- 1 Repeat Example 16 7 P 125 For A Conflict Given With Feasible Set H F F F F2 0 F 2f 4 V Example 1 1 (6.57 KiB) Viewed 50 times
- 1 Repeat Example 16 7 P 125 For A Conflict Given With Feasible Set H F F F F2 0 F 2f 4 V Example 1 2 (31.23 KiB) Viewed 50 times
- 1 Repeat Example 16 7 P 125 For A Conflict Given With Feasible Set H F F F F2 0 F 2f 4 V Example 1 3 (52.05 KiB) Viewed 50 times
- 1 Repeat Example 16 7 P 125 For A Conflict Given With Feasible Set H F F F F2 0 F 2f 4 V Example 1 4 (51.86 KiB) Viewed 50 times
V Example 16.7 Consider conflict with disagrement payoff vector (0,0) and feasible set H = {(f₁, f2)| f₁, f2 ≥ 0, f₂ ≤ 1- fi} 1. Nash solution 8(i)=1-f² 1 max (1 - - 0) (₁ - 0) s.to 0≤fi≤1 Objective fi-fi is zem at endpoints f₁ = 0 and f₁ = 1. Derivative: 1-3ff = f = fi = ≈ 0.5774, f2 = g(fi) = 3 So solution is (0.5774,0.6667). 1 H 213 ≈ 0.6667 125
2. Nonsymmetric Nash solution s. to 0≤ fi ≤ 1. By differentiation afi¹(1-f¹ + fi(1-a)(1-fi) (-2f₁) = 0 Divide by fi(1-fi) to have a(1-fi)-2fi(1-a) = 0 a-af-2ff +2ffa = 0 @= =fi(2-a) a 2 - 2a fi=2=h= = f₂=1-fi = 2-a 2-a In the special case of a-, the symmetric Nash solution is obtained: fi = = = 7 2- 3 f₂=1-f=1- 3. Area monotonic solution Total area: maximize fi(1-fi)¹-* f₁ [a₁-a² = [1 - f²) df - انت ته است || (f₁, 1-f²) 1 3 1 0 Cofio Game theory 126
80 fo 1, Newton's method: So solution is (0.60,0.64). 4. Equal sacrifice solution -انه -fi(1-fi) + (₁-³) = 6 - ² + [1-4]} f²) df (1-5), 2 1 + ² = A + {{{ A 3 3 hi-fi 2 2 3 3f₁-3f1 +4-6f₁ +2fi h=f³+3f-2 2 0, h'=3f² +3 h h' 1 2 6 0.6667 0.2964 4.3335 0.5983 0.0091 4.0739 0.5961 0.0001 4.0660 (1.1) 1 1-f² = 1- (1 - i) fi+f-1 = 0 fi f2 = 1 - fi So solution is (0.62,0.62). 5. Kalai-Smorodinsky solution fi = |||| = 1-f₁ -1+√1+4 −1+√5 2 2 = 0.62 S = 1, so f₁ (1) +0 -0 fi+fi-1 ≈ 0.62 0, same as above Game theory = 0 V 127