import numpy as np import matplotlib.pyplot as plt from matplotlib import rcParams import matplotlib.colors as colors from scipy.integrate import solve_ivp from sympy import * # baryzentrisches Dreiecks-Koordinatensystem def xy(vx): return np.array([vx[1]+vx[2]/2,vx[2]]) # Definition der Funktionen g_x und g_y def g(xy, D): m = 3 x = np.array([1-xy[0]-xy[1]/2,xy[0]-xy[1]/2,xy[1]]) dx_dt = [] for i in range(m): dx_dt.append(sum(D[i,j]*x[i]*x[j] for j in range(m)) - sum(sum(D[k,j]*x[k]*x[j] for j in range(m)) for k in range(m))*x[i]) return [dx_dt[1]+dx_dt[2]/2,dx_dt[2]] # Vektorisiertes DGL-System mit numpy def DGLsys(t, x, D): x = np.asarray(x) Dx = D @ x u = np.dot(x, Dx) return x * (Dx - u) # Alternative Definition des DGL-Systems #def DGLsys(t,x,D): # m=3 # dx_dt = [] # for i in range(m): # dx_dt.append(sum(D[i,j]*x[i]*x[j] for j in range(m)) - sum(sum(D[k,j]*x[k]*x[j] for j in range(m)) for k in range(m))*x[i]) # return dx_dt # Funktion zur Berechnung der Nashgleichgewichte def find_nash_equilibria(D): equilibria = [] # Berechnung der reinen Nash-Gleichgewichte for i in range(D.shape[0]): for j in range(D.shape[0]): if D[i, j] == max(D[:, j]) and D[j, i] == max(D[:, i]): equilibria.append({'type': 'pure', 's': (i+1,j+1)}) #Berechnung der gemischten Nash-Gleichgewichte (Interior-gemischte und Boundary-gemischte) Loes_GN = [] x1,x2,x3,y1,y2,y3 = symbols('x_1,x_2,x_3,y_1,y_2,y_3') xs = Matrix([x1,x2,x3]) ys = Matrix([y1,y2,y3]) Dollar_A = transpose(xs)*D*ys Dollar_As = Dollar_A.subs(x3,1-x1-x2).subs(y3,1-y1-y2)[0] Dollar_As_1 = Dollar_A.subs(x1,0).subs(x3,1-x2).subs(y3,1-y1-y2)[0] Dollar_As_2 = Dollar_A.subs(x2,0).subs(x3,1-x1).subs(y3,1-y1-y2)[0] Dollar_As_3 = Dollar_A.subs(x3,0).subs(x2,1-x1).subs(y3,1-y1-y2)[0] GemNash_Eq1 = Eq(Dollar_As.diff(x1),0) GemNash_Eq2 = Eq(Dollar_As.diff(x2),0) GemNash_Eq_1 = Eq(Dollar_As_1.diff(x2),0) GemNash_Eq_2 = Eq(Dollar_As_2.diff(x1),0) GemNash_Eq_3 = Eq(Dollar_As_3.diff(x1),0) Bed=Eq(1,y1+y2+y3) Loes_GN.append(solve([GemNash_Eq1,GemNash_Eq2,Bed])) Bed_a=Eq(0,y1) Bed_b=Eq(1,y2+y3) Loes_GN.append(solve([GemNash_Eq_1,Bed_a,Bed_b])) Bed_a=Eq(0,y2) Bed_b=Eq(1,y1+y3) Loes_GN.append(solve([GemNash_Eq_2,Bed_a,Bed_b])) Bed_a=Eq(0,y3) Bed_b=Eq(1,y1+y2) Loes_GN.append(solve([GemNash_Eq_3,Bed_a,Bed_b])) for l in Loes_GN: if l and 0 <= l[y1] < 1 and 0 <= l[y1] < 1 and 0 <= l[y1] < 1 : equilibria.append({'type': 'mixed', 's*': (l[y1],l[y2],l[y3])}) return equilibria # Loesen der DGL und plotten der Populationsentwicklung def solve_and_plot(D, x_init, t_end=15, n_points=1000): # Groessenfestlegung der Labels usw. im Bild rcParams.update({ 'text.usetex' : True, 'figure.figsize' : [8,6], 'axes.titlesize' : 14, 'axes.labelsize' : 16, 'xtick.labelsize' : 14 , 'ytick.labelsize' : 14 }) # Weitere Festlegungen fehler = 10**(-13) t_eval = np.linspace(0, t_end, n_points) # Loesung der DGL fuer eine Anfangspopulation x_init Loes = solve_ivp(DGLsys, [0, t_end], x_init, args=(D, ), t_eval=t_eval, rtol=fehler, atol=fehler) # Fuer die Darstellung des Feldliniendiagramms streamplot Y, X = np.mgrid[0:1:100j, 0:1:100j] gXY = g([X,Y],D) # Die Farbe wird die Geschwindigkeit der Aenderung des Populationsvektors anzeigen colorspeed = np.sqrt(gXY[0]**2 + gXY[1]**2) # Plotten des Bildes fig, ax = plt.subplots(figsize=(10, 8)) ax.set_xlabel(r"$x$") ax.set_ylabel(r"$y$") strm = ax.streamplot(X, Y, gXY[0], gXY[1], linewidth=1,density=[2, 2],norm=colors.Normalize(vmin=0.,vmax=0.4), color=colorspeed, cmap=plt.cm.jet) ax.fill([0,0.5,0], [0,1,1], facecolor='black') ax.fill([1,0.5,1], [0,1,1], facecolor='black') ax.fill([0,1,1,0], [0,0,-0.02,-0.02], facecolor='black') ax.fill([0,1,1,0], [1,1,1.02,1.02], facecolor='black') ax.scatter(xy(x_init)[0],xy(x_init)[1], s=50, marker='o', c="black") ax.plot(xy(Loes.y[0:3])[0], xy(Loes.y[0:3])[1],c="black",linewidth=2) # Erzeugung der nebenstehenden Farblegende colorbar cbar=plt.colorbar(strm.lines, ax=ax, aspect=20) cbar.set_label(r'$\left| \vec{v} \right| =\sqrt{{g_x}^2 + {g_y}^2}$',size=20) ax.set_xlim(0, 1) ax.set_ylim(-0.02, 1.02) # Berechnung und Terminalausgabe der Nashgleichgewichte equilibria = find_nash_equilibria(D) print("Nash-Gleichgewichte:") for eq in equilibria: if eq['type'] == 'pure': print(f"Reines: Spieler 1 Strategie {eq["s"][0]}, Spieler 2 Strategie {eq["s"][1]}") # Kennzeichnung der reinen Nashgleichgewichte im Bild if eq["s"][0] == 1 and eq["s"][1] == 1: plt.scatter(0,0, s=150, marker='h', c="red") if eq["s"][0] == 2 and eq["s"][1] == 2: plt.scatter(1,0, s=150, marker='h', c="red") if eq["s"][0] == 3 and eq["s"][1] == 3: plt.scatter(0.5,1, s=150, marker='h', c="red") else: print(f"Gemischtes: s*={eq["s*"]}") plt.scatter(xy(eq["s*"])[0],xy(eq["s*"])[1], s=100, marker='^', c="red") #Speichern der Bilder als (.png , benötigt dvipng unter Linux)- und .pdf-Datei saveFig="./2x3-Spiel.png" plt.savefig(saveFig, dpi=100,bbox_inches="tight",pad_inches=0.05,format="png") saveFig="./2x3-Spiel.pdf" plt.savefig(saveFig,bbox_inches="tight",pad_inches=0.05,format="pdf") plt.show() # Definition der Auszahlungsmatrizen der 19 Zeeman-Klassen der evolutionaeren symmetrischen (2x3)-Spiele D_1 = np.array([[0,2,-1],[-1,0,2],[2,-1,0]]) D_2 = np.array([[0,3,-1],[1,0,1],[3,-1,0]]) D_3 = np.array([[0,1,1],[-1,0,3],[1,1,0]]) D_4 = np.array([[0,6,-4],[-3,0,5],[-1,3,0]]) D_5 = np.array([[0,1,1],[1,0,1],[1,1,0]]) D_6 = np.array([[0,3,-1],[3,0,-1],[1,1,0]]) D_7 = np.array([[0,1,3],[-1,0,5],[1,3,0]]) D_8 = np.array([[0,1,-1],[-1,0,1],[-1,1,0]]) D_9 = np.array([[0,-1,3],[-1,0,3],[1,1,0]]) D_10 = np.array([[0,1,1],[-1,0,1],[-1,-1,0]]) D_11 = np.array([[0,1,1],[1,0,1],[-1,-1,0]]) D_12 = np.array([[0,1,-1],[1,0,1],[1,-1,0]]) D_13 = np.array([[0,-1,-1],[1,0,1],[-1,1,0]]) D_14 = np.array([[0,-1,1],[-1,0,1],[-1,-1,0]]) D_15 = np.array([[0,-1,-1],[1,0,-1],[-1,-1,0]]) D_16 = np.array([[0,-1,-1],[1,0,-3],[-1,-3,0]]) D_17 = np.array([[0,1,-1],[-3,0,1],[-1,1,0]]) D_18 = np.array([[0,1,-3],[1,0,-1],[-3,-1,0]]) D_19 = np.array([[0,-3,-1],[-3,0,-1],[-1,-1,0]]) #Festlegung der Anfangspopulation, Endzeit x_0 = [0.06, 0.9, 0.04] t_e = 15 # Loesen der DGL und plotten der Populationsentwicklung solve_and_plot(D_3, x_0, t_e)