Question

Two field generals, 1 and 2, are positioned to attack an enemy stronghold in the valley below. The two armies are separated by a treacherous canyon and communication between the two commanders is impossible. The generals know that if they coordinate their attack, both moving their troops into the valley at dawn, they will defeat the enemy, earning a value v > 0. If one of the two sides attacks alone, the other will not be able to re-enforce the attacker’s troops in time to prevent a defeat at the enemy’s hands, losing v. A stalemate where no general attacks is worth 0. Suppose all these elements of the strategic problem are common knowledge.

1. (5 pts) Solve for the (pure and mixed strategy) Nash equilibria.

2. (5 pts) Consider a modified version of this game. Suppose that General 1 must first decide whether or not to attack, and then General 2 decides whether or not to attack based upon General 1’s decision. Solve for the (pure) Nash equilibria and the Subgame Perfect Nash Equilibria (solve through backward induction).

Answer 1

1. To find pure strategy Nash equilibrium of this game we need to draw represent this game in normal form. Below I have drawn the normal form of the game in matrix form.

To find the pure strategy Nash equilibrium of this game we can use underlying the payoff method. In this method we underline the best strategy profile for each player when other player is playing a given strategy. There are two strategy profiles from which player can choose to play, either attack or don't attack.

Let's first underline the payoffs for general 1.

When general 2 chooses to attack, general one can either choose to attack and don't attack. If he attacks he gets payoff of V>0 and if he chooses to don't attack he gets the payoff of 0. So the best strategy for general 1 is to attack hence we underline V.

When general 2 chooses to don't attack. Again general 1 can choose either to attack or don't attack. If he chooses to attack he gets the payoff of -V and if he chooses to don't attack he gets the payoff of 0. So the best strategy profile for general 1 is to don't attack, and hence we underline 0.

Now let's underline the payoffs for general 2.

When general 1 chooses to attack, general 2 can either choose to attack or don't attack. If he chooses to attack he gets the payoff of V>0 and if he chooses to don't attack he gets the payoff of 0. So the best strategy for general 2 is to attack, and hence we underline V.

When general 1 chooses to don't attack. Again general 2 can choose to either attack or don't attack. If he chooses to attack he gets the payoff of -V and if he chooses don't attack he gets the payoff of 0. So the best strategy for general 2 in this case is don't attack and hence we underline 0.

The nash equilibrium strategy profile is one whose both payoff gets underlined. Here as you can see we have two pure strategy Nash equilibrium. One is when both attacks and another one is when both don't attack.

To find the mixed strategy Nash equilibrium of this game, we need to find the non zero probabilities with which each player plays both strategy profiles.

Let general 1 plays attack with probability P and don't attack with probability 1-p. Similarly let general 2 plays attack with probability q and don't attack with probability 1-q.

The purpose of each player is to randomize other player in playing both strategies. In other words each player chooses its probability as to make other player indifferent between playing either strategies.

General 1 will choose P such as it makes general 2 indifferent between playing attack and don't attack.

Payoff from attack = payoff from don't attack.

pV + (1-p) (-V) = p×0 + (1-p)×0.

pV + pV -V = 0

2pV = V

2p = 1

p = 1/2.

And 1-p = 1-1/2 =1/2.

Similarly general 2 will choose q such that it makes general 1 indifferent between playing attack and don't attack.

Payoff from attack = payoff from don't attack

qV + (1-q) (-V) = q×0 + (1-q)×0

qV + qV - V = 0

2qV = V

2q = 1

q= 1/2.

And 1-q = 1-1/2 = 1/2.

The mixed strategy Nash equilibrium of this will be (p, 1-p) and (q, 1-q) which we found above as (1/2, 1/2) and (1/2, 1/2).

2. In this there is a slight modification in the game, before both generals were choosing whether to attack or not simultaneously but now general 1 chooses first whether to attack or not then general 2 decides whether to attack or not.

In the above diagram I have shown who this game will be represented in sequential form. As you can see that general 1 first decides whether to attack or not and then general 2 will decides based on what general has choosen first.

We have to find the pure strategy Nash equilibrium of this game by using backward induction. In backward induction we start underlying the payoffs from backwards.

Let's see how its done. Let's underline the payoffs for general 2.

When general 1 plays attack. General two can choose either to attack or don't attack. If he chooses to attack he gets payoff of V and if he chooses to don't attack he gets 0. So the best for him is to choose attack and hence we underline attack.

When general 1 chooses not to attack. Again general 2 can choose either to attack or don't attack. If he chooses to attack he gets -V and if he choose to don't attack he gets 0. The best for him is to play don't attack and hence we underline 0.

And the game is reduced to only for general 1 to decide. And now general 1 can choose either to attack or don't attack. If he chooses to attack he gets V and if he chooses to don't attack he gets 0. So the best strategy for him is to play attack and get the payoff of V. Hence we underline V and the we get equilibrium of this game. ​​​​​​

The pure strategy Nash equilibrium of this game is (V, V) that is general 1 and general 2 chooses to attack.