Backgrounds and Previous Researches
This paper focuses on how human make decisions in different circumstances. Expected utility theory [1] was a dominant narrative in behavioural economics on how humans made choices. It was believed that people tend to select the choice that maximizes their expected utility. In 1979, Kahneman and Tversky proposed a new model called prospect theory [2]. Later in 1992, a renewed version known as cumulative prospect theory was published [3]. It shows that people attitudes towards risks change among different circumstances. Prospect theory can be used as a model to describe human behaviours and predict its outcomes. The most obvious application includes financial market and insurance [4], [5], portfolio allocation [6], attitude toward tax evasion and tax audit [7]. Examples of other applications are motivation to reenter entrepreneurship [11], marketing strategy [12], [13] and security of drone delivery system [14]. Game theory is one of the mathematical disciplines that concerns with human decision making. Nash [15] developed a concept called the Nash Equilibrium. His work focused on non-cooperative games, where each player acted independently without a coalition. Nash Equilibrium is useful for analyzing the stability of the game. To model human decision making, several authors have combined risk preference pattern from prospect theory with game theory. For instance, a player following behaves according to prospect theory yields a better result than a player who only maximizes their utility [16]. [17] verifies the existence of equilibrium of the system with prospect theory risk preference. [18] studies the geometry of such a system. This paper will propose a system of non-cooperative game where each player behaves according to prospect theory and study the stability property of this system.
Problem Formulation
When comparing the equal amount of change, losses loom larger than gain. As this graph is asymmetric, the slope near the origin is steeper for loss than for gain. In our non-cooperative system, player will try to move to the state where their payoff is increasing. From the playerbs perspective, decreasing (resp., increasing) payoff resembles a loss (gain). The player will try to move quickly (slowly) towards the state where their payoff is increasing.
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| Fig.1: Prospect Theory [24] |
Conclusion
This paper proposed a model for a two-player non-cooperative system in which the playerbs strategy is based on prospect theory. To analyze the stability, first, we determine the regions of in state space where the payoff is increasing and decreasing. Consider a case where the boundary of those two regions is a hyperbolic curve. In the neighborhood of the Nash Equilibrium, we linearize this curve. We can approximate our dynamics to be a piecewise linear dynamics which depends on the current sensitivity profile. The stability near around Nash Equilibrium can be examined using the integral of radial growth rate.
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