1. (Simultaneous vs. sequential moves) Two generals must choose between the actions Fight and not Fight, which we will label F and F. If one of the Generals chooses F while the other chooses F, then a war ensues and is won by the General who chooses F. If both Generals choose F once again a War ensues, but this time both Generals win with a probability of 1 /2 . The payoff for winning the war is 6 while the payoff for losing is 0. Finally, if both Generals choose then Peace ensues and both Generals receive a payoff of 4. However, General 1 has taken Prisoners of War (POWs) from General 2s army. If both Generals choose F these POWs will be released and General 2 will pay no additional cost. However, General 2 will pay an additional cost of -3 any time there is War, representing the idea that if at least one General chooses F these POWs will not be released (and may be mistreated...) (a) (5 points) Assume the Generals move simultaneously. Based on the payoffs above, build the 22 payoff matrix associated with the simultaneous move game. Find the games Nash Equilibrium. (Note that if both choose F, then they both win with probability 1 /2 , meaning that you need to calculate expected payoffs for the profile (F, F)) (b) (2 points) Now assume that General 1 moves first, and construct the Extensive Form Game Tree to represent the game. (c) (5 points) Use Backward Induction to find the games SPEs when General 1 moves first. (Tree with arrows is sufficient) (d) (2 points) Now assume that General 2 moves first, and construct the Extensive Form Game Tree to represent the game. (e) (5 points) Use Backward Induction to find the games Subgame Perfect Nash Equilibria when General 2 moves first. (Tree with arrows is sufficient) Now assume General 2 successfully rescues his or her hostages. The game remains exactly the same, except that General 2 no longer pays an additional cost of -3 whenever there is a War. (f) (5 points) For the game without POWs, assume the Generals move simultaneously and build the associated 22 payoff matrix. Find the Nash Equilibrium. (g) (2 points) Now assume that General 1 moves first, and construct the Extensive Form Game Tree to represent the game. (h) (5 points) Use Backward Induction to find the games Subgame Perfect Nash Equilibria when General 1 moves first. (Tree with arrows is sufficient) (e) (4 points) Write a short paragraph about why the SPE in (1c) is different from that in (1e) and (1h), and how is that difference related to the credibility of promises or the lack of commitment?.

1. 1. (Simultaneous vs. sequential moves) Two generals must choose between the actions Fight and
not Fight, which we will label F and F. If one of the Generals chooses F while the other chooses
F, then a war ensues and is won by the General who chooses F. If both Generals choose F once
again a War ensues, but this time both Generals win with a probability of 1 /2 . The payoff for
winning the war is 6 while the payoff for losing is 0. Finally, if both Generals choose then Peace
ensues and both Generals receive a payoff of 4. However, General 1 has taken Prisoners of War
(POWs) from General 2s army. If both Generals choose F these POWs will be released and
General 2 will pay no additional cost. However, General 2 will pay an additional cost of -3 any
time there is War, representing the idea that if at least one General chooses F these POWs will
not be released (and may be mistreated...)
(a) (5 points) Assume the Generals move simultaneously. Based on the payoffs above, build the
22 payoff matrix associated with the simultaneous move game. Find the games Nash
Equilibrium. (Note that if both choose F, then they both win with probability 1 /2 , meaning that
you need to calculate expected payoffs for the profile (F, F))
(b) (2 points) Now assume that General 1 moves first, and construct the Extensive Form Game
Tree to represent the game.
(c) (5 points) Use Backward Induction to find the games SPEs when General 1 moves first. (Tree
with arrows is sufficient)
(d) (2 points) Now assume that General 2 moves first, and construct the Extensive Form Game
Tree to represent the game.
(e) (5 points) Use Backward Induction to find the games Subgame Perfect Nash Equilibria when
General 2 moves first. (Tree with arrows is sufficient)
Now assume General 2 successfully rescues his or her hostages. The game remains exactly the
same, except that General 2 no longer pays an additional cost of -3 whenever there is a War.
(f) (5 points) For the game without POWs, assume the Generals move simultaneously and build
the associated 22 payoff matrix. Find the Nash Equilibrium.
(g) (2 points) Now assume that General 1 moves first, and construct the Extensive Form Game
Tree to represent the game.
(h) (5 points) Use Backward Induction to find the games Subgame Perfect Nash Equilibria when
General 1 moves first. (Tree with arrows is sufficient)
(e) (4 points) Write a short paragraph about why the SPE in (1c) is different from that in (1e) and
(1h), and how is that difference related to the credibility of promises or the lack of commitment?