Comparing nn-dice fixing the sum of the faces

These notes describe some results on dice comparisons when changing the numbers on the faces while the sum of all the face stay the same.Comment: 7 pages. there are some minor changes, in particular in the last section. Also, it is in the DMTCS forma

The Probability of Intransitivity in Dice and Close Elections

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $\{1,\ldots,n\}$ and conditioned on the average of faces equal to $(n+1)/2$ are intransitive with asymptotic probability $1/4$. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to $0$, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $H\in(0,1)$. Second, we pose an analogous model in the context of Condorcet voting. We consider $n$ voters who rank $k$ alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the $k$ alternatives is equal to $2^{-k(k-1)/2}$, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of "close to tied" for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.Comment: Ver3: 45 pages, additional details and clarifications; Ver2: 43 pages, additional co-author, major revision; Ver1: 23 page

Epistemic Decision Theory’s Reckoning

Epistemic decision theory (EDT) employs the mathematical tools of rational choice theory to justify epistemic norms, including probabilism, conditionalization, and the Principal Principle, among others. Practitioners of EDT endorse two theses: (1) epistemic value is distinct from subjective preference, and (2) belief and epistemic value can be numerically quantified. We argue the first thesis, which we call epistemic puritanism, undermines the second

Epistemic Decision Theory's Reckoning

Epistemic decision theory (EDT) employs the mathematical tools of rational choice theory to justify epistemic norms, including probabilism, conditionalization, and the Principal Principle, among others. Practitioners of EDT endorse two theses: (1) epistemic value is distinct from subjective preference, and (2) belief and epistemic value can be numerically quantified. We argue the first thesis, which we call epistemic puritanism, undermines the second

Epistemic Decision Theory’s Reckoning

Epistemic decision theory (EDT) employs the mathematical tools of rational choice theory to justify epistemic norms, including probabilism, conditionalization, and the Principal Principle, among others. Practitioners of EDT endorse two theses: (1) epistemic value is distinct from subjective preference, and (2) belief and epistemic value can be numerically quantified. We argue the first thesis, which we call epistemic puritanism, undermines the second

Representation Theorems and the Grounds of Intentionality

This work evaluates and defends the idea that decision-theoretic representation theorems can play an important role in showing how credences and utilities can be characterised, at least in large part, in terms of their connection with preferences. Roughly, a decision-theoretic representation theorem tells us that if an agent’s preferences satisfy constraints C, then that agent can be represented as maximising her expected utility under a unique set of credences (modelled by a credence function 3el) and utilities (modelled by a utility function Des). Such theorems have been thought by many to not only show how credences and utilities can be understood via their relation to preferences, but also to show how credences and utilities can be naturalised—that is, characterised in wholly non-mental, non-intentional, and nonnormative terms. There are two broad questions that are addressed. The first (and more specific) question is whether any version of characterisational representationism, based on one of the representation theorems that are currently available to us, will be of much use in directly advancing the long-standing project of showing how representational mental states can exist within the natural world. I answer this first question in the negative: no current representation theorem lends itself to a plausible and naturalistic interpretation suitable for the goal of reducing facts about credences and utilities to a naturalistic base. A naturalistic variety of characterisational representationism will have to await a new kind of representation theorem, quite distinct from any which have yet been developed. The second question is whether characterisational representationism in any form (naturalistic or otherwise) is a viable position—whether, in particular, there is any value to developing representation theorems with the goal of characterising what it is to have credences and utilities in mind. This I answer in the affirmative. In particular, I defend a weak version of characterisational representationism against a number of philosophical critiques. With that in mind, I also argue that there are serious drawbacks with the particular theorems that decision theorists have developed thus far; particularly those which have been developed within the four basic formal frameworks developed by Savage, Anscombe and Aumann, Jeffrey, and Ramsey. In the final part of the work, however, I develop a new representation theorem, which I argue goes some of the way towards resolving the most troubling issues associated with earlier theorems. I first show how to construct a theorem which is ontologically similar to Jeffrey’s, but formally more similar to Ramsey’s—but which does not suffer from the infamous problems associated with Ramsey’s notion of ethical neutrality, and which has stronger uniqueness results than Jeffrey’s theorem. Furthermore, it is argued that the new theorem’s preference conditions are descriptively reasonable, even for ordinary agents, and that the credence and utility functions associated with this theorem are capable of representing a wide range of non-ideal agents—including those who: (i) might have credences and utilities only towards non-specific propositions, (ii) are probabilistically incoherent, (iii) are deductively fallible, and (iv) have distinct credences and utilities towards logically equivalent propositions

Uncertainty in Individual and Social Decisions: theory and experiments

In most decisions we have to choose between options that involve some uncertainty about their outcomes and their effect on our well-being. Casual observation and carefully controlled studies suggest that, in making these decisions, we often deviate from the benchmark of expected income maximization. This should not come as a surprise. Our well-being is affected by many factors, and the outside observer does not know the importance of various dimensions of the outcome to the decision maker. Even if goals are well defined, it is far from obvious that we succeed in choosing what is best for us. The psychological literature has shown deviations from optimal behavior in simple decision tasks, and we may expect similar deviations to occur in more complex real life problems. In real life situations, however, experience and market interaction will help to restrain suboptimal behavior. This thesis examines deviations from expected income maximization in situations involving uncertainty. We focus on deviations generated by social factors