A differential model of $N$ player games concerning ethical dilemmas

TL;DR

The paper addresses how ethical versus unethical behavior can be modeled in $N$-player games by developing two analytical frameworks: a polytope-geometry approach that encodes extremal ethical states as convex polytopes and a differential-dynamics approach that tracks time-evolving ethical markers via coupled ODEs. It applies these methods to two- and three-player scenarios, including the Prisoner’s Dilemma analogue and external perturbations by a third party (Carl), and discusses extensions to general $N$-player settings with stability and phase-space analyses. Key contributions include explicit polytope constructions for ethical states and a rich set of dynamical regimes (equilibria, oscillations, chaos) arising from cross-couplings and perturbations, along with exemplified phase portraits and perturbation scenarios. The results provide a rigorous mathematical toolkit for understanding how ethical behavior may emerge, stabilize, oscillate, or be driven by external influences in organizational decision-making contexts.

Abstract

Ethics play an important role in determining the behavior of an individual under certain circumstances. Ethical or unethical behavior can be treated as a strategy of a player in a pay-off game. In this paper, we present two analytical solutions to studying time evolution of behavior of an individual from ethics perspective. We also present the effect of a third player as a perturbation to a two player game and develop a general approach for a $N$ player game. We demonstrate geometric modeling of behavioral characteristics of individuals as polytopes residing in $D$ dimensional space. We treat three player and two player games using set of differential equations that lead to time evolution of phase trajectories which reveal about the interdependencies and self dependencies of each player. We also demonstrate the effect of strategies of each player on other players in cardinal games.

Figures (27)

• Figure 2: Time evolution of behavioral parameters of Alice and Bob in the simple $\phi$ matrix scheme.
• Figure 3: An ethical polytope obtained with four vertices representing four independent extremal events for ethical behavior of Alice, Bob and the third party C with AND condition.
• Figure 4: An ethical polytope obtained with four vertices representing four independent extremal events for ethical behavior of Alice, Bob and the third party C with OR condition.
• Figure 5: Combined ethical polytope of AND and OR conditions. AND condition polytope is represented with blue edges, OR condition polytope is represented with red edges and common edges are represented with green.
• Figure : (a)