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Subject: Optimal choice of Tactical Chits rss

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Olivier Clementin
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In GMT Brandywine and Saratoga, player simultaneously pick Tactical chits to solve combat situations. The comparison of the chits gives a drm which is applied to the combat.

I assuming no player wishes to retreat (i.e. they place no value on the No Combat result) and that both player are risk neutral (i.e. the utility derived from a certain drm is proportional to the drm). The risk neutrality is necessary for the calculation, but it is usually a pretty good approximation.

Per the minimax theorem, there is an optimal choice of tactical chit, which is the following:

1. No leader present on both sides

pick at random 33.3% of the time Skirmish, 33.3% of the time Echelon, 33.3% of the time Stand

2. Both players have a leader and an open flank

Pick 20% Skirmish 15% Echelon 17.5% Stand, 10% Frontal Assault 10% Commit Reserve 12.5% Turn Flank, 15% Refuse Flank

3. Both players have a leader and no open flank

11.1% Skirmish 44.4% Echelon 11.1% Stand, 22.2% Frontal Assault 11.1% Commit Reserve

4. Only one player has a leader and an open flank

50% Frontal Assault 50% Turn Flank

his unhappy leaderless opponent picks 25% Echelon et 75% Stand

5. Only one player has a leader but no open flank

11.1% Skirmish 55.5% Echelon 33.3% Frontal Assault

his leaderless opponent 22.2% Skirmish, 44.4% Echelon, 33.3% Stand

Note that in cases 4 and 5 the optimal strategy is not unique for the player with a leader.
 
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Olivier Clementin
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Re:Optimal choice of Tactical Chits
Assuming both player use the minimax optimal strategy shown above, it is possible to calculate directly the probabilty of each outcome.

1. No leader present on both sides

33.3% +1 drm
33.3% no drm
33.3% -1 drm

2. Both players have a leader and an open flank

6.1% +2 drm
26.3% +1 drm
35.4% no rm
26.3% -1 drm
6.1% -2 drm

3. Both players have a leader and no open flank

2.5% +2 drm
18.5% +1 drm
58.0% no rm
18.5% -1 drm
2.5% -2 drm

4. Only one player has a leader and an open flank

12.5% +2 drm
37.5% +1 drm
37.5% no drm
12.5% -1 drm

5. Only one player has a leader but no open flank

7.3% +2 drm
23.6% +1 drm
42.0% no drm
27.1% -1 drm
 
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