Game Transformations that preserve Nash Equilibrium sets and/or Best Response sets

In the literature on simultaneous non-cooperative games, it is well-known that a positive affine (linear) transformation (PAT) of the utility payoffs do not change the best response sets and the Nash equilibrium set. PATs have been successfully used to expand the classes of 2-player games for which we can compute a Nash equilibrium in polynomial time. We investigate which game transformations other than PATs also possess one of the following properties: (i) The game transformation shall not change the Nash equilibrium set when being applied on an arbitrary game. (ii) The game transformation shall not change the best response sets when being applied on an arbitrary game. First, we prove that property (i) implies property (ii). Over a series of further results, we derive that game transformations with property (ii) must be positive affine. Therefore, we obtained two new and equivalent characterisations with game theoretic meaning for what it means to be a positive affine transformation. All our results in particular hold for the 2-player case of bimatrix games.Comment: 18 pages, 0 figure

Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterise the vertices of these polytopes. We define the combinatorial problem “Another completely labeled Gale string” whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful

An Empirical Study on Computation of Exact and Approximate Equilibria

The computation of Nash equilibria is one of the central topics in game theory, which has received much attention from a theoretical point of view. Studies have shown that the problem of finding a Nash equilibrium is PPAD-complete, which implies that we are unlikely to find a polynomial-time algorithm for this problem. Naturally, this has led to a line of work studying the complexity of finding approximate Nash equilibria. This thesis examines the computation of such approximate Nash equilibria within several classes of games from an empirical perspective. In this thesis, we address the computation of approximate Nash equilibria in bimatrix and polymatrix games. For both of these game classes, we provide a library of implementations of algorithms for the computation of exact and approximate Nash equilibria, as well as a suite of game generators which were used as a base for our empirical analysis of the algorithms. We investigate the trade-off between quality of approximation produced by the algorithms and the expected runtime. We provide some insight into the inner workings of the state-of-the-art algorithm for computing ε-Nash equilibria, presenting worst-case examples found for our provided suite of game generators. We then show lower bounds on these algorithms. In the case of polymatrix games, we generate this lower bound from a real-world application of game theory. For bimatrix games, we provide a robust means of generating lower bounds for approximation algorithms with the use of genetic algorithms

Elevating Repositioning Costs: Strategy Dynamics and Competitive Interactions

Research summary: This article proposes an approach for modeling competitive interactions that incorporates the costs to firms of changing strategy. The costs associated with strategy modifications, which we term “repositioning costs,” are particularly relevant to competitive interactions involving major changes to business strategies. Repositioning costs can critically affect competitive dynamics and, consequently, the implications of strategic interaction for strategic choice. While the literature broadly recognizes the importance of such costs, game-theoretic treatments of major strategic change, with very limited exceptions, have not addressed them meaningfully. We advocate greater recognition of repositioning costs and illustrate with two simple models how repositioning costs may facilitate differentiation and affect the value of a firm\u27s capability to reduce repositioning costs through investments in flexibility. Managerial summary: This article illustrates how the decision to make a strategic change is affected by both the cost to the firm of making the various strategy modifications, as well as the cost to its rivals of changing their strategies in response. These “repositioning costs” are important because they shape the responses each competitor would likely make to a move by the other competitor, and should be anticipated when considering an initial change to one\u27s own strategy. The paper shows how repositioning costs can be used strategically to facilitate differentiation, and to assess the value of potential investments in flexibility

Skill and strategy in games

This thesis consists of two parts. Part I deals with relative skill and the role of random factors in games. Part II is devoted to the computation of optimal strategies in two interesting classes of games: poker and take-and-guess games.

Skill and Strategy in Games.

This thesis consists of two parts. Part I deals with relative skill and the role of random factors in games. Part II is devoted to the computation of optimal strategies in two interesting classes of games: poker and take-and-guess games.

Neural networks as a learning paradigm for general normal form games

This paper addresses how neural networks learn to play one-shot normal form games through experience in an environment of randomly generated game payoffs and randomly selected opponents. This agent based computational approach allows the modeling of learning all strategic types of normal form games, irregardless of the number of pure and mixed strategy Nash equilibria that they exhibit. This is a more realistic model of learning than the oft used models in the game theory learning literature which are usually restricted either to repeated games against the same opponent (or games with different payoffs but belonging to the same strategic class). The neural network agents were found to approximate human behavior in experimental one-shot games very well as the Spearman correlation coefficients between their behavior and that of human subjects ranged from 0.49 to 0.8857 across numerous experimental studies. Also, they exhibited the endogenous emergence of heuristics that have been found effective in describing human behavior in one-shot games. The notion of bounded rationality is explored by varying the topologies of the neural networks, which indirectly affects their ability to act as universal approximators of any function. The neural networks' behavior was assessed across various dimensions such as convergence to Nash equilibria, equilibrium selection and adherence to principles of iterated dominance