Question

Question 2 (24 points) A Rebel Force of guerilla fighters seeks to inflict damage on a Con- ventional Army of the government, while the Conventional Army would like to destroy the Rebel Force. The two sides play a game in which each must decide whether to locate their forces in the Hills or in the Valley. The Rebels can inflict the most damage from the Valley, but if the Army is also in the Valley, it will be able to force the Rebels into open combat, where they are most vulnerable. By contrast, the Rebel Force is safer in the Hills, although it is also less capable of inflicting damage from that location. The payoff matrix for this game is shown below. Army Valley Hills Rebel Force Valley 1,4 4,1 Hills 3,2 2, 3 A. Is this a zero-sum game? Briefly explain. B. Assume the Rebel Force and the Army move simultaneously. Find the best-response rules for the Rebel Force and the Army when mixed strategies are considered. C. Depict the two best-response rules on one diagram. Find all (pure and mixed-strategy) Nash equilibria. D. For the mixed-strategy Nash equilibrium calculate the players' expected payoffs.

Answer 1

A) Zero-sum game is a situation when one person's gains are equivalent to another person's loss by the same amount, so the net benefit should be zero.

In this question, it is not a zero-sum game since the Rebel forces and Army who located their forces in the valley have obtained a payoff of 1 and 4 respectively. The net payoff is not zero.

Instead of a zero-sum game, it should be -4 and 4 (for instance) for the Rebel forces and Army who located their forces in the valley.

Similarly, you can figure for the rest of the cases, the net benefit is not zero.

☺ Army B) И V 1,4 4, I Rebel for 1- n И 3,2 2,3 Rebel forces expected payoff to the mised strategy (a,,da o p q li)+(1-2)(4)] + (1-6) [3(q) + 2(1-q)] þ [q+4-49] + Llap) [3q +2-2q] p [-3q +4 ] + (-p) [q+Q] 2p6, (0, 4₂) + (1-8) 6, (H, 2₂) B, (42) or Best response of Rebel force when Conventional army mining their strategy c.,). 1) if B, CV, K ₂) > E, (Tg & then, p=1 -39+ 4 > q+2 27 49 . B, (02) = 2 2.) if &, V, 22 )= E(T ,z) then, osphl q=b 3) if 8, ( Ugdz / < E, (T, Kzl then p=0 q> %

2 Conventional army's expected payoff to the mined strategy 14,9%) q [p (4) + (1-6) (2)] + (1-q) [p (1) + (1-P) (3)] q[up+2-2p]+(1-q)[$+3-2p] q [2p+] + (-ql [-2p+3] q (4,9U) + (-q) 62(d,,H) (1) if 8266,9U) > €218.gh) then q=1 2p+2 -2p+3 48 > B.(6,J = p> Yu e) if 62(6,9V) = E(Xigh) then ocq<I p=t 3) Jef Ez (d, gul <8₂(4,9 HI then q= p < Y4

c) 12 I b ) = f, g, ) mined strategy Wash equilibrium = (h. £) z I j p ll.pl q tq D) Players expected payoffs to p [3q +4] + (-p) [q+2] (á) [-3 (2) +47 + 27 (6+2] ↑ [3+4]+ 2 [ 2 ] 25/ [Rebel torney] Conventional carry =) q [2p+2]+(1-q) E-2p+3] [244}+23+ (1)[443] 512