The geometry of moral decision making

TL;DR

This work presents an integrated, information‑geometric framework for bounded rationality that unites utilitarian optimization with deontological regularisation via Markov kernels. It develops a rich geometric view of decision rules, rate‑distortion tradeoffs, and geodesic solutions on the probability simplex, including the Boltzmann‑Gibbs family as a gradient flow of the utility. The main contributions include analyses of multiplier‑robust and constraint‑robust control problems, the utility expansion path, and an application to legal theory showing how a regulator can set the coupling constant $\beta$ to balance rights protection with utilitarian aims. The framework has implications for autonomous agents design and formal reasoning about rights restrictions under uncertainty, highlighting the normative role of the coupling parameter in governance.

Abstract

We show how (resource) bounded rationality can be understood as the interplay of two fundamental moral principles: deontology and utilitarianism. In particular, we interpret deontology as a regularisation function in an optimal control problem, coupled with a free parameter, the inverse temperature, to shield the individual from expected utility. We discuss the information geometry of bounded rationality and aspects of its relation to rate distortion theory. A central role is played by Markov kernels and regular conditional probability, which are also studied geometrically. A gradient equation is used to determine the utility expansion path. Finally, the framework is applied to the analysis of a disutility model of the restriction of constitutional rights that we derive from legal doctrine. The methods discussed here are also relevant to the theory of autonomous agents.

Figures (9)

• Figure 1: Left panel: Ought set $\mathfrak{O}_x\subset A$ of allowed actions in situation $x$. A particular norm or law corresponds to a co-section $\sigma:X\rightarrow Y$ with $y=\sigma(x)$. $\Gamma_X$ global co-section. Right panel: Utility of all actions. Maximum utility $u_x^*$ achieved at $a^*$, but not eligible because $a^*\notin\mathfrak{O}_x$.
• Figure 2: Hierarchical clustering and filtration with corresponding bracketing $\{\{\{\{A\},\{B\}\},\{C\}\},\{\{D\},\{E\}\}\}$. The root node corresponds to the trivial algebra $\mathcal{F}_{\top}$, and the leaf nodes correspond to the power set algebra $\mathcal{F}_{\bot}$.
• Figure 3: Conditional Markov kernel
• Figure 4: Left panel \ref{['fig:Information foliatation']}: Submanifolds $\mathscr{M}_{{\mathbb P}}$, ${\color{red}\mathscr{E}_r}$ and ${\color{red}\mathscr{E}_{r'}}$, $r<r'$, of $\Delta(X\times Y)$. Black contour lines surround the coloured areas corresponding to the different values of the free energy difference $F_{\beta}:={\mathbb E}_{\pi}[U]-\frac{1}{\beta}I(\pi)$. The blue dashed lines show contours of constant mutual information. Right panel \ref{['fig:Rate distortion']}: Submanifolds $\mathscr{M}_{{\mathbb P}}$ (fibre over ${\mathbb P}$), ${\color{red}\mathscr{E}_{\perp\!\!\!\perp}}$ (exponential family of product measures) and $\mathscr{E}_0({\mathbb P})$ of $\Delta(X\times Y)$. ${\color{red}\mathscr{E}_0({\mathbb P})}=\mathscr{M}_{{\mathbb P}}\cap{\color{red}\mathscr{E}_{\perp\!\!\!\perp}}$. Section ${\color{blue}\sigma_{K_C}}:\Delta(X)\rightarrow\Delta(X\times Y)$, $\mu\rightarrow\mu\rtimes K_C$ induced by channel $K_C$, with image ${\color{blue}\mathscr{M}_{K_C}}$, and ${\color{blue}\sigma_{K_C}({\mathbb P})}:=\mathscr{M}_{{\mathbb P}}\cap{\color{blue}\mathscr{M}_{K_C}}$.
• Figure 5: Set of product measures ${\color{red}\mathscr{E}_0({\mathbb P})}$ ( red diagonal) in $\mathscr{M}_{{\mathbb P}}$ and ${\color{teal}\mathscr{E}_0(\nu)}$ ( dark green diagonal) in ${\color{teal}\mathscr{M}_{\nu}}$. Left panel \ref{['fig:couplings']}: $m$-convex set of couplings of ${\mathbb P}$ and $\nu$ in $\Delta(X\times Y)$, given by the intersection $\Gamma({\mathbb P},\nu)={\color{teal} \mathscr{M}_{\nu}}\cap{\color{red}\mathscr{M}_{{\mathbb P}}}$, and ${\color{red}\mathscr{E}_0({\mathbb P})}\cap{\color{teal}\mathscr{E}_0(\nu)}={\mathbb P}\otimes\nu$. Right panel \ref{['fig:e_geodesics']}: Rear view of the probability simplex $\Delta(2\times 2)$. The blue dashed lines show contours of constant mutual information. The thick blue curves represent three different $e$-geodesics in $\Delta(2\times 2)$ that start and end in $\mathscr{M}_{{\mathbb P}}$ but are not otherwise included.

Theorems & Definitions (35)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4: Bayes' disintegration/factorisation
• proof
• Lemma 4.1
• proof