I am a complete noob at game theory and trying to complete an exercise which asks to solve the extensive form of a game using backward induction. The game has players 1 and 2 with 2 possible actions each (1 has H and L, 2 has G and B). We have the payoff matrices and we also know that player 1 starts.
Here are the payoff matrices (each cell is formatted as "(payoff for player 1, payoff for player 2)"). Payoff matrices
Player 2
| | G | B |
|---|---------|---------|
| H | (5,4) | (-5,2) |
Player 1 |---|---------|---------|
| L | (2,-2) | (0,0) |
I was able to do this and find the Subgame-Perfect Nash equilibrium easily, and I also found a website (http://app.test.logos.bg/) that calculates it which gets the same result : < H >; < G, B >. SPNE of first scenario
Please tell me if my way of drawing the extensive form of this game is not the right one. It is possible that I reached the correct answer with a wrong resolution. Additionally, if there exists any better convention for writing SPNE's, I would appreciate any correction, because "G,B" looks ambiguous to me. I guess I want to say that player 2 should play "G" in a game following the equilibrium (which happens if player 1 plays H), but player 2 should play "B" if player 1 does not play according to the equilibrium.
Finally, my problem comes with the 2nd subexercise : we now have to consider that player 1 starts by deciding whether to play the game or not (I call these actions A for choosing to play and D for not playing). If he chooses D, the payoffs are 1 for both players. If he chooses to play, the game starts with player 2 playing first by choosing between G and B and then player 1 ends the game by choosing between H and L.
For me, backward induction on this game is possible as well and just results in the following SPNE : < A, H, L >; < G >, but once again that notation looks ambiguous to me. The move "A" is not a problem but "H,L" are ambiguous because H has to be played only in a game following the equilibrium path, otherwise L should be played.
Am I doing this right ?
Lastly, the reason I am sceptical about the second subexercise is because the website returns an error when I try to get the SPNE after drawing my tree. Here is a screenshot of the tree + the error in the top-right-corner : Error for 2nd exercise
Sorry if this post wasn't clear at all, it is my first ever on such forums. I can try and make it clearer if needed. Thank you in advance for your kind help !