RogerBW's Blog

One of the more complex aspects of En Garde! is gambling, and correct choices are not obvious. So I worked it out in detail.

Let's look at it in financial terms first. The procedure is this:

• You wager an amount S.
• The house rolls a d6; call this value C.
• You decide to cut or stay.
• If you cut, you lose S/2.
• If you stay, you roll a d6. If this is higher than C, you win S; equal or lower, you lose S.

If you stay, your chance of a win is 1 - C/6, and of a loss C/6. Therefore your expected winnings are S × (1 - C/3). If you cut, your expected winnings are -S/2. Therefore from a purely financial point of view you should cut on 5+; on average you'll still lose, but you'll only lose 1/18 of your stake in the long term.

(Cut on 1+, expected loss is 1/2; 2+, 11/36; 3+, 1/6, 4+, 1/12, 5+, 1/18; 6+, 1/12; not at all, 1/6.)

However, there's also status to consider, because social climbing is the goal of the game. When you win, you gain a status point (SP). When you lose or cut, you lose an SP. If gambling at the club, there's a house divisor D: the total amount you staked in that session, divided by that value and rounded down, is awarded as SP. But, if you cut, you halve the staked value for that bet. (You can bet a maximum of nine times in a session.)

Let's call the house divisor D, and ignore the rounding - so each bet also gets you S/D SP from the house divisor, or S/D/2 if you cut. Then SP gains/losses are:

• cut: +S/D/2 - 1
• lose: +S/D -1
• win: +S/D +1

Now your expected SP gain if you stay is S/D + 1 - C/3. So the break-even cut threshold is

• S/D/2 - 1 = S/D + 1 - C/3

which becomes

• C = S/D × 3/2 + 6
• S/D = C × 2/3 - 4

If D is not a consideration, for example gambling at a bawdyhouse, the cut threshold for maximising SP is 6 (because for anything less you might win), and it only goes up from there; for 1/3 (the house limit of 50 and gambling divisor of 150 at Red Philip's) it's 6.5. In other words, to maximise SP at a club, you should bet as much as you are able and never cut. (As long as you can afford it…)

It's the whole-number multiple of the divisor that counts, so your total amount staked should be an even multiple of D. For example, at the Blue Gables with a limit of 150 and divisor of 200, it may make sense to stake 134, so that each three bets gets you 2SP from S/D but you slightly limit your losses compared with staking 150 each time.