I've been doing a lot of thinking about game theory and how it relates to pitch selection and swing rates. I finally decided to run some numbers to find the baselines for swinging, pitch selection, and strike throwing based on the ball/strike count.
The rate at which pitchers throw strikes aligns perfectly with the average run expectancy in each count. However, batters' swing rates are not likewise dictated by run expectancy. Instead, batters like to swing more the deeper they get in the count.
Batters swing 74% of the time on full counts, by far the highest percentage of any count. At the other end, they swing at only 6% of 3-0 pitches.
Pitchers simply aren't good enough at throwing strikes on 3-0 to warrant batters mixing their strategy between swinging and taking. Pitchers only hit the zone about 60% of the time 3-0, whereas they would need to hit it at least 70% of the time to make batters consider swinging I believe. Strangely, batters are eight times as likely to swing on 3-1 as they do 3-0. I think straight takes on 3-1 might be a viable strategy at times.
We already know and accept that batter's don't act completely rationally on the first pitch. Some players just don't like swinging 0-0, so they don't, and that's that. Yet they up their swing rates from 27% on 0-0 to 40% on 1-0, even though pitchers have similar pitch selections and locations and more importantly, the reward of taking is greater.
There is a 50/50 split between fastballs and off-speed pitches on 0-2 and 1-2 counts. Naturally, fastballs are thrown in the zone at a higher frequency. What's odd is that batters swing at more off-speed pitches on those counts.
The big question is, How much do batters learn from pitch to pitch? The deeper into his repertoire a pitcher must go, the greater the advantage is for the batter. There are probably advantages to taking pitches besides drawing balls. I don't think this applies to the full count, though, which might be why the swing rate is too damn high.
Here's the relevant data. I should note that I used the same strike zone model for all counts, which means that more pitches would be called strikes on 3-0 than listed as being in the zone, and fewer strikes would be called than listed on 0-2.