Russian Roulette (2)

...please see the post Russian Roulette to catch up...


Obviously, the further down the chain your turn is, the better. But how much better? An interesting exercise is to create tables showing your edge against the field for two to five people taking turns in 1st to 5th place when everyone is 1) on equal footing and are forced to spin and 2) where one player uses the strategic advantage against all the others...


For a two person game with one bullet, if I go last then my expectation can be represented by


$E[game\ |\ go\ last]=\frac{1}{6}*1+\frac{5}{6}*(\frac{1}{6}*-1+\frac{5}{6}*E[game\ |\ go\ last])$
$E[game\ |\ go\ last]=\frac{1}{9}$


Conversely, if I go first then my expectation can be represented by


$E[game\ |\ go\ first]=\frac{1}{6}*-1+\frac{5}{6}*E[game\ |\ go\ last]$
$E[game\ |\ go\ first]=-\frac{1}{9}$


Using a similar analysis via the symmetry inherent to the game, I have created tables below describing the $E[game\ |\ number\ of\ bullets\ \&\ turn\ number]$ where all people are forced to spin (x-axis: number of bullets; y-axis: turn number):

2 Players

      1     2     3     4     5
1 -0.09 -0.20 -0.33 -0.50 -0.71

2  0.09  0.20  0.33  0.50  0.71


3 Players

      1     2     3     4     5
1 -0.19 -0.42 -0.71 -1.08 -1.51

2  0.01  0.05  0.14  0.31  0.58
3  0.18  0.37  0.57  0.77  0.93


4 Players

      1     2     3     4     5
1 -0.29 -0.66 -1.13 -1.70 -2.34

2 -0.07 -0.11 -0.07  0.10  0.44
3  0.11  0.26  0.47  0.70  0.91
4  0.25  0.51  0.73  0.90  0.98



5 Players

      1     2     3     4     5
1 -0.39 -0.92 -1.58 -2.35 -3.17

2 -0.16 -0.28 -0.29 -0.12  0.31
3  0.03  0.15  0.35  0.63  0.88
4  0.19  0.43  0.68  0.88  0.98
5  0.33  0.62  0.84  0.96  1.00

Knowing that the strategic alternative adds value to the informed player when 2, 3 and 4 bullets are used, below you can find tables describing the $E[game\ using\ strategic\ advantage\ |\ number\ of\ bullets\ \&\ turn\ number]$ against all others who are forced to spin (x-axis: number of bullets; y-axis: turn number): 

2 Players

      2     3     4 

1 -0.11 -0.25 -0.47

2  0.33  0.50  0.60


3 Players

      2     3     4
1 -0.33 -0.65 -1.06

2  0.25  0.40  0.47
3  0.50  0.70  0.82


4 Players

      2     3     4
1 -0.59 -1.09 -1.69

2  0.14  0.27  0.32
3  0.43  0.64  0.77
4  0.62  0.82  0.92


5 Players

      2     3     4
1 -0.86 -1.55 -2.34

2  0.02  0.13  0.16
3  0.35  0.57  0.72
4  0.43  0.68  0.88
5  0.71  0.89  0.97