The other night at men's bridge I pushed to game twice when by rights I should have stopped a trick short of game. I got to wondering what the math of this decision is.
The scoring for this type of bridge is that a non-vulnerable game is worth 300, a vulnerable game 500, and 50 points are awarded for a part score.
Say it is 50-50 whether you make 4 spades or 3 spades. Bidding 3 spades yields you 140 half the time, and 170 half the time, for an expected gain of 155. Bidding 4 spades when not-vulnerable yields you 420 half the time, and a minus 50 half the time, for an expected gain of 185. If vulnerable, you make 620 half the time, and lose 100 half the time, for an expected gain of 260. (This sounds great, but remember if you are doubled and go down the penalties can be severe.)
Hence, one can conclude that if you have a 50-50 shot, go for it. The reality in the men's group is that there is such a variation in skill level, that the decision really rests on your evaluation of the relative skill levels of you and your partner, compared to your opponents.