Question

d a town, which has two r town is attacked by two divisions of army I, either one along each road or two along the same road. The defenders, ary II, can either put all three divisions to guard one road or two to guard one road and one the other. Whichever army has more divisions on a given road will command the road. If there are equal numbers of defenders road then half the time the defenders will hold the road and half the time the attackers will enter the city. List all strategies and calculate the payoff matrix of the attackers' chances of entering the town. and attackers on the same

Answer 1

There can be 3 strategies for attackers (army I),

1. One division attack on each road
2. Both division attack on road 1
3. Both division attack on road 2

There can be 4 strategies for defenders (army II),

1. All 3 divisions guard on road 1
2. All 3 divisions guard on road 2
3. Two divisions guard on road 1 and one division guard on road 2.
4. Two divisions guard on road 2 and one division guard on road 1.

The attackers enter the town whenever the number of divisions on any of the roads is higher than that of defenders. If the number of divisions are equal on both roads, the payoff will be 0.5.

For Strategy 1 of Attacker, (One division attack on each road), for Strategy 1 and 2 of Defenders one road will be empty, So, the probability to enter the town is 1. For Strategy 3 and 4 of Defenders, one road will have 2 divisions of defenders and other road have 1 division, so, the probability to enter the town is 0.5 from the road with 1 division of defenders.

Similarly, we can calculate the payoff matrix for other strategies.

The payoff matrix is,

Attacker \ Defender	Strategy 1	Strategy 2	Strategy 3	Strategy 4
Strategy 1	1	1	0.5	0.5
Strategy 2	0	1	0.5	1
Strategy 3	1	0	1	0.5