Game Theory: Analyzing Strategies in a Number Guessing Game

Strategies for Player 2 in a Number Guessing Game

Imagine a game where Player 1 secretly chooses a number between 1 and 4. Player 2 then attempts to guess this number, losing $1 for each incorrect guess. After each guess, Player 1 informs Player 2 whether the guess was correct, too high, or too low.

This article analyzes the various strategies Player 2 can employ in this game. We begin by listing all possible strategies and then use the Z2 group of symmetries to reduce the complexity of the payoff matrix.

Strategies for Player 2:

1. Guess 1, 2, 3, 4 in order
2. Guess 4, 3, 2, 1 in order
3. Guess randomly
4. Guess 2, 1, 4, 3 in order
5. Guess 3, 1, 2, 4 in order
6. Guess 1, 3, 2, 4 in order
7. Guess 3, 4, 1, 2 in order
8. Guess 2, 4, 1, 3 in order
9. Guess 4, 2, 1, 3 in order
10. Guess 1, 4, 3, 2 in order
11. Guess 2, 3, 4, 1 in order
12. Guess 4, 1, 3, 2 in order
13. Guess 3, 2, 1, 4 in order
14. Guess 1, 2, 4, 3 in order

Payoff Matrix:

| Strategy | Guesses | Payoff if correct | Payoff if incorrect | Total payoff | |---|---|---|---|---| | 1 | 1, 2, 3, 4 | 3 | -3 | | | 2 | 4, 3, 2, 1 | 3 | -3 | | | 3 | Random | 3 | -3 | | | 4 | 2, 1, 4, 3 | 3 | -3 | | | 5 | 3, 1, 2, 4 | 3 | -3 | | | 6 | 1, 3, 2, 4 | 3 | -3 | | | 7 | 3, 4, 1, 2 | 3 | -3 | | | 8 | 2, 4, 1, 3 | 3 | -3 | | | 9 | 4, 2, 1, 3 | 3 | -3 | | | 10 | 1, 4, 3, 2 | 3 | -3 | | | 11 | 2, 3, 4, 1 | 3 | -3 | | | 12 | 4, 1, 3, 2 | 3 | -3 | | | 13 | 3, 2, 1, 4 | 3 | -3 | | | 14 | 1, 2, 4, 3 | 3 | -3 | |

Using the Z2 Group of Symmetries

The Z2 group consists of two elements: the identity permutation 'e' (which keeps the order of guesses the same) and the permutation 'σ' (which reverses the order of guesses). We can use these symmetries to group the strategies into orbits.

| Strategy | Guesses | Payoff if correct | Payoff if incorrect | Total payoff | Z2 Orbit | Z2-stabilizer | Reduced payoff | |---|---|---|---|---|---|---|---| | 1 | 1, 2, 3, 4 | 3 | -3 | | {1} | {e} | | | 2 | 4, 3, 2, 1 | 3 | -3 | | {2} | {σ} | | | 3 | Random | 3 | -3 | | {3} | {e, σ} | | | 4 | 2, 1, 4, 3 | 3 | -3 | | {4, 10} | {e} | | | 5 | 3, 1, 2, 4 | 3 | -3 | | {5, 6, 9, 12} | {e} | | | 7 | 3, 4, 1, 2 | 3 | -3 | | {7, 11, 13} | {σ} | | | 8 | 2, 4, 1, 3 | 3 | -3 | | {8} | {σ} | |

Reduced Payoff Matrix

By averaging the payoffs within each orbit, we can construct a reduced payoff matrix:

| Orbit | Guesses | Payoff if correct | Payoff if incorrect | Total payoff | |---|---|---|---|---| | {1} | 1, 2, 3, 4 | 3 | -3 | | | {2} | 4, 3, 2, 1 | 3 | -3 | | | {3} | Random | 3 | -3 | | | {4, 10} | 2, 1, 4, 3 | 3 | -3 | | | {5, 6, 9, 12} | 1, 3, 2, 4 | 3 | -3 | | | {7, 11, 13} | 3, 4, 1, 2 | 3 | -3 | | | {8} | 2, 4, 1, 3 | 3 | -3 | |

This reduced matrix provides a clearer understanding of the strategic landscape in this number guessing game. We can now analyze the payoffs for each strategy group to determine the optimal strategy for Player 2.