Analysis Stratego
The game where higher cards beat lower cards, except 1 against 5, which is a tie (both cards are replaced)
Solution of the 1,2,4,5 Game: In the first move, play 1 with probability 0.3268, play never 2, Play 4 with probability 0.2693,
and play 5 with probability 0.4039.
For the 1,2,3,4,5 game we need games beyond 3*n. This is done, using
an online applet. Here is the data:
| 1234 | 1235 | 1245 | 1345 | 2345 |
| 1234 | 0 | -0.25 | -0.431 | -0.5 | -0.75 |
| 1235 | 0.25 | 0 | -0.573 | -0.643 | -0.318 |
| 1245 | 0.431 | 0.573 | 0 | -0.454 | -0.161 |
| 1345 | 0.5 | 0.643 | 0.454 | 0 | 0 |
| 2345 | 0.75 | 0.318 | 0.161 | 0 | 0 |
|
| 1234 | 1235 | 1245 | 1345 | 2345 |
| 12345 | 1 | 0.749 | 0.602 | 0.494 | 1 |
|
| 1 | 2 | 3 | 4 | 5 |
| 1 | 0 | -1 | -1 | -1 | 0.75 |
| 2 | 1 | 0 | -0.494 | -0.494 | -0.494 |
| 3 | 1 | 0.494 | 0 | -0.602 | -0.602 |
| 4 | 1 | 0.494 | 0.602 | 0 | -0.749 |
| 5 | -0.75 | 0.494 | 0.602 | 0.749 | 0 |
|
Solution of the 1,2,3,4,5 Game: At the beginning, play
5 with probability 0.4,
1 or 4 both with probability 0.3,
but never play 2 or 3 at the beginning.
May 23-25, 2007
Complete Strategy:
| Ann | Beth |
1: | 2: | 3: | 4: | 5: |
--- |
1: | 2: | 3: | 4: | 5: --- |
exp |
| 12345 | 12345 |
0.3 | | | 0.3 | 0.4 |
|
0.3 | | | 0.3 | 0.4 |
0 |
|
| 12345 | 1235 |
0.15 | | 0.59 | | 0.26 |
|
1: | 2: | 3: | 4: | 5: |
0.749 |
| 12345 | 1245 |
| 0.65 | | 0.03 | 0.32 |
|
1: | 2: | 3: | 4: | 5: |
0.602 |
| 12345 | 1345 |
0.31 | | | 0.31 | 0.38 |
|
1: | 2: | 3: | 4: | 5: |
0.494 |
|
| 12345 | 125 |
0.2 | 0.58 | | | 0.22 |
|
1: | 2: | 3: | 4: | 5: |
0.893 |
| 12345 | 135 |
0.25 | | 0.52 | | 0.23 |
|
1: | 2: | 3: | 4: | 5: |
0.85 |
| 12345 | 145 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
0.72 |
|
| 12345 | 15 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
1 |
| Your Hand | Opponent |
1: | 2: | 3: | 4: | 5: |
--- |
1: | 2: | 3: | 4: | 5: --- |
exp |
| 1234 | 1234 |
| | | 1 | |
|
| | | 1 | |
0 |
| 1234 | 2345 |
0.25 | | | 0.75 | |
|
| | | 0.75 | 0.25 |
-0.75 |
| 1235 | 1235 |
0.3268 | | 0.2693 | | 0.4039 |
|
0.3268 | | 0.2693 | | 0.4039 |
0 |
| 1235 | 1245 |
0.2542 | | 0.5702 | | 0.1756 |
|
| 0.6184 | | 0.0403 | 0.3413 |
-0.573 |
| 1235 | 1345 |
0.2417 | | 0.4463 | | 0.3121 |
|
0.0631 | | 0.6241 | | 0.3128 |
-0.643 |
| 1235 | 2345 |
0.1478 | 0.0418 | 0.6626 | | 0.1478 |
|
| 0.4289 | 0.1067 | 0.1235 | 0.3409 |
-0.318 |
| 1245 | 1245 |
0.3268 | | | 0.2693 | 0.4039 |
|
0.3268 | | | 0.2693 | 0.4039 |
0 |
| 1245 | 1345 |
0.2561 | 0.5313 | | 0.0154 | 0.1973 |
|
0.2676 | | 0.062 | 0.2604 | 0.41 |
-0.453 |
| 1245 | 2345 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| 1345 | 1345 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| 1345 | 2345 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| 2345 | 2345 |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
|
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| Your Hand | Opponent |
1: | 2: | 3: | 4: | 5: |
--- |
1: | 2: | 3: | 4: | 5: --- |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |
| You | Opponent |
1: | 2: | 3: | 4: | 5: |
|
1: | 2: | 3: | 4: | 5: |
exp |