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

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

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

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

Fast algorithms for rank-1 bimatrix games

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r − 1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component

Nash Codes for Noisy Channels

This paper studies the stability of communication protocols that deal with transmission errors. We consider a coordination game between an informed sender and an uninformed decision maker, the receiver, who communicate over a noisy channel. The sender's strategy, called a code, maps states of nature to signals. The receiver's best response is to decode the received channel output as the state with highest expected receiver payoff. Given this decoding, an equilibrium or "Nash code" results if the sender encodes every state as prescribed. We show two theorems that give sufficient conditions for Nash codes. First, a receiver-optimal code defines a Nash code. A second, more surprising observation holds for communication over a binary channel which is used independently a number of times, a basic model of information transmission: Under a minimal "monotonicity" requirement for breaking ties when decoding, which holds generically, EVERY code is a Nash code.Comment: More general main Theorem 6.5 with better proof. New examples and introductio

10171 Abstracts Collection -- Equilibrium Computation

From April 25 to April 30, 2010, the Dagstuhl Seminar 10171 ``Equilibrium Computation\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available