The influence of mutations: an evolutionary inspection game with three strategic actors

Abstract

Suboptimal business decisions lead to corporate cost increases. The basis of the following study is a game theoretical model of Fandel and Trockel (Eur J Oper Res 226:85–93, 2013a), which analyses the relationship between bonuses and financial penalties in a three-person inspection game and the measures that counteract suboptimal decisions. In the present article we investigate from evolutionary perspectives whether the strategic behaviour of the actors described in an inspection game can be invaded by mutants and what risks emerge as a result. In a first step each of the three decision variables of the players will be discussed. It will become apparent that corporate optimal behaviour is realised when the actions of the business management or the controlling department are fixed. In a second step it will be shown that in games with three strategic variables mutations can undermine the solutions. In a third step we will investigate the model in consideration of monotonic payment and monotonic positive payment functions and divide the area of the solutions into octants to which we will allocate the influence of the mutations and demonstrate the circumstances under which a solution tending towards optimal corporate behaviour can be generated.

Appendix

The calculation basis for the phase diagram in Fandel and Trockel’s three-person inspection model (2013a) for the manager is:

$$\pi^{D} \left( {p_{m} = 1,p_{a} ,p_{h} } \right) > \pi^{D} \left( {p_{m} = 0,p_{a} ,p_{h} } \right)$$

$$B_{D} > - p_{h} \cdot \left( {Q_{D} + B_{D} } \right) - p_{a} \cdot \left( {Q_{D} + B_{D} } \right) + p_{a} \cdot p_{h} \cdot \left( {Q_{D} + B_{D} } \right) + B_{D} + L$$

$$p_{h} > \frac{{L - p_{a} \cdot \left( {Q_{D} + B_{D} } \right)}}{{\left( {1 - p_{a} } \right) \cdot \left( {Q_{D} + B_{D} } \right)}}$$

for the controller:

$$\pi^{C} \left( {p_{m} , p_{a} , p_{h} = 1} \right) > \pi^{C} \left( {p_{m} , p_{a} , p_{h} = 0} \right),$$

$$p_{m} < \frac{{B_{C} - K_{C} + p_{a} \cdot Q_{C} }}{{\left( {1 - p_{a} } \right) \cdot B_{C} }}$$

and for the corporate management:

$$\pi^{U} \left( {p_{m} , p_{h} , p_{a} = 1,p_{h} } \right) > \pi^{U} \left( {p_{m} , p_{a} = 0,p_{h} } \right),$$

$$p_{m} < \frac{{\left( {1 - p_{h} } \right) \cdot \left( {Q_{D} + Q_{C} + B_{D} } \right) - K_{U} }}{{\left( {1 - p_{h} } \right) \cdot \left( {Q_{D} + B_{D} } \right) + p_{h} \cdot B_{C} }}.$$

If one now observes Eq. (34), $\dot{p}_{m} = p_{m} \left( {1 - p_{m} } \right)\left( {p_{h} - p_{h}^{*} } \right)\left( {1 - p_{a} } \right)\left( {Q_{D} + B_{D} } \right)$, it must be recognised that the first, second and fourth multiplier in each case is greater than or equal to zero and the fifth multiplier is greater zero. For Eq. (34) this means that if in the third multiplier p _h  > p _h * applies, the growth rate is $\dot{p}_{m} > 0$ or that if the case p _h  < p _h * applies, the growth rate is $\dot{p}_{m} < 0$. By analogy one recognises by Eq. (36) that the first, second and fourth multiplier in each case is greater than or equal to zero and the fifth is greater zero. Thus for Eq. (36) it can be determined that if in the third multiplier p _m  < p _m * applies, the growth rate $\dot{p}_{h} > 0$, and vice versa. If one now considers Eq. (38) $\dot{p}_{a} = p_{a} \left( {1 - p_{a} } \right)\left( {p_{m}^{*} - p_{m} } \right)\left[ {p_{h} B_{C + } \left( {1 - p_{h} } \right)\left( {B_{D} + Q_{D} } \right)} \right]$, it will be recognised that the first and the second multiplier in each case is greater than or equal to zero and the fourth multiplier is greater zero. For Eq. (38) this means that if in the third multiplier p _m  < p _m * applies, the growth rate is $\dot{p}_{a} > 0$, and vice versa.