Could the backward induction controversy be a metaphorical problem?

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Abstract

The backward induction controversy in game theory emerged and practically ended in the 1990s. The protagonists however did not converge to an agreement for explaining the controversy. Why is this the case if opposing sides had access to the same modeling techniques and empirical facts? In this paper I offer an explanation to this controversy and to its unsettled end. It is argued that the answer is not to be found in the modeling claims made by the opposing protagonists but in the tacit metaphors they operate under. Metaphor in Greek means the transfer of meaning from one domain to another. Aristotle defines metaphor as giving a “thing a name that belongs to something else” (Poetica 1457b). The meaning of metaphors has not changed much since then in contrast to models which are comparatively new and still not well understood modes of scientific reasoning. Metaphors provide an independent standard of evaluation from models. The aim of this paper is to utilize game theory to provide a case study that illustrates how modeling, by itself, is not sufficient to overcome the controversies it generates. Metaphor, as a distinct mode of reasoning, could however successfully explain them. The controversy of backward induction in game theory in particular can be used as testing ground for this hypothesis. The paper frames the controversy in terms of metaphor choice to provide a common qualitative reasoning framework to those involved in it. This results in the identification of three different domains—mathematical logic, game theory, and the economic world–each connected to the other via different metaphors. The controversy on backward induction is placed in, and tentatively explained by, this framework.

Posted for comments on 16 May 2016, 10:23 am.

Comments (2)

  • Johanna Thoma says:

    The appeal to metaphors has been popular in economic methodology for a long time, but sceptics may wonder about the value added by analysing economic discourse in terms of the metaphors used. And so it is potentially a very valuable addition to the literature to provide a case study of a debate in economics that can be illuminated by such an analysis. And this is what the paper at hand aims to do. It aims to show that a deadlock in the economic/philosophical literature, namely the backward induction controversy, can be explained by showing that the opposing sides use different constitutive metaphors that underlie their inquiry.

    While I think the paper contains some interesting discussion of the literature on metaphors in economics, and of the backward induction debate, I don’t think it should be published, at least not in its present form. This is because two central argumentative moves in the paper appear to me mistaken, and in any case undermotivated.

    First, Mabsout provides two arguments why we should look to metaphors to explain the backward induction deadlock. First, he suggests that it would be argumentatively circular to explain a deadlock in modelling by appeal to a model. But this is seems obviously mistaken. It would be circular to use a model to provide an argument in favour of models in general, but why should it be circular to use a model to explain a deadlock between specific classes of models? The second argument is based on the claim that we have a clearer definition of what a model is than we have of what a metaphor is. In fact, the paper spends a lot of time trying to establish that this is so. However, first, given the substantial amount of controversy in philosophy about what metaphors are (one need only take a glance at the Stanford Encyclopedia article), it is not clear to me at all that this claim is true. But more importantly, even if it were true, it is not clear what this shows. Just because we have a clearer definition of one concept than of another does not make it explanatorily more relevant.

    My second main concern has to do with the metaphorical explanation of the deadlock itself. I think the paper points to some interesting metaphors used by those who are swayed by appeal to cases and context in game theory, such as the metaphor of the ‘trembling hand’. The paper opposes this with Aumann’s appeal to mathematical logic as a constitutive metaphor. I am not convinced that it even makes sense to think of mathematical logic as a metaphor. Surely all game theorists are convinced that their models should abide by the rules of logic. If they question the relevance of Aumann’s proof, they do so because they think Aumann’s model does not apply to the cases they are interested in, that the elements of the model don’t map onto the world. Perhaps this is rooted in his using the wrong metaphor, but it doesn’t seem to be rooted in his using mathematical logic.

    I have some additional concerns about the paper. One is that it takes the author a long time, namely until page 7 to mention a concrete example of metaphor in economics. More generally, it proceeds at an extremely high level of abstraction, that makes many of the claims about metaphors hard to assess. In terms of structure, the lengthy discussion of the literature before turning to the case study is unhelpful (partly because it remains unclear what Mabsout’s own position on the various contentious points in the debate is). It is not clear that all of the discussion here is really relevant to Mabsout’s main concerns, and in any case, the discussion would be more fruitful after the reader is already engaged in the backward induction case study.

    The presentation of the backward induction debate was interesting, but I would have hoped for a clearer statement of what Aumann’s proof establishes and how. Moreover, I had never thought of the backward induction debate as centering around the question of whether rationality and common knowledge of rationality are inconsistent with each other. In fact I don’t see exactly what the inconsistency is supposed to be. This may just be a question of clarification, however.

    Johanna Thoma
    Department of Philosophy, Logic, and Scientific Method
    London School of Economics

  • Ramzi Mabsout says:

    I wish to thank Prof Thoma for her insightful comments on my manuscript (ms). I will use this opportunity to restate the aim of the ms and then proceed to engage her comments. A debate on backward induction in game theory took place in the mid 90s between opponents that had access to the same information (there were no information asymmetries between them): All participants had access to the same game theoretic modelling techniques and empirical knowledge on whether backward induction was in fact empirically corroborated or not. Reading through this exchange I could not find a justification for the disagreement to persist unless there were elements external to the exchange that were not being identified, possibly, theoretical vested interests, professional status, etc.

    Before adopting any of the latter extra-scientific explanations, I claim to have gathered sufficient evidence that those participating in this exchange operated under different constitutive metaphors. As noted by Prof Thoma, I could have used some specific (not general) model to explain the controversy. Another concept I may have used is that of paradigms. Let me briefly offer some reasons why I did not focus on the latter two and employed metaphors instead. First, with respect to paradigms, I noted that constitutive metaphors have a close relationships with paradigms. While Kuhnian (1963) type paradigms could offer a good explanation of the debate, especially with regards to the incommensurability between paradigms, paradigms are incomplete and offer only a partial picture of this controversy. Even if we agree the opponents operate under different paradigms, we still need to fill in the blanks within each paradigm and the relationship between paradigms. More flesh is offered by Lakatos’s (1970) extension of Kuhn (and Popper) in his methodology of scientific research program (MSRP): we could conceive Aumann defending the hard core of game theory from the questions (negative heuristics) used by Binmore and Samuelson. Clearly paradigm and MSRP can be used in tandem to complete the explanation I offer.

    One of Thoma’s critiques is that even if metaphors are better understood than models, “it is not clear what this shows. Just because we have a clearer definition of one concept than of another does not make it explanatorily more relevant.” I insist in arguing that metaphors are better understood than models which form a much broader categories of scientific tools (cf Morgan 2012). Metaphors—as figures of speech—have been used in oral and written language for thousands of years. This usage—in terms of a mapping from one domain to another—has also been consistent through time and cultures (Artistotle’s definition makes perfect sense today, 2000 years later). I therefore disagree there is controversy on the definition of metaphors. The ms reviews the literature on metaphors in science and posits that while there are different degrees of complexities in metaphors , the definition used, that of a mapping between two domains, is all that is needed to motivate my argument. We have accordingly a solid nod in our web of beliefs and it may be employed to explain other possibly looser parts, or so I argue. My argument ultimately rests on its explanatory power. Can metaphors explain the controversy? I believe they can and a valid counter-argument needs to engage the substance of the explanation not question it in general terms.

    Thoma’s strongest critique is a counter-factual, namely, why not use specific models to explain a controversy in modelling instead of metaphors? As she, points out, “It would be circular to use a model to provide an argument in favor of models in general, but why should it be circular to use a model to explain a deadlock between specific classes of models?” Let me first note that we could as well label metaphors models however this would add an unnecessary conceptual layer: why refer to metaphors as models if we can directly refer to them as metaphors? There are additional arguments to be made here. The first is that the controversy is a controversy in modelling and game theoretic modelling in specific. However nothing in the paper militates against the usage of models except that no one has done that before (for this specific controversy at least) and I remain open to consider such an alternative. But this is not the aim of the ms and while a game theoretic model of the controversy could be a worthwhile pursuit that falls under the economics of science, I do not object and think that such an effort could complement this contribution rather than substitute it.

    Another concern is that the epistemology of models is an open empirical field as noted above. Many different things have been called models and there does not appear to be consensus among those working on the epistemology of models on their usage across and within disciplines. In fact, a choice has to be made as to which kind of model one should use generating substantive difficulties such as model choice (what model to naturalize on) and reflexivity. I briefly discussed the problem of reflexivity in the ms. One of course may not agree that reflexivity is a problem. But if it were, then a tool emerging from a different disciplinary field—in our case the metaphor—would avoid the potential critique of reflexivity by offering an external vantage point on the controversy.

    Thoma further notes “I am not convinced that it even makes sense to think of mathematical logic as a metaphor.” The ms does not argue that mathematical logic is a metaphor but instead that there is a unificational metaphorical mapping between game theory and mathematical logic (for Aumann). This unificational metaphor is rejected by Aumann’s opponents who argue the world interferes with this mapping and it cannot be a unificational metaphor: both mathematics and the world constrain game theory which acts as hinge between the two. In the revised version of the ms, some related ambiguities will dissipate by introducing Popper’s (1972) three worlds since they are quite similar to the three spheres I incorporated to explain the controversy. According to Popper, World 1 contains physical stuff such as trees, oceans and humans; World 2 is the world of our subjective thoughts; and World 3 is the world of objective knowledge. The interconnections of the three world is complex but World 2 acts as a hinge between World 1 and 3. World 1 and World 3 have some autonomy over World 2. For example, World 1 is independent of World 2 (Popper is a realist after all), that is the world of our subjective thoughts does not constitute World 1. But World 3 is also independent from World 2 even though it is constituted by it. World 3, according to Popper, is the world of knowledge without a knower, it is a man-made library of all accumulated knowledge, including true and false theories, logic, geometry etc. On the one hand, for Aumann, game theory generates coherence (a unificational metaphorical mapping) between World 2 and 3 and whatever contradicts it (from either World 2 or World 1) is irrelevant. Binmore and Samuelson, on the other hand, claim game theory generates mapping in both directions of World 2 connecting all three worlds (without seeking exclusive between coherence between World 3 and World 2).

    There are other organizational concerns Prof Thoma expresses—such as the delay in introducing concrete examples of metaphors in economics and the lengthy discussion before the case study. Both of these can be incorporated in a revised version of the ms to improve readability.

    Lakatos, I. (1970) Falsification and the methodology of scientific research programmes. In Criticism and the Growth if Knowledge, I. Lakatos and A. Musgrave (eds). Cambridge: Cambridge University Press.
    Kuhn, T. (1962) The Structure of Scientific Revolutions. 1st ed, Chicago: Chicago University Press.
    Popper, K. (1972) Objective Knowledge. An Evolutionary Approach. Oxford: Clarendon Press.

    Ramzi Mabsout
    Department of Economics
    American University of Beirut

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