Originally posted by forkedknight
One item where I think we read the rules differently is regarding who rolls / how many dice are rolled.
I read the rules as meaning that each player rolls their own dice, whereas from reading your post, I get the impression there is only a single dice roll.
With one roll per player, I think it would be possible to change the wager for each value to make it more fair, and potentially into a reasonable game
I wasn't sure, the thread title said "die wager" and not "dice wager" implying only one die and the analysis in the other posts seemed to indicate one die. It makes more sense with two dice rolls though - whoever rolls highest wins but not if they've gone bust. If player A bets $6 and B bets $N then there are N ties, (36 - N^2) cases where A rolls greater than N (and wins as B is lower or bust) or B rolls greater than N (and goes bust), N(N - 1)/2 cases where A wins and the same number of cases where B wins. It's easier to keep track of B's winnings:
A bets $6 and rolls Ra and B bets $N and rolls Rb; B's winnings per hand = Wb, the profit = Pb
(Ra > N) | (Rb > N) Wb = 0; Pb = -N
(Ra = Rb) & (Rb <= N) Wb = 3 + N/2; Pb = 3 - N/2
(Ra > Rb) & (Rb <= N) Wb = 0; Pb = -N
(Ra < Rb) & (Rb >= N) Wb = 6 + N; Pb = 6
B pays out 36N. <x> = expectation of x
36<Wb> = (3 + N/2)N + (6 + N)N(N - 1)/2
36<Pb>/N = (6 + N)N/2 - 36
<Pb> = N[N(N + 6)/2 - 36]/36
which is negative definite for N < 6 (<Pb>(5) = -42.5/36 ~ -$1.18). In other words the best strategy is just to bet to the maximum. So adding a second die doesn't change anything. Having a rule where you always win if you roll what you bet (if your opponent misses) doesn't do enough as if B bets $5 he gains $5.50 when both roll a 5 but still loses to A's 6 as A has hit as well for an average gain of $0.15, not enough to overturn the deficit. I agree one could try tinkering with the bet sizes to correct this, but it needs to force the players to play adaptively.
I like the idea of only one die roll. Each player has two unknowns, the other player's bet and the die roll. Really rolling two dice is the same as rolling one bigger one, it just makes the range of possibilities bigger without changing anything fundamental. One would then have to work out a win table so over the 90 possible non-trivial cases (where A and B do not bet the same, or a rule where A bets first and if B bets the same B can bet again with knowledge of A's bet) so that any bet would always have negative expectation against at least one other bet - for example betting one less than the opponent could be made to be the best move. With betting 2 less one gets break-even but betting $1 against $6 is a disaster (6 counting as 1 less than 1 for this). That way if A tries to bet $6 every time then B can just bet $5. That way a strategy needs to be responsive and the game becomes interesting.