50,897 research outputs found
Uncomputability and Undecidability in Economic Theory
Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity
Merging of opinions in game-theoretic probability
This paper gives game-theoretic versions of several results on "merging of
opinions" obtained in measure-theoretic probability and algorithmic randomness
theory. An advantage of the game-theoretic versions over the measure-theoretic
results is that they are pointwise, their advantage over the algorithmic
randomness results is that they are non-asymptotic, but the most important
advantage over both is that they are very constructive, giving explicit and
efficient strategies for players in a game of prediction.Comment: 26 page
A novel result on the revenue equivalence theorem
This paper gives two examples to break through the revelation principle. Furthermore, the revenue equivalence theorem does not hold.Quantum game theory; Algorithmic Bayesian mechanism; Revelation principle; Revenue equivalence theorem.
An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u
Informational Substitutes
We propose definitions of substitutes and complements for pieces of
information ("signals") in the context of a decision or optimization problem,
with game-theoretic and algorithmic applications. In a game-theoretic context,
substitutes capture diminishing marginal value of information to a rational
decision maker. We use the definitions to address the question of how and when
information is aggregated in prediction markets. Substitutes characterize
"best-possible" equilibria with immediate information aggregation, while
complements characterize "worst-possible", delayed aggregation. Game-theoretic
applications also include settings such as crowdsourcing contests and Q\&A
forums. In an algorithmic context, where substitutes capture diminishing
marginal improvement of information to an optimization problem, substitutes
imply efficient approximation algorithms for a very general class of (adaptive)
information acquisition problems.
In tandem with these broad applications, we examine the structure and design
of informational substitutes and complements. They have equivalent, intuitive
definitions from disparate perspectives: submodularity, geometry, and
information theory. We also consider the design of scoring rules or
optimization problems so as to encourage substitutability or complementarity,
with positive and negative results. Taken as a whole, the results give some
evidence that, in parallel with substitutable items, informational substitutes
play a natural conceptual and formal role in game theory and algorithms.Comment: Full version of FOCS 2016 paper. Single-column, 61 pages (48 main
text, 13 references and appendix
- …