To infinity and beyond: A general framework for scaling economic theories

Gonczarowski, Yannai; Kominers, Scott Duke; Shorrer, Ran I. Article To infinity and beyond: A general framework for scaling economic theories Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Gonczarowski, Yannai; Kominers, Scott Duke; Shorrer, Ran I. (2025) : To infinity and beyond: A general framework for scaling economic theories, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 20, Iss. 2, pp. 511-542, https://doi.org/10.3982/TE5878 This Version is available at: https://hdl.handle.net/10419/320292 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 20 (2025), 511–542 1555-7561/20250511 To infinity and beyond: A general framework for scaling economic theories Yannai A. Gonczarowski Department of Economics and Department of Computer Science, Harvard University Scott Duke Kominers Entrepreneurial Management Unit, Harvard Business School, Department of Economics and Center of Mathematical Sciences and Applications, Harvard University, and a16z crypto Ran I. Shorrer Department of Economics, Penn State University Many economic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. We provide a principled framework for scaling results from such models by removing these finiteness assumptions. Our sufficient conditions are on the theorem statement only, and not on its proof. This results in short proofs, and even allows us to use the same argument to scale similar theorems that were proven using distinctly different tools. Yannai A. Gonczarowski: [email protected] Scott Duke Kominers: [email protected] Ran I. Shorrer: [email protected] An earlier version of this paper, entitled “To Infinity and Beyond: Scaling Economic Theories via Logical Compactness,” appeared as a one-page abstract in the Proceedings of the 21st ACM Conference on Economics and Computation. We thank David Ahn, Bob Anderson, Morgane Austern, Archishman Chakrabortyz, Chris Chambers, Yunseo Choi, Henry Cohn, Piotr Dworczak, Andrew Ellis, Tamás Fleiner, Drew Fudenberg, Wayne Gao, Jerry Green, Joseph Halpern, Ron Holzman, Ravi Jagadeesan, M. Ali Khan, David Laibson, Rida Laraki, Bar Light, Elliot Lipnowski, Ce Liu, George Mailath, Michael Mandler, Paul Milgrom, Ankur Moitra, Yoram Moses, Juan Pereyra, Marek Pycia, Debraj Ray, John Rehbeck, Phil Reny, Joseph Root, Ariel Rubinstein, Dov Samet, Chris Shannon, Tomasz Strzalecki, Sergiy Verstyuk, Shing-Tung Yau, Bill Zame, and numerous seminar audiences for helpful comments. Gonczarowski was supported in part by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities; his work was supported in part by ISF grants 1435/14, 317/17, and 1841/14 administered by the Israeli Academy of Sciences; by the United States–Israel Binational Science Foundation (BSF grant 2014389); and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant 740282), and under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant number 337122. Kominers gratefully acknowledges the support of the National Science Foundation (grant SES-1459912), as well as the Ng Fund and the Mathematics in Economics Research Fund of the Harvard Center of Mathematical Sciences and Applications. Shorrer was supported by a grant from the United States–Israel Binational Science Foundation (BSF grants 2016015 and 2022417). Part of this work was conducted during the Simons Laufer Mathematical Sciences Institute Fall 2023 program on the Mathematics and Computer Science of Market and Mechanism Design, which was supported by the National Science Foundation under Grant DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778. Parts of the work of Gonczarowski were carried out while at the Hebrew University of Jerusalem, at Tel Aviv University, and at Microsoft Research. ©2025 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5878 512 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) We demonstrate the versatility of our approach via an array of examples from revealed-preference theory. Keywords. Revealed preferences, infinite models. JEL classification. C02, D00. “More is good... all is better.” —Ferengi Rule of Acquisition #242, Star Trek 1. Introduction In economic theory, we frequently make finiteness assumptions for simplicity and/or tractability—and those assumptions can play a real role in the analysis. Of course, the real world is itself finite, so there is in some sense no “loss” from assuming finiteness in our models. But finiteness assumptions nevertheless sometimes lead to conceptual problems—if our understanding of economic theory hinges on finiteness, then our models may not quite tell the whole story. For example, in decision theory, revealed preference analysis seeks to understand what we can infer about agents from their choice behavior. While a list of observed choices is always finite, if we make parametric assumptions such as homotheticity, then each data point becomes infinitely many data points. Even without such assumptions, we would like to reason about possible demand functions—defined everywhere—that are consistent with the data, and this requires conjecturing about behavior over an infinite dataset. Furthermore, theorizing about observing infinite datasets lets us separate the limitations of inference about agents’ preferences that are just imposed by data finiteness from those that are inherent even with access to every possible observation. As another example, if a game-theoretic finding is true only when the set of agents is finite, then there is an implicit discontinuity, possibly relying on an edge effect or a specific starting condition that may not be robust to small frictions or perturbations.1Thus finite-market results that also hold in infinite markets are in some sense more robust. In this paper, we present a general framework for strengthening results that assume finiteness, by scaling them to infinite settings.2Our approach, by relying on results in propositional logic, implicitly leverages topological properties of the space of theorem statements rather than any features or techniques from their proofs. As such, it allows us to prove results along the lines of “if a certain statement holds when assuming finiteness (regardless of how one would prove it), then—due merely to the structure of this statement—it must hold even if the finiteness assumption is dropped.” Our methods allow us to relax various finiteness assumptions, such as dataset size and market size. In this paper, we focus on applications to decision theory, where the infinity that we tackle is the infinity of data. To demonstrate the versatility of our approach 1For an example of a different kind of discontinuity—between a finite and a continuum setting—see the work of Miralles and Pycia (2017), showing that a continuum model may rule out important phenomena that are observed in the finite models that converge to it. 2As a side note: economic theory sometimes also turns to infinite models when their finite analogues are hard to analyze—for example, to smooth out integer effects. That is not our focus here. Theoretical Economics 20 (2025) To infinity and beyond 513 across disparate fields, we also demonstrate an application to game theory, where the infinity that we tackle is that of the market size. In Section 2, we state and prove our general scaling lemma that we apply throughout the paper. Section 3presents a “warm up,” applying our approach to scale a fundamental result for which the proof of the finite case is considerably simpler than that of the infinite case. Specifically, the result that we scale is that any dataset satisfying the strong axiom of revealed preferences (SARP) is rationalizable, for which the finite case is simple and the infinite case is usually proven by appealing to Zorn’s lemma. In Section 4,we proceed with using a proof very similar to the simple proof from the warm-up, to relax a finiteness assumption in a result for which the infinite case has not been previously proven. Specifically, we prove a novel infinite-data version of Masatlioglu, Nakajima, and Ozbay’s (2012) characterization of limited-attention rationalizability. The finitecase proofs for this result and for the warm-up are starkly different. Nonetheless, the statements of these two results are similar, and this enables us to scale them using essentially the same proof. We furthermore show that since our approach only relies on theorem statements (and not on how they are proven) it can be used to conditionally scale a rich family of not-yet-proven results (i.e., conditional on the finite case being true). At first glance, our approach might seem limited to proving results that are discrete in nature (see discussion below). Nonetheless, in Section 5we use our approach to prove results regarding objects that are nondiscrete (coming from a continuum space). Specifically, here we reprove Reny’s (2015) infinite-data version of Afriat’s (1967)theorem, where utilities come from a continuum space, as well as Caplin, Dean, and Leahy’s (2017) infinite-data version of Caplin and Dean’s (2015) characterization of having a costly information acquisition representation, where priors come from a continuum space. Again, our scaling proofs for both of these theorems are nearly identical. In Section 6, we discuss limitations of our approach in the context of decision theory. In Section 7, we conclude with an application to a different field and with a different notion of infinity, reproving the existence of a Nash equilibrium in infinite games on graphs. We then discuss limitations of our approach more broadly. As already mentioned, some of the results that we prove in this paper are novel to this work. Other results that we (re)prove have already been obtained using other, very different methods, which allows us to compare and contrast prior proof techniques with ours. As our illustrative applications demonstrate, proofs that use our framework have several notable features. First, they use one tool rather than having to choose from various setting-specific tools. Second, the proof structure is modular: our conditions for scaling the finite-case result to the infinite case depend only on the statement of the finite-case result and are completely agnostic to the argument/methods used to prove that result. Furthermore, the proofs are robust in that even their dependence on the details of the model is quite weak, and essentially the same proof can sometimes be used in quite different models. 514 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) 1.1 Technique Our general approach, formulated via Lemma 1in Section 2, allows us to scale problems provided that they have what we call a well description, and that this well description satisfies what we call the finite-subset property. We define these concepts precisely in Section 2.2, after reviewing preliminaries of propositional logic in Section 2.1.Here,we provide an informal description and an illustrative application (which we later formalize): showing that infinite datasets that satisfy SARP are rationalizable. Awell description specifies for each problem a (potentially infinite) set of individually finite logical statements over Boolean variables, such that the problem has a solution if and only if there is an assignment of truth values to these Boolean variables under which all these statements hold simultaneously.3For example,4in a revealedpreferences setting, we can encode a rationalizing preference order using a set of Boolean variables {agtb}(“agreater than b”), each being Tr u e if abfor the corresponding aand b. We can then express all the required properties of a rationalizing order (completeness, transitivity, antisymmetry, and consistency with whichever outcomes are revealed preferred to others) using logical statements phrased in terms of these variables (infinitely many such statements, but each statement individually finite). An assignment of truth values to the variables under which all statements hold simultaneously corresponds exactly to a solution (a rationalizing order), and vice versa. In particular, for each problem, our set of logical statements has such an assignment if and only if the problem has a solution and, therefore, this is a well description. The preceding well description captures finite and infinite problems equally: the only difference that arises is in the cardinalities of the sets of Boolean variables and logical statements. When the problem is infinitary (e.g., with an infinite dataset or with infinitely many agents), the associated set of logical statements is infinite as well. Yet, each of the logical statements we construct is nonetheless individually finite, that is, it contains only finitely many of the Boolean variables. Fix a well description, and call the set of logical statements associated with each problem “the description of the problem.” The well description satisfies the finite-subset property if every finite subset of the description of any problem belongs to the description of a problem that has a solution. In our example, we identify such a problem that has a solution by restricting the original problem to the data points “mentioned” in the given finite subset. The given finite subset is indeed part of the well description of the restricted problem, and since this problem is finite, it can be solved by known existence results for finite problems, so long as we verify that it “inherits” from the infinite problem any properties required by these results (namely, for our example, satisfying SARP). Lemma 1then guarantees the existence of an appropriate solution of the infinite problem. We prove Lemma 1using logical compactness (see Section 2.1), a central result in the theory of propositional logic. While the above example demonstrates the applicability of 3The reader may think of an assignment of truth values to the Boolean variables as a “state of the world.” In the language of mathematical logic, such an assignment under which all statements hold is called a model for the statements (not to be confused with the standard economic concept of a “model”). 4We further elaborate on this example in Section 3. Theoretical Economics 20 (2025) To infinity and beyond 515 our approach to existence results of inherently discrete objects, we also show how to use this approach to scale economic results that go beyond discrete solutions into infinite settings. 2. Framework In this section, we provide a brief introduction to propositional logic (in Section 2.1),5 and use it to state and prove our main technical lemma (in Section 2.2). 2.1 Propositional logic preliminaries In propositional logic, we work with a set of Boolean variables, and study the truth values of statements—called formulae—made up of those variables. We construct formulae by conjoining variables with simple logical operators such as or,not,andimplies.Variables are abstract, and do not have meaning on their own—but we can imbue them with “semantic” meaning by introducing formulae that reflect the structure of economic (or other) problems. Once given semantic meaning, the truth or falsity of statements in our propositional logic model imply the corresponding results in the associated economic model. We start by formalizing the idea of (well-formed propositional) formulae. To define the set of formulae at our disposal, we first introduce a basic (finite or infinite) set of (Boolean) variables. In each section of this paper, we introduce a different set of variables built around the economic setting that we model in that section. Once we have introduced a (finite or infinite) set Vof variables, we can define the set of all well-formed formulae inductively: •‘φ’ is a well-formed formula for every variable φ∈V. •‘¬φ’ is a well-formed formula for every well-formed formula φ. •‘(φ∨ψ)’, ‘(φ∧ψ)’, ‘(φ→ψ)’, and ‘(φ↔ψ)’ are well-formed formulae for every two well-formed formulae φand ψ. Example 1. We could start with a set of four variables V={P,Q,R,S}. Then each of the following is a well-formed formula: ‘P’(1)‘(P∨Q)’(2)‘¬(P∧Q)’(3)‘((P∧R)→S)’(4)♦ We sometimes abuse notation by omitting parentheses and writing, for example, ‘φ∨ψ∨ξ’ when any arbitrary placement of parentheses in the formula (e.g., ‘((φ∨ψ)∨ξ)’or‘ (φ∨(ψ∨ξ))’) will not make a difference. We sometimes abuse notation even further by writing, for example, ‘10 i=1φi’ to mean ‘φ1∨φ2∨···∨φ10’(once 5For a more in-depth look at propositional logic primitives and at the compactness theorem, see a textbook on mathematical logic (e.g., Enderton (2001), Gonczarowksi and Nisan (2022)). Propositional logic formulae are also used for stating Boolean satisfiability problems. 516 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) again, only when the precise placement of omitted parentheses is of no consequence to our analysis). We note that while well-formed formulae can be arbitrarily long, each well-formed formula is always finite in length. Thus, for example, a disjunction ‘φ1∨φ2∨···’of infinitely many formulae is not a well-formed formula. We therefore take special care when we claim that formulae of the form ‘φ∈φ’ are well-formed, as this is true only if is finite. Amodel is a mapping from the set Vof all variables to Boolean values, that is, each variable is mapped either to being Tr u e or to being False. This induces a truth value for every formula ‘φ’whereφ∈V. A model also induces a truth value for all other formulae, defined inductively as follows: •‘¬φ’isTr u e if and only if φis False; •‘(φ∨ψ)’isTr u e if and only if either or both of φand ψis Tr u e ; •‘(φ∧ψ)’isTr u e if and only if both φand ψare Tr u e ; •‘(φ→ψ)’isTr u e if and only if either φis False or ψis Tr u e or both (i.e., ‘(φ→ψ)’is False if and only if both φis Tr u e and ψis False); and •‘(φ↔ψ)’isTr u e if and only if φand ψare either both Tr u e or both False. Example 2. Given the concept of truth values, we can reinterpret the formulae (1)–(4) as follows: ‘P’“P[is Tr u e ],” (1) ‘(P∨Q)’“Por Q[is Tr u e ],” (2) ‘¬(P∧Q)’“not(Pand Q[are both Tr u e ]),” (3) ‘((P∧R)→S)’“Pand R[both being Tr u e ],impliesS[being Tr u e ].” (4) The formula in (2) is Tr u e in a model if and only if either ‘P’or‘Q’(orboth)areTr u e in that model; the formula in (3) is Tr u e in a model unless both ‘P’and‘Q’areTr u e in that model; and the formula in (4) is Tr u e in a model unless both ‘P’and‘R’areTr u e in that model while ‘S’isFalse in that model. ♦ We say that a formula is satisfied by a model if it is Tr u e under that model. For example, each of the formulae (1), (2), and (4) is satisfied by the model that assigns value Tr u e to all variables, however, the formula (3) is not satisfied by that model. We say that a (possibly infinite) set of formulae is satisfied by a model if every formula in the set is satisfied by the model. For example, the set of the formulae (1)–(4) is satisfied by the model that assigns value Tr u e to all variables except Q. We say that a (possibly infinite) set of formulae is satisfiable,orthatithas a model, if it is satisfied by some model. For example, the set containing ‘P’and‘¬P’ is not satisfiable. Theoretical Economics 20 (2025) To infinity and beyond 517 Clearly, if a (finite or infinite) set of formulae is satisfiable, then every subset of  is also satisfiable (by the same model), and in particular every finite subset of is satisfiable; the compactness theorem for propositional logic gives a surprising and nontrivial converse to this statement. Theorem 1 (Compactness Theorem for Propositional Logic (Gödel,1930;Malcev, 1936)). A set of formulae is satisfiable if (and only if) every finite subset ⊆is satisfiable. 2.2 A scaling lemma for economic theories In this section, we use propositional logic to derive a sufficient condition for scalability of an economic theorem to infinite cases. This condition, formalized in Lemma 1,isat the heart of all of our proofs. Let Sbe a set to which we refer as the set of (potential) solutions.LetPbeaset to which we refer as the set of (economic) problems, whose solutions (if such exist) are in S. For example, for the consumer choice rationalization example from Section 1.1,S is the set of all consumer preferences over some set of objects, and a problem P∈Pis to rationalize a specific dataset D. Define I:P×S→{Tr u e ,False}such that I(P,S)is Tr u e if and only if Sis a solution for P. NotethatgivenaproblemP, even if we can easily determine for any given S whether Sis a solution of P(i.e., whether I(P,S)is Tr u e ), it may not be clear just from examining P(and I) whether or not it has any solution (i.e., whether there exists S∈S such that I(P,S)is Tr u e ). For example, while it is easy to describe when given preferences rationalize a given dataset, it is not immediate from examining a dataset whether there exist preferences that rationalize it. Similarly, while it is easy to describe when a given strategy profile constitutes a Nash equilibrium in a given game, it is not immediate from examining a game whether it admits a Nash equilibrium. Regardless of the economic setting, given a problem, our goal will be to ascertain whether a solution for it indeed exists. We say that the set Pof problems is a set of well-describable problems if for every P∈Pthere exists a set Pof well-formed formulae such that Phas a solution if and only if Phas a model. We call a collection (P)P∈Pof such sets a well description of P. Example 3. Consider a set Pof problems where for each problem P=(X,D)∈P,the set of solutions of Pconsists of all strict preferences over the universe Xthat rationalize the given choice data D. To well describe P, we may use the variable of the form agtbfrom Section 1.1 (where agtbhas the semantic interpretation “ais preferred to b”). Specifically, for each P∈P, one may have the set of formulae Pthat consists of: (i) for all distinct aand bsuch that in the given choice data, ais chosen from a menu that contains b,theformula‘ agtb’, requiring that the preferences (that correspond to any model of the formulae) rationalize the given choice data D; (ii) for all distinct a,b∈X,theformula‘ agtb∨bgta’, requiring that the preferences be complete; 518 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) (iii) for all distinct a,b∈X,theformula‘¬(agtb∧bgta)’, requiring that the preferences be antisymmetric; (iv) for all distinct a,b,c∈X,theformula‘ (agtb∧bgtc)→agtc’, requiring that the preferences be transitive. By construction, (P)P∈Pis a well description of P.6(The preceding formulae correspond exactly with the formulae (1)–(4) from Section 2.1 upon taking P=agtb,Q=bgta, R=bgtc,andS=agtc.) ♦ Given a well description of P, we say that a problem P∈Psatisfies the finite-subset property (with respect to the given well description of P) if for every finite subset ⊂P there exists a problem P∈Pthat has a solution and for which ⊆P.ByTheorem1, we then have the following. Lemma 1 (Scaling Lemma). Let Pbe a set of well-describable problems and let (P)P∈P be a well description of P.LetP∈P.IfPsatisfies the finite-subset property, then Phas a solution. Proof.LetP∈Pbe a problem satisfying the finite-subset property. By well describability, it is enough to show that Pis satisfiable. By Theorem 1, it is therefore enough to show that every finite ⊂Pis satisfiable. Let be such a finite subset. Since Psatisfies the finite-subset property, there exists P∈Pthat has a solution such that ⊆P. Since Phas a solution, by our well-describability assumption, we have that Pis satisfiable by some model. Since ⊆P, the same model also satisfies ,andsois satisfiable as required. As demonstrated in Section 1.1, in many cases of interest, the existence of a solution for an appropriate Pfor any can be established by finite-case theorems (e.g., on rationalizability of finite datasets or stable matching in finite markets). Thus, by Lemma 1,we obtain existence of solutions for the infinite case of such problems as well. In this paper, we demonstrate the applicability of Lemma 1to a wide range of economic problems. 3. Warm-up:Rational choice functions We begin with a classic revealed preference setup. Let Xbe a (possibly infinite) set of goods. A menu is a finite subset of X.Adataset D⊆{(S,a)∈2X×X|a∈S}consists of the (unique) respective choices made by an agent in a (possibly infinite) set of menus. Apair(S,a)∈Dis interpreted to mean that the agent selected a∈Swhen presented with the menu S. We say that a dataset Dis rationalized by a strict preference relation (complete, antisymmetric, and transitive) over X, if for every (S,a)∈D, the agent’s choice ais the maximal element from Saccording to .Adatasetisrationalizable if it is rationalized by some strict preference relation over X. 6In fact, an even stronger property holds: the set of solutions of Pis in one-to-one correspondence with the set of models of P(see Section 3for more details on this correspondence). While such a one-to-one correspondence holds in many of our applications, this is not required for our arguments. Theoretical Economics 20 (2025) To infinity and beyond 525 WefirstclaimthateverymodelthatsatisfiesDcorresponds to a solution for D. Fix a model for D. For every ¯ x∈Rm +and every n∈N,letvn∈Vnbe the value such that utilityn ¯ x,vnis Tr u e in the model (well-defined by the first and second formula types above), and define u(¯ x)=limn→∞ vn(well-defined, e.g., by the third formula type above since vnis a Cauchy sequence). The resulting utility function u is a limit of nondecreasing quasiconcave functions (by the fourth and fifth formula types above) that weakly rationalize the data (by the seventh formula type above). Hence, uitself is a nondecreasing quasiconcave function that weakly rationalizes the data. Furthermore, for every ¯ x,¯ y∈Rm +such that ¯ x¯ y,thereexisttworational number vectors “in between” them, that is, there exists k∈Nsuch that ¯ x ¯ qk 1¯ qk 2¯ y. Therefore, we have that u(¯ x)≤u(¯ qk 1)≤u(¯ qk 2)−2−k−1<u (¯ qk 2)≤u(¯ y) (the second inequality stems from that inequality holding for almost all functions of which uis the limit, by the sixth formula type above), so uis strictly increasing when all coordinates strictly increase. Finally, since uweakly rationalizes Dand is also strictly increasing when all coordinates strictly increase, then ualso rationalizes D. Second, we claim that if Dhas a solution, then Dhas a model. Fix a solution u for D,andlet¯ u(¯ x)1/4+(1/2π)·arctan(u(¯ x)) +k:¯ qk 2≤¯ x2−k−1for every ¯ x∈Rm +.As this transformation of utilities is strictly monotone, the resulting function ¯ ustill rationalizes the data, and is quasiconcave, nondecreasing, and strictly increasing when all coordinates strictly increase. Furthermore, the sum of the first two summands is in [0, 1/2], and so is the third summand, so the overall sum is in [0, 1]. Finally, due to the third summand, ¯ u(¯ qk 2)>¯ u(¯ qk 1)+2−k−1for every k∈N. Using ¯ u, we can therefore construct a model for D(by setting each utilityn ¯ x,vto be Tr u e if and only if v= ¯ u(x)εn), and so Dhas a model. To sum up, (D)D∈Pis a well description of P. Finite-subset property: Let D∈P.Let⊂Dbe a finite subset. Since is finite, there are only finitely many formulae of the above seventh type (the only formula type that depends on the dataset) in .LetD⊂Dbe the set of datapoints that induce these formulae. By definition, ⊆D. Furthermore, Dsatisfies GARP since any subdataset of Dsatisfies GARP, and hence, by Theorem 7,Dis rationalizable. Therefore, Dsatisfies the finite-subset property. Thus, by Lemma 1,Dis rationalizable. A natural question is why the same argument cannot be used to scale Theorem 7 while maintaining concavity rather than quasiconcavity. The short answer is that—due to the requirement that ube strictly monotone, and the inherent need to make each formula finite—our proof of Theorem 8relies heavily on the fact that quasiconcavity, unlike concavity, is maintained under weakly monotone transformations (such as the mapping of uto ¯ u); we discuss this further in Section 6. 5.1 Additional application of the same proof: Rational inattention Our proof of Theorem 8is quite a bit more flexible than one might imagine. In Appendix B, we use essentially the same well description to scale—from finite to infinite 526 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) datasets—the seminal result of Caplin and Dean (2015) in quite a different rationalizability domain: a state-dependent stochastic choice dataset has a costly information acquisition representation if and only if it satisfies the No Improving Action Switches (NIAS) and No Improving Attention Cycles (NIAC) conditions. Caplin, Dean, and Leahy (2017) recently proved the infinite version of this result via a novel proof that diverges from Caplin and Dean’s proof of the finite case.18 We reprove this result using essentially the same well description as in our proof of Theorem 8, despite the differences between the two settings considered, and despite the fact that neither the original proofs of the finite versions nor the original proofs of the infinite versions of any of these quite different theorems share any common core technique. The main difference between the two well descriptions is that this application does not require strict monotonicity. Therefore, the sixth formula type of the above well description is not required, and the proof that a solution implies a model is simpler as it does not require carefully “massaging” the function uinto ¯ uas above. 6. Remarks and limitations In the preceding sections, we demonstrated the versatility of our approach across several revealed-preference settings. Our approach can be used to scale many additional finite-data results—like those of Cattaneo, Ma, Masatlioglu, and Suleymanov (2020)on the existence of random attention representation, or Filiz-Ozbay and Masatlioglu (2023) on progressive random choice—to encompass infinite datasets.19 In addition to scaling a wide array of finite-data results to encompass infinite datasets, our approach can also be used to adapt finite-data rationalization results to support parametric restrictions, as in Hu, Li, Quah, and Tang (2021), since such restrictions often translate into infinitely many constraints. Our approach, however, is not without limitations. In this section, we provide some remarks on limitations of our proofs within revealed preference. A more high-level discussion of settings in which our approach is not applicable is provided in Section 7.2. Afriat’s theorem (Theorem 7) guarantees that finite demand datasets satisfying GARP can be rationalized using a concave utility function. But there are well-known examples of quasiconcave utility functions whose full (infinite) demand dataset (which satisfies GARP since it is derived from the choices of a utility function) cannot be rationalized using a concave utility function. In Section 5, we reproved the main result of Reny (2015) that unified these settings—i.e., any demand dataset, finite or infinite, that satisfies GARP can be rationalized using a quasiconcave utility function (Theorem 8). By Lemma 1, the existence of a counterexample, together with the correctness of Afriat’s theorem for finite datasets, implies that the existence of a concave rationalizing utility function (as guaranteed by Afriat’s theorem) has no well description that satisfies the finite-subset property. This might seem puzzling since a simple modification to the fourth formula type in our proof (which imposes quasiconcavity) can be used to impose 18de Oliveira, Denti, Mihm, and Ozbek (2017) provide a similar result for infinite datasets of a different kind. 19We thank Yusufcan Masatlioglu for proposing these applications. Theoretical Economics 20 (2025) To infinity and beyond 527 concavity (as in our similar scaling proof in Appendix B), and so should seemingly result in a well description as required. The answer to this puzzle is that this well description does not, in fact, satisfy the finite-subset property. Specifically, the sixth formula type in our proof makes a stronger monotonicity requirement than the monotonicity that is guaranteed by Afriat’s theorem and, therefore, Afriat’s theorem cannot be used to show that the finite-subset property required by Lemma 1holds. To address this issue, a natural approach would be to change the monotonicity requirement that we use to require only strict monotonicity, as guaranteed by Afriat’s theorem. But it is not possible to well-describe strict monotonicity with our variables (since strict inequalities are not preserved in the limit). Our way around this limitation was to make a stronger requirement that is well describable. But in order to use Afriat’s theorem to show that the finite-subset property holds, we had to relax the concavity requirement (recall that our proof applied monotonic transformations to the utility function; while these transformations do not preserve concavity, they do preserve quasiconcavity). We note that while this may appear to be an artefact of using Lemma 1, the existence of the abovementioned counterexample guarantees that no other approach could circumvent this issue. The tradeoff between strengthening monotonicity and weakening concavity, so that well describability and the finite-subset property are satisfied, sheds some new light on what breaks in the infinite case, which at first glance might look like an issue with concavity, but at a deeper look reveals itself to be an issue with strict monotonicity. This affords some degree of intuition for “why” the concavity assumption in Theorem 7 must be relaxed to quasiconcavity when scaling it to infinite datasets. We note that when we use a very similar proof in Appendix Bto scale a result by Caplin and Dean (2015), we do require concavity rather than merely quasiconcavity. This is possible because in that result only weak (rather than strict) monotonicity (in information) is required, which can be well described without being strengthened. Contrasting these two proofs provides yet another example of the power of our approach to tangibly pinpoint why certain conditions can be maintained when some theorems are scaled but not when others are. Meanwhile, Theorem 8illustrates some of the limitations of our framework. A propositional logic formulation precludes the use of quantifiers (e.g., “there exists apositive gap by which the utility from ¯ q2is greater than the utility from ¯ q1”), and also precludes infinitely long formulae such as infinite disjunctions (e.g., “the utility from ¯ q2is greater than the utility from ¯ q1by at least one of the following infinitely many positive gaps”). This prohibits the well description of certain properties of interest (e.g., strict monotonicity, unless strengthened) without the use of variables that refer to infinitely many objects. But such variables oftentimes hinder the ability to invoke finite theorems to show that the finite-subset property holds. The requirement of strict monotonicity underlies another well-known counterexample. While a strict preference order over a finite set of objects can always be represented by a utility function, the same need not be true when the set of objects is un- 528 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) countable.20 Accordingly, any attempt to use our strengthened monotonicity requirement to scale the finite case is of course bound to fail when the set of objects is uncountable. It is instructive to consider how it would fail. Recall that our strengthened monotonicity requires fixed positive gaps between various utility values. In the case of Afriat’s theorem, requiring countably many such gaps sufficed, and hence the required gap lengths could be chosen so that their sum is finite. By contrast, scaling the existence of a utility representation for strict preferences to uncountable sets would involve requiring uncountably many positive gaps. This means that the sum of lengths of required gaps would be infinite, and so some objects would not be associated with a finite utility. 7. Beyond revealed preferences Our approach is not in any way limited to revealed preferences. In this section, we illustrate its applicability in another domain: noncooperative game theory. We furthermore provide examples where our framework is inapplicable.21 7.1 Nash equilibria in games on infinite graphs In this section, we turn to the setting of games on graphs (see, e.g., Kearns (2007), and the references therein), which includes overlapping generations models, even with infinite time. We use Lemma 1to show the existence of a Nash equilibrium in games on infinite graphs. Our result here is covered by Peleg (1969) (who directly scales the seminal existence result of Nash (1951)), but we give a new proof that uses the same principled approach that we use throughout this paper. Here, we use Lemma 1to scale the existence of arbitrarily good approximate Nash equilibria, and then show that the existence of such approximate equilibria implies the existence of an exact equilibrium. This two-step proof strategy is chosen for convenience: with additional variables, it is easy to encode the second step of the proof into the logical formulation just like we did in Section 5.22 In a game on a graph, there is a (potentially infinite) set of players I, each having a finite set of pure strategies Si.Eachplayeri∈Iis linked to a finite set of neighbors N(i)⊂Iwith i∈N(i), and her utility only depends on the strategies played by players in the set N(i).23 This setting occurs, for example, in infinite-horizon overlapping generations models, where at each point in time there are only finitely many players alive, and a player’s utility depends only on the behavior of contemporary players. For each player i,wedenotebyi(Si)the set of mixed strategies (i.e., distributions over pure strategies) of player i.Amixed-strategy profile (σi)i∈Iis a specification of a 20For example, lexicographic preferences over R2or any strict preference order over 2R. 21An in-preparation companion paper presents applications from matching theory, some of which were in our original working paper (Gonczarowski, Kominers, and Shorrer,2023); see also Choi (2024). 22The converse does not hold for the analysis in that section, though: The proof there hinges on a full infinite sequence of approximations being encoded by a single model. 23Readers familiar with the work of Peleg (1969) will note that even on graphs, Peleg’s assumptions are weaker than those stated here. Our analysis can be generalized to cover such weaker assumptions. Theoretical Economics 20 (2025) To infinity and beyond 529 mixed strategy σi∈ifor every player i∈I. A mixed-strategy profile (σi)i∈Iis a Nash equilibrium if for every i∈Iand every possible deviating strategy σ i∈i,itholdsthat ui(σN(i))≥ui(σ i,σN(i)\{i}). Games on finite graphs have finitely many players and finitely many strategies per player; hence, the seminal analysis of Nash (1951) implies that they have Nash equilibria. Theorem 9 (follows from Nash (1951)). Every game on a finite graph has a Nash equilibrium. Our main result of this section is that Nash equilibria are guaranteed to exist even in games on infinite graphs. Theorem 10 (follows from Peleg (1969)). Every game on a (possibly infinite) graph has a Nash equilibrium. As already noted, we prove Theorem 10 by first using Lemma 1to prove the existence of arbitrarily good approximate Nash equilibria, and then showing that the existence of such approximate Nash equilibria implies Theorem 10.Foragivenε>0, a mixed-strategy profile (σi)i∈Iis an ε-Nash equilibrium if for every i∈Iand every possible deviating strategy σ i∈i,itholdsthatui(σN(i))≥ui(σ i,σN(i)\{i})−ε. Lemma 2. For any ε>0, every (possibly infinite) game on a graph has an ε-Nash equilibrium. Proof.Letε>0. For each player i∈I, it will be convenient to consider the space of profiles of mixed strategies of players in N(i)as a metric space with the ∞metric. Note that this metric space is compact. As each player ihas a continuous utility function whose domain is this compact metric space, players’ utility functions are uniformly continuous by the Heine–Cantor theorem. Thus, there exists ˆ δi>0 that assures that if two profiles of mixed strategies of players in N(i)are less than ˆ δiapart, then the utilities they yield to idiffers by no more than ε/2. For each player i, choose δimin{ˆ δj|j∈N(i)}>0. Recall that idenotes the space of player i’s mixed strategies, and let δi i⊂ibe a finite set of strategies that includes all of i’s pure strategies, and includes for any mixed strategy in ia strategy that is at most δiaway from it; such a set exists by the compactness of i. We prove the lemma by proving that the given game admits an ε-Nash equilibrium in which each player iplays a strategy in δi i. We prove this using Lemma 1. Definition of P:Let Pbe all the subsets of I.Asolution for I∈Pis a strategy profile for Ithat is a ε-Nash equilibrium in the induced game between all players in I(where all other “players” play any arbitrary strategy), in which each player i∈Iplays a strategy in δi i. Well describability: We introduce a variable plays(i,σi)for every player i∈Iand discretized strategy σi∈δi i. In what follows, for each I∈Pwe define a set Iof formulae over these variables so that models of Iare in one-to-one correspondence with 530 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) the (not-yet-proven-to-be-nonempty) set of solutions for I. The correspondence is obtained by endowing the variable plays(i,σi)with the semantic interpretation “iplays the strategy σi.” That is, it maps a model for Ito the strategy profile such that for every i∈I,wehavethatiplays the strategy σiif and only if the variable plays(i,σi)is Tr u e in that model. For every player iand every profile σN(i)\{i}of mixed strategies for N(i)\{i}, we define the set of ε-best responses of i: BRε i(σN(i)\{i})σiui(σi,σN(i)\{i})≥max σ i∈iuiσ i,σN(i)\{i}−ε. We define the set Ito consist of the following formulae: (i) for all i∈I, the (finite!) formula ‘σ∈δi i plays(i,σ)’, requiring that iplays some (discretized) strategy (this formula is finite because δi iis); (ii) for all i∈Iand all distinct σi,σ i∈δi i,theformula‘plays(i,σi)→¬plays(i,σ i)’, requiring that the strategy that player iplays be unique; (iii) for all i∈Iand all profiles σ=(σj)j∈N(i)\{i}∈×j∈N(i)\{i}δi iof discretized mixed strategies of N(i)\{i}, the (finite!) formula ‘ j∈N(i)\{i} plays(j,σj)→ σi∈δi i∩BRε i(σ) plays(i,σi)’, requiring for ito ε-best respond to the strategies played by the other players. By construction, (I)I∈Pis a well description of P. Finite-subset property: Let ⊂Ibe a finite subset. Since is finite, it “mentions” (through variables used) only finitely many players; denote the set of these players by I⊂I. By definition, ⊆I. Consider the induced game on Iobtained by having each player i∈I\Imechanically play some fixed strategy in δi i.ByTheorem9,this game has a Nash equilibrium. By choosing for each player i∈Ia closest strategy in δi i to the one she plays at this Nash equilibrium, each player’s utility changes by at most ε/2 (by uniform continuity), and so does the utility attainable by best responding. Therefore, since we started with a Nash equilibrium, it is assured that each player is now playing an ε-best response, so the resulting strategy profile is a solution to I. Therefore, Isatisfies the finite-subset property. Thus, by Lemma 1, there exists an ε-Nash equilibrium in the grand game (among all player in I), as required. Now, we can use Lemma 2to prove Theorem 10 by way of a diagonalization argument. Proof of Theorem 10. Since eachplayer in the graph has finitely many neighbors, every connected component of the graph consists of at most countably many players. As it is enough to show the existence of a Nash equilibrium in each connected component separately (we use the axiom of choice here), let us focus on one connected component. Theoretical Economics 20 (2025) To infinity and beyond 531 By Lemma 2, there exists a sequence (σn)∞ n=1of 1 n-Nash equilibria in the game on this connected component. Since each of the at-most-countably-many coordinates of each element in this sequence lies in [0, 1], we can choose a subsequence (a “diagonal subsequence”) that converges in all coordinates; let σ∗denote the limit of that subsequence. We claim that σ∗is a Nash equilibrium. To see this, note that for every i∈Iand σ i∈i,wehaveforthenth elements of the sequence that uiσn N(i)≥uiσ i,σn N(i)\{i}−1 n. By the continuity of ui, this means that for every i∈Iand σ i∈i,wehave uiσ∗ N(i)≥uiσ i,σ∗ N(i)\{i}, so no player has a profitable deviation under the profile σ∗.Hence,σ∗is indeed a Nash equilibrium—and in particular, we see that a Nash equilibrium exists in the game, as desired. 7.2 Nonapplications When using our framework, one faces an inherent tension. On the one hand, each formula in a well description must use finitely many variables. On the other hand, to be able to use finite results to establish the finite-subset property, each variable must be semantically related only to a finite set of elements in the economic problem. At first glance, this seems to preclude applications in which the desired solution has a parameter with an infinite domain, since requiring that the parameter take some value would require an infinite disjunction. Indeed, it is simpler to handle parameters with finite domains (which, as we have seen, naturally occur in many applications). Nevertheless, we have successfully applied Lemma 1also to settings with infinite-domain parameters, such as utilities (Section 5), costs/prices (Appendix B), or probabilities (Section 7.1). Still, as the following examples demonstrate, in seemingly similar problems, this approach could not possibly work, since the infinite case has no solution. Example 4 (Splitting the Dollar). A dollar must be split between a set Iof agents. A solution is an efficient and envy-free division. When |I|<∞, splitting the dollar equally is a solution. But the case I=Nhas no solution. ♦ Example 5 (Higher Number Wins). Two players can state a number in S⊆R. The player whose stated number is higher wins a prize (which is shared in case of a tie). A solution is a pure-strategy Nash equilibrium. When |S|<∞, a solution exists (each player states maxS), but the case S=Nhas no solution. ♦ What are the limitations of our approach that prevent it from covering the preceding two examples? Our approach for well-describing problems with infinite-domain parameters is to “encode” these parameters via a sequence of values, each from a finite 532 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) domain. Specifically, we have encoded each of the abovementioned parameters using a sequence of increasingly fine discretizations. Two features are critical for the success of this approach: First, that the desiderata on the encoded parameter can be imposed by individually finite formulae on the discretizations. For example, in Section 5, a utility function weakly rationalizes the data if and only if each of its discretizations weakly rationalize the data. This allowed us to represent a utility function that weakly rationalizes the data by a sequence of discretizations that each weakly rationalizes the data. The second critical feature is that not only each parameter value can be encoded by such a sequence of values (discretizations), but also each such sequence from any valid model encode a valid parameter value (i.e., in the parameter domain). In Section 5, since the sequence of discretized utilities is pointwise increasing and bounded, it has a finite limit. In our scaling of the rational inattention result of Caplin and Dean (2015) in Appendix B, since discretized costs are increasing, they have a limit, which is in R≥0∪{∞}—the domain of costs in that problem. For each of the above examples, there is no encoding that has both of these features. In “splitting the dollar,” if, for example, we encode each player’s allocation using discretizations, then there is no set of individually finite formulae on the discretizations that holds if and only if the limit division is efficient. Hence, the first feature is missing. And, any encoding that has the first feature would lose the second one. In “higher number wins,” if, for example, we encode each player’s number using discretizations, then there is no set of individually finite formulae on the discretizations that holds if and only if the limit is finite. Hence, the second feature is missing. And, any encoding that has the second feature would lose the first one. 8. Related literature In decision theory, both finiteand infinite-data models are important and common.24 Reny (2015) showed how to unify these two approaches in the setting of Afriat (1967). In our view, our main contribution to this literature is in generalizing beyond any specific setting by providing a way to systematically unify these approaches. In Section 4, we proved that the result of Masatlioglu, Nakajima, and Ozbay (2012) scales to infinite datasets using essentially the same proof we used to reprove the classic result of Richter (1966)andHansson (1968) that SARP suffices for rationalization by strict preferences. Like the original theorem of Masatlioglu, Nakajima, and Ozbay, our infinite version of that theorem applies to full datasets. de Clippel and Rozen (2021) provide an analogous theorem for finite datasets that need not necessarily be full; our proof can be used to similarly scale their theorem to infinite datasets. To our knowledge, we are the first to use propositional logic as a general tool for scaling results in economics. It is worth mentioning within this context, though, the work of Holzman (1984), who used logical compactness to relax topological conditions of Fishburn (1984).25 24For a recent example of a treatment of finite and infinite datasets, see Aguiar, Hjertstrand, and Serrano (2022). 25Logical compactness is frequently used to scale existence results in mathematics from finite settings to infinite ones (see, e.g., de Bruijn and Erdös (1951) and Halmos and Vaughan (1950)). Theoretical Economics 20 (2025) To infinity and beyond 533 Our approach was stated using propositional logic, but, in fact, Lemma 1generalizes to well descriptions using first-order logic as well. Propositional logic is a special case of first-order logic. Importantly, the former does not use quantifiers (i.e., ∀and ∃). We chose to focus on this special case to simplify the exposition, since we were not able to identify any economic application in which the added generality would be beneficial.26 Other papers have used first-order (rather than propositional) logic and nonstandard analysis to unify, refine, and scale results in economic theory. Examples include the work of Anderson (1978), Brown and Khan (1980), Anderson (1991), Khan (1993), Blume and Zame (1994), Halpern (2009), and Halpern and Moses (2016). Chambers, Echenique, and Shmaya (2014) used compactness in first-order logic to formalize the notion of the empirical content of a model. Like us, Chambers, Echenique, and Shmaya (2014) study applications to revealed preference theory (see also Chambers, Echenique, and Shmaya (2017)), however they deal with different questions from us, and use different techniques. Hellman and Levy (2019) use (still different) tools from mathematical logic to prove conceptually related, yet incomparable, results: while our paper scales certain finite results to infinite settings, their paper scales certain countably infinite results to uncountably infinite settings. Specifically, they give sufficient conditions to scale certain existence results that are known to hold whenever there are countably many possible states of the world into scenarios with uncountably many possible states of the world. Their results are incomparable to any of our results, and even to our existence-in-largemarket results, first because they always assume that the number of agents is finite (an infinite number of agents, even with only two possible types for each, would already result in an uncountably infinite set of possible states of the world to begin with), and second, because they require that the theorems that they scale be already known to hold for the countably infinite, rather than only the finite case. We have been asked about the relation to various theorems in topology. Lemma 1is stated in terms of logical propositions and its proof relies on logical compactness. Logical propositions can be translated into closed sets in an application-specific topological (product) space, in which setting logical compactness follows from Tychonoff’s theorem on topological compactness. In other words, Lemma 1can be proved using Tychonoff’s theorem, and its statement can be translated to the language of topology. However, in our view, the resulting lemma would be harder to directly formulate and the conditions would be harder to verify. And while topological compactness or the language of nets are stronger and more general approaches, in the domains we study, they often introduce technical issues that can render arguments incorrect in subtle ways (e.g., matchings may converge to an object that is not a matching). We therefore view the methodological part of our contribution as introducing a unifying approach that is simple and intuitive to 26Well-describing economic problems using the full generality of first-order logic is challenging. For example, fixing sets of objects (e.g., men and women) is not straightforward. In fact, by the (upward) Löwenheim–Skolem theorem, if a first-order theory has an infinite model (a model with an infinite domain), then it has a model of any larger cardinality, which implies that first-order theories cannot bound the cardinality of their infinite models. Hence, constants would have to play an important role in the well description. 534 Gonczarowski, Kominers, and Shorrer Theoretical Economics 20 (2025) work with, and that does not require us to look for the “right” topological space or apply topological reasoning directly.27 9. Discussion This paper provides a novel, principled approach for scaling economic theory results from finite models to infinite ones. We identify a sufficient condition for scaling a result: A result can be scaled if it is well describable with a description satisfying the finitesubset property. The bulk of this paper is dedicated to demonstrating that many results in revealed-preference theory meet our condition and, therefore, hold even with infinite datasets. We also demonstrate a game-theoretic application that focuses on a different “type” of scaling to infinity: allowing an infinite (rather than finite) number of players. Our approach is not without limitations, and may fail where other approaches can succeed. That said, we have curated an array of applications showing that it has merit in decision theory and beyond in proving novel results, as well as consolidating and shortening proofs of previously known results, in a way that often sheds new light on them. We view the main contribution of this paper to be a methodologicalone: a new, easyto-use, and versatile tool for the economic theory toolbox. We hope that readers of this paper will be able to further leverage our approach. Appendix A: Proof of Theorem 5 Proof of Theorem 5.AswithTheorem3, the “only if” direction is immediate, so we prove the “if” direction using Lemma 1. Definition of P:Fixing X,letPbe the set of all pairs (X,D)such that X⊆Xand Dis a full dataset over Xthat satisfies WARP-LA. A solution for a pair (X,D)∈Pis a pair comprising a strict preference order over Xand an attention filter that together rationalize D. Well describability: We introduce a variable agtbfor every pair of distinct a,b∈X and a variable attn(S,T)for each pair of finite sets S,Tsuch that ∅=T⊆S⊆X.Inwhat follows, for each (X,D)∈Pwe define a set (X,D)of formulae over these variables so that models of (X,D)are in one-to-one correspondence with the (not-yet-proven-tobe-nonempty) set of solutions for (X,D). The correspondence is obtained by endowing the variable agtbwith the semantic interpretation “ais preferred to b(when both are attention attracting),” and the variable attn(S,T)with the semantic interpretation “Tis the set of attention-attracting elements when the menu is S.” That is, it maps a model for (X,D)to the preference such that for every distinct a,b∈X,wehavethatabif and only if the variable agtbis Tr u e in that model and to the attention filter such that for every S,Tsuch that ∅=T⊆S⊆X,wehavethat(S)=Tif and only if the variable attn(S,T)is Tr u e in that model. We define the set (X,D)to consist of the following formulae: 27Once a proof is derived using Lemma 1, it is of course possible to then translate it to a topological statement and attempt to achieve greater generality, if/when such generality is of interest. Theoretical Economics 20 (2025) To infinity and beyond 541 Gonczarowksi, Yannai A. and Noam Nisan (2022), Mathematical Logic Through Python. 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Manuscript received 14 October, 2023; final version accepted 7 August, 2024; available online 7 August, 2024.