Continuity and catastrophic risk

Suppose that a decision-maker’s aim, under certainty, is to maximize some continuous value, such as lifetime income or continuous social welfare. Can such a decision-maker rationally satisfy what has been called ‘continuity for easy cases’ while at the same time satisfying what seems to be a widespread intuition against the full-blown continuity axiom of expected utility theory? In this note I argue that the answer is ‘no’: given transitivity and a weak trade-off principle, continuity for easy cases violates the anti-continuity intuition. I end the note by exploring an even weaker continuity condition that is consistent with the aforementioned intuition.

Non-human primates satisfy utility maximization in compliance with the continuity axiom of Expected Utility Theory

ABSTRACTExpected Utility Theory (EUT), the first axiomatic theory of risky choice, describes choices as a utility maximization process: decision makers assign a subjective value (utility) to each choice option and choose the one with the highest utility. The continuity axiom, central to EUT and its modifications, is a necessary and sufficient condition for the definition of numerical utilities. The axiom requires decision makers to be indifferent between a gamble and a specific probabilistic combination of a more preferred and a less preferred gamble. While previous studies demonstrated that monkeys choose according to combinations of objective reward magnitude and probability, a concept-driven experimental approach for assessing the axiomatically defined conditions for maximizing subjective utility by animals is missing. We experimentally tested the continuity axiom for a broad class of gamble types in four male rhesus macaque monkeys, showing that their choice behavior complied with the existence of a numerical utility measure as defined by the economic theory. We used the numerical quantity specified in the continuity axiom to characterize subjective preferences in a magnitude-probability space. This mapping highlighted a trade-off relation between reward magnitudes and probabilities, compatible with the existence of a utility function underlying subjective value computation. These results support the existence of a numerical utility function able to describe choices, allowing for the investigation of the neuronal substrates responsible for coding such rigorously defined quantity.SIGNIFICANCE STATEMENTA common assumption of several economic choice theories is that decisions result from the comparison of subjectively assigned values (utilities). This study demonstrated the compliance of monkey behavior with the continuity axiom of Expected Utility Theory, implying a subjective magnitude-probability trade-off relation which supports the existence of numerical subjective utility directly linked to the theoretical economic framework. We determined a numerical utility measure able to describe choices, which can serve as a correlate for the neuronal activity in the quest for brain structures and mechanisms guiding decisions.

Decision Making and Games with Vector Outcomes

In this paper, we study decision making and games with vector outcomes. We provide a general framework where outcomes lie in a real topological vector space and the decision maker’s preferences over outcomes are described by a preference cone, which is defined as a convex cone satisfying a continuity axiom. Further, we define a notion of utility representation and introduce a duality between outcomes and utilities. We provide conditions under which a preference cone is represented by a utility and is the dual of a set of utilities. We formulate a decision-making problem with vector outcomes and study optimal choices. We also consider games with vector outcomes and characterize the set of equilibria. Lastly, we discuss the problem of equilibrium selection based on our characterization.

Tarski Geometry Axioms – Part II

Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

Original Position Models, Trade-offs and Continuity

John Harsanyi has offered an argument grounded in Bayesian decision theory that purports to show that John Rawls's original position analysis leads directly to utilitarian conclusions. After explaining why a prominent Rawlsian line of response to Harsanyi's argument fails, I argue that a seemingly innocuous Bayesian rationality assumption, the continuity axiom, is at the heart of a fundamental disagreement between Harsanyi and Rawls. The most natural way for a Rawlsian to respond to Harsanyi's line of analysis, I argue, is to reject continuity. I then argue that this Rawlsian response fails as a defence of the difference principle, and I raise some concerns about whether it makes sense to posit the discontinuities needed to support the other elements of Rawls's view, although I suggest that Rawls may be able to invoke discontinuity to vindicate part of his first principle of justice.

UNACCEPTABLE RISKS AND THE CONTINUITY AXIOM

Consider a sequence of outcomes of descending value, A > B > C > . . . > Z. According to Larry Temkin, there are reasons to deny the continuity axiom in certain ‘extreme’ cases, i.e. cases of triplets of outcomes A, B and Z, where A and B differ little in value, but B and Z differ greatly. But, Temkin argues, if we assume continuity for ‘easy’ cases, i.e. cases where the loss is small, we can derive continuity for the ‘extreme’ case by applying the axiom of substitution and the axiom of transitivity. The rejection of continuity for ‘extreme’ cases therefore renders the triad of continuity in ‘easy’ cases, the axiom of substitution and the axiom of transitivity inconsistent.As shown by Arrhenius and Rabinowitz, Temkin's argument is flawed. I present a result which is stronger than their alternative proof of an inconsistency. However, this result is not quite what Temkin intends, because it only refers to an ordinal ranking of the outcomes in the sequence, whereas Temkin appeals to intuitions about the size of gains and losses. Against this background, it is argued that Temkin's trilemma never gets off the ground. This is because Temkin appeals to two mutually inconsistent conceptions of aggregation of value. Once these are clearly separated, it will transpire, in connection with each of them, that one of the principles to be rejected does not appear plausible. Hence, there is nothing surprising or challenging about the result; it is merely a corollary to Expected Utility Theory.

Uncertainty behind the Veil of Ignorance

This article argues that the decision problem in the original position should be characterized as a decision problem under uncertainty even when it is assumed that the denizens of the original position know that they have an equal chance of ending up in any given individual's place. It supports this claim by arguing that (a) the continuity axiom of decision theory does not hold between all of the outcomes the denizens of the original position face and that (b) neither us nor the denizens of the original position can know the exact point at which discontinuity sets in, because the language we employ in comparing different outcomes is ineradicably vague. It is also argued that the account underlying (b) can help proponents of superiority in value theory defend their view against arguments offered by Norcross and Griffin.