Axiomatic decomposition of a zero-sum game: the penalty shoot-out case

Abstract

The game of soccer has offered matter of wide scientific analysis about the effective application of the game theory in real-life. The field observations have often detected divergent behaviors from theoretical predictions. The basic problem comes from the fact that it is difficult to build scientific models reflecting reality as closely as possible. Axiomatic Design offers us a powerful tool of rational decomposition of a real and complex issue into elementary components. Independence Axiom guarantees that game decomposition will define a set of elementary actions logically consistent and free of redundancies. At the same time, Information Axiom can allow to select among alternative strategies, those that they predict the actions with a higher probability rate of success. In this paper, it is suggested the use of the Axiomatic Design methodology in the Collectively Exhaustive and Mutually Exclusive (CEME) mode, as a tool of analysis of the penalty shoot-out in extra time. This methodology allows to define the game strategies for goalkeepers and penalty takers. It will be analyzed both, the case when the opponents' behavior is well known and the situation when the statistics about the opponents are unknown. Axiomatic Design allows the process of decomposition to be simplified, enabling the selection of optimal game strategies. These strategies correspond to Nash’s equilibrium solutions when you already know about your opponents' game behavior. On the contrary, when penalty takers whose behavior is unknown, then it is always possible to define a strategy corresponding to the Bayesian equilibrium game solutions.