Question

Represent the following strategic situation as an extensive-form game. Janet is a contestant on a popular game show, and her task is to guess behind which door Liz, another contestant, is standing. With Janet out of the room, Liz chooses a door behind which to stand-either door A or door B. The host, Monty, observes this choice. Janet, not having observed Liz's choice, then enters the room. Monty says to Janet either "Red" or "Green" (which sounds silly, of course, but it is a silly game show). After hearing Monty's statement, Janet picks a door (she says either "A" or "B"). If she picks the correct door, then she wins \(\$ 100\). If she picks the wrong door, then she wins nothing. Liz wins \(\$ 100\) if Janet picks the wrong door and nothing if she picks the correct door. (Thus, Liz would like to hide from Janet, and Janet would like to find Liz.) Monty likes the letter A. If Janet selects door A, then this selection makes Monty happy to the tune of 10 units of utility. If she selects door B, then Monty receives 0 utility units.

Short answer

Represent the situation with a tree where Liz chooses first, followed by Monty's remark, and finally Janet's decision, showing all payoffs.

Step-by-step solution

Identify Players and Choices

In this game, there are three players: Janet, Liz, and Monty. Liz makes the first move by choosing door A or door B. Then, Monty observes Liz's choice and announces either "Red" or "Green" to Janet, though his announcement is unrelated to Liz's actual choice. Finally, Janet, not having any indication of Liz's choice, picks door A or door B.

Determine Payoffs for Each Player

The payoffs depend on Janet's choice aligned with Liz's hiding strategy. If Janet picks the correct door, she wins \(\\(100\), and Liz wins nothing. If Janet picks the wrong door, Liz wins \(\\)100\) and Janet wins nothing. Additionally, Monty's utility depends on Janet's final decision: he gains 10 utility points if she picks door A and 0 if she picks door B.

Construct the Game Tree

Draw the extensive-form game tree. Start with Liz's decision node where she chooses between door A and door B. After Liz's choice, Monty observes and moves next. His choice ("Red" or "Green") doesn’t change based on Liz’s hiding spot but needs to be shown in branches under each of Liz’s initial decisions. Finally, Janet has two decision nodes under each of Monty's announcements, where she chooses either door A or B.

Assign Outcomes to Each Path

For each sequence of choices (Liz's initial position, Monty's announcement, Janet's final guess), assign the payoffs: (Janet's payoff, Liz's payoff, Monty's payoff). For example, if Liz stands behind door A, Monty says "Red", and Janet picks door A, then the outcome is (100, 0, 10).

Analyze Strategies

Identify the best strategies for each player given the potential moves of others. Liz wants to hide effectively, Monty’s aim influences Janet’s choice subtly without directly informing her, and Janet needs to make a best guess without accurate information about Liz’s location.

Key concepts

Extensive-form Game

In game theory, an extensive-form game is a representation of a strategic scenario where decisions are made in a sequential manner. This structure allows us to visualize the order of players’ moves, their available actions at each step, and the resulting outcomes. In the provided scenario, we observe three players: Janet, Liz, and Monty. Each player makes decisions in a specific sequence, beginning with Liz. This type of game is helpful in analyzing scenarios where timing and the order of actions impact the outcomes substantially. Extensive-form games are typically depicted using a game tree, showing all possible decision points (called decision nodes) and paths leading to the possible payoffs.

Player Strategies

Player strategies are the plans or actions each player takes within the game to achieve their highest possible payoff. In our game, each participant has their agenda. Liz makes a choice first, deciding between door A or B, aiming to stay hidden from Janet to maximize her payoff.
Monty makes the second move, but his choice of saying 'Red' or 'Green' remains unrelated to Liz’s initial action, acting more as a narrative tactic rather than a strategic play. He hopes Janet chooses door A to maximize his utility. Lastly, Janet makes her move with the strategy of maximizing her winnings by guessing correctly. She needs to predict Liz’s move and balance it with Monty’s seemingly random indication. Understanding the strategy of each player helps to predict the game’s outcome and form appropriate responses.

Payoff Structure

The payoff structure in a game tells us what each player will earn from the game's possible outcomes. For Janet, the goal is to maximize her winnings. Successfully identifying Liz’s location yields a $100 payoff, while an incorrect guess results in no winnings. Liz, on the contrary, wins $100 if Janet fails to find her. This tug-of-war shapes their opposing strategies.

• Janet’s correct pick: (100, 0, utility for Monty).
• Janet’s wrong pick: (0, 100, utility for Monty).

Monty's payoff is linked to Janet's final choice: he receives 10 units of happiness if she picks door A and none for door B. The cumulative payoff structure allows us to evaluate each player’s incentives and results.

Decision Nodes

Decision nodes are crucial elements in extensive-form games, representing points at which players make decisions. The game tree starts with Liz’s decision node, as she chooses to hide behind either door A or B. Following Liz’s decision, Monty reaches his decision node. At his point in the game, he decides to say either "Red" or "Green." However, since his announcement is not tied to her actual hiding place, this decision affects the game's flow without revealing information.
Finally, Janet faces her decision nodes under possible scenarios, having to choose either door A or B based on Monty's hint. Decision nodes help in mapping the game’s progression and provide insight into the available strategies by branching out into potential outcomes, which is essential for planning strategic moves.