2019 53rd Asilomar Conference on Signals, Systems, and Computers, 2019
Recent literature interprets the predictions of "black-box" machine learning models (Neural Netwo... more Recent literature interprets the predictions of "black-box" machine learning models (Neural Networks, Random Forests, etc.) by approximating these models in terms of simpler models such as piecewise linear or piecewise constant models. Existing literature does not provide guarantees on whether these approximations reflect the nature of the predictive model well, which can result in misleading interpretations. We provide a tractable dynamic programming algorithm that partitions the feature space into subsets and assigns a local model (constant/linear model) to provide piecewise constant/piecewise linear interpretations of an arbitrary predictive model. When approximation loss (between the interpretation and the predictive model) is measured in terms of mean squared error, our approximation is optimal; for more general loss functions, our interpretation is approximately optimal. Therefore, in both cases it probably approximately correctly (PAC) learns the predictive model. Experiments with real and synthetic data show that it provides significant improvements (in terms of mean squared error) over competing approaches. We also show real use cases to establish the utility of the proposed approach over competing approaches.
Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspi... more Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspire to make choices for society — that is, for themselves and for others — as if in the original position, behind a veil of ignorance that prevents them from knowing their own social and economic positions in society. While the original position is a purely hypothetical situation, developed as a thought experiment, the main result of this paper is that (under certain assumptions) prefer-ences — hence choices — behind the veil of ignorance are determined by preferences in front of the veil of ignorance. This linkage between preferences behind and in front of the veil of ignorance has implica-tions for distributive theories of justice and for theories of choice. ∗We are grateful to Raymond Fisman, Douglas Gale and Daniel Markovitz for helpful conversations, and to seminar audiences at Bocconi, Brown, Caltech, Columbia, EUI,
For every prediction we might wish to make, we must decide what to observe (what source of inform... more For every prediction we might wish to make, we must decide what to observe (what source of information) and when to observe it. Because making observations is costly, this decision must trade off the value of information against the cost of observation. Making observations (sensing) should be an active choice. To solve the problem of active sensing we develop a novel deep learning architecture: Deep Sensing. At training time, Deep Sensing learns how to issue predictions at various cost-performance points. To do this, it creates a different presentation at each of a variety of different performance levels, each associated with a particular set of measurement rates (costs). This requires learning how to estimate the value of real measurements vs. inferred measurements, which in turn requires learning how to infer missing (unobserved) measurements. To infer missing measurements, we develop a Multi-directional Recurrent Neural Network (M-RNN). An M-RNN differs from a bi-directional RNN in that it sequentially operates across streams in addition to within streams, and because the timing of inputs into the hidden layers is both lagged and advanced. At runtime, the operator prescribes a performance level or a cost constraint, and Deep Sensing determines what measurements to take and what to infer from those measurements, and then issues predictions. To demonstrate the power of our method, we apply it to two real-world medical datasets with significantly improved performance.
We study sequential votes on proposals in two-period environments. If the first proposal is rejec... more We study sequential votes on proposals in two-period environments. If the first proposal is rejected, a second proposal is made; if both proposals are rejected, the status quo prevails. We use a 2x2 design that distinguishes the environments. Voters can be näıve or sophisticated; the electorate in the second period can be the same as in the first period or different. A risk-neutral agenda Setter has incomplete information about Voter preferences. Voters have quadratic loss. We identify environments with policy adjustment (different proposals in the two periods) and environments with no policy adjustment (identical proposals in the two periods). We identify environments where the Setter learns from a rejection and uses the information to tailor the second period proposal. Importantly, sophistication often limits, or entirely obviates, the Setter’s ability to exploit the opportunity to make two proposals. Results complement the work of Romer and Rosenthal (1979) on näıve Voters.
This paper presents a dynamic model to study the impact on economic outcomes of individualism (ti... more This paper presents a dynamic model to study the impact on economic outcomes of individualism (time spent working alone) and collectivism (complementary time spent working with others) in different societies during the Malthusian/Agricultural Era (after the Neolithic Age and before the Industrial Revolution). The model is driven by three forces: a greater degree of collectivism provides a higher safety net for low quality workers, a greater degree of individualism allows high quality workers to leave larger bequests, a larger population creates more congestion and lowers productivity. Although these forces seem simple, they interact in a very complicated way. The predictions of the model suggest that a greater degree of individualism leads to greater per capita income and to greater income inequality. Some historical evidence is consistent with these predictions.
This paper presents a dynamic model to study the impact on the economic outcomes in different soc... more This paper presents a dynamic model to study the impact on the economic outcomes in different societies during the Malthusian Era of individualism (time spent working alone) and collectivism (complementary time spent working with others). The model is driven by opposing forces: a greater degree of collectivism provides a higher safety net for low quality workers but a greater degree of individualism allows high quality workers to leave larger bequests. The model suggests that more individualistic societies display smaller populations, greater per capita income and greater income inequality. Some (limited) historical evidence is consistent with these predictions.
Campaigns for political office often center on the "character" of the candidates. Moral issues as... more Campaigns for political office often center on the "character" of the candidates. Moral issues aside, there is one aspect of a candidate's character that voters clearly should care about: attitude toward risk. If there is a linkage between the candidate's attitudes toward risk in the private domain and in the public domain then (this aspect of) the candidate's character provides important information about (future) policymaking decisions. This paper formalizes this issue and identifies such linkage. The strength of the link depends on the amount the voter observes/infers and on the degree of rationality the voter ascribes to the candidate. (JEL Classification Numbers: D72, D81.) * The results reported here were previously distributed in a paper titled "Choosing a President: Does Character Matter?" We thank Chris Chambers for pointing out an error in our thinking and Robert Powell for detailed comments and suggestions. We are also grateful to
Campaigns for political office often center on the "character" of the candidates. Moral issues as... more Campaigns for political office often center on the "character" of the candidates. Moral issues aside, there is one aspect of a candidate's character that voters clearly should care about: attitude toward risk. If there is a linkage between the candidate's attitudes toward risk in the private domain and in the public domain then (this aspect of) the candidate's character provides important information about (future) policymaking decisions. This paper formalizes this issue and identifies such linkage. The strength of the link depends on the amount the voter observes/infers and on the degree of rationality the voter ascribes to the candidate. (JEL Classification Numbers: D72, D81.) * The results reported here were previously distributed in a paper titled "Choosing a President: Does Character Matter?" We thank Chris Chambers for pointing out an error in our thinking and Robert Powell for detailed comments and suggestions. We are also grateful to
Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspi... more Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspire to make choices for society-that is, for themselves and for others-as if in the original position, behind a veil of ignorance that prevents them from knowing their own social and economic positions in society. While the original position is a purely hypothetical situation, developed as a thought experiment, the main result of this paper is that (under certain assumptions) preferences-hence choices-behind the veil of ignorance are determined by preferences in front of the veil of ignorance. This linkage between preferences behind and in front of the veil of ignorance has implications for distributive theories of justice and for theories of choice.
This project tests for the effect of social status in a laboratory experimental market. We consid... more This project tests for the effect of social status in a laboratory experimental market. We consider a special "box design" market in which a vertical overlap in supply and demand ensure that there are multiple equilibrium prices. We manipulate the relative social status of our subjects by awarding high status to a subset of the group based one of two procedures. In the first, a subject's score on a trivia quiz determines his or her status; in another, subjects are assigned randomly to a higher-status or lower-status group. In both treatments we find that average prices are higher in markets where higher-status sellers face lowerstatus buyers, and lower when buyers have higher status than sellers. Across all sessions, the higher-status side of the market captures a greater share of the surplus, earning significantly more than their lowerstatus counterparts.
Let X be a compact subset of the complex plane and let A be an algebra of functions analytic near... more Let X be a compact subset of the complex plane and let A be an algebra of functions analytic near X which contains the polynomials and is complete in its natural topology. This paper is concerned with determining the spectrum of A and describing A in terms of its spectrum. It is shown that the spectrum of A is, formed from the disjoint union of certain compact subsets of C (suitably topologized) by making certain identifications. A is closed under differentiation exactly when no identifications need be performed, and then A admits a simple, complete description. In particular, if X is connected, then the completion of A is merely the restriction to X of the algebra of all functions analytic near the union of X with some of the bounded components of C-X.
Let X be a compact subset of C m and let έ?(X) be the space of germs on X of functions holomorphi... more Let X be a compact subset of C m and let έ?(X) be the space of germs on X of functions holomorphic near X, equipped with its natural locally convex inductive limit topology. The object of this paper is to give, under a mild topological assumption on X, an internal description of this topology, and in particular, of the bounded sets and convergent sequences. These results follow from a general extendibility theorem. Surprisingly, the topological assumption on X is necessary, and examples are constructed which illustrate this point. A related local extendibility result is also established.
Normal form games are nearly compact and continuous (NCC) if they can be understood as games play... more Normal form games are nearly compact and continuous (NCC) if they can be understood as games played on strategy spaces that are dense subsets of the strategy spaces of larger compact games with jointly continuous payoffs. There are intrinsic algebraic, measure theoretic, functional analysis, and finite approximability characterizations of NCC games. NCC games have finitely additive equilibria, and all their finitely additive equilibria are equivalent to countably additive equilibria on metric compactifications. The equilibrium set of an NCC game depends upper hemicontinuously on the specification of the game and contains only the limits of approximate equilibria of approximate games.
In this short note we discuss how posterior implementation, due to Green and La¤ont (1987), relat... more In this short note we discuss how posterior implementation, due to Green and La¤ont (1987), relates to ex-post implementation for which Jehiel. et. al (2006) have established a strong impossibility result.
This paper offers a new approach to the study of economic problems usually modeled as games of in... more This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual bestresponse criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence.
This paper de¯nes a general equilibrium model with exchange and club formation. Agents trade mult... more This paper de¯nes a general equilibrium model with exchange and club formation. Agents trade multiple private goods widely in the market, can belong to several clubs, and care about the characteristics of the other members of their clubs. The space of agents is a continuum, but clubs are¯nite. It is shown that (i) competitive equilibria exist, and (ii) the core coincides with the set of equilibrium states. The central subtlety is in modeling club memberships and expressing the notion that membership choices are consistent across the population.
2019 53rd Asilomar Conference on Signals, Systems, and Computers, 2019
Recent literature interprets the predictions of "black-box" machine learning models (Neural Netwo... more Recent literature interprets the predictions of "black-box" machine learning models (Neural Networks, Random Forests, etc.) by approximating these models in terms of simpler models such as piecewise linear or piecewise constant models. Existing literature does not provide guarantees on whether these approximations reflect the nature of the predictive model well, which can result in misleading interpretations. We provide a tractable dynamic programming algorithm that partitions the feature space into subsets and assigns a local model (constant/linear model) to provide piecewise constant/piecewise linear interpretations of an arbitrary predictive model. When approximation loss (between the interpretation and the predictive model) is measured in terms of mean squared error, our approximation is optimal; for more general loss functions, our interpretation is approximately optimal. Therefore, in both cases it probably approximately correctly (PAC) learns the predictive model. Experiments with real and synthetic data show that it provides significant improvements (in terms of mean squared error) over competing approaches. We also show real use cases to establish the utility of the proposed approach over competing approaches.
Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspi... more Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspire to make choices for society — that is, for themselves and for others — as if in the original position, behind a veil of ignorance that prevents them from knowing their own social and economic positions in society. While the original position is a purely hypothetical situation, developed as a thought experiment, the main result of this paper is that (under certain assumptions) prefer-ences — hence choices — behind the veil of ignorance are determined by preferences in front of the veil of ignorance. This linkage between preferences behind and in front of the veil of ignorance has implica-tions for distributive theories of justice and for theories of choice. ∗We are grateful to Raymond Fisman, Douglas Gale and Daniel Markovitz for helpful conversations, and to seminar audiences at Bocconi, Brown, Caltech, Columbia, EUI,
For every prediction we might wish to make, we must decide what to observe (what source of inform... more For every prediction we might wish to make, we must decide what to observe (what source of information) and when to observe it. Because making observations is costly, this decision must trade off the value of information against the cost of observation. Making observations (sensing) should be an active choice. To solve the problem of active sensing we develop a novel deep learning architecture: Deep Sensing. At training time, Deep Sensing learns how to issue predictions at various cost-performance points. To do this, it creates a different presentation at each of a variety of different performance levels, each associated with a particular set of measurement rates (costs). This requires learning how to estimate the value of real measurements vs. inferred measurements, which in turn requires learning how to infer missing (unobserved) measurements. To infer missing measurements, we develop a Multi-directional Recurrent Neural Network (M-RNN). An M-RNN differs from a bi-directional RNN in that it sequentially operates across streams in addition to within streams, and because the timing of inputs into the hidden layers is both lagged and advanced. At runtime, the operator prescribes a performance level or a cost constraint, and Deep Sensing determines what measurements to take and what to infer from those measurements, and then issues predictions. To demonstrate the power of our method, we apply it to two real-world medical datasets with significantly improved performance.
We study sequential votes on proposals in two-period environments. If the first proposal is rejec... more We study sequential votes on proposals in two-period environments. If the first proposal is rejected, a second proposal is made; if both proposals are rejected, the status quo prevails. We use a 2x2 design that distinguishes the environments. Voters can be näıve or sophisticated; the electorate in the second period can be the same as in the first period or different. A risk-neutral agenda Setter has incomplete information about Voter preferences. Voters have quadratic loss. We identify environments with policy adjustment (different proposals in the two periods) and environments with no policy adjustment (identical proposals in the two periods). We identify environments where the Setter learns from a rejection and uses the information to tailor the second period proposal. Importantly, sophistication often limits, or entirely obviates, the Setter’s ability to exploit the opportunity to make two proposals. Results complement the work of Romer and Rosenthal (1979) on näıve Voters.
This paper presents a dynamic model to study the impact on economic outcomes of individualism (ti... more This paper presents a dynamic model to study the impact on economic outcomes of individualism (time spent working alone) and collectivism (complementary time spent working with others) in different societies during the Malthusian/Agricultural Era (after the Neolithic Age and before the Industrial Revolution). The model is driven by three forces: a greater degree of collectivism provides a higher safety net for low quality workers, a greater degree of individualism allows high quality workers to leave larger bequests, a larger population creates more congestion and lowers productivity. Although these forces seem simple, they interact in a very complicated way. The predictions of the model suggest that a greater degree of individualism leads to greater per capita income and to greater income inequality. Some historical evidence is consistent with these predictions.
This paper presents a dynamic model to study the impact on the economic outcomes in different soc... more This paper presents a dynamic model to study the impact on the economic outcomes in different societies during the Malthusian Era of individualism (time spent working alone) and collectivism (complementary time spent working with others). The model is driven by opposing forces: a greater degree of collectivism provides a higher safety net for low quality workers but a greater degree of individualism allows high quality workers to leave larger bequests. The model suggests that more individualistic societies display smaller populations, greater per capita income and greater income inequality. Some (limited) historical evidence is consistent with these predictions.
Campaigns for political office often center on the "character" of the candidates. Moral issues as... more Campaigns for political office often center on the "character" of the candidates. Moral issues aside, there is one aspect of a candidate's character that voters clearly should care about: attitude toward risk. If there is a linkage between the candidate's attitudes toward risk in the private domain and in the public domain then (this aspect of) the candidate's character provides important information about (future) policymaking decisions. This paper formalizes this issue and identifies such linkage. The strength of the link depends on the amount the voter observes/infers and on the degree of rationality the voter ascribes to the candidate. (JEL Classification Numbers: D72, D81.) * The results reported here were previously distributed in a paper titled "Choosing a President: Does Character Matter?" We thank Chris Chambers for pointing out an error in our thinking and Robert Powell for detailed comments and suggestions. We are also grateful to
Campaigns for political office often center on the "character" of the candidates. Moral issues as... more Campaigns for political office often center on the "character" of the candidates. Moral issues aside, there is one aspect of a candidate's character that voters clearly should care about: attitude toward risk. If there is a linkage between the candidate's attitudes toward risk in the private domain and in the public domain then (this aspect of) the candidate's character provides important information about (future) policymaking decisions. This paper formalizes this issue and identifies such linkage. The strength of the link depends on the amount the voter observes/infers and on the degree of rationality the voter ascribes to the candidate. (JEL Classification Numbers: D72, D81.) * The results reported here were previously distributed in a paper titled "Choosing a President: Does Character Matter?" We thank Chris Chambers for pointing out an error in our thinking and Robert Powell for detailed comments and suggestions. We are also grateful to
Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspi... more Theories of justice in the spirit of Harsanyi and Rawls argue that fair-minded people should aspire to make choices for society-that is, for themselves and for others-as if in the original position, behind a veil of ignorance that prevents them from knowing their own social and economic positions in society. While the original position is a purely hypothetical situation, developed as a thought experiment, the main result of this paper is that (under certain assumptions) preferences-hence choices-behind the veil of ignorance are determined by preferences in front of the veil of ignorance. This linkage between preferences behind and in front of the veil of ignorance has implications for distributive theories of justice and for theories of choice.
This project tests for the effect of social status in a laboratory experimental market. We consid... more This project tests for the effect of social status in a laboratory experimental market. We consider a special "box design" market in which a vertical overlap in supply and demand ensure that there are multiple equilibrium prices. We manipulate the relative social status of our subjects by awarding high status to a subset of the group based one of two procedures. In the first, a subject's score on a trivia quiz determines his or her status; in another, subjects are assigned randomly to a higher-status or lower-status group. In both treatments we find that average prices are higher in markets where higher-status sellers face lowerstatus buyers, and lower when buyers have higher status than sellers. Across all sessions, the higher-status side of the market captures a greater share of the surplus, earning significantly more than their lowerstatus counterparts.
Let X be a compact subset of the complex plane and let A be an algebra of functions analytic near... more Let X be a compact subset of the complex plane and let A be an algebra of functions analytic near X which contains the polynomials and is complete in its natural topology. This paper is concerned with determining the spectrum of A and describing A in terms of its spectrum. It is shown that the spectrum of A is, formed from the disjoint union of certain compact subsets of C (suitably topologized) by making certain identifications. A is closed under differentiation exactly when no identifications need be performed, and then A admits a simple, complete description. In particular, if X is connected, then the completion of A is merely the restriction to X of the algebra of all functions analytic near the union of X with some of the bounded components of C-X.
Let X be a compact subset of C m and let έ?(X) be the space of germs on X of functions holomorphi... more Let X be a compact subset of C m and let έ?(X) be the space of germs on X of functions holomorphic near X, equipped with its natural locally convex inductive limit topology. The object of this paper is to give, under a mild topological assumption on X, an internal description of this topology, and in particular, of the bounded sets and convergent sequences. These results follow from a general extendibility theorem. Surprisingly, the topological assumption on X is necessary, and examples are constructed which illustrate this point. A related local extendibility result is also established.
Normal form games are nearly compact and continuous (NCC) if they can be understood as games play... more Normal form games are nearly compact and continuous (NCC) if they can be understood as games played on strategy spaces that are dense subsets of the strategy spaces of larger compact games with jointly continuous payoffs. There are intrinsic algebraic, measure theoretic, functional analysis, and finite approximability characterizations of NCC games. NCC games have finitely additive equilibria, and all their finitely additive equilibria are equivalent to countably additive equilibria on metric compactifications. The equilibrium set of an NCC game depends upper hemicontinuously on the specification of the game and contains only the limits of approximate equilibria of approximate games.
In this short note we discuss how posterior implementation, due to Green and La¤ont (1987), relat... more In this short note we discuss how posterior implementation, due to Green and La¤ont (1987), relates to ex-post implementation for which Jehiel. et. al (2006) have established a strong impossibility result.
This paper offers a new approach to the study of economic problems usually modeled as games of in... more This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual bestresponse criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence.
This paper de¯nes a general equilibrium model with exchange and club formation. Agents trade mult... more This paper de¯nes a general equilibrium model with exchange and club formation. Agents trade multiple private goods widely in the market, can belong to several clubs, and care about the characteristics of the other members of their clubs. The space of agents is a continuum, but clubs are¯nite. It is shown that (i) competitive equilibria exist, and (ii) the core coincides with the set of equilibrium states. The central subtlety is in modeling club memberships and expressing the notion that membership choices are consistent across the population.
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