Recursive Inspection Games

We consider a sequential inspection game where an inspector uses a limited number of inspections over a larger number of time periods to detect a violation (an illegal act) of an inspectee. Compared with earlier models, we allow varying rewards to the inspectee for successful violations. As one possible example, the most valuable reward may be the completion of a sequence of thefts of nuclear material needed to build a nuclear bomb. The inspectee can observe the inspector, but the inspector can only determine if a violation happens during a stage where he inspects, which terminates the game; otherwise the game continues. Under reasonable assumptions for the payoffs, the inspector's strategy is independent of the number of successful violations. This allows to apply a recursive description of the game, even though this normally assumes fully informed players after each stage. The resulting recursive equation in three variables for the equilibrium payoff of the game, which generalizes several other known equations of this kind, is solved explicitly in terms of sums of binomial coefficients. We also extend this approach to non-zero-sum games and, similar to Maschler (1966), "inspector leadership" where the inspector commits to (the same) randomized inspection schedule, but the inspectee acts legally (rather than mixes as in the simultaneous game) as long as inspections remain.Comment: final version for Mathematics of Operations Research, new Theorem

Optimal Lower Bounds for Projective List Update Algorithms

The list update problem is a classical online problem, with an optimal competitive ratio that is still open, known to be somewhere between 1.5 and 1.6. An algorithm with competitive ratio 1.6, the smallest known to date, is COMB, a randomized combination of BIT and the TIMESTAMP algorithm TS. This and almost all other list update algorithms, like MTF, are projective in the sense that they can be defined by looking only at any pair of list items at a time. Projectivity (also known as "list factoring") simplifies both the description of the algorithm and its analysis, and so far seems to be the only way to define a good online algorithm for lists of arbitrary length. In this paper we characterize all projective list update algorithms and show that their competitive ratio is never smaller than 1.6 in the partial cost model. Therefore, COMB is a best possible projective algorithm in this model.Comment: Version 3 same as version 2, but date in LaTeX \today macro replaced by March 8, 201

Bernhard von Stengel: Supermarket pricing tricks

Bernhard von Stengel goes shopping and uncovers the pricing tricks not every consumer manages to detect. Bernhard von Stengel is a Professor in LSE’s Department of Mathematics. He teaches abstract mathematics, optimisation and game theory. He co-authored the Game Theory Explorer Software

A mathematician takes issue with supermarket price promotion gambits

Bernhard von Stengel goes shopping and uncovers the pricing tricks not every consumer manages to detect

Alternative Exercise Technologies to Fight against Sarcopenia at Old Age: A Series of Studies and Review

The most effective physiologic mean to prevent sarcopenia and related muscle malfunction is a physically active lifestyle, or even better, physical exercise. However, due to time constraints, lack of motivation, or physical limitations, a large number of elderly subjects are either unwilling or unable to perform conventional workouts. In this context, two new exercise technologies, whole-body vibration (WBV) and whole-body electromyostimulation (WB-EMS), may exhibit a save, autonomous, and efficient alternative to increase or maintain muscle mass and function. Regarding WB-EMS, the few recent studies indeed demonstrated highly relevant effects of this technology on muscle mass, strength, and power parameters at least in the elderly, with equal or even higher effects compared with conventional resistance exercise. On the contrary, although the majority of studies with elderly subjects confirmed the positive effect of WBV on strength and power parameters, a corresponding relevant effect on muscle mass was not reported. However, well-designed studies with adequate statistical power should focus more intensely on this issue

Complexity of searching an immobile hider in a graph

AbstractWe study the computational complexity of certain search-hide games on a graph. There are two players, called searcher and hider. The hider is immobile and hides in one of the nodes of the graph. The searcher selects a starting node and a search path of length at most k. His objective is to detect the hider, which he does with certainty if he visits the node chosen for hiding. Finding the optimal randomized strategies in this zero-sum game defines a fractional path covering problem and its dual, a fractional packing problem. If the length k of the search path is arbitrary, then the problem is NP-hard. The problem remains NP-hard if the searcher may freely revisit nodes that he has seen before. In that case, the searcher selects a connected subgraph of k nodes rather than a path of k nodes. If k is logarithmic in the number of nodes of the graph, then the problem can be solved in polynomial time. This is shown using a recent technique called color-coding due to Alon, Yuster and Zwick. The same results hold for edges instead of nodes, that is, if the hider hides in an edge and the searcher searches k edges on a path or on a connected subgraph

Predictive Minds: LLMs As Atypical Active Inference Agents

Large language models (LLMs) like GPT are often conceptualized as passive predictors, simulators, or even stochastic parrots. We instead conceptualize LLMs by drawing on the theory of active inference originating in cognitive science and neuroscience. We examine similarities and differences between traditional active inference systems and LLMs, leading to the conclusion that, currently, LLMs lack a tight feedback loop between acting in the world and perceiving the impacts of their actions, but otherwise fit in the active inference paradigm. We list reasons why this loop may soon be closed, and possible consequences of this including enhanced model self-awareness and the drive to minimize prediction error by changing the world.Comment: 6 page