Mathematically, when should you make a game try (and which one)?

My partner and I have had some disagreements on when game tries should be made, which one, and when they should be accepted (eg, should you accept with shortness?). Being a fan of programming and simulations, I decided to do some analysis of the different approaches, and find out how they compare.

However, I seem to have hit a mental roadblock and would appreciate some advice.

It is a well known fact that at IMPs, if you are deciding whether to bid a major suit game or not, you want to be bidding 45% games non-vulnerable and 38% games vulnerable.

This isn't entirely accurate - for example, if you're guaranteed to go down 2 tricks every time game doesn't make, the odds change considerably. However, it a reasonable rule of thumb.

The calculation is quite simple. Suppose the auction has started 1 - 3. If the opponents are in 3, you will gain 6 IMPs bidding a non-vulnerable game if it makes, and lose 5 IMPs if it is down one (not taking into account cases where only 8 tricks are available). Conversely, if the opponents are in 4, the same numbers work in reverse. This means your choice is independent of your opponents choice.

I can confirm these results by dealing myself a given 1 hand, and simulating many 3 hands for my partner. For each one, I can calculate the difference in score between bidding 4 and passing, and therefore whether or not I should accept. This even takes into account the 8 trick cases.

Suppose instead the auction has started 1 - 2.

If the probability of making game at this stage is, say, 50%, that doesn't necessarily mean you should bid it directly. In the extremal case, if you had a game try available where game makes every time it is accepted, and fails every time it isn't, then clearly using it is better than jumping to game.

Let's say we've agreed on a game try approach, and I'm trying to decide whether I should pass, make a 3 game try, make a 3 game try, or bid 4S. I can simulate many possible hands, detect which ones partner would accept each game try on, and figure out *our* score for each hand.

But what should I compare this to at the other table?

I could assume the other table bids 4 always, and figure out which of the above approaches gives the highest average IMPs.

Or, I could assume they pass 2 always, and do the same.

Or, I could assume the opposition are always in the 'optimal' contact (ie game if and only if it makes).

None of these are even close to accurate representations of the actual score each approach would receive - and due to the non-transitive nature of bridge scoring aren't certain to provide the same results.

Even comparing the two different game tries to each other doesn't work, since other tables won't necessarily use the same methods.

Like there is with deciding whether to bid grand slams, part of it must involve estimating how likely the opposition are to be in each contract. But this seems immensely complicated, especially when we need to consider not an overall probability, but probabilities for each of the different cases (when we make an accept a game try, when we make and reject, etc).

On top of this, I was wanting to compare different approaches to accepting game tries, and that runs into identical issues.

TLDR; if I have a choice between passing, making multiple game tries, and bidding game, how do I determine what option to take?

While it is a major part in real life - ignore the 'information hiding' aspect (as well as eg the chance the opposition may balance if we pass).