For the Warfare extensive game, consider the assessment (s, ) where

s(RE1) = Dont

s(RE2) = Day

s(RE3) = Day

s(GT1) = (1/2)Lake + (1/2)Forest

s(GT2) = Ambush

s(GT3) = Ambush

(RE1) = (Forest, Lake, Attack)

(RE2) = (Lake, Forest, Attack)

(GT1) = (1/2)(Forest) + (1/2)(Lake).

a. Show that (s, ) is sequentially rational.

b. Show that (s, ) satisfies

(h\I) = Pr(h reached given s used) / Pr(I reached given s used)

for each h in each information set I satisfying Pr(I reached given s used) > 0 in the original game G. Explain how this differs from the consistency of beliefs condition.

c. Explain why (s, ) is not a weak sequential equilibrium.

d. Discuss whether (s, ) is a reasonable solution.

The game tree and utility table is shown below.

Utility Terminal History Outcome (Forest, Forest) RE loses a battle to raid the village (Lake, Lake) (Don't Attack) Status quo for GT; RE a future threat (Forest, Lake, Ambush, Day) RE raids village but GT wins a battle (Lake, Forest, Ambush, Day) (Forest, Lake, Attack, Night) RE raids village and both lose a battle (Lake, Forest, Attack, Night) (Forest, Lake, Ambush, Night) RE raids village & leaves without casualties (Forest, Lake, Attack, Day) (Lake, Forest, Ambush, Night) (Lake, Forest, Attack, Day) 2 2 1 1 1 0 0 -2 -2 -2 11 11 Day Don't 1 - a - b (-1,1) (0,1) 1-c Ambush 1-f 1 1 9 Night (2,-2) C Lake GT2 RE2 1-e 1 Forest Day (2,-2) 1 al 1-C 1 1 1 Attack f 9 Forest Night (-2,2) (1,0) e RE1 C I GT1 Lake 1 (-2,2) Day d (0,1) 1- e Ambush 1 1 1 1-9 r Lake Night d (2,-2) b 1 1 p GT3 Forest RE3 Day (2,-2) 1-d Attack 9 1 1 (RE, GT) Night d (1,0) Utility Terminal History Outcome (Forest, Forest) RE loses a battle to raid the village (Lake, Lake) (Don't Attack) Status quo for GT; RE a future threat (Forest, Lake, Ambush, Day) RE raids village but GT wins a battle (Lake, Forest, Ambush, Day) (Forest, Lake, Attack, Night) RE raids village and both lose a battle (Lake, Forest, Attack, Night) (Forest, Lake, Ambush, Night) RE raids village & leaves without casualties (Forest, Lake, Attack, Day) (Lake, Forest, Ambush, Night) (Lake, Forest, Attack, Day) 2 2 1 1 1 0 0 -2 -2 -2 11 11 Day Don't 1 - a - b (-1,1) (0,1) 1-c Ambush 1-f 1 1 9 Night (2,-2) C Lake GT2 RE2 1-e 1 Forest Day (2,-2) 1 al 1-C 1 1 1 Attack f 9 Forest Night (-2,2) (1,0) e RE1 C I GT1 Lake 1 (-2,2) Day d (0,1) 1- e Ambush 1 1 1 1-9 r Lake Night d (2,-2) b 1 1 p GT3 Forest RE3 Day (2,-2) 1-d Attack 9 1 1 (RE, GT) Night d (1,0)