Optimal Guessing Strategies in a Number Guessing Game

There are 24 permutations of the numbers 1 through 4:

'1 2 3 4' '1 2 4 3' '1 3 2 4' '1 3 4 2' '1 4 2 3' '1 4 3 2' '2 1 3 4' '2 1 4 3' '2 3 1 4' '2 3 4 1' '2 4 1 3' '2 4 3 1' '3 1 2 4' '3 1 4 2' '3 2 1 4' '3 2 4 1' '3 4 1 2' '3 4 2 1' '4 1 2 3' '4 1 3 2' '4 2 1 3' '4 2 3 1' '4 3 1 2' '4 3 2 1'

For each permutation, we can determine the strategy that P2 employs. For example, if the permutation is '1 2 3 4', then P2 can use the strategy of guessing in order: 1, 2, 3, 4. If the permutation is '1 3 2 4', then P2 can use the strategy of guessing in order: 1, 3, 2, 4. We can continue this process for all 24 permutations to determine the 14 different strategies that P2 can employ:

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

The payoff matrix is as follows, where the rows represent P1's number and the columns represent P2's strategy:

| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 12| 13| 14| |-------|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | 1 | 0 | -1| -2| -3| -1| -2| -1| -2| -3| -3| -2| -3| -2| -3| | 2 |-1 | 0| -1| -2| -2| -3| -1| -2| -3| -3| -3| -2| -2| -3| | 3 |-2 | -1| 0| -1| -3| -3| -2| -2| -1| -2| -3| -3| -2| -2| | 4 |-3 | -2| -1| 0| -2| -2| -3| -3| -3| -1| -2| -3| -3| -2|

Each cell represents the payoff for P2 given P1's number and P2's strategy. For example, if P1's number is 2 and P2 employs strategy 3 (guessing in order: 1, 3, 2, 4), then P2 will receive a payoff of -1 (losing one dollar) because the second guess was incorrect.