Andy, Brad, and Carly are playing a new online video game: Zombie Civil War. Each has an army of 100 zombies and must decide how to allocate them to battle each of the other two players’ armies. Three simultaneous battles are occurring: one between Andy and Brad, one between Andy and Carly, and one between Brad and Carly. Let Ab denote how many zombie soldiers Andy allocates to his battle with Brad, with the remaining 100 - Ab soldiers in Andy’s zombie army assigned to the battle with Carly. Bc denotes the number of zombie soldiers that Brad allocates to his battle with Carly, and 100 - Bc zombies go to his battle with Andy. Ca is the the number of zombie soldiers that Carly allocates to the battle with Andy, and 100 - Ca in her battle with Brad. To see how payoffs are determined, consider Andy. If Ab > 100 - Bc, so that Andy has more zombies than Brad in the Andy-Brad battle, then Andy wins the battle and receives w points where w > 2. If Ab = 100 - Bc, so that Andy and Brad have the same number of zombies in their battle, then it is a tie and Andy earns 2 points. If Ab < 100 - Bc, so that Andy has fewer zombies when battling Brad, then Andy loses the battle and receives 0 points. Payoffs are similarly determined for Brad and Carly. (This determination of payoffs can be thought of as resulting from the following process: Each zombie army on the battlefield forms a line and the two lines face each other. The zombies in the front of the line attack and eat each other until both are destroyed. The winner of the battle is the one with at least one zombie still standing on the battlefield. Sadly, this description is not too different from how wars were fought in much of the eighteenth and nineteenth centuries.) The payoff to a player is the sum of points from the two battles in which he or she is engaged. A strategy is an allocation of all 100 zombie soldiers, so a strategy profile can be presented as a triple, (Ab,Bc,Ca), with the remaining soldiers used in a player’s other battle. In answering these questions, note that the answer may depend on the value of w.
a. Is (Ab,Bc,Ca) = (50,50,50) a Nash equilibrium?
b. Is (Ab,Bc,Ca) = (70,70,70) a Nash equilibrium?
c. Is (Ab,Bc,Ca) = (40,60,50) a Nash equilibrium?
d. Consider (Ab,Bc,Ca) = (x,x,x). Find values for x such that this is an NE.
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