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Judgment aggregation is a mathematical theory of collective decision-making. It concerns the methods whereby individual opinions about logically interconnected issues of interest can, or cannot, be aggregated into one collective stance. Aggregation problems have traditionally been of interest for disciplines like economics and the political sciences, as well as philosophy, where judgment aggregation itself originates from, but have recently captured the attention of disciplines like computer science, artificial intelligence and multi-agent systems. Judgment aggregation has emerged in the last decade as a unifying paradigm for the formalization and understanding of aggregation problems. Still, no comprehensive presentation of the theory is available to date. This Synthesis Lecture aims at filling this gap presenting the key motivations, results, abstractions and techniques underpinning it

The paper provides an analysis of the voting method known as delegable proxy voting, or liquid democracy. The analysis first positions liquid democracy within the theory of binary aggregation. It then focuses on two issues of the system: the occurrence of delegation cycles; and the effect of delegations on individual rationality when voting on logically interdependent propositions. It finally points to proposals on how the system may be modified in order to address the above issues.

Liquid democracy is a proxy voting method where proxies are delegable. We propose and study a game-theoretic model of liquid democracy to address the following question: when is it rational for a voter to delegate her vote? We study the existence of pure-strategy Nash equilibria in this model, and how group accuracy is affected by them. We complement these theoretical results by means of agent-based simulations to study the effects of delegations on group's accuracy on variously structured social networks.

Chess involves the capacity to reason iteratively about potential intentional choices of an opponent and therefore involves high levels of explicit theory of mind [ToM] (i.e. ability to infer mental states of others) alongside clear, strategic rule-based decision-making. Functional magnetic resonance imaging was used on 12 healthy male novice chess players to identify cortical regions associated with chess, ToM and empathising. The bloodoxygen- level-dependent (BOLD) response for chess and empathising tasks was extracted from each ToM region. Results showed neural overlap between ToM, chess and empathising tasks in righthemisphere temporo-parietal junction (TPJ) [BA40], left-hemisphere superior temporal gyrus [BA22] and posterior cingulate gyrus [BA23/31]. TPJ is suggested to underlie the capacity to reason iteratively about another's internal state in a range of tasks. Areas activated by ToM and empathy included righthemisphere orbitofrontal cortex and bilateral middle temporal gyrus: areas that become active when there is need to inhibit one's own experience when considering the internal state of another and for visual evaluation of action rationality. Results support previous findings, that ToM recruits a neural network with each region sub-serving a supporting role depending on the nature of the task itself. In contrast, a network of cortical regions primarily located within right- and left-hemisphere medial-frontal and parietal cortex, outside the internal representational network was selectively recruited during the chess task. We hypothesize that in our cohort of novice chess players the strategy was to employ an iterative thinking pattern which in part involved mentalizing processes and recruited core ToM related regions.