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Infinite Populations, Choice and Determinacy infinite populations, choice and determinacy tadeusz litak informatik 8 FAU erlangen-nurnberg¨ 1/54 By all means mathematicians should learn all the pure mathematics they want. But they must also learn applications. Get to be an expert in either the social sciences or in computer science. John Kemeny why am I giving this talk? 2/54 why am I giving this talk? By all means mathematicians should learn all the pure mathematics they want. But they must also learn applications. Get to be an expert in either the social sciences or in computer science. John Kemeny 2/54 3/54 Logics for Social Behaviour Workshop: 10 - 14 November 2014, Leiden, the Netherlands Scientific • Frederik Herzberg, Bielefeld U Organizers • Alexander Kurz, U Leicester • Alessandra Palmigiano, TU Delft Invited • Samson Abramsky, U Oxford Speakers • Bob Coecke, U Oxford The Lorentz Center is an international • Martin Hyland, U Cambridge center in the sciences. Its aim is to • Philippe Mongin, CNRS Paris organize workshops for scientists in an atmosphere that fosters collaborative • Ariel Rubinstein, U Tel Aviv work, discussions and interactions. • Carl Wagner, U Tennessee For registration see: www.lorentzcenter.nl • Viktor Winschel, U Mannheim Academic fencing in Göttingen, Germany (1847). Poster design: SuperNova Studios . NL a few years ago I was invited to some meetings on Logics for Social Behaviour ... but seriously, why? 4/54 Logics for Social Behaviour Workshop: 10 - 14 November 2014, Leiden, the Netherlands Scientific • Frederik Herzberg, Bielefeld U Organizers • Alexander Kurz, U Leicester • Alessandra Palmigiano, TU Delft Invited • Samson Abramsky, U Oxford Speakers • Bob Coecke, U Oxford The Lorentz Center is an international • Martin Hyland, U Cambridge center in the sciences. Its aim is to • Philippe Mongin, CNRS Paris organize workshops for scientists in an atmosphere that fosters collaborative • Ariel Rubinstein, U Tel Aviv work, discussions and interactions. • Carl Wagner, U Tennessee For registration see: www.lorentzcenter.nl • Viktor Winschel, U Mannheim Academic fencing in Göttingen, Germany (1847). Poster design: SuperNova Studios . NL but seriously, why? a few years ago I was invited to some meetings on Logics for Social Behaviour ... 4/54 I I didn’t know much about social choice at that time or set theory, for that matter. and it’s not that I know that much more now btw I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later 5/54 I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later I I didn’t know much about social choice at that time or set theory, for that matter. and it’s not that I know that much more now btw 5/54 I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later I I didn’t know much about social choice at that time or set theory, for that matter. and it’s not that I know that much more now btw I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings 5/54 6/54 I and now I got invited to give a talk in Freiburg about that piece . I . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove I and then I got invited to submit a paper to a special issue of Studia Logica ... 7/54 I . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . 7/54 I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . I . so, without further ado, what is this industry about? 7/54 I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . I . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations 7/54 I the problem lies in how they handle them and what they prove I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . I . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide 7/54 I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . I . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove 7/54 intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis voting theory... ??? . analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations whence infinite populations? idealized markets uncountably many agents when studying an idealized notion of perfect competition (Aumann 1964) the essential idea . is that the economy under consideration has a “very large” number of participants, and that the influence of each individual participant is “negligible” . one can integrate over a continuum, and changing the integrand at a single point does not affect the value of the integral 8/54 social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis voting theory... ??? . analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations whence infinite populations? idealized markets uncountably many agents when studying an idealized notion of perfect competition (Aumann 1964) the essential idea . is that the economy under consideration has a “very large” number of participants, and that the influence of each individual participant is “negligible” . one can integrate over a continuum, and changing the integrand at a single point does not affect the value of the integral intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams 8/54 voting theory... ??? . analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations whence infinite populations? idealized markets uncountably many agents when studying an idealized notion of perfect competition (Aumann 1964) the essential idea . is that the economy under consideration has a “very large” number of participants, and that the influence of each individual participant is “negligible” . one can integrate over a continuum, and changing the integrand at a single point does not affect the value of the integral intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis 8/54 whence infinite populations? idealized markets uncountably many agents when studying an idealized notion of perfect competition (Aumann 1964) the essential idea . is that the economy under consideration has a “very large” number of participants, and that the influence of each individual participant is “negligible” . one can integrate over a continuum, and changing the integrand at a single point does not affect the value of the integral intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis voting theory..
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