4. 4 pts, class 22 in pitch perfect, the barden bellas are preparing for their next a cappella competition. lets say that beca and aubrey are having a disagreement about who gets to perform a solo. they decide to settle the argument by playing three rounds of rock, paper, scissors. in the first round, beca plays rock and loses to aubrey. if shes using the psychological strategies we discussed in class, what should beca play if she wants to beat aubrey in the second round? explain why.
#4) - 2: beca played rock and lost, which means aubrey played paper in the first round. so, shes likely to play paper again. what should beca play to beat aubreys paper?
becca should play paper because after losing to rock aubrey may think to reuse rock and has a better chance at beating aubrey because of that.
ec. (optional extra credit, +1 pt) in the big bang theory, sheldon and raj play an expanded version of rock, paper, scissors called rock, paper, scissors, lizard, spock, which adds two more options in order reduce the probability of ties. the rules are as follows:
- scissors cuts paper
- paper covers rock
- rock crushes lizard
- lizard poisons spock
- spock smashes scissors
- scissors decapitates lizard
- lizard eats paper
- paper disproves spock
- spock vaporizes rock
- (and as it always has) rock crushes scissors
if sheldon and raj pick the same option, they tie and their payoffs are both 0. if they pick different options, the winner’s payoff is 1 and the loser’s payoff is -1. use this information to complete the game matrix below.
raj
sheldon rock paper scissors lizard spock
rock
paper
scissors
lizard
spock

Question

Accepted Answer

### Part 4 (Rock, Paper, Scissors Dispute)
#### Brief Explanations:
In Rock, Paper, Scissors, the rules are: Rock crushes Scissors, Scissors cuts Paper, Paper covers Rock. In the first round, Beca played Rock and lost, so Aubrey must have played Paper (since Paper covers Rock). If Aubrey uses a psychological strategy (like repeating a choice she thinks worked or following a pattern), she may play Paper again. To beat Paper, the option that beats Paper is Scissors? Wait, no—wait, Paper is beaten by Scissors? Wait, no: Scissors cuts Paper, so Scissors beats Paper. Wait, but let's recheck: Rock loses to Paper (Paper covers Rock). So to beat Paper, you need Scissors? Wait, no—wait, the options are Rock, Paper, Scissors. So:

- Rock vs Paper: Paper wins (covers Rock)
- Paper vs Scissors: Scissors wins (cuts Paper)
- Scissors vs Rock: Rock wins (crushes Scissors)

So if Aubrey played Paper in round 1 (to beat Beca’s Rock), and if Aubrey is likely to repeat Paper (psychological strategy: sticking to a winning choice), then Beca needs to play Scissors to beat Paper (since Scissors cuts Paper). Wait, the original student answer said Paper, but that’s incorrect. Let's correct:

Step 1: Determine Aubrey’s first move. Beca played Rock and lost, so Aubrey’s move was Paper (because Paper beats Rock).
Step 2: Predict Aubrey’s second move. Using psychological strategy (e.g., “if it worked once, do it again”), Aubrey may play Paper again.
Step 3: Find the move that beats Paper. In Rock-Paper-Scissors, Scissors beats Paper (Scissors cuts Paper).

So Beca should play Scissors to beat Aubrey’s predicted Paper.

### Extra Credit (Rock, Paper, Scissors, Lizard, Spock Game Matrix)
To complete the game matrix, we analyze each of Sheldon’s moves (rows) against Raj’s moves (columns) using the rules:

#### Rules Recap:
- Scissors cuts Paper
- Paper covers Rock
- Rock crushes Lizard
- Lizard poisons Spock
- Spock smashes Scissors
- Scissors decapitates Lizard
- Lizard eats Paper
- Paper disproves Spock
- Spock vaporizes Rock
- Rock crushes Scissors

For each cell (Sheldon’s move, Raj’s move):
- If same: Payoff = 0 (tie)
- If different: Sheldon’s payoff is 1 (win) or -1 (lose) based on rules.

Let’s fill each row (Sheldon’s move) against each column (Raj’s move):

#### 1. Sheldon: Rock
- Raj: Rock → Tie → Payoff: 0
- Raj: Paper → Paper covers Rock → Raj wins → Sheldon’s payoff: -1
- Raj: Scissors → Rock crushes Scissors → Sheldon wins → Payoff: 1
- Raj: Lizard → Rock crushes Lizard → Sheldon wins → Payoff: 1
- Raj: Spock → Spock vaporizes Rock → Raj wins → Sheldon’s payoff: -1

#### 2. Sheldon: Paper
- Raj: Rock → Paper covers Rock → Sheldon wins → Payoff: 1
- Raj: Paper → Tie → Payoff: 0
- Raj: Scissors → Scissors cuts Paper → Raj wins → Sheldon’s payoff: -1
- Raj: Lizard → Lizard eats Paper → Raj wins → Sheldon’s payoff: -1
- Raj: Spock → Paper disproves Spock → Sheldon wins → Payoff: 1

#### 3. Sheldon: Scissors
- Raj: Rock → Rock crushes Scissors → Raj wins → Sheldon’s payoff: -1
- Raj: Paper → Scissors cuts Paper → Sheldon wins → Payoff: 1
- Raj: Scissors → Tie → Payoff: 0
- Raj: Lizard → Scissors decapitates Lizard → Sheldon wins → Payoff: 1
- Raj: Spock → Spock smashes Scissors → Raj wins → Sheldon’s payoff: -1

#### 4. Sheldon: Lizard
- Raj: Rock → Rock crushes Lizard → Raj wins → Sheldon’s payoff: -1
- Raj: Paper → Lizard eats Paper → Sheldon wins → Payoff: 1
- Raj: Scissors → Scissors decapitates Lizard → Raj wins → Sheldon’s payoff: -1
- Raj: Lizard → Tie → Payoff: 0
- Raj: Spock → Lizard poisons Spock → Sheldon wins → Payoff: 1

#### 5. Sheldon: Spock
- Raj: Rock → Spock vaporizes Rock → Sheldon wins → Payoff: 1
- Raj: Paper → Paper disproves Spock → Raj wins → Sheldon’s payoff: -1
- Raj: Scissors → Spock smashes Scissors → Sheldon wins → Payoff: 1
- Raj: Lizard → Lizard poisons Spock → Raj wins → Sheldon’s payoff: -1
- Raj: Spock → Tie → Payoff: 0

Now, let’s tabulate the matrix (rows: Sheldon’s move; columns: Raj’s move):

| Sheldon \ Raj | Rock | Paper | Scissors | Lizard | Spock |
|---------------|------|-------|----------|--------|-------|
| Rock          | 0    | -1    | 1        | 1      | -1    |
| Paper         | 1    | 0     | -1       | -1     | 1     |
| Scissors      | -1   | 1     | 0        | 1      | -1    |
| Lizard        | -1   | 1     | -1       | 0      | 1     |
| Spock         | 1    | -1    | 1        | -1     | 0     |

Let’s verify a few cells:
- Sheldon: Rock vs Raj: Lizard → Rock crushes Lizard → Sheldon wins (1) ✔️
- Sheldon: Lizard vs Raj: Paper → Lizard eats Paper → Sheldon wins (1) ✔️
- Sheldon: Spock vs Raj: Rock → Spock vaporizes Rock → Sheldon wins (1) ✔️
- Sheldon: Scissors vs Raj: Lizard → Scissors decapitates Lizard → Sheldon wins (1) ✔️

### Final Answers
#### Part 4 (Beca’s Move):
Beca should play **Scissors** to beat Aubrey’s predicted Paper (since Scissors cuts Paper).

#### Extra Credit (Game Matrix):
The completed matrix is as shown above. Each cell’s payoff (Sheldon’s) is filled based on the rules.

### Part 4 Correction (Original Student Mistake)
The original student answer was incorrect. The correct reasoning is:
- Aubrey’s first move: Paper (beats Rock).
- Predict Aubrey’s second move: Paper (psychological strategy: repeat winning move).
- Move to beat Paper: Scissors (Scissors cuts Paper).

### Final Answer for Part 4:
Beca should play Scissors.

### Final Answer for Extra Credit:
The completed game matrix is:

| Sheldon \ Raj | Rock | Paper | Scissors | Lizard | Spock |
|---------------|------|-------|----------|--------|-------|
| Rock          | 0    | -1    | 1        | 1      | -1    |
| Paper         | 1    | 0     | -1       | -1     | 1     |
| Scissors      | -1   | 1     | 0        | 1      | -1    |
| Lizard        | -1   | 1     | -1       | 0      | 1     |
| Spock         | 1    | -1    | 1        | -1     | 0     |

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QUESTION IMAGE

Question

Explanation:

Part 4 (Rock, Paper, Scissors Dispute)

Extra Credit (Rock, Paper, Scissors, Lizard, Spock Game Matrix)

Rules Recap:

1. Sheldon: Rock

2. Sheldon: Paper

3. Sheldon: Scissors

4. Sheldon: Lizard

5. Sheldon: Spock

Answer:

Extra Credit (Rock, Paper, Scissors, Lizard, Spock Game Matrix)

Rules Recap:

1. Sheldon: Rock

2. Sheldon: Paper

3. Sheldon: Scissors

4. Sheldon: Lizard

5. Sheldon: Spock

Final Answers

Part 4 (Beca’s Move):

Extra Credit (Game Matrix):

Part 4 Correction (Original Student Mistake)

Final Answer for Part 4:

Final Answer for Extra Credit: