Question

Two generals are preparing for a battle. Each must choose how many units to bring or whether they should just pass on the battle and not fight. The generals must make their decisions simultaneously. Each general can bring any number of units between 1 and 5, or he can pass. If either general passes or if they both bring the same number of units, then each general has a payoff of zero. If one general brings more units than the other, then the one with more units receives a payoff of 1 and the other general receives a payoff of -1.

1.)Write out the normal form of this game (It will be a 6x6 matrix).

2.)What are the pure strategy Nash equilibria of this game?

Answer 1

Given : Each general can bring any number of units between 1 and 5, or he can pass.

If one general brings more units than the other, then the one with more units receives a payoff of 1 and the other general receives a payoff of -1.

Based on the given facts in the question, we have the following matrix.

1. So the below matrix is the normal form :

	First general chooses
	Units	0(pass)	1	2	3	4	5
Second General Chooses	0(pass)	0	(-1, 1)	(-1, 1)	(-1, 1)	(-1, 1)	(-1, 1)
	1	(1, -1)	0	(-1, 1)	(-1, 1)	(-1, 1)	(-1, 1)
	2	(1, -1)	(1, -1)	0	(-1, 1)	(-1, 1)	(-1, 1)
	3	(1, -1)	(1, -1)	(1, -1)	0	(-1, 1)	(-1, 1)
	4	(1, -1)	(1, -1)	(1, -1)	(1, -1)	0	(-1, 1)
	5	(1, -1)	(1, -1)	(1, -1)	(1, -1)	(1, -1)	0

2. Now, we know that Nash equilibrium is a concept within game theory where the optimal outcome of a game is where there is no incentive to deviate from their initial strategy.

In simple words, if Nash equilibrium is the situation when even the opponent knows the other plan , he has no incentive to change his plan.

As we can see the colored part (yellow) in the matrix, when first general brings all 5 units , then second general also has to bring 5 units other wise first general will win.

So pure, Nash equilibrium in this game is when both general brings all 5 units.