Optimal Off-line Experimentation for Games

Many business situations can be called “games” because outcomes depend on multiple decision makers with differing objectives. Yet, in many cases, the payoffs for all combinations of player options are not available, but the ability to experiment off-line is available. For example, war-gaming exercises, test marketing, cyber-range activities, and many types of simulations can all be viewed as off-line gaming-related experimentation. We address the decision problem of planning and analyzing off-line experimentation for games with an initial procedure seeking to minimize the errors in payoff estimates. Then, we provide a sequential algorithm with reduced selections from option combinations that are irrelevant to evaluating candidate Nash, correlated, cumulative prospect theory or other equilibria. We also provide an efficient formula to estimate the chance that given Nash equilibria exists, provide convergence guarantees relating to general equilibria, and provide a stopping criterion called the estimated expected value of perfect off-line information (EEVPOI). The EEVPOI is based on bounded gains in expected utility from further off-line experimentation. An example of using a simulation model to illustrate all the proposed methods is provided based on a cyber security capture-the-flag game. The example demonstrates that the proposed methods enable substantial reductions in both the number of test runs (half) compared with a full factorial and the computational time for the stopping criterion.

Golden-Rule Level of the Employment

The competitive economy, over a long time scale, would produce a large number of general equilibria, each of which can be regarded as a possible microstate of this economy. Then by the principle of maximum entropy, we can obtain the most probable macrostate which in the case of perfect competition involving a single industry will lead to a Solow-type aggregate production function. By this aggregate production function, one can make clear how labors match firms on the balanced growth path. Here, we prove that when the capital stock of a society arrives at the golden-rule level on the balanced growth path, the social employment will reach the best level at which every firm on average employs an optimal amount of workers.