Optimal Strategies for Guessing a Number 1-4: A Game Theory Analysis

This article explores the strategies available to a player (P2) attempting to guess a number between 1 and 4 chosen by another player (P1). P2 makes repeated guesses until they correctly identify the number, incurring a $1 penalty for each incorrect guess. After each guess, P1 informs P2 if the guess was correct, too high, or too low.

Identifying Strategies

We can list all 14 possible strategies P2 can employ by considering permutations of the numbers 1 through 4:

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

Payoff Matrix

The payoff matrix represents the outcome for each strategy based on the number chosen by P1:

| Strategy \ Outcome | Correct | Too High | Too Low | |---|---|---|---| | Guess 1, 2, 3, 4 | 4 | -3 | -3 | | Guess 4, 3, 2, 1 | 4 | -3 | -3 | | Guess 1, 3, 2, 4 | 2 | -1 | -1 | | Guess 4, 2, 3, 1 | 2 | -1 | -1 | | Guess 1, 4, 2, 3 | 2 | -1 | -1 | | Guess 3, 2, 1, 4 | 2 | -1 | -1 | | Guess 2, 1, 4, 3 | 2 | -1 | -1 | | Guess 3, 1, 4, 2 | 2 | -1 | -1 | | Guess 2, 4, 1, 3 | 2 | -1 | -1 | | Guess 3, 4, 1, 2 | 2 | -1 | -1 | | Guess 2, 3, 1, 4 | 2 | -1 | -1 | | Guess 4, 1, 3, 2 | 2 | -1 | -1 | | Guess 2, 4, 3, 1 | 1 | -2 | -2 | | Guess 3, 1, 2, 4 | 1 | -2 | -2 |

Reducing the Payoff Matrix

We can utilize the Z2 = {e, σ : σ^2 = e} group of symmetries to simplify the payoff matrix. This group represents a symmetry where a strategy is equivalent to its reverse. For instance, strategies 1 and 2 are symmetric, as are strategies 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Strategy 13 and 14 are their own symmetries.

We can combine the payoff values for symmetric strategies, reducing the matrix to:

| Strategy \ Outcome | Correct | Too High | Too Low | |---|---|---|---| | Guess 1, 2, 3, 4 | 4 | -3 | -3 | | Guess 1, 3, 2, 4 | 2 | -1 | -1 | | Guess 2, 4, 3, 1 | 1 | -2 | -2 | | Guess 3, 1, 2, 4 | 1 | -2 | -2 | | Guess 4, 1, 3, 2 | 2 | -1 | -1 |

This reduced matrix provides a clearer picture of the relative effectiveness of different strategies. For instance, strategies 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 are all symmetric and perform similarly, while strategies 13 and 14 are less effective but are their own symmetries.