Arthur Paul Pedersen

M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

Samuel Allen Alexander, Arthur Paul Pedersen

A fundamental question asked in modal logic is whether a given theory is consistent. But consistent with what? A typical way to address this question identifies a choice of background knowledge axioms (say, S4, D, etc.) and then shows the assumptions codified by the theory in question to be consistent with those background axioms. But determining the specific choice and division of background axioms is, at least sometimes, little more than tradition. This paper introduces **generic theories** for propositional modal logic to address consistency results in a more robust way. As building blocks for background knowledge, generic theories provide a standard for categorical determinations of consistency. We argue that the results and methods of this paper help to elucidate problems in epistemology and enjoy sufficient scope and power to have purchase on problems bearing on modalities in judgement, inference, and decision making.

Lawrence S. Moss, Arthur Paul Pedersen

This paper studies strict majority reasoning in finite electorates using so-called $\textit{social decision frames}$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.

Lawrence S. Moss, Arthur Paul Pedersen

This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) <doi: 10.48550/arXiv.2606.23853> introduce a coherence criterion that characterizes exactly when qualitative majority judgments are representable by a finitely additive measure. The question addressed here is whether that coherence criterion can be replaced, in the finite setting, by any bounded finite fragment. We prove that it cannot. For every $k\ge 1$, we construct a maximal standard frame whose shortest coherence violation has length exactly $2k+2$. Hence there is no uniform finite bound on the incoherence index of social decision frames, resolving Conjecture 5.7 stated by Moss and Pedersen (2026). The construction is geometric, in the sense that it proceeds via orthogonality and dimension in rational vector spaces, and self-contained: it isolates a symmetric family of half-sized voting blocs and extends it to a maximal frame in which every shorter balanced obstruction is excluded. Along the explicit infinite sequence of universe sizes obtained in the construction, this also establishes the middle-layer family predicted by Conjecture B.25 by Moss and Pedersen (2026). Together with the soundness and completeness theorem for the Moss-Pedersen minimal logic for strict majorities, this establishes that measurable social decision frames are not finitely axiomatizable in that language.

Arthur Paul Pedersen, Samuel Allen Alexander

We advance a general theory of coherent preference that surrenders restrictions embodied in orthodox doctrine. This theory enjoys the property that any preference system admits extension to a complete system of preferences, provided it satisfies a certain coherence requirement analogous to the one de Finetti advanced for his foundations of probability. Unlike de Finetti's theory, the one we set forth requires neither transitivity nor Archimedeanness nor boundedness nor continuity of preference. This theory also enjoys the property that any complete preference system meeting the standard of coherence can be represented by utility in an ordered field extension of the reals. Representability by utility is a corollary of this paper's central result, which at once extends Hölder's Theorem and strengthens Hahn's Embedding Theorem.

Samuel Alexander, Arthur Paul Pedersen

Researchers have formalized reinforcement learning (RL) in different ways. If an agent in one RL framework is to run within another RL framework's environments, the agent must first be converted, or mapped, into that other framework. Whether or not this is possible depends on not only the RL frameworks in question and but also how intelligence itself is measured. In this paper, we lay foundations for studying relative-intelligence-preserving mappability between RL frameworks. We define two types of mappings, called weak and strong translations, between RL frameworks and prove that existence of these mappings enables two types of intelligence comparison according to the mappings preserving relative intelligence. We investigate the existence or lack thereof of these mappings between: (i) RL frameworks where agents go first and RL frameworks where environments go first; and (ii) twelve different RL frameworks differing in terms of whether or not agents or environments are required to be deterministic. In the former case, we consider various natural mappings between agent-first and environment-first RL and vice versa; we show some positive results (some such mappings are strong or weak translations) and some negative results (some such mappings are not). In the latter case, we completely characterize which of the twelve RL-framework pairs admit weak translations, under the assumption of integer-valued rewards and some additional mild assumptions.

Ted Fujimoto, Arthur Paul Pedersen

Single-agent reinforcement learning algorithms in a multi-agent environment are inadequate for fostering cooperation. If intelligent agents are to interact and work together to solve complex problems, methods that counter non-cooperative behavior are needed to facilitate the training of multiple agents. This is the goal of cooperative AI. Recent research in adversarial machine learning, however, shows that models (e.g., image classifiers) can be easily deceived into making inferior decisions. Meanwhile, an important line of research in cooperative AI has focused on introducing algorithmic improvements that accelerate learning of optimally cooperative behavior. We argue that prominent methods of cooperative AI are exposed to weaknesses analogous to those studied in prior machine learning research. More specifically, we show that three algorithms inspired by human-like social intelligence are, in principle, vulnerable to attacks that exploit weaknesses introduced by cooperative AI's algorithmic improvements and report experimental findings that illustrate how these vulnerabilities can be exploited in practice.

Eric Pacuit, Arthur Paul Pedersen, Jan-Willem Romeijn

In this extended abstract, we carefully examine a purported counterexample to a postulate of iterated belief revision. We suggest that the example is better seen as a failure to apply the theory of belief revision in sufficient detail. The main contribution is conceptual aiming at the literature on the philosophical foundations of the AGM theory of belief revision [1]. Our discussion is centered around the observation that it is often unclear whether a specific example is a "genuine" counterexample to an abstract theory or a misapplication of that theory to a concrete case.