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Formalizing Arrow’s theorem
"... Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of for ..."
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Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of formalization (encoding proofs in the computer) in general. Keywords: formalization of mathematics, Mizar, social choice theory, Arrow’s theorem, GibbardSatterthwaite theorem, proof errors.
Using Theorema in the Formalization of Theoretical Economics
 Proceedings CICM 2011, pages 58–73, 2011. + 11] [KW11] [Lan11] [Lib10
"... Abstract. Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can—at least in principle—also be carried through in formal syste ..."
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Abstract. Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can—at least in principle—also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization. Concretely we use Theorema to show properties previously proved on paper by two of the coauthors for pillage games with three agents. Of particular interest is some pseudocode which summarizes the results previously shown. Since the computation involves infinite sets the pseudocode is in several ways noncomputational. However, in the presence of appropriate lemmas, the pseudocode has sufficient computational content that Theorema can compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions. 1
FirstOrder Logic Formalisation of Arrow’s Theorem
, 2009
"... Arrow’s Theorem is a central result in social choice theory. It states that, under certain natural conditions, it is impossible to aggregate the preferences of a finite set of individuals into a social preference ordering. We formalise this result in the language of firstorder logic, thereby reduc ..."
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Arrow’s Theorem is a central result in social choice theory. It states that, under certain natural conditions, it is impossible to aggregate the preferences of a finite set of individuals into a social preference ordering. We formalise this result in the language of firstorder logic, thereby reducing Arrow’s Theorem to a statement saying that a given set of firstorder formulas does not possess a finite model. In the long run, we hope that this formalisation can serve as the basis for a fully automated proof of Arrow’s Theorem and similar results in social choice theory. We prove that this is possible in principle, at least for a fixed number of individuals, and we report on initial experiments with automated reasoning tools.
Logic and Social Choice Theory
"... We give an introduction to social choice theory, the formal study of mechanisms for collective decision making, and highlight the role that logic has taken, and continues to take, in its development. The first part of the chapter is devoted to a succinct exposition of the axiomatic method in social ..."
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We give an introduction to social choice theory, the formal study of mechanisms for collective decision making, and highlight the role that logic has taken, and continues to take, in its development. The first part of the chapter is devoted to a succinct exposition of the axiomatic method in social choice theory and covers several of the classical theorems in the field. In the second part we then outline three areas of recent research activity: logics for social choice, social choice in combinatorial domains, and judgment aggregation.
Automated Analysis of Social Choice Problems: Approval Elections with Small Fields of Candidates
"... We analyse the incentives of a voter to vote insincerely in an election conducted under the system of approval voting. Central to our analysis are the assumptions we make on how voters deal with the uncertainty stemming from the fact that a tiebreaking rule may have to be invoked to determine the u ..."
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We analyse the incentives of a voter to vote insincerely in an election conducted under the system of approval voting. Central to our analysis are the assumptions we make on how voters deal with the uncertainty stemming from the fact that a tiebreaking rule may have to be invoked to determine the unique election winner. Because we only make very weak assumptions in this respect, it is impossible to obtain general positive results. Instead, we conduct a finegrained analysis using an automated approach that reveals a clear picture of the precise conditions under which insincere voting can be ruled out. At the methodological level, this approach complements other recent work involving the application of techniques originating in computer science and artificial intelligence in the domain of social choice theory. 1
A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory ⋆
"... Abstract Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them m ..."
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Abstract Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machinechecked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickrey’s 1961 theorem on the properties of secondprice auctions. Based on our formalisation experience, taking an auction designer’s perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox. 1
Developing an Auction Theory Toolbox
"... Abstract. Auctions allocate trillions of dollars in goods and services every year. Auction design can have significant consequences, but its practice outstrips theory. We seek to advance auction theory with help from mechanised reasoning. To that end we are developing a toolbox of formalised represe ..."
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Abstract. Auctions allocate trillions of dollars in goods and services every year. Auction design can have significant consequences, but its practice outstrips theory. We seek to advance auction theory with help from mechanised reasoning. To that end we are developing a toolbox of formalised representations of key facts of auction theory, which will allow auction designers to have relevant properties of their auctions machinechecked. As a first step, we are investigating the suitability of different mechanised reasoning systems (Isabelle, Theorema, and TPTP) for reproducing a key result of auction theory: Vickrey’s celebrated 1961 theorem on the properties of second price auctions – the foundational result in modern auction theory. Based on our formalisation experience, we give tentative recommendations on what system to use for what purpose in auction theory, and outline further steps towards a complete auction theory toolbox. 1