Indian policemans dilemma: A game theoretic model

Kaushik Majumdar, Systems Science and Informatics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India; E-mail: kmajumdar@isibang.ac.in

Abstract: This paper focuses on a one person game called Indian policeman’s dilemma (IPD). It represents the internal conflict between emotion and profession of a typical Indian police officer. We have ‘split’ the game to be played independently by different personality modules of the same player. Each module then appears as an independent individual player of the game. None of the players knows the exact payoff values of any of the others. Only greater than or less than type of inequalities among the payoff values across the players are to be inferred probabilistically. There are two Nash equilibrium (NE) points in this game signifying two completely opposing behavior by the policeman involved. With the help of the probabilistic inequalities probable propensities of the different behaviors have been determined. The model underscores the need for new surveys and data generation. A design of one such survey to measure professionalism of the police personnel has been outlined.

Keywords: Bayesian analysis, criminal law, illegal behavior and the enforcement of law, litigation process, model construction and estimation.

1. Introduction

   One person games have been discussed by Von Neumann and Morgenstern (Von Neumann & Morgenstern 1953). In a bridge each two player team has been treated as a single player and the two players in the same team as two representatives of that single player (p. 53). This has been termed ‘splitting’ of a player. More specifically various games of ‘patience’ and ‘solitaire’ are considered one person games. Indian policeman’s dilemma1 (IPD) is a one person game where the police officer, who happens to be the only player involved, is spilt between two personality tracts namely, ‘Professionalism’ and ‘Emotionality’. It has been assumed that the officer makes various moves either from purely professional consideration or purely emotional consideration. This makes the Professionalism and Emotionality mutually independent. The game has been designed as a two player game played by Professionalism and Emotionality of the same officer. Like Bridge and Poker there is no chance move involved in the IPD. However that does not mean that everything is certain in this game. The payoff values of the players against the strategies of the opponents are unknown and have to be inferred

1 Policeman’s dilemma refers to the following situation: A devastating fire is engulfing a few vehicles following a road accident. Police and rescuers who have reached the scene realize that not everybody can be saved from being burnt alive despite the best efforts. One motorist who was sure to die in the inferno begged an armed policeman to shoot him so that he could have a less painful death. Euthanasia is illegal, but not killing in such a situation is inhuman. This is called policeman’s dilemma or PD, which can be modeled as a one person constant sum game with two mutually conflicting strategies.

2 probabilistically from the events in the environment. In this sense IPD is a game of incomplete information. We will show that in order to reach a solution to the game it will be enough to know about the dominance relationship between certain pairs of payoff values of the two players. This game has another significance. It represents an oriental model. While physical, mathematical, chemical, biological, earth and atmospheric sciences are same in the east and in the west, the social sciences tend to be different due to the influence of culture between orient and occident. IPD represents a professional dilemma of the police personnel in many third world countries, particularly India. In more general form it is probably true of any professional human being, when the person has to resolve the conflict between gains through professional commitments and gains through emotional commitments. The utility function here is the maximization of personal gain. Von Neumann and Morgenstern (Von Neumann & Morgenstern 1953) have quantified the gain (p. 86), but in case of the IPD the quantification of personal gain is impossible or extremely difficult to make. Here we haven’t tried to quantify the gains at all, but analyzed the game only in terms of gain dominance. In the next section we describe the game of IPD. In section 3 we give a normal form representation of it. In section 4 we determine its two Nash equilibrium points. In section 5 we probabilistically determine several inequalities among the payoff values, because the actual payoff values are not known. In section 6 we propose a professionalism index for the Indian police personnel. In section 7 we give the complete solution

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