An Alternative Model for the Theory of Games
Joseph E Johnson
University of South Carolina
Last modified: June 19, 2007
Traditional game-theory consists of n players who try to select optimal strategies from a set of alternatives in order to maximize their expected payoff from an n-dimensional payoff array. Substantial computational difficulties arise from more than two players, non-zero sum games, and games where alliances can be formed among players. But perhaps the greatest difficulty is one of realistically allocating the payoff array for practical applications in business, medical, warfare and other applications.
We propose a different framework for games that is like the standard framework in that n players still each seek to maximize their profit by their responses but here the responses are to be the optimal (‘truest’) responses to valid meaningful questions. The payoff is dynamically constructed in terms of information functions for optimizing the expected truth of these responses. The questions posed to the players are to have: (1) a unique computer comparable alphanumeric response that is considered to be ‘true’ or ‘correct’ (within a predetermined level of accuracy such as numerical error or spelling correction allowance). Examples would include: “What is the name of the third largest moon of Jupiter?” and “What will be the Dow Jones Industrial Average value on July 1, 2009?”. By ‘computer comparable’ we normally mean that the correct response is to be a unique series of printable characters unbroken by spaces (i.e. one word or one number). (2) The questions must not be too difficult for some of the players to forward a valid answer within the given limits. The answers to the questions will be estimated by self-correcting expert consensus algorithms by computing (a) E the probability of each response to question in terms of the probabilities of correctness, Pi, of those choosing that response and the probabilities of error (1- Pi) by those choosing another response. The functions Pi are conversely computed in terms of the estimated optimal answers. These two sets of nonlinear coupled equations are solved iteratively for self-consistent solutions (an expert consensus model). (3) The questions also must not be trivial. (4) The questions for a given ‘game’ are also to come from a well defined domain of knowledge (e.g. optimal pharmaceuticals for Alhimezers, quantum mechanics of nuclei, …). If different domains of knowledge were utilized for questions, then the Pi would be too inconsistent. Optionally this model allows for values and reliabilities to be assigned to questions from experts whose weighted estimates come from a similar set of self-consistent equations for (a) the values assigned to questions and (b) the experts ability to consistently estimate the questions ‘value’. This results in a coupled dual game for ‘value’ of the ‘true’ answers. The taxonomy of domain knowledge is managed using a new kind of modular database that allows extremely rapid development and leads to an XML database which can be automatically converted to a relational database with a network of joins. This new model easily supports n-person nonzero sum games with alliances and is immediately applicable to real world problems.