Two philosophical questions arise about rationality in centipede games that are logically prior to attempts to apply the formal tools of game theory to this topic. First, given that the players have common knowledge of mutual rationality and common knowledge that they are each motivated solely to maximize their own profits, is there a backwards-induction argument that employs only familiar non-technical concepts about rationality, leads to the conclusion that the first player is rationally obligated to end the game at the (...) first step, is deductively valid, employs premises all of which are prima facie highly plausible, and is prima facie sound and )? Second, if there is such an argument, then is it actually sound, or is it instead defective somehow despite being prima facie sound? Addressing these two questions is our project. We present a backwards-induction argument that is prima facie sound; we argue that it is an instance of the notorious sorites paradox, and hence that the concepts of rational obligatoriness and rational permissibility are vague; and we briefly address the potential consequences of all this for the foundations of game theory and decision theory. (shrink)

The backward induction argument purports to show that rational and suitably informed players will defect throughout a finite sequence of prisoner's dilemmas. It is supposed to be a useful argument for predicting how rational players will behave in a variety of interesting decision situations. Here, I lay out a set of assumptions defining a class of finite sequences of prisoner's dilemmas. Given these assumptions, I suggest how it might appear that backward induction succeeds and why it is actually fallacious. Then, (...) I go on to consider the consequences of adopting a stronger set of assumptions. Focusing my attention on stronger sets that, like the original, obey the informedness condition, I show that any supplementation of the original set that preserves informedness does so at the expense of forcing rational participants in prisoner's dilemma situations to have unexpected beliefs, ones that threaten the usefulness of backward induction. (shrink)

We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.

Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic and the philosophy of science to (...) investigate the applicability of game theory in such fields as economics, philosophy and strategic consultancy. I prove new mathematical theorems about the beliefs, desires and rationality principles of individual human beings, and explore in detail the logical form of game theory as it is used in explanatory and normative contexts. I argue that game theory reduces to rational choice theory if used as an explanatory device, and that game theory is nonsensical if used as a normative device. A provocative account of the history of game theory reveals that this is not bad news for all of game theory, though. Two central research programmes in game theory tried to find the ultimate characterisation of strategic interaction between rational agents. Yet, while the Nash Equilibrium Refinement Programme has done badly thanks to such research habits as overmathematisation, model-tinkering and introversion, the Epistemic Programme, I argue, has been rather successful in achieving this aim. "The 'epistemic' approach to game theory has emerged over the past twenty-five years. What is this approach? How does it differ from the conventional equilibrium-based approach to game theory? What have been its strengths and weaknesses to date? To find out, read this comprehensive and excellently written account". Adam Brandenburger, J. P. Valles Professor of Business Economics and Strategy, Stern School of Business, New York University "Reading Boudewijn de Bruin's book should be rewarding both for game theorists interested in the conceptual foundations of their discipline and for philosophers who want to learn more about formal analysis of strategic interaction. It provides an in-depth logical study of the currently dominant epistemic approaches to non-cooperative games, with an eye both to the attractions and to the serious challenges facing the Epistemic Programme". Wlodek Rabinowicz, Professor of Practical Philosophy, Department of Philosophy, Lund University . (shrink)

The paper is essentially a short version Spohn "Strategic Rationality" which emphasizes in particular how the ideas developed there may be used to shed new light on the iterated prisoner's dilemma (and on iterated Newcomb's problem).

The paper argues that the standard decision theoretic account of strategies and their rationality or optimality is much too narrow, that strategies should rather condition future action to future decision situations (a point of view already developed in my Grundlagen der Entscheidungstheorie, sect. 4.4), that practical deliberation must therefore essentially rely on a relation of superiority and inferiority between possible future decision situations, that all this allows to substantially broaden the theory of practical rationality, that a long list of points (...) attended to in the literature can be subsumed under the broadened perspective (including a novel view on the iterated prisoner's dilemma and on iterated Newcomb's problem, which, however, is revised in Spohn (2003) "Dependency Equilibria and the Causal Structure of Decision and Game Situation"), and that the task to complete and systematize this list indeed forms a fruitful research programme. (shrink)

The backward induction controversy in game theory flared up and then practically ended within a decade – the 1990s. The protagonists, however, did not converge on an agreement about the source of the controversy. Why was this the case, if opposing sides had access to the same modelling techniques and empirical facts? In this paper I offer an explanation for this controversy and its unsettled end. The answer is not to be found in the modelling claims made by the opposing (...) protagonists, but in the tacit metaphors they operate under. Aristotle defined 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, scientific tools. The controversy of backward induction in game theory provides a test bed for the explanatory power of metaphors. This paper frames the controversy in terms of metaphor choice to provide a common framework for the protagonists. This results in the identification of three different domains – mathematical logic, game theory and the world – each connected to the other via different metaphors. The controversy around backward induction is placed in, and tentatively explained by, this framework. (shrink)

In this study I analyse the performance of a democratic decision-making rule: the weighted majority rule. It assigns to each voter a number of votes that is proportional to her stakes in the decision. It has been shown that, for collective decisions with two options, the weighted majority rule in combination with self-interested voters maximises the common good when the latter is understood in terms of either the sum-total or prioritarian sum of the voters’ well-being. The main result of my (...) study is that this argument for the weighted majority rule — that it maximises the common good — can be improved along the following three main lines. (1) The argument can be adapted to other criteria of the common good, such as sufficientarian, maximin, leximin or non-welfarist criteria. I propose a generic argument for the collective optimality of the weighted majority rule that works for all of these criteria. (2) The assumption of self-interested voters can be relaxed. First, common-interest voters can be accommodated. Second, even if voters are less than fully competent in judging their self-interest or the common interest, the weighted majority rule is weakly collectively optimal, that is, it almost certainly maximises the common good given large numbers of voters. Third, even for smaller groups of voters, the weighted majority rule still has some attractive features. (3) The scope of the argument can be extended to decisions with more than two options. I state the conditions under which the weighted majority rule maximises the common good even in multi-option contexts. I also analyse the possibility and the detrimental effects of strategic voting. Furthermore, I argue that self-interested voters have reason to accept the weighted majority rule. (shrink)