Question

Suppose Andy and Brian play a guessing game. There are two slips of paper; one is black and the other is white. Each player has one of the slips of paper pinned to his back. Neither of the players observes which of the two slips is pinned to his own back. (Assume that nature puts the black slip on each player's back with probability \(1 / 2\).) The players are arranged so that Andy can see the slip on Brian's back, but Brian sees neither his own slip nor Andy's slip. After nature's decision, the players interact as follows. First, Andy chooses between \(\mathrm{Y}\) and \(\mathrm{N}\). If he selects \(\mathrm{Y}\), then the game ends; in this case, Brian gets a payoff of 0 , and Andy obtains 10 if Andy's slip is black and \(-10\) if his slip is white. If Andy selects \(\mathrm{N}\), then it becomes Brian's turn to move. Brian chooses between \(\mathrm{Y}\) and \(\mathrm{N}\), ending the game. If Brian says \(\mathrm{Y}\) and Brian's slip is black, then he obtains 10 and Andy obtains 0 . If Brian chooses \(Y\) and the white slip is on his back, then he gets \(-10\) and Andy gets 0 . If Brian chooses \(\mathrm{N}\), then both players obtain 0 . (a) Represent this game in the extensive form. (b) Draw the Bayesian normal-form matrix of this game.

Short answer

The game tree and Bayesian matrix show each player's payoffs for strategy combinations.

Step-by-step solution

Understand the Game

First, we need to understand the roles and the decision points of both Andy and Brian. Andy can see Brian's slip and must decide between \( Y \) and \( N \). If Andy chooses \( Y \), the game ends immediately with payoffs depending on Andy's own slip color. If Andy chooses \( N \), Brian then decides between his own \( Y \) and \( N \), where payoffs for \( Y \) depend on Brian's slip color. If both choose \( N \), the payoff is 0 for both.

Create the Extensive Form

The extensive form is a game tree describing each player's decision at each stage. 1. Start with Nature's move, placing either a Black or White slip with probability \( \frac{1}{2} \) each, for both Andy and Brian.2. From each initial Nature move, Andy then makes his decision to choose \( Y \) or \( N \).3. If Andy chooses \( Y \), the game ends with payoff depending on Andy's slip color. If \( N \,\) proceeds to Brian's move.4. For Brian's move, show both possible slip color scenarios and his decision between \( Y \) and \( N \).5. If Brian chooses \( Y \), the payoff depends on his slip color; if \( N \), both get 0.

Construct Bayesian Normal Form

Convert the extensive form into a normal form matrix. Since we have incomplete information (each doesn't know their own slip color), treat it as a Bayesian game:- Each player has two possible types based on slip color (Black or White).- Write down each player's strategies: \( (Y,N) \) for Andy and \( (Y,N) \) for Brian.- Calculate payoffs for each combination of strategies and types using expected values, considering slip color probabilities (1/2). Fill the matrix with these results.

Populate the Payoff Values

For each possible strategy profile, calculate the payoffs:- Andy knows Brian's slip, so his decision impacts both players' payoffs. Assign payoffs like "10 or -10 for Andy and 0 for Brian" if Andy chooses \( Y \).- If Andy opts \( N \,\) consider Brian's likely choices - determine payoffs "0 for both" if Brian chooses \( N \) or "dependent on Brian's slip" if choosing \( Y \).- Capture these payoff results into the Bayesian matrix.

Key concepts

Extensive Form Representation

The extensive form representation is a way to visually depict a strategic game using a tree diagram. Each node represents a point of decision or chance, and each line (or branch) from a node maps out possible moves or choices. In the game setup with Andy and Brian, it's critical to understand how these choices play out graphically.
- **Nature's Move:** At the start, Nature decides which slips are on Andy's and Brian's backs, with equal probability for black or white slips. This is not visible to the players initially.
- **Andy's Decision:** The next decision point in the tree comes when Andy chooses between 'Y' (ending the game) or 'N' (passing the decision to Brian). The payouts from 'Y' depend on Andy's slip color.
- **Brian's Decision:** If the game continues to Brian, he will need to choose 'Y' or 'N' with partial information. This choice branches again, with the resulting payoffs dependent on Brian's slip color.
This step-by-step formation helps in visualizing the sequential nature of decisions and outcomes, clearly showing where uncertainties and strategic decisions occur.

Bayesian Games

Bayesian games introduce elements of incomplete information where players have limited information about other players, including their types. Types refer to different possible characteristics or private information that a player might have, which affects their payoff. In this guessing game:

• Each player's type is determined by the color of the slip on their back, black or white.
• Andy has the advantage of seeing Brian's slip, so his strategies adapt to Brian's type.
• Brian cannot see Andy's slip or his own, making his strategic thinking more complex, requiring him to make decisions based on probability and expectations.

This game becomes a Bayesian game because players manage their actions without full knowledge of both their own type and the type of the opponent.

Decision-Making Process

The decision-making process in game theory involves analyzing potential outcomes for each possible choice at every decision point. Let's break it down based on our scenario with Andy and Brian.
**Andy's Process:**
- **Observation:** Andy sees Brian's slip.
- **Action:** Andy chooses 'Y' to end the game or 'N' to pass the decision to Brian. He considers the slip color and potential payoffs when deciding.
**Brian's Process:**
- **Lack of Information:** Brian doesn't know either slip color, increasing reliance on probability.
- **Action:** When it's his turn, he decides between 'Y' (risking a high gain or loss) or 'N' (guaranteeing no gain/loss), based on expected outcomes from his incomplete information.
Each player's process is marked by forward-thinking strategy that anticipates and minimizes risks, even in the face of missing information.

Payoff Matrix

A payoff matrix is a table that helps to list all possible outcomes of a game, considering the strategies of all players and their respective payoffs. When transforming a game from extensive form to normal (matrix) form in the context of Bayesian games, important considerations are made:

• **Players' Strategies:** The matrix reflects strategies like `(Y, N)` for Andy and `(Y, N)` for Brian, across all possible combinations of slip colors.
• **Outcome Scenarios:** Each cell in the matrix corresponds to a specific strategy combination and shows the respective payoffs like "10 or -10 for Andy, dependent on his decision of 'Y' or 'N'."
• **Probabilities:** Incorporate the slip color probabilities to calculate expected payoffs, given each player's potential decision.
  This matrix serves as a critical tool that synthesizes the game's complex decisions and randomness into a format that allows for clearer strategic evaluation and decision-making.