642 research outputs found
Ordinal Games
We study strategic games where players' preferences are weak orders which need not admit utility representations. First of all, we ex- tend Voorneveld's concept of best-response potential from cardinal to ordi- nal games and derive the analogue of his characterization result: An ordi- nal game is a best-response potential game if and only if it does not have a best-response cycle. Further, Milgrom and Shannon's concept of quasi- supermodularity is extended from cardinal games to ordinal games. We ¯nd that under certain compactness and semicontinuity assumptions, the ordinal Nash equilibria of a quasi-supermodular game form a nonempty complete lattice. Finally, we extend several set-valued solution concepts from cardinal to ordinal games in our sense.Ordinal Games, Potential Games, Quasi-Supermodularity, Rationalizable Sets, Sets Closed under Behavior Correspondences
Randomized approximation algorithms : facility location, phylogenetic networks, Nash equilibria
Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions
Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability
The purpose of this thesis is twofold. The first and major part is devoted to
sensitivity analysis of various discrete optimization problems while the second
part addresses methods applied for calculating measures of solution stability
and solving multicriteria discrete optimization problems.
Despite numerous approaches to stability analysis of discrete optimization
problems two major directions can be single out: quantitative and qualitative.
Qualitative sensitivity analysis is conducted for multicriteria discrete optimization
problems with minisum, minimax and minimin partial criteria. The main
results obtained here are necessary and sufficient conditions for different stability
types of optimal solutions (or a set of optimal solutions) of the considered
problems.
Within the framework of quantitative direction various measures of solution
stability are investigated. A formula for a quantitative characteristic called
stability radius is obtained for the generalized equilibrium situation invariant
to changes of game parameters in the case of the H¨older metric. Quality of the
problem solution can also be described in terms of robustness analysis. In this
work the concepts of accuracy and robustness tolerances are presented for a
strategic game with a finite number of players where initial coefficients (costs)
of linear payoff functions are subject to perturbations.
Investigation of stability radius also aims to devise methods for its calculation.
A new metaheuristic approach is derived for calculation of stability
radius of an optimal solution to the shortest path problem. The main advantage
of the developed method is that it can be potentially applicable for
calculating stability radii of NP-hard problems.
The last chapter of the thesis focuses on deriving innovative methods based
on interactive optimization approach for solving multicriteria combinatorial
optimization problems. The key idea of the proposed approach is to utilize
a parameterized achievement scalarizing function for solution calculation and
to direct interactive procedure by changing weighting coefficients of this function.
In order to illustrate the introduced ideas a decision making process is
simulated for three objective median location problem.
The concepts, models, and ideas collected and analyzed in this thesis create
a good and relevant grounds for developing more complicated and integrated
models of postoptimal analysis and solving the most computationally challenging
problems related to it.Siirretty Doriast
Stake-governed tug-of-war and the biased infinity Laplacian
In tug-of-war, two players compete by moving a counter along edges of a
graph, each winning the right to move at a given turn according to the flip of
a possibly biased coin. The game ends when the counter reaches the boundary, a
fixed subset of the vertices, at which point one player pays the other an
amount determined by the boundary vertex. Economists and mathematicians have
independently studied tug-of-war for many years, focussing respectively on
resource-allocation forms of the game, in which players iteratively spend
precious budgets in an effort to influence the bias of the coins that determine
the turn victors; and on PDE arising in fine mesh limits of the constant-bias
game in a Euclidean setting.
In this article, we offer a mathematical treatment of a class of tug-of-war
games with allocated budgets: each player is initially given a fixed budget
which she draws on throughout the game to offer a stake at the start of each
turn, and her probability of winning the turn is the ratio of her stake and the
sum of the two stakes. We consider the game played on a tree, with boundary
being the set of leaves, and the payment function being the indicator of a
single distinguished leaf. We find the game value and the essentially unique
Nash equilibrium of a leisurely version of the game, in which the move at any
given turn is cancelled with constant probability after stakes have been
placed. We show that the ratio of the players' remaining budgets is maintained
at its initial value ; game value is a biased infinity harmonic
function; and the proportion of remaining budget that players stake at a given
turn is given in terms of the spatial gradient and the -derivative of
game value. We also indicate examples in which the solution takes a different
form in the non-leisurely game.Comment: 69 pages with four figures. Updated to include discussion of the
economics literature of tug-of-wa
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