Strictly monotonic multidimensional sequences and stable sets in pillage games

Let $S \subset \mathbb{R}^n$ have size $|S| > \ell^{2^n-1}$. We show that there are distinct points $\{x^1,..., x^{\ell+1}\} \subset S$ such that for each $i \in [n]$, the coordinate sequence $(x^j_i)_{j=1}^{\ell+1}$ is strictly increasing, strictly decreasing, or constant, and that this bound on $|S|$ is best possible. This is analogous to the \erdos-Szekeres theorem on monotonic sequences in $\real$. We apply these results to bound the size of a stable set in a pillage game. We also prove a theorem of independent combinatorial interest. Suppose $\{a^1,b^1,...,a^t,b^t\}$ is a set of $2t$ points in $\real^n$ such that the set of pairs of points not sharing a coordinate is precisely $\{\{a^1,b^1\},...,\{a^t,b^t\}\}$. We show that $t \leq 2^{n-1}$, and that this bound is best possible

Asymmetric majority pillage games

This paper studies pillage games (Jordan in J Econ Theory 131.1:26-44, 2006, “Pillage and property”), which are well suited to modelling unstructured power contests. To enable empirical test of pillage games’ predictions, it relaxes a symmetry assumption that agents’ intrinsic contributions to a coalition’s power is identical. In the three-agent game studied: (i) only eight configurations are possible for the core, which contains at most six allocations; (ii) for each core configuration, the stable set is either unique or fails to exist; (iii) the linear power function creates a tension between a stable set’s existence and the interiority of its allocations, so that only special cases contain strictly interior allocations. Our analysis suggests that non-linear power functions may offer better empirical tests of pillage game theory

Efficient sets are small

AbstractWe introduce efficient sets, a class of sets in Rp in which, in each set, no element is greater in all dimensions than any other. Neither differentiability nor continuity is required of such sets, which include: level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, mean–variance frontiers, production possibility frontiers, and Pareto efficient sets. By Lebesgue’s density theorem, efficient sets have p-dimensional measure zero. As Lebesgue measure provides an imprecise description of small sets, we then prove the stronger result that each efficient set in Rp has Hausdorff dimension at most p−1. This may exceed its topological dimension, with the two notions becoming equivalent for smooth sets. We apply these results to stable sets in multi-good pillage games: for n agents and m goods, stable sets have dimension at most m(n−1)−1. This implies, and is much stronger than, the result that stable sets have m(n−1)-dimensional measure zero, as conjectured by Jordan

Sufficient conditions for unique stable sets in three agent pillage games

Pillage games (Jordan, 2006a) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we characterise the core for any number of agents; for three agents, all anonymous pillage games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility

Quality Control in Criminal Investigation

Edited by Xabier Agirre Aranburu, Morten Bergsmo, Simon De Smet and Carsten Stahn, this 1,108-page book offers detailed analyses on how the investigation and preparation of fact-rich cases can be improved, both in national and international jurisdictions. Twenty-four chapters organized in five parts address, inter alia, evidence and analysis, systemic challenges in case-preparation, investigation plans as instruments of quality control, and judicial and prosecutorial participation in investigation and case-preparation. The authors include Antonio Angotti, Devasheesh Bais, Olympia Bekou, Gilbert Bitti, Leïla Bourguiba, Thijs B. Bouwknegt, Ewan Brown, Eleni Chaitidou, Cale Davis, Markus Eikel, Shreeyash Uday Lalit, Moa Lidén, Tor-Geir Myhrer, Trond Myklebust, Matthias Neuner, Christian Axboe Nielsen, Gilad Noam, Gavin Oxburgh, David Re, Alf Butenschøn Skre, Usha Tandon, William Webster and William H. Wiley, in addition to the four co-editors. There are also forewords by Fatou Bensouda and Manoj Kumar Sinha, and a prologue by Gregory S. Gordon.The book follows from a conference at the Indian Law Institute in New Delhi, and is the main outcome of the third leg of a research project of the Centre for International Law Research and Policy (CILRAP) known as the 'Quality Control Project'. Other books produced by the project are Quality Control in Fact-Finding (Second Edition, 2020) and Quality Control in Preliminary Examination: Volumes 1 and 2 (2018). Covering three distinct phases - documentation, preliminary examination and investigation - the volumes consider how the quality of each phase can be improved. Emphasis is placed on the nourishment of an individual mindset and institutional culture of quality control.bookExploring the Frontiers of International La