According to the following game tree, and if the entrant and incumbent both only care about their own monetary payoff, what is/are the game's Nash equilibrium?

Entrant - stays out (2,15)
- enters
If the entrant enters,
Incumbent - co operates (5,5)
- punishes (-3,9)

So what is the games nash equilibrium/equilibria, and what is the game's subgame perfect nash equilibrium/equlibria?

(I wasn't sure what was the difference too)

Thank you!

To find the Nash equilibrium in a game, you need to identify a strategy or a combination of strategies for each player where no player has an incentive to unilaterally deviate from their chosen strategy. In other words, no player can improve their outcome by changing their strategy while the other players keep theirs unchanged.

In this game, the entrant has two options: stay out or enter. The incumbent also has two options: cooperate or punish. Let's analyze these options to identify the Nash equilibrium.

First, let's consider the entrant's decision. If the entrant stays out, they receive a payoff of 15, and if they enter, they receive a payoff of 2. The entrant will choose to enter because 2 is better than 15. Thus, the entrant's strategy is to enter.

Next, let's consider the incumbent's decision, assuming the entrant enters. The incumbent has two options: cooperate (payoff of 5) or punish (payoff of 9). Since 9 is better than 5, the incumbent will choose to punish. Therefore, the incumbent's strategy is to punish.

Now, we have the following strategies:
- Entrant: Enter
- Incumbent: Punish

To determine if this strategy combination is a Nash equilibrium, we need to check if any player has an incentive to deviate from their chosen strategy.

If the entrant deviates from their strategy and stays out, they receive a payoff of 15, which is worse than the payoff of 2 if they stay with their current strategy. Therefore, the entrant has no incentive to deviate.

If the incumbent deviates from their strategy and cooperates instead of punishing, they receive a payoff of 5, which is worse than the payoff of 9 if they stay with their current strategy. Hence, the incumbent has no incentive to deviate.

Therefore, (Enter, Punish) is the Nash equilibrium for this game.

Now, let's talk about subgame perfect Nash equilibrium (SPNE). A subgame is a smaller part of the overall game that arises when a player makes a decision. In this game, the only subgame occurs when the entrant enters and the incumbent decides whether to cooperate or punish.

To find the SPNE, we need to analyze the subgame. In this subgame, the incumbent's dominant strategy is to punish because it leads to a higher payoff regardless of the entrant's decision. However, it is important to note that subgame perfect equilibrium only exists if the strategy chosen by the incumbent in the subgame is consistent with the overall strategic choice that ensures the highest combined payoff for both players.

In this case, the strategy combination (Enter, Punish) is also a subgame perfect Nash equilibrium because the incumbent's strategy to punish aligns with the overall chosen strategy in the game.

So, in summary:
- Nash Equilibrium: (Enter, Punish)
- Subgame Perfect Nash Equilibrium: (Enter, Punish)

Both the Nash equilibrium and the subgame perfect Nash equilibrium are (Enter, Punish) in this game.