Decisions: Week 4

Transcript

1. D S D M William Lowe Data Science Lab, Hertie School Week : . .
2. S e role of utilities in making decisions How to think about your future How to think about other people’s future How people actually think about the future Making decisions in groups Case study Trouble with group decision making
3. P Representation eorem (Von Neumann & Morgenstern, ) If you have preferences over outcomes that are → complete → re exive → transitive then they can be represented numerically with a utility function: U(.)
4. P Representation eorem (Von Neumann & Morgenstern, ) If you have preferences over outcomes that are → complete → re exive → transitive then they can be represented numerically with a utility function: U(.) Outcomes of actions are uncertain. Represent them as P(Oi A = a) → a ‘lottery’ in the decision theory terminology Pick the value of A with the highest expected value EU(a) = i U(Oi)P(Oi A = a)
5. I : C It’s not clear that P(Oi A = a) is quite what we want here. Strictly it might be better to work with P(OA=a ) a.k.a. P(Oi do(a)) the distribution of O given that we do or set A to a, not just see A = a (Pearl )
6. I : C It’s not clear that P(Oi A = a) is quite what we want here. Strictly it might be better to work with P(OA=a ) a.k.a. P(Oi do(a)) the distribution of O given that we do or set A to a, not just see A = a (Pearl ) An expected utility formulated this way gives us causal decision theory → Probably that was the one you had in mind to start with! (I hope) For an example of where we might get di erent answers depending on this choice, see any discussion of “Newcomb’s Problem”
7. F Many (all) policy decision problems have future consequences → How to weigh them relative to the present consequences?
8. F Many (all) policy decision problems have future consequences → How to weigh them relative to the present consequences? Simple theory: → A reward r τ time steps into the future is worth U(r )D(τ) → where D is the discount function → in simple cases D(τ) = exp(−λτ) → where λ is a discount rate
9. F is works rather like a (negative) interest rate, δτ with ≤ δ ≤
10. T Case: climate change → How much are future (not yet existing) lives (and their quality) worth relative to current ones?
11. T Case: climate change → How much are future (not yet existing) lives (and their quality) worth relative to current ones? Two (separable?) considerations: → e e ect of an action on the existing population (or its quality of life) t years in the future → e e ect on populations that would exist (or the quality of their lives) t years in the future, some of whom would not exist (or whose life quality would be di erent) depending on the action How to think about this problem? (Note: however you feel about discount rates, policy behaviour embodies an implicit rate)
12. P As far as we are aware no animals discount exponentially → Why?
13. P As far as we are aware no animals discount exponentially → Why? Irrationality?
14. P As far as we are aware no animals discount exponentially → Why? Irrationality? Maybe. Maybe not. Let’s see what they do instead
15. H Human discounting is usually not exponential but hyperbolic and time sensitive Something more like ( + kτ) with unintuitive and apparently irrational consequences → preference reversal! For an interpretation in terms of ‘weakness of the will’ (Ainslie, )
16. E One way to understand what may be happening is to return to the last lecture on representing uncertainty
17. E One way to understand what may be happening is to return to the last lecture on representing uncertainty (Sozou ) argues → Exponential discounting implies a belief in a known and constant hazard rate λ → hazard rate: the probability that the future good fails to appear as expected at t + τ despite being there at t → So consider priors over λ
18. E One way to understand what may be happening is to return to the last lecture on representing uncertainty
19. E One way to understand what may be happening is to return to the last lecture on representing uncertainty And now posteriors over λ e exponential prior generates discounting behaviour that is hyperbolic in exactly the functional form we saw before What does the exponential prior mean? And what does this have to do with marshmallows?
20. M Americans eat these things. Nobody really knows why
21. S ? Mischel and Ebbesen ( ) gave children the following situation: → Favourite snack on a chair → Told they could eat it → But if they waited minutes then they could have a two
22. S ? Mischel and Ebbesen ( ) gave children the following situation: → Favourite snack on a chair → Told they could eat it → But if they waited minutes then they could have a two Results → A small minority ate the snack immediately → / delayed long enough to get the second → Age predicted ability to delay Follow up in : “preschool children who delayed grati cation longer [...] were described [...] as adolescents who were signi cantly more competent” Follow up in : SAT was a predictor
23. E Self control with snack predicts SAT and other positive outcomes! or Reliable parental environment makes it rational to expect rewards to come and make self-control a better idea Please don’t make graphs like this (it was , but still)
24. T We need to weight the future somehow → ere are mathematical problems with putting positive weight on the in nite future → Rational choices for the discount rate seem...under-constrained → For personal utilities the problem can seem straightforward → For utilities that range over others (real or potential) things get...di cult → Unfortunately these are the important policy cases → Temporal discounting involves questions of policy, psychology, and to fairness
25. T We need to weight the future somehow → ere are mathematical problems with putting positive weight on the in nite future → Rational choices for the discount rate seem...under-constrained → For personal utilities the problem can seem straightforward → For utilities that range over others (real or potential) things get...di cult → Unfortunately these are the important policy cases → Temporal discounting involves questions of policy, psychology, and to fairness Now, about all those people we were making decision about. Maybe they should be involved in the process?
26. G What is the ‘will’ of the people? (government, company, organization)
27. G What is the ‘will’ of the people? (government, company, organization) Operationally: How to aggregate individual preferences?
28. C Turns out this is not so straightforward. Consider the problem of choosing a pope (Maltzman et al., )
29. C Turns out this is not so straightforward. Consider the problem of choosing a pope (Maltzman et al., ) From until the pope was chosen by → God → a / vote majority among ≤ cardinals → All of the above
30. C Turns out this is not so straightforward. Consider the problem of choosing a pope (Maltzman et al., ) From until the pope was chosen by → God → a / vote majority among ≤ cardinals → All of the above In John Paul II changed the rules (Apostolic Constitution, ‘Universi Dominici Gregis’) → a simple majority and the possibility of a runo vote
31. C R . . . B XVI
32. P Pope selection had been historically di cult and controversial → e conclave in produced two outcomes (the pope and the anti-pope) → a cardinal grabbed the papal coat and tried to run o with it, a erwards reigning as Victor VI → Deadlock in Viterbo prompted the locals to remove the roof and put the cardinals on bread and water is hasn’t happened recently, but it could...
33. C
34. D wins for against margin M vs B M + M vs R R + B vs M M + B vs R B + R vs B B + R vs M R + Here there is no Condorcet winner
35. C Complete and transitive preferences over ≥ options can lead to cycles in aggregate preferences, i.e. majority preferences are cyclical Speci c example: → ere need be no overall winner using a majority voting rule
36. C Generalized by Ken Arrow into his Impossibility eorem (Arrow, ) Any aggregation method for preferences that has all the following properties → Non-dictatorial: e wishes of multiple cardinals should be taken into consideration → Unrestricted Domain: Voting must account for all individual preferences / cover all the options → Pareto Optimal: Unanimous individual preferences must be respected: If every cardinal prefers cardinal A over cardinal B, cardinal A should be the pope → Independent of Irrelevant Alternatives: If a candidate is removed or drops out, then the others’ order should not change can have cycles. (You will see lots of slightly di erent versions of these, depending on the proof)
37. T John-Paul II adjusted the rules in to → Remove ’unrestricted domain’ → Allow a bare majority a er rounds of voting
38. T John-Paul II adjusted the rules in to → Remove ’unrestricted domain’ → Allow a bare majority a er rounds of voting In principle, changing the number of deliberation cycles ( ) before a supermajority was no longer does not matter
39. T John-Paul II adjusted the rules in to → Remove ’unrestricted domain’ → Allow a bare majority a er rounds of voting In principle, changing the number of deliberation cycles ( ) before a supermajority was no longer does not matter Anecdotally, with cardinals, some abstaining → Round : Ratzinger gets votes → Round : Ratzinger gets votes (but no supermajority) → Round : and a clear winner
40. A Interpretation: → Cycles can be real → Cycles can represent the lack of a unique behavioural prediction Note: → We are not guaranteed cycles. at depends on a lot of things, including the individual’s preferences who are being aggregated
41. A Interpretation: → Cycles can be real → Cycles can represent the lack of a unique behavioural prediction Note: → We are not guaranteed cycles. at depends on a lot of things, including the individual’s preferences who are being aggregated Most theorists get around this result by assuming more structure on preferences → i.e. structured utility functions
42. I e most natural form of structured utility function is spatial How and how fast does utility decrease from the ideal point? → Linear: x − x∗ → Quadratic: (x − x∗ ) → Gaussian: exp (x−x∗) σ Interpretations: → Gaussian implies an ‘alienation from indi erence’ and is prpbably empirically more accurate (Carroll et al., )
43. I Black ( ) showed that if preferences were → single dimensional → single peaked then no choices are well behaved. Special case: → Majority vote converges on the median voter
44. I We can have preferences on several dimensions at once In this (radially symmetrical) case, majority rule also converges on the median voter → yay! But not so fast. It doesn’t work without the radial structure (Plott, ) McKelvey ( ) showed that in general multi-dimensional settings there need be ‘no stable equilibrium’ at all. i.e. cycling
45. S e McKelvey’s ‘chaos theorem’ shows that by careful choice of comparisons, a smart agenda controller can o er a series of votes that move naive spatial voters to support anything she wants ese seem to be rather dumb voters who have not taken a social choice course. Can we design a system where it’s best to be as honest as these folk?
46. I : S Not usually
47. I : S Not usually Gibbard-Satterthwaite theorem (informally): For group of ≥ individuals and ≥ alternatives (with no constraints on preferences and a non-dictatorial aggregation function) there is always some set of preferences it’s worth lying about (Gibbard, ; Satterthwaite, )
48. I : S Not usually Gibbard-Satterthwaite theorem (informally): For group of ≥ individuals and ≥ alternatives (with no constraints on preferences and a non-dictatorial aggregation function) there is always some set of preferences it’s worth lying about (Gibbard, ; Satterthwaite, ) ere’s just no avoiding strategy...
49. I : S Not usually Gibbard-Satterthwaite theorem (informally): For group of ≥ individuals and ≥ alternatives (with no constraints on preferences and a non-dictatorial aggregation function) there is always some set of preferences it’s worth lying about (Gibbard, ; Satterthwaite, ) ere’s just no avoiding strategy... Riker ( ) noted that it was o en possible to add a dimension in order to manipulate the outcome. More generally ‘the heresthetic’ is a type of political strategy, emphasizing → agenda control → strategic voting → manipulations of dimensions
50. G Being a rational group and making rational group decisions may be harder than you think Lots of possibilities for making it easier → Restrict the choice domain: dictators, committees, departments, runo s → Induce structured utilities, e.g. elite leadership, psychological heuristic
51. G Being a rational group and making rational group decisions may be harder than you think Lots of possibilities for making it easier → Restrict the choice domain: dictators, committees, departments, runo s → Induce structured utilities, e.g. elite leadership, psychological heuristic Shot: e concept of rational group preferences may just be a mistake (or an impossibility)? Chaser: You are o en, in a psychological sense, a group decision maker and may su er the same problems making decisions (Utility elicitation may be as hard as probability elicitation)
52. R Ainslie, G. ( ). ‘Breakdown of will’. Cambridge University Press. Arrow, K. J. ( ). ‘Social choice and individual values’. John Wiley & Sons. Black, D. ( ). ‘On the rationale of group decision-making’. Journal of Political Economy, ( ), – . Carroll, R., Lewis, J. B., Lo, J., Poole, K. T. & Rosenthal, H. ( ). ‘ e structure of utility in spatial models of voting’. American Journal of Political Science. Gibbard, A. ( ). ‘Manipulation of voting schemes: A general result’. Econometrica, ( ), . Maltzman, F., Schwartzberg, M. & Sigelman, L. ( ). ‘Vox populi, vox dei, vox sagittae’. PS - Political Science & Politics, ( ), – . McKelvey, R. D. ( ). ‘Intransitivities in multidimensional voting models and some implications for agenda control’. Journal of Economic eory, ( ), – .
53. R Mischel, W. & Ebbesen, E. B. ( ). ‘Attention in delay of grati cation’. Journal of Personality and Social Psychology, ( ), – . Plott, C. R. ( ). ‘A notion of equilibrium and its possibility under majority rule’. e American Economic Review, ( ), – . Riker, W. H. ( ). ‘ e art of political manipulation’. Yale University Press. Satterthwaite, M. A. ( ). ‘Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions’. Journal of Economic eory, ( ), – . Von Neumann, J. & Morgenstern, O. ( ). ‘ eory of games and economic behavior’. Princeton University Press.