Armies 1 and 2 are fighting over an island. Analyze this situation as an extensive game and, using the notion of subgame perfect equilibrium, predict the winner as a function of k and l.
a. Army 1 wins if k > l
b. Army 2 wins if k > l
c. Army 1 wins if k < l
d. Army 2 wins if k < l
The extensive game can be represented as follows:
- Army 1 chooses whether to fight or retreat.
- If Army 1 fights, then Army 2 chooses whether to fight or retreat.
- If both armies fight, the outcome is determined by the relation of k and l.
- If k > l, Army 1 wins.
- If k < l, Army 2 wins.
- If Army 1 retreats, Army 2 wins.
In this game, Army 1 has the first-mover advantage and can choose whether to fight or retreat. If Army 1 chooses to fight, then Army 2 can decide whether to fight or retreat based on the knowledge of k and l. If both armies fight, the outcome depends on the comparison of k and l.
In a subgame perfect equilibrium, we need to look at the entire game tree and ensure that at every possible decision point, players are making their best response to each other's actions.
Considering the given options, we can see that the correct prediction would be:
a. Army 1 wins if k > l
This is because if Army 1 chooses to fight, then Army 2's best response, if k > l, would be to retreat as they would lose in a direct confrontation. Therefore, in the subgame perfect equilibrium, Army 1 would win if k > l.