Fig 1.
Game Tree for the harassment bribery game.
Condition for (a) Symmetric liabilities: po = pc (b) Asymmetric liabilities: po > 0 and pc = 0.
Fig 2.
Equilibrium population structure for the five-strategy model for variable punishment p, and prosecution rate k ‘with refund’ (A, D) and ‘without-refund’ (B, E) in the asymmetric liability and symmetric liability scenarios (C, F).
Shades of white and black colors denote the equilibrium abundance of officers of type O1 and O2. Shade of green and blue and red colors denote the stationary frequencies of C1, C2 and C3 categories of citizens. The values of other parameters are c = 1, v = 1, b = 0.4, t = 0.1. The initial condition corresponds to xC1 = 1/3, xC2 = 1/3, xC3 = 1/3, xO1 = 1/2, xO2 = 1/2 i.e. all strategies are initially equally abundant in the population.
Fig 3.
Equilibrium population structure for the five-strategy model as a function of bribe amount b, and cost of complaining t, ‘with refund’ (A, D) and ‘without-refund’ (B, E) in the asymmetric liability and symmetric liability scenarios (C, F).
Shades of white and black colors denote the equilibrium abundance of officers of type O1 and O2. Shade of green and blue and red colors denote the stationary frequencies of C1, C2 and C3 categories of citizens. The values of other parameters are: c = 1, v = 1, po = 2 k = 0.4. The initial condition corresponds to xC1 = 1/3, xC2 = 1/3, xC3 = 1/3, xO1 = 1/2, xO2 = 1/2.
Fig 4.
Equilibrium population structure for the five-strategy model as a function of punishment p, and bribe amount b in ‘with refund’ (A, D) and ‘without-refund’ (B, E) asymmetric liability and symmetric liability scenario (C, F).
Shades of white and black colors denote the equilibrium abundance of O1 and O2 type of officers. Shade of green and blue and red colors denote the stationary frequencies of C1, C2 and C3 categories of citizens. The values of other parameters are: c = 1, v = 1, k = 0.4, t = 0.1. The initial condition corresponds to xC1 = 1/3, xC2 = 1/3, xC3 = 1/3, xO1 = 1/2, xO2 = 1/2.
Fig 5.
Equilibrium population for the four-strategy model while varying of bribe amount b, and cost of complaining t with refund (A, C) and without-refund (B, D) for asymmetric liability scenario.
Shades of white and black colors denote the equilibrium abundance of O1 and O2 type of officers. Shade of white and cyan colors denote the stationary frequencies of C1 and C2 type of citizens. The values of other parameters are: c = 1, v = 1, po = 2, k = 0.6. The initial condition corresponds to xC1 = 0.5, xC2 = 0.5, xO1 = 0.5, xO2 = 0.5
Fig 6.
Equilibrium population for the four-strategy model as a function of bribe amount b, and punishment p with refund (A, C) and without-refund (B, D) for asymmetric liability scenario.
Shades of white and black color denote the equilibrium abundance of O1 and O2 type of officers. Shades of white and cyan color denote the stationary frequencies of C1 and C2 type of citizens. The values of other parameters are: c = 1, v = 1, k = 0.6. The initial condition corresponds to xC1 = 0.5, xC2 = 0.5, xO1 = 0.5, xO2 = 0.5
Fig 7.
Equilibrium population for the four-strategy model as a function of punishment k and prosecution rate k with refund (A, C) and without-refund (B, D) for asymmetric liability scenario.
Shades of white and black color denote the equilibrium abundance of O1 and O2 type of officers. Shades of white and cyan color denote the stationary frequencies of C1 and C2 type of citizens. The values of other parameters are: c = 1, v = 1, b = 0.4, t = 0.1. The initial condition corresponds to xC1 = 0.5, xC2 = 0.5, xO1 = 0.5, xO2 = 0.5.
Fig 8.
Equilibrium population for the four strategy model as a function of punishment p and cost of complaining t with refund (A, C) and without-refund (B, D) for asymmetric liability scenario.
Shades of white and black color denote the equilibrium abundance of O1 and O2 type of officers. Shades of white and cyan color denote the stationary frequencies of C1 and C2 type of citizens. The values of other parameters are: c = 1, v = 1, k = 0.6, b = 0.4. The initial condition corresponds to xC1 = 0.5, xC2 = 0.5, xO1 = 0.5, xO2 = 0.5.
Fig 9.
Phase diagram of the four-strategy model with asymmetric liability for variable initial conditions.
Panel A-C corresponds to situations ‘with refund’, ‘without refund’ and ‘no complaining cost without refund’ respectively. Each point in this phase plot (simplex) specifies the population structure of officers and citizens. Arrows represents the direction of the change in frequency of a strategy in the phase space. Red represents fast dynamics and blue represents slow dynamics, close to fixed points. The values of the parameters are: c = 1, v = 1, po = 1.5, k = 0.4, b = 0.4, t = 0.1 for panel A and B. Parameter values are: c = 1, v = 1, po = 3, k = 0.4, b = 0.4, t = 0 for panel C.
Fig 10.
Time evolution of the total fraction of honest and corrupt officers in the population for stochastic ABS (A & B) and deterministic simulation (C & D) for pe = 0 (A & C) and pe = 0.5 (B & D).
Other values of parameters: c = 1, v = 1, po = 1.3, pc = 0, k = 0.4, b = 0.4, r = 0; t = 0.1. Number of officers in ABS: N0 = 100; Number of pure citizens in ABS: NC = 100.
Fig 11.
Time evolution of the frequencies of each of the different categories of officers officers in the partially symmetrized game.
The panels show the outcome of the stochastic ABS (A & B) and deterministic simulations (C & D) for pe = 0 (A & C) and pe = 0.5 (B & D) case. Other values of parameters are: c = 1, v = 1, po = 1.3, pc = 0, k = 0.4, b = 0.4, r = 0, t = 0.1. Number of officers in ABS: N0 = 100; Number of pure citizens in ABS: NC = 100.
Fig 12.
Equilibrium population for the alternative strategy exploration model as a function of punishment p and prosecution rate k with refund (A, C) and without-refund (B, D) for asymmetric liability scenario.
Shades of white and black color denote the equilibrium abundance of O1 and O2 type of officers. Shades of white and cyan color denote the stationary frequencies of C1 and C2 type of citizens. The values of other parameters are: c = 1, v = 1, b = 0.4, t = 0.1. The initial condition corresponds to xC1 = 0.5, xC2 = 0.5, xO1 = 0.5, xO2 = 0.5