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- Harsanyi’s aggregation; source-dependent utility
- Harsanyi’s aggregation
dynamic consistency: favors abandoning RCLA when time is physical. This can be caused by an intrinsic value of information, even if no better decisions can be made with it. Derive logical relations between preference or dispreference for information and quasi-convexity/concavity of prior/posterior preferences. Use term recursivity for what Luce calls consequence monotonicity, what Segal calls compound independence, what is similar to what was called substitution, etc. %} Grant, Simon, Atsushi Kajii, & Ben Polak (1998) “Intrinsic Preference for Information,” Journal of Economic Theory 83, 233–259. {% dynamic consistency; information aversion; assume that timing of resolution of uncertainty matters (is crucial for their SAIL = Single-Act-Information-Loving). Have results on betweenness and RDU very similar to what Sarin & I get with sequential consistency, implying EU in one stage but not the other. We are not aware of logical relations between the results. %} Grant, Simon, Atsushi Kajii, & Ben Polak (2000) “Temporal Resolution of Uncertainty and Recursive Non-Expected Utility Models,” Econometrica 68, 425–434. {% dynamic consistency; value of information; DC = stationarity: end of §4 before appendix %} Grant, Simon, Atsushi Kajii, & Ben Polak (2000) “Preference for Information and Dynamic Consistency,” Theory and Decision 48, 263–286. {% source-dependent utility; game theory for nonexpected utility & dynamic consistency: Dixit & Skeath (1999) suggested that with high stakes more risk averse strategies in a two-outcome game is more plausible, but EU says the heigth of stakes shouldn’t matter. This paper shows that giving up RCLA and using recursive utility, and not other aspects of nonEU, can resolve the paradox. The authors are in fact using (as they properly reference) the Kreps & Porteus (1978) model. %} Grant, Simon, Atsushi Kajii, & Ben Polak (2001) “ “Third Down with a Yard to Go”: Recursive Expected Utility and the Dixit-Skeath Conundrum,” Economics Letters 73, 275–286. {% Harsanyi’s aggregation; source-dependent utility: this paper characterizes the Kreps & Porteus (1978) model, well known nowadays for its use in the KMM smooth ambiguity model, and also analyzed by Grant, Kajii, & Polak (2001). It does so, however, not for the Anscombe-Aumann model, but for the more general Harsanyi (1955) model, but the latter in an extended sense. To see the former point (Harsanyi 1955 more general than AA): Harsanyi has a set of outcomes X, with generic element x. Can write x as (x1,…,xn) with xj denoting what x means for individual j. If y = (y1,…,yn) has yj ~j xj then we identify xj and yj. That way, Harsanyi’s X becomes an arbitrary subset of a product set X1 ... Xn. A Harsanyi probability distribution over X thus becomes an Anscombe-Aumann probability distribution over X1 ... Xn. In this way Anscombe & Aumann (1963) becomes a corollary of Harsanyi (1955). Whereas Harsanyi, implicitly, has 1/n probabilities over being individual i, in which case different subjective (endogenous) weights for different individuals can be interpreted as different welfare weights rather than probabilities, this paper adds an extra structure, making it different (not more or less general than Harsanyi): it additionally assumes probability distributions over the set I of individuals. Thus the choice set is a product set (I) (X), where designates set of probability distributions. On p. 1953, beginning of §6, the authors write that Harsanyi worked with (I X), deviating some from the (X) that I assumed above. I took weights over I in Harsanyi as endogenous and not exogenous. Harsanyi does not write very explicitly about domain, and one can view it in different ways. %} Grant, Simon, Atsushi Kajii, Ben Polak, & Zvi Safra (2010) “Generalized Utilitarianism and Harsanyi’s Impartial Oberver Theorem,” Econometrica 78, 1939–1971. {% Harsanyi’s aggregation: generalize Harsanyi by using only subset of lotteries, involving less imagination of the social planners, by considering only lotteries over the identities the observer may assume independent of the social alternative. %} Grant, Simon, Atsushi Kajii, Ben Polak, & Zvi Safra (2012) “Generalized Representation Theorem for Harsanyi’s (‘Impartial’) Observer,” Social Choice and Welfare 39, 833–846. {% For the same preference domain as in their Econometrica (2010) model, they provide a representation with a dual treatment of the stages (intersecting with the Econometrica paper only in EU), dealing with Fleurbaey’s objection to Harsanyi, getting inequality aversion ex post. %} Grant, Simon, Atsushi Kajii, Ben Polak, & Zvi Safra (2012) “Equally-Distributed Equivalent Utility, Ex Post Egalitarianism and Utilitarianism,” Journal of Economic Theory 147, 1545–1571. {% %} Grant, Simon & Edi Karni (2004) “A Theory of Quantifiable Beliefs,” Journal of Mathematical Economics 40, 515–546. {% %} Grant, Simon & Edi Karni (2005) “Why Does It Matter that Beliefs and Valuations Be Correctly Represented?,” International Economic Review 46, 917–934. {% Application of ambiguity theory; Intro nicely relates ambiguity of decision theory to linguistic ambiguity. %} Grant, Simon, Jeffrey J. Kline, & John Quiggin (2014) “A Matter of Interpretation: Ambiguous Contracts and Liquidated Damages,” Games and Economic Behavior 85, 180–187. {% This paper considers robustness of experiments w.r.t. small probabilistic perturbations. For example, a decision maker exhibiting the typical Allais paradox might in fact maximize EU, be almost indifferent between the options provided, but slightly misperceive the probabilities. A theory is developed of experiments robust against this (being in topological interiors), and many paradoxes are discussed using this criterion. %} Grant, Simon, Jeffrey J. Kline, Idione Meneghel, John Quiggin, Rabee Tourky (2016) “A Theory of Robust Experiments for Choice under Uncertainty,” Journal of Economic Theory 165, 124–151. {% game theory can/cannot be seen as decision under uncertainty: the paper models game theory as Savagean decision under uncertainty. The authors consider general games. They define states of nature that describe all uncertainties, being not only moves by nature but also all moves of players. For a player an information event is one that she can observe and condition strategy choice on. The authors emphatically assume NO randomization device, which I like. This is the main novelty relative to some preceding general modelings of game theory with ambiguity (discussed on p. 669). They only assume general preferences of players over outcomes, that can be state-dependent (Axiom A1 on p. 648). In fact, they only assume preferences over own strategies when the strategy choices of all others are fixed (sounds normal-form like). This fits with the idea that players choose their strategies independently and cannot influence each other, but not with the idea that in a meta-sense players can still influence each other (“if I come to conclude that x is optimal for me then player 2 will come to conclude that y is optimal for him”). Without further info, it also does not (yet) allow for comparisons of different equilibria. Outcomes can be general combinations of strategies. A difficulty is that the revealed preference approach to observe preferences over own strategies given strategy choices of all others does not work well in games. It involves problematic thought experimennts as in Aumann & Drèze (2008): “imagine that I can only choose between x and y, but my opponents continue to think that I can choose from all my strategies.” The authors write that they will not discuss this issue. The authors derive the existence of an equilibrium (Theorem 1, p. 656), which requires richness, more or less a continuum of states. They assume such Savage-type richness of nonatomicity. As they emphasize, their model does not use randomization and does not need expected utility in any sense and can allow for general ambiguity attitudes (p. 642 end of 1st para). The absence of randomization and absence of commitment to expected utility for risk add to the generality of their approach. They do assume many conditionings on events that are observable to a player and there assume something like Savage’s sure-thing principle, or backward induction (p. 650 l. 1). So it is not a universal sure-thing principle, but still it is a restriction. It is useful to have a general framework for game theory without commitment to randomization and expected utility, allowing for ambiguity all over the place, and this is the first paper to do so. There is a price to pay of complex richness, general complexity of model, and still a sure-thing principle at some places. P. 643, end of 2nd para: “It allows us, as well as behooves us, to model equilibrium behavior without the usual technical paraphernalia of convexity or monotonicity of strategies and preferences, and the related praxis that seems to have arisen more from considerations of analytical tractability rather than motivated by, for example, behavioral properties of the underlying preferences.” %} Grant, Simon, Idione Meneghel, & Rabee Tourky (2016) “Savage Games,” Theoretical Economics 11, 641–682. {% Nice generalization of Machina & Schmeidler (1992) by using P4 and a weaker analog of Savage’s P2. %} Grant, Simon, Hatice Özsoy & Ben Polak (2008) “Probabilistic Sophistication and Stochastic Monotonicity in the Savage Framework,” Mathematical Social Sciences 55, 371–380. {% They weaken a central axiom in Machina & Schmeidler’s (1995) probabilistic sophistication model in the Anscombe-Aumann setup to stochastic monotonicity (independence when one of the prospects is degenerate). %} Grant, Simon & Ben Polak (2006) “Bayesian Beliefs with Stochastic Monotonicity: An Extension of Machina and Schmeidler,” Journal of Economic Theory 130, 264–282. {% Propose a generalization of mean-variance where the combination of mean and variance is linear. The main contribution: it goes for uncertainty/ambiguity rather than for risk. Assume Anscombe-Aumann (AA). The mean is mean AA-EU. Instead of variance they take a generalized dispersion measure, satisfying conditions specified below. The measure of dispersion is the subjective EU an agent would be willing to give up to achieve constant EU over the state space. A generalization relaxing constant absolute uncertainty aversion will be in Chambers, Grant, Polak, & Quiggin (2014 JET). A probability measure on the state space S is derived subjectively à la Savage (or AA). The model is very general and encompasses Siniscalchi’s (2009) vector utility, variational, multiplier, and many other models. The authors share with variational a sort of constant absolute uncertainty aversion. They point out that absolute uncertainty aversion need not always be constant, but they just focus on this case. They axiomatize it in general, given a few inequalities specified below. P. 1363 penultimate para (& p. 1367 5th para): in the models assumed to be special cases, they incorporate Choquet expected utility, apparently implicitly assuming AA. P. 1365: the general form is V(f) = E(Uf) (Uf)) where E(Uf) denotes the subjective AA EU, and captures dispersion about E(Uf), and (0) = 0 for acts with constant k-utility level at every state. P. 1366 lists axioms. A4 is unrestricted solvability and implies unbounded utility. A5 is constant absolute uncertainty aversion: f + (1)x z + (1)x f + (1)y z + (1)y for constant acts x and y, and also constant act z. The latter is immaterial, and could have been any act g, as the authors point out p. 1366 bottom. Hence the axiom is equivalent to weak certainty independence. P. 1367 para -4 (also p. 1364 2nd para): without further assumptions, the model is too general to, for instance, have identifiable. Theorem 1 is called too general to be very useful. (P. 1372 3rd para: in general, any is possible and is completely unidentifiable.) They next consider properties called desirable such as uncertainty aversion (A6 p. 1368: convexity, or A6*: preference for complete hedges, or A7 (p. 1368): certainty betweenness, or A.8 (p. 1368): Siniscalchi’s complementary independence, and positivity of , properties that rule out likelihood insensitivity (inverse-S) and, hence, will not work well empirically. Theorem 2 (p. 1368) gives the equivalent properties of . P. 1373: is identifiable if local smoothness. Problem is that this is a mathematical nontestable condition. P. 1374 5th para: Siniscalchi’s symmetry makes identifiable. P. 1375 considers (2nd order) probabilistic sophistication. 2nd order because we have not only on S but also the AA objective probabilities. %} Grant, Simon & Ben Polak (2013) “Mean-Dispersion Preferences and Constant Absolute Uncertainty Aversion,” Journal of Economic Theory 148, 1361–1398. {% %} Grant, Simon & John Quiggin (1997) “Strategic Trade Policy under Uncertainty: Sufficient Conditions for the Optimality of ad Valorem, Specific and Quadratic Trade Taxes,” International Economic Review 38, 187–204. {% On social security investments. Equity premium puzzle. Do only EU, where their novelty is to introduce a government that can commit agents to payments. %} Grant, Simon & John Quiggin (2002) “The Risk Premium for Equity: Implications for the Proposed Diversification of the Social Security Fund,” American Economic Review 92, 1104–1115. {% A footnote points out that Epstein & Zhang (2001, Econometrica, second part of Corollary 7.4(a) on p. 287) is incorrect. %} Grant, Simon & John Quiggin (2005) “Increasing Uncertainty: A Definition,” Mathematical Social Sciences 49, 117–141. {% About Babyloniers and so on. %} Grauer, Hans (1990) “Die Unendlichkeit in der Mathematik,” Mathematische Semesterberichte 37, 153–156. {% %} Gravel, Nicholas, Thierry Marchant, & Arunava Sen (2011) “Comparing Societies with Different Numbers of Individuals on the Basis of their Average Advantage.” In Marc Fleurbaey, Maurice Salles & John A. Weymark (2010) Social Ethics and Normative Economics, 261–277, Springer, Berlin. {% They consider orderings of finite subsets of a set, and characterize average utility maximization: {x1,…,xn} --> (U(x1) + … + U(xn))/n, where n is variable. Note the braces and not brackets around the n objects! These are sets and not arrays. So, each element can appear only once. In this sense it is different than generalized quasilinear means. %} Gravel, Nicolas, Thierry Marchant, & Arunava Sen (2012) “Uniform Expected Utility Criteria for Decision Making under Ignorance or Objective Ambiguity,” Journal of Mathematical Psychology 56, 297–315. {% Conditions under which, for the aggregation of individual utilities, welfarism must be its special case of utilitarianism, under unanimity. Welfarism was defined by Sen (1977) and sounds much like Fishburn’s marginal independence. %} Gravel, Nicolas & Patrick Moyes (2013) “Utilitarianism or Welfarism: Does It Make a Difference?,” Social Choice and Welfare 40, 529–551. {% information aversion: poem of 1742; ends with: “where ignorance is bliss, ‘Tis folly to be wise” %} Gray, Thomas (1742) “Ode on a Distant Prospect of Eton College,” {% Nice early (1960!) application of decision analysis to drilling oil. First part is descriptive, considerations made with actual decisions, and second part is prescriptive, doing an actual decision analysis. He assessed utility functions using the PE method (hypothetical) of many oil prospectors. One person, William Beard of the Beard Oil Company, had a utility function that could very well be approximated by ln(y + 150,000) on the domain [150,000, 800,000]. A simplified didactical version is in Winkler (1972, Example 5.10). Seems he measured the risky utility function of the owner of an oil exploration company twice, three months between, finding greater risk aversion the second time but with reasons of changed circumstances to justify the change. %} Grayson, C. Jackson Jr. (1979) “Decisions under Uncertainty: Drilling Decisions by Oil and Gas Operators.” Arno Press, New York; first version 1960, Harvard Business School. {% Seems to already have derived Schmeidler’s 1986 representation theorem for Choquet integral functionals, according to Denneberg (1994). An earlier and more general result was given by Anger (1977). %} Greco, Gabriele (1982) “Sulla Rappresentazione di Funzionali Mediante Integrali,” Rend. Sem. Mat. Univ. Padova 66, 21–42. {% They propose a generalization of the Choquet integral that can be interpreted as having state-dependent utility or as having outcome-dependent weighting function. They cite Green & Jullien (1988) and Segal (1989) for a similar functional for decision under risk. They do not know Chew & Wakker (1996) who, more generally, consider such functionals also for a state space and who consider connected topological spaces (in their appendix) generalizing the reals, and allow for nonlinear, continuous, utility functions. This paper concerns the special case of the Chew & Wakker (1989) functional for the reals and with utility the identity. This paper takes the functional as primitive when axiomatizing its form, whereas Chew & Wakker (1996) did it with the represented preference relation as primitive. Chew & Wakker also point out that 1992-prospect theory is a special case, but, unlike this paper (§9), do not note that the Sugeno integral is also a special case. P. 15 l. -3 correctly points out that the functional in itself is too general to be very useful. They also analyze the Möbius transform (§8.1), and bipolar generalizations. I next show briefly how the characterization provided in this paper in Theorem 1 is related to Theorem B1 of Chew & Wakker (1993). Their main characterizing condition, cardinal tail independence (p. 9) implies ordinal independence of Chew & Wakker (Remark A1). The other axioms in Theorem B1 of Chew & Wakker (1993) are implied readily, mainly by the assumed existence of the functional. Thus this Theorem B1 implies the existence of the functional of Chew & Wakker, and all that remains to be proved is that their utility function is the identity, which follows from cardinal tail independence. %} Greco, Salvatore, Benedetto Matarazzo, & Silvio Giove (2011) “The Choquet Integral with Respect to a Level Dependent Capacity,” Fuzzy Sets and Systems 175, 1–35. {% Into 2nd page or so, about the Sugeno integral: “It appears, however, that this operator has some unpleasant limitations: the most important is the so called co-commensurability; i.e., the evaluation with respect to each considered criterion should be defined on the same scale.” %} Greco, Salvatore, Benedetto Matarazzo, & Roman Slowinski (2001) “Conjoint Measurement and Rough Set Approach for Multicriteria Sorting Problems in Presence of Ordinal Criteria.” In Alberto Colorni, Massimo Paruccini, Bernard Roy (eds.) A-MCD-A Aide Multicritère à la Décision (Multiple Criteria Decision Aiding) EUR Report, 117–144, Joint Research Centre, The European Commission, Ispra. {% %} Greco, Salvatore, Benedetto Matarazzo, & Roman Slowinski (2008) “Case-Based Reasoning Using Gradual Rules Induced from Dominance-Based Rough Approximations.” In Guoyin Wang, Tianrui Li, Jerzy W. Grzymala-Busse, et al. (eds.) Rough Sets and Knowledge Technology. Lecture Notes in Computer Science, 268–275, Springer, Heidelberg, Germany. {% Bipolar is the mathematical way of saying sign dependence. %} Greco, Salvatore, Radko Mesiar, & Fabio Rindone (2016) “Generalized Bipolar Product and Sum,” Fuzzy Optimization and Decision Making 15, 2131. {% Distinguish between necessary preferences, that are felt with certainty, and possible preferences. Sets of additive value functions represent it. Similar to Gilboa, Maccheroni, Marinacci, & Schmeidler (2010). %} Greco, Salvatore, Vincent Mousseau, & Roman Slowinski (2010) “Multiple Criteria Sorting with a Set of Additive Value Functions,” European Journal of Operational Research 207, 1455–1470. {% %} Greco, Salvatore, Vincent Mousseau, & Roman Slowinski (2009) “The Possible and the Necessary for Multiple Criteria Group Decision,” {% Generalizes PT by dropping gain-loss separability. So no additive decomposability between gains and losses. %} Greco, Salvatore & Fabio Rindone 2014) “The Bipolar Choquet Integral Representation,” Theory and Decision 77, 1–29. {% %} Green, Donald P., Daniel Kahneman, & Howard C. Kunreuther (1994) “How the Method and Scope of Public Funding Affects Willingness to Pay for Public Goods,” Public Opinion Quarterly 58, 48–67. {% Argue that of the three concepts states, consequences, acts, it is not self-evident that the former two are given first and that then the third is a mapping from the first to the second. Do a kind of state-dependent version of AA; argue in favor of EU. %} Green, Edward J. & Kent Osband (1991) “A Revealed Preference Theory for Expected Utility,” Review of Economic Studies 58, 677–696. {% %} Green, H.A. John (1961) “Direct Additivity and Consumers’ Behaviour,” Oxford Economic Papers 13, 132–136. {% dynamic consistency %} Green, Jerry R. (1987) ““Making Book against Oneself,” The Independence Axiom, and Nonlinear Utility Theory,” Quarterly Journal of Economics 102, 785–796. {% %} Green, Jerry R. & Bruno Jullien (1988) “Ordinal Independence in Non-Linear Utility Theory,” Journal of Risk and Uncertainty 1, 355–387. (“Erratum,” 1989, 2, 119.) {% %} Green, Jerry R., Lawrence J. Lau, & Herakles M. Polemarchakis (1978) “A Theory on the Identification of the von Neumann-Morgenstern Utility Function from Asset Demands,” Economics Letters 1, 217–220. {% Seems that they use hypothetical choices; no assumptions needed about utility functions (even though they might not have realized this) they do use the assumption of linear utility in arguing that the intercept changes as the amounts change, while keeping the ratio of amounts constant. It is not the ratio of amounts that they should hold constant, but the ratio of utilities. Median data reject exponential and hyperbolic discounting; there is decreasing impatience but not hyperbolic discounting. %} Green, Leonard, Nathanael Fristoe, & Joel Myerson (1994) “Temporal Discounting and Preference Reversals in Choice between Delayed Outcomes,” Psychonomic Bulletin and Review 1, 383‑389. {% No new experiment; seems that they don't fit data at the individual level, only at group level. %} Green, Leonard, Joel Myerson (1993) “Alternative Frameworks for the Analysis of Self-Control,” Behavior and Philosophy 21, 37–47. {% Survey of intertemporal choice together with risky choice. They consider only one nonzero outcome and mostly take linear utility. Then risk attitude is entirely driven by probability weighting, which the authors also call discounting. They consider exponential functions exp(-bx), hyperbolic functions A/(1+kx), and what they call hyperbola-like A/(a+kx)s. In intertemporal context they take time t for x, and in risky choice they take odds ratio p/(1-p) for x (then the hyperbola-like family is the same as the one used by Goldstein & Einhorn (1987). Why odds ratio would be the analog for time is not clear to me, even if it does cover the same range. So different behavior of utility for one than for the other (a finding presented in several places) is not clear to interpret, the more so as transaction costs work differently for one than for the other. The authors find that both for intertemporal choice and for risky choice the hyperbola perform better than exponential, and the extra parameter s improves the fit. From no more than this usefulness of extra parameter s for time as for risk the authors again and again derive the far-fetched conclusion that the mechanisms for time are the same as for risk, making this the main message of their paper. They say they find support for inverse-S but this is little surprise if only functions are fit that are inverse-S. P. 774 claims that hyperbola-like functions fit well at individual level for ALL individuals. P. 774: when they find that the extra parameter s is worthwhile both for children and for elderly people this is what they conclude: “These findings demonstrate that the hyperbola-like discounting function (Equation 3) is extremely general in that it describes temporal discounting in individuals from childhood to old age.” Variation in payoff (p. 781, top of 2nd column) amounts to tests of constant relative risk aversion. Download 7.23 Mb. Share with your friends:
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