The 50/50 Conundrum

One of the complaints I hear most often about baseball board games like Strat-O-Matic is that they utilize a 50/50 model. This is a game mechanic where the result of the at bat is randomly read from the batter’s card half the time and the pitcher’s card the other half. Thus, the classic batter vs. pitcher duel isn’t a duel at all; it’s only ever influenced by one player or the other (and, of course, the occasional fielder).

Growing up, this bothered me, too. In fact, it bothered me most whenever I played Statis Pro Baseball and the pitcher on the mound had a large PB range (e.g., 2-9). I would look longingly at all the doubles and homeruns available on the batter’s card and know the moment I read the next fast action card and realized the result would come from the pitcher’s card they were all for naught.

I noticed it less in Strat-O-Matic because it utilized the 50/50 model and at least that seemed fair. Nevertheless, in critical situations my favorite pitcher or batter was reduced to being a mere spectator.

These days I don’t play board games very often but out of a sense of nostalgia in part driven by the work I’ve put into my own game, Trivia Challenge Baseball, I’ve been watching a lot of YouTube videos produced by folks who do and a few of them complain about it, too— or at least tout those games that include batter-pitcher interactions for each at bat as vaguely superior in this regard.

It got me thinking: does it really matter? The first thing I thought was that if you take a Strat-O-Matic batter and pitcher card and tape them together, you essentially have a single card where the pitcher and batter both influence the results of the at bat. Certainly no one would argue it wouldn’t produce the same results the two cards produced separately before they were taped together.

Of course, some might argue the 50/50 model is still in place in this example, since that batter columns are still 1 thru 3 and the pitcher columns 4 thru 6.

If we wanted to, we could cut out every dice roll result from the pitcher and batter cards and randomly assign them to a column (1 thru 6) on a new card. Indeed, we could physically create this sort of card with a pair of scissors, glue and the grotesque indifference required to defile Strat-O-Matic cards! Such a Frankenstein-like card, when constructed, would also produce the same results. None of the probabilities would have changed.

This would seem to suggest that the influence the pitcher has on the batter and vice-versa is still present in a 50/50 model and that our belief it isn’t is simply a misconception on our part.

If you aren’t buying any of this, that’s OK. Being skeptical is a good thing. So let’s explore things further. We’ll start with a definition: When are two sets of results the same?

I would posit they are the same if they both produce the “same” distribution and measures of central tendency, including streaks. To see what I mean, let’s consider a simple example: a batter’s hits. A hit, of course, is anything from a single to a homerun. I’ll assign a 1 to every hit and a 0 to every out. We’ll consider the following two sets representing 10 at bats:

Set 1 = {1, 0, 0, 1, 0, 0, 0, 1, 0, 1}

Set 2 = {1, 1, 1, 0, 0, 0, 0, 0, 1, 0}

Obviously a set of 10 outcomes is on the small side but don’t worry, I’m not making a scientific point here. Right now, I’m just trying to settle on a definition of “same.”

Note that both sets have the same mean and variance. They have the same mode and median. But they don’t quite look the same. Set 2 appears more “streaky”. When I think of the 50/50 model, I am convinced the mean is the same as the mean obtained from samples involving the combined contributions of the batter and pitcher, but I wonder about streaks.

To explore this further, imagine a .400 hitter facing a pitcher who allowed batters to hit just .150. (Though slightly more extreme, this is a little like pitting Ted Williams from 1941 against Pedro Martinez from 2000 when Ted batted .406 and Pedro allowed opposing hitters to bat just .167). Let’s assume the league average is .250. Thus, for our .400 hitter to bat his average in a 50/50 model, his card would need to represent a .550 average (since .550 × ½ + .250 × ½ = .400). Likewise, our superstar pitcher would allow just a .050 average on his card (.050 × ½ + .250 × ½ = .150). When facing each other, we expect our mythical batter to hit .300, since  .550 × ½ + .050 × ½ = .300.

I chose to use these fictional players for this example precisely because their cards are so different. I think intuitively we might be tempted to believe our .400 batter will be less streaky in a 50/50 model where his probability of getting a hit goes from .550 to .050 depending on whose card is read.

To test this theory, I’ll run a simulation where our .400 batter faces off against our superstar pitcher over a series of 100 million at bats. We will record the number of hits our batter accumulates over those at bats and calculate a few statistics. Note the combined results referred to below are those compiled by a .300 hitter while the split results are those obtained by randomly reading each result from the batter or pitcher card (i.e., our .400 batter and .050 pitcher).

Combined Results	Statistic	Split Results
.3001	Mean	.3001
6,303,213	2 Hit Streaks	6,303,685
1,894,404	3 Hit Streaks	1,892,426
568,379	4 Hit Streaks	566,823
170,349	5 Hit Streaks	169,673
51,232	6 Hit Streaks	51,198
15,470	7 Hit Streaks	15,332
4,610	8 Hit Streaks	4,611
1,373	9 Hit Streaks	1,355
457	10 Hit Streaks	405

* Note: Steaks are defined as consisting of exactly the number of hits shown. For example, every 3 hit streak includes a 2 hit streak but these were not counted as 2 hit streaks. Put another way, all streaks are bracketed by outs.

Not surprisingly, the means are the same rounded to 4 decimals and very close to the .300 average we expected (which isn’t a surprise given the size of the sample). Even more encouraging are the streaks, which are very similar between the two sets.

If you’re the meticulous type and looked closely at the numbers in the table, you may have noticed the combined results seem to include more streaks, even though the sums are close. There are a couple of items worth noting. First, this was not true of other data sets I ran. Second, remember we are viewing the raw totals—not the percentages— and they tend to accentuate differences between the two sets.

The Law of Large Numbers governing probabilities does not imply differences will vanish as the sample size increases, only that the average (or mean) will tend to get closer to the expected value as the sample size increases. The next two graphs provide evidence of this fact. The first graph shows that the difference between the expected number of hits and the number of hits observed appears to be growing while the second graph shows how the difference between the calculated batting average and the expected batting average (.300) seems to be shrinking.

So where does this leave us? Well, for one thing, it seems to indicate that on an at bat-by-at bat basis the results from a 50/50 model are indistinguishable from those produced by models that simultaneously account for the pitcher and batter. This is a powerful statement and should be enough to end the argument.

And yet…

Perhaps, like me, you’re having a hard time getting over the fact that individual at bats are controlled by a single player. If so, I urge you to read the above paragraph again and let it sink in. We aren’t talking above averages or long term trends. If you take any set of results from a 50/50 model— including sets with just one, two or three at bats— you won’t be able to tell which model produced it.

The truth is, the batter-pitcher interactions do exist for every at bat in the 50/50 model and I can prove it. It’s the little white die you roll in Strat-O-Matic, for example, to determine which card to read. It may not look or feel like the sort of interaction you get with, say, Payoff Pitch Baseball or Replay Baseball, for example, but it’s there and produces the same outcomes.

I’ve identified the interactive mechanism but I said I’d prove it and I haven’t yet. I’ll do so by running another simulation featuring our two ballplayers, only this time I won’t roll a little white die to determine whose card to read, I’ll simply alternate between the two cards. In other words, I will use a “deterministic” 50/50 model.

Looking at the results, it is clear at a glance that the deterministic model does not produce the same results. Notice the means are the same but nothing else. It produces the kind of results critics of a 50/50 model would be correct to criticize. But, as we’ve seen, these aren’t the results 50/50 models produce.

Combined Results	Statistic	Split Results
.3000	Mean	.3000
6,298,221	2 Hit Streaks	1,924,607
1,889,300	3 Hit Streaks	748,722
567,063	4 Hit Streaks	52,790
169,950	5 Hit Streaks	20,544
51,095	6 Hit Streaks	1,436
15,234	7 Hit Streaks	578
4,493	8 Hit Streaks	31
1,358	9 Hit Streaks	12
419	10 Hit Streaks	0

Still not convinced? How about we roll for it!