      6. cancellation — discarding shared components
      7. simplification — rounding
      8. elimination of dominated options
   5. probability weighting function — the estimation that people do on probabilities (magnification of small values in the expense of larger values)
   6. 4 fold pattern of risk preferences
      1. risk seeking over low p gains — lotteries
      2. risk averse over low p losses — insurance
      3. risk averse over high p losses — certainty effect
      4. risk seeking over high p losses — end of the day effect
   7. certainty effect
      1. insurance which does not pay with a very small probability
   8. end of the day effect
      1. people make large bets on the last day to “break even’’
   9. Shortcomings
      1. average choices on average
      2. not for every one
      3. not even for someone over the domain of time
      4. no insight — just better predictions
      5. does not scale. (portfolio example, mulistage risky decisions)
   10. Dynamic decision contexts are defined by:
       1. sequential choices are made and they are interspersed with updated information about the environment
       2. the decision environment changes over time (depending on the choices of DM)
   11. normative approaches
       1. maximize expected payoff
       2. dynamic programming
   12. descriptive approaches
       1. case studies
       2. experimental methods (behavioral laboratories)
       3. human behaviour is compared to the normative approach
       4. identification of systematic departures from optimality — existent cognitive mechanisms can be accounted for
       5. ⇒ people predictably irrational (underlying assumption)
   13. optimal stopping
       1. example: secretary problem
       2. target: select a maximum value
       3. full information games
          1. perfectly known distribution
          2. draw a number from [0,20] — given the relative order of the number that you encountered so far but not the number
       4. partial information
          1. known distribution but not its parameters
          2. german tank problem
       5. no information
          1. iid random variables with unknown distribution
          2. secretary problem
          3. Sometimes called the Sultan’s Dowry Problem, the Marriage Problem, or the Fussy Suitor Problem
          4. optimal decision policy — threshold (e^{-1})
          5. generalized secretary problem
          6. persistent bias for stopping too soon
          7. probability distortion and not risk aversion is the (assumed) answer
       6. Other dynamic decision problem
          1. sell a good over a time interval where offers in a specific range arrive with a probability
   14. Kelly criterion
       1. always bet the same proportion (2p – 1)
       2. maximizes the expected geometric mean
       3. poor criterion in prediction decision makers behavior
       4. people have strong urge to adjust their investments
       5. evidence of learning to adjust less over rounds
       6. if we charge people when they adjust, we help them
   15. sequential investment problem
       1. optimal policy: bet max or not according to whether we are “above or below the expectation’’
   1. Heuristics as rule of thumb.
   2. gambler’s fallacy (law of small numbers)
   3. hot hand fallacy (law of small numbers)
   4. stars
   5. availability heuristics (how easy it is to recall a piece of information)
   6. anchoring and adjustment
      1. factorial where presented the larger number first or last
      2. use of early values to predict
   7. framing effect (the influence of questions)
   8. confirmation bias
   9. are people really irrational — heuristics evolved for some reason. Carefully designed corner case are examined.
   1. public goods game
      1. selfish players take no interest
      2. social optimal is far off
      3. many people are pro-social
      4. design vector according to the answers
   2. overconfidence quiz
      1. attentional
         1. selective memory/information search: confirmation bias
         2. selective encoding: excuses/rationalizations
         3. selective encoding : rewarded events are remembered better
      2. motivational
         1. need to appear competent and confident to tothers and oneself. confidence and optimism help to get things done.
   1. bordeaux and regression
   2. wisdom of crowds
   3. risky decision theory and gambling
   4. rush of winning
   5. highly reinforcing

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