Signaling Game: Beer and Quiche Payoffs and Extensive Form Representation

Question

I am working on a signaling game scenario named "Beer and Quiche" and need some assistance with understanding the payoffs and representing the game in an extensive form.

In this game, there are two types of senders: type 1 ("coward") with a probability of 0.1, and type 2 ("bad-tempered") with a probability of 0.9. The sender's message corresponds to their choice of breakfast, either beer or quiche. The receiver's action involves deciding whether to engage in a duel with the sender.

The payoffs have the following characteristics:

• Both sender types receive a payoff of 1 for having their preferred breakfast.
• Both sender types receive an additional payoff of 2 for avoiding the duel.
• The receiver's payoffs are 1 and -1 for dueling with the coward and bad-tempered sender types, respectively.
• All other payoffs are zero.

I am struggling with two main aspects:

1. Understanding Payoffs: I'm having trouble grasping the interplay between the sender and receiver payoffs based on their choices and actions.

2. Extensive Form Representation: I'm unsure how to accurately represent this game in an extensive form diagram. I would appreciate guidance on how to properly structure the game tree considering the sender's types, messages, receiver's actions, and associated payoffs.

Could someone provide a step-by-step explanation of the payoffs' implications in the context of the game and help me create a clear extensive form representation? Any insights, explanations, or diagrams would be greatly appreciated. Thank you!

Edit:

Is this extensive representation matching the criteria:

Sender
├── Coward (0.1)
│   ├── Beer
│   │   ├── Duel (0, 1)
│   │   └── No Duel (3, 0)
│   └── Quiche
│       ├── Duel (1, 1)
│       └── No Duel (4, 0)
└── Bad tempered (0.9)
    ├── Beer
    │   ├── Duel (1, -1)
    │   └── No Duel (4, 0)
    └── Quiche
        ├── Duel (0, -1)
        └── No Duel (3, 0)