**BASEBALL**

Lately I’ve seen a lot of debate about what makes a good third base coach on message boards. Specifically Gene Lamont has seemed to run into a lot of outs recently. This got me thinking about how to evaluate third base coaches. I quickly decided that would be very hard for a number of reasons. Very few runners are nabbed at the plate over a season. Whether to send someone depends on that player’s speed, the outfielder’s position and the strength of the outfielder’s arm.

It is easy, however to know when you should send someone based on the probability they will be safe at the plate in the coach’s estimation. You can do this with run expectancy. For instance the Tigers recently had a situation with no outs and a runner on first. A double was hit and Peralta the runner on first was out trying to score. Should Lamont have sent him?

If Lamont had done nothing the Tigers would likely have scored 2.050 runs that inning. Had Peralta been safe the Tigers would have likely scored 2.170 runs. They already had one run and would be expected to score 1.170 more with a runner on second and nobody out. Since he was out they were expected to score 0.721 runs with a man on second and one out. Early in a game you want to maximize the number of runs scored. So you can make a calculation based on the two outcomes being equally helpful to the team on average:

Runs Holding Peralta = Chance Peralta is Safe x Runs if Safe + Chance Peralta is Out x Runs if Out

2.050 = 2.170p + 0.721(1-p) with p the chance that Peralta would be safe at the plate in Lamont’s mind.

Solving for p gives 92%. Lamont should have thought there was a 92% chance to score Peralta before he sent him. It didn’t look like that good a chance on TV.

I get these run expectencies from tangotiger.net. They are based on historical record, not mathematical calculations- though the two systems rarely differ much. They include all data from 1993-2010. This might make the expectencies high, but since they are higher for all situations it probably about washes out. It does not take personnel into account. Clearly you will score more runs on average if Cabrera is on deck than if Kelly is on deck.

Now you can repeat the calculation for any situation. If H is the runs expected with holding the runner and S is the runs expected if the runner is safe and O is the runs expected with out you can calculate that p = (H-O)/(S-O). I did that in the chart below. The chart below assumes no other runner advances. For instance on a sacrifice fly only the runner from third will tag and try to advance. If there is a single with runners on first and second and under two outs, then after a play at the plate there will still be runners on first and second after the play. No one will go first to third. The chart below has the results.

Notice that with one out at the start of the play you almost always want to send the runner from third when a sacrifice fly makes the second out. The runner only needs a 35% chance to be safe. Given that a throw has to be online you might even want V. Martinez to challenge Choo on a shallow fly ball.

With two outs you try to score the runner from first on a double if it is better than 50-50 (46%) that he will make it. Because of the base-out state before and after the same holds for runners on 2-3 when a fly out makes the second out.

Lamont’s situation was the one where you had to be surest the run would score. This makes sense intuitively given that many different balls in play from either of the next two batters would score the runner from third anyway. I think Lamont should have held Peralta on that play. But, having a too conservative base coach can cost runs over a season, too.

One thing I think is worth considering is that if you send a runner from 2nd home on a single (we’ll say with 0 out), the defense then has to make a decision as well, if they try to throw him out and succeed they get runner on 2nd, 1 out. If they fail, it is runner on 2nd, 0 out. Which is simply to say, there is a break even point for the offense as well as the defense. You need to be 95% sure you’ll make it to send the runner in the above situation, but you also need to be (made up number) 60% sure you’ll get the runner out to make the throw. There is a little game theory, it’d be interesting to see how those play together.

Yes, and my chart is a little oversimplified. If a play is made at the plate in the situation you describe there is a chance that the runner from first advances to second and a chance they choose to stay at first. The chart just assumes they don’t. You are assuming the runner does advance. It would be easy to apply the formula assuming the runner does advance to see how different the Send? number looks. Then you can pick a number in the middle to use as the rule of thumb for sending.