Over at Twinsgeek there is an interesting set of data that is provided to discuss late game decisions:
"historically what percentage of games were won by the home team carrying various leads or deficits into the top of the ninth.
Leading by 5 - win 99.7% of the time
Leading by 4 - win 98.8% of the time
Leading by 3 - win 98.0% of the time
Leading by 2 - win 94.5% of the time
Leading by 1 - win 86.6% of the time
Tied - win 52.2% of the time
Losing by 1 - win 15.2% of the time
Losing by 2 - win 6.3% of the time
Losing by 3 - win 2.9% of the time
Losing by 4 - win 1.3% of the time
Losing by 5 - win 0.6% of the time"
I think the really interesting part of this data is that it clearly shows that not all runs are created equally. While the data here is only for one particular situation, I think the pattern is likely to apply regardless of the inning.
In a tie game, the first run that scores in the bottom of the 8th raises a team's chances of winning by 66%,more than all the additional runs combined. The second run raises the team's chances by 10% and the third run raises the team's chances by 4%. If the home team starts the bottom of the 8th down by a run, the first run it scores raises its chances of winning by 243%.
On the other hand, when down by two runs, scoring the first run only increases the chances of winning by 141% and when down by 2 runs only 117%.
What this demonstrates, of no surprise to traditionalists, is that in a close game, playing for one run is by far and away the better strategy than playing for a big inning. If you are down by more than a run, then playing for the big inning makes more sense.
One other note. I calculated the increases based on the formula (chances after run scores- chances before run scores)/(chances before run scores). That is to say, the change in the probability a team will win. You an argue whether the percentage increase is more appropriate than just the straight number of additional games a team would win. But the pattern is the same either way. And, of course, these are averages. It may well be that in close games where both teams are scoring runs in bunches, the extra runs become more important. Or that a single run is more important in a pitchers duel. In fact, both of those are likely true. Those are the limits of all "probabilities" in baseball. There are so many variables that every situation is different and the average has only limited value for a specific decision or player.