Ninth Inning Rally Chart

I rarely use this space to discuss game rules for the game I created, Baseball Trivia Challenge, but figured I’d make a rare exception. I sent the game to several folks to play test and have received some excellent feedback. The particular bit of feedback I want to discuss here concerns teams that enter the ninth inning trailing by a large number of runs. The reason this is a problem is because, regardless of their ability, players in Baseball Trivia Challenge cannot score more than 4 runs per at bat.

Both Earle Shamblin from Tabletop Baseball + and Nick Hawes came up with the same excellent house rule to deal with these situations. Their solution is to let the ninth batter continue the at bat as long as runs are scored. In other words, the at bat will continue unless one of three things happen: 1) the d20 die roll indicates no runs are possible; 2) the batting team’s manager answers the trivia question incorrectly; 3) the pitching team’s manager answers the trivia question correctly. It’s an innovative solution to a problem I didn’t recognize existed.

If you’ve read many of my posts you may have noticed that I’m a little over-the-top when it comes to mathematics and realism. So while I love their solution, I feel I must point out that the realism of the game will be affected if you implement this rule. Simply put, teams will score more runs than they should and relief pitcher ERAs could skyrocket. If that isn’t a problem for you, I think it’s the best solution to this problem because it puts the most pressure on both “managers.”

Another option that solves the realism issue is to use the Ninth Inning Rally Chart shown below and included with the game. I created it only after Earle and Nick pointed out the need for it, so they deserve most of the credit for it, too. You may use this chart whenever your team is down by no more than 9 runs in the ninth inning.* Simply state your intention to do so and roll the dice. Observe the rating of the player indicated (batter, pitcher, fielder) then locate the number in the cell corresponding to the player rating and the number of runs behind. If the d20 is less than or equal to the number shown, read the next trivia question to whosoever turn it is to answer. If you answer correctly or your opponent gets it wrong, you score the runs required to tie the game.

That’s all there is to it! Feel free to use either rule! Both approaches add a much-needed element of excitement to games that might otherwise be decided!

*Note that, in theory, you could use this chart to play an entire game where you selected how many runs you want to “try for” each at bat.

RTG RUNS BEHIND
1 2 3 4 5 6 7 8 9
0
1 1 1
2 2 1 1 1
3 3 2 1 1 1 1
4 4 2 1 1 1 1 1 1
5 5 3 2 1 1 1 1 1 1
6 6 3 2 2 1 1 1 1 1
7 7 4 2 2 1 1 1 1 1
8 8 4 3 2 2 1 1 1 1
9 9 5 3 2 2 2 1 1 1
10 10 5 3 3 2 2 1 1 1
11 11 6 4 3 2 2 2 1 1
12 12 6 4 3 2 2 2 2 1
13 13 7 4 3 3 2 2 2 1
14 14 7 5 4 3 2 2 2 2
15 15 8 5 4 3 3 2 2 2
16 16 8 5 4 3 3 2 2 2
17 17 9 6 4 3 3 2 2 2
18 18 9 6 5 4 3 3 2 2
19 19 10 6 5 4 3 3 2 2
20 20 10 7 5 4 3 3 3 2
21 20 11 7 5 4 4 3 3 2
22 20 11 7 6 4 4 3 3 2
23 20 12 8 6 5 4 3 3 3
24 20 12 8 6 5 4 3 3 3
25 20 13 8 6 5 4 4 3 3
26 20 13 9 7 5 4 4 3 3
27 20 14 9 7 5 5 4 3 3
28 20 14 9 7 6 5 4 4 3
29 20 15 10 7 6 5 4 4 3
30 20 15 10 8 6 5 4 4 3
31 20 16 10 8 6 5 4 4 3
32 20 16 11 8 6 5 5 4 4
33 20 17 11 8 7 6 5 4 4
34 20 17 11 9 7 6 5 4 4
35 20 18 12 9 7 6 5 4 4
36 20 18 12 9 7 6 5 5 4
37 20 19 12 9 7 6 5 5 4
38 20 19 13 10 8 6 5 5 4
39 20 20 13 10 8 7 6 5 4
40 20 20 13 10 8 7 6 5 4
41 20 20 14 10 8 7 6 5 5
42 20 20 14 11 8 7 6 5 5
43 20 20 14 11 9 7 6 5 5
44 20 20 15 11 9 7 6 6 5
45 20 20 15 11 9 8 6 6 5
46 20 20 15 12 9 8 7 6 5
47 20 20 16 12 9 8 7 6 5
48 20 20 16 12 10 8 7 6 5
49 20 20 16 12 10 8 7 6 5
50 20 20 17 13 10 8 7 6 6
51 20 20 17 13 10 9 7 6 6
52 20 20 17 13 10 9 7 7 6
53 20 20 18 13 11 9 8 7 6
54 20 20 18 14 11 9 8 7 6
55 20 20 18 14 11 9 8 7 6
56 20 20 19 14 11 9 8 7 6
57 20 20 19 14 11 10 8 7 6
58 20 20 19 15 12 10 8 7 6
59 20 20 20 15 12 10 8 7 7
60 20 20 20 15 12 10 9 8 7
61 20 20 20 15 12 10 9 8 7
62 20 20 20 16 12 10 9 8 7
63 20 20 20 16 13 11 9 8 7
64 20 20 20 16 13 11 9 8 7
65 20 20 20 16 13 11 9 8 7
66 20 20 20 17 13 11 9 8 7
67 20 20 20 17 13 11 10 8 7
68 20 20 20 17 14 11 10 9 8
69 20 20 20 17 14 12 10 9 8
70 20 20 20 18 14 12 10 9 8
71 20 20 20 18 14 12 10 9 8
72 20 20 20 18 14 12 10 9 8
73 20 20 20 18 15 12 10 9 8
74 20 20 20 19 15 12 11 9 8
75 20 20 20 19 15 13 11 9 8
76 20 20 20 19 15 13 11 10 8
77 20 20 20 19 15 13 11 10 9
78 20 20 20 20 16 13 11 10 9
79 20 20 20 20 16 13 11 10 9
80 20 20 20 20 16 13 11 10 9

When Winning Really Matters…

A while back, I was involved in a fun discussion on the Tabletop Baseball+ Facebook group page about baseball game mechanics when a friend of mine posted the following: “When I’ve got a player who will ultimately hit .220, I will use him as much as I have to but certainly not in clutch situations, even though he was used that way during the season.”

It’s an intriguing point. As a kid playing replays in the late seventies and early eighties, I didn’t much care if the lineups for the teams I was playing were optimized to score the maximum number of runs, just as I didn’t care if the relief pitcher I called in from the bullpen was particularly skilled at his craft. My goal was not to win— I was managing both teams, after all— but rather to faithfully recreate games from that season. Indeed, without cable television (much less the Internet and Baseball Reference or Retrosheet to guide me), trying to settle on a lineup was hard enough!

That all changes when you’re playing to win. All of a sudden that .220 hitter who was relied on to perform in the clutch and didn’t (he did, after all, bat just .220) is no longer a viable option in those situations. Indeed, unless his defense is extraordinary, he is probably not the best choice to start, either.

In this article, I’ll consider position players who either started and put up rotten numbers as well as position players who did not start and put up extraordinary numbers and discuss ways to handle both.

For the most part, this discussion will be limited to dealing with these situations as they apply to one-offs or a short series of games (like you might encounter when playing in a tournament, for example), not an entire season.

We’ll start with the player whose playing time was limited but nonetheless put up monster numbers, a situation I like to refer to as “The Oscar Gamble Problem” because it first came to my attention when I was a kid playing Strat-O-Matic baseball with the 1979 card set.

The Oscar Gamble Problem: The Hot Hitting Reserve

For those too young to remember, Oscar Gamble was a journeyman outfielder who played for seven different teams during a 17 year major league career. By most accounts, Gamble, a lifetime .265 hitter, enjoyed his best year in the majors playing for the Chicago White Sox in 1977, when he finished 29th in the voting for American League MVP. But I can assure you his best season actually occurred two years later when he split time between Texas and New York and didn’t receive a single MVP vote.

Don’t take my word for it. Check out the stats. In 64 games for Texas, Gamble managed to hit a career-best .335 (his previous best was .297 during the aforementioned 1977 season) and compiled enough extra-base hits to record a .522 slugging percentage. But even these gaudy statistics paled in comparison to those he compiled during his brief stint in The Big Apple later that same year, where he batted an incredible .389 and belted 11 homeruns in just 113 at bats*. His slugging percentage was .735 and his OPS a dazzling 1.187, stratospheric numbers shared by players named Ruth, Williams and Bonds.

For players of tabletop games at the time, this presented a real problem. Gamble wasn’t just a good backup outfielder for New York, he was easily their best player. If Jackson was, as he once claimed, “the straw that stirred the drink” in New York, Gamble was the drink. His statistics were clearly superior to Jackson’s and anyone else on the team for that matter. The decision to play him in left field over Piniella was a no-brainer, despite Piniella’s solid stats: .297 BA, 137 hits, and 69 RBI (third best on the team behind Jackson and Nettles).

In my experience, there are a couple of ways to deal with situations like this. One technique is to divide position players into three groups: starters, bench players, and emergency players and refer to the table below.

Pct. Games Played* Description
60% to 100% Starters. No restrictions on how they are used.
40% to less than 60% Bench Players. Can be used as pinch-hitters and defensive replacements.
Less than 40% Should be relegated to emergency situations only!

* You should feel free to adjust these percentages as you see fit.

Since Gamble played only 36 games for New York (22%), he will only be available in emergency situations (e.g., if and when Piniella is injured, etc.). Of course, you can play around with these percentages as you see fit but it’s a good idea to place at least some restrictions on these sorts of players. While less effective for Texas, in the game I created, Baseball Trivia Challenge, Gamble is worth more than 0.5 runs per game to the New York lineup, a figure that could result in as many as 10 additional wins over the course of a 162 game season or a critical victory in a 7-game series.

A second approach is to simply divide the number of games the player played by the number of games his team played during the season, convert that number to a d20 dice roll, and roll to see if the player is available before each game. For example, using this approach, Gamble would need to roll a 4 or less to be eligible to play.

While this approach will generally work to assure each player appears in the appropriate number of games, it isn’t particularly realistic, as players aren’t often shuffled in and out of the lineup every other game for no apparent  reason. Also, in the case of Oscar Gamble the main reason he missed so many games is because he wasn’t with the team until early October. As we’ll see throughout this discussion, there are no perfect solutions.

Next, we’ll consider the starter who compiled unimpressive numbers at the plate. For this example, we may as well consider the standard-bearer in this regard.

The Mendoza Line: What to do about Poor Hitting Starters

Mario Mendoza was a slick-fielding shortshop who struggled at the plate, compiling a lifetime .215 batting average. The term “Mendoza Line” was coined by his Mariner teammates, Tom Paciorek and Bruce Bochte, and was intended as a harmless joke. Today, it is frequently used to define the threshold of mediocre hitting, which is generally defined as a .200 average. If you’re below the Mendoza Line, you’re not long for “The Bigs”, or so the saying goes.

This is a harder case. The rules we discussed earlier to deal with hot-hitting reserves assures they are unlikely to be available to start in place of a weak-hitting starter, but do not prevent other players who played at least 60% of their team’s games from starting in his stead. For example, Larry Milbourne, who played 65 games at shortstop and was a better hitter, could be used to replace Mendoza for the ’79 M’s.

One approach to stop this from happening is to assign starters and backups at each position based on games played.

In the case of Menoza and Milbourne, this approach would work well, since Mendoza played in 148 games and Milbourne only 65; however, in cases when the difference between the starter and backup is much smaller, it may not work as well if it works at all. For example, Leon Roberts played 67 games in left field for Seattle in 1979, but Tom Paciorek (47 games), John Hale (34), Dan Meyer (31) and Joe Simpson (27) all saw significant time at the position.

A second approach— to create d20 ranges for each player and roll to see who plays that game— has all the drawbacks mentioned earlier. In addition, it is not only a work intensive solution, it takes control away from the manager.

A third approach would be to utilize the d20 ranges described for “hot” players but instead of rolling to see who will play that game, roll to see who is “available” to play during the series or tournament. You could eliminate those players deemed “unavailable” entirely. Thus, it would be quite likely that Gamble would not only be unavailable to start, he would be unavailable period!

This situation is precisely the case described by my friend and is very difficult to deal with. In cases when the player is clearly in the lineup due to his defensive prowess, it is not unreasonable to conclude he is more valuable than other players at that position as a result. It isn’t unreasonable, but it may not be true. Indeed, I don’t believe it to be true in the game I created, where Mendoza is a much better fielder than Milbourne but fielding plays are nonetheless relatively rare.

Part of the problem get’s to the heart of the issue my friend described. Managers in a tabletop baseball game not only know the statistics each player compiled that year, but exactly how good they are according to the ratings in the game. For example, tabletop baseball game managers know that George Brett, who led the American League with a .329 average in 1990, batted just .255 the following year, but managers John Wathan and Hal McRae, who played with Brett and witnessed his brilliance, would have had no idea. Even if they assumed Brett wasn’t the .305 hitter he was over his lifetime, they might at least assume he was a .290 hitter or, at worst, a .280 hitter. But .255? Never!

If anyone has a great way of handling situations like this, post your ideas below!

* Starting in about 1973, Gamble consistently showed good power leading to above average slugging percentages. Near the end in his career in 1984, Gamble still managed to slug 10 homeruns in just 125 at bats despite hitting just .184.

 

Pitcher Fatigue In Strat-O-Matic

When I was a kid, I played a lot of sports board games, from Strat-O-Matic (MLB baseball, NFL football, NBA basketball, NHL hockey and even a card and dice NCAA football game), to APBA (NBA basketball and Saddle Racing), to Statis-Pro (MLB baseball, NBA basketball). I played Speed Circuit (an under-rated game in my opinion), Avalon Hill’s USAC Auto Racing, Win, Place & Show, Paydirt… it’s a long list. In addition to all of that, I once played a World War II war game by Metagaming Concepts called “Hitler’s War.” I bought it because it was a “pocket game” and therefore not very expensive.

I am a huge fan of replays (even though I never made it past a few games when I played regularly as a kid) so usually when I played any of the sports games I owned I played solitaire and didn’t really care who won. I was much more focused on the stats, which I assiduously scribbled down on a scoresheet or, more often, a sheet of notebook paper.

Metagaming

Hitler’s War was different. I played against my uncle and brother and I really wanted to win. Indeed, I wanted to win so much I spent the night before preparing. And I don’t mean merely thinking through scenarios, I mean calculating the dice probabilities associated with various strategies. I was a “metagamer” before I even understood what that meant.

These days, I view metagamers and metagaming in a pretty negative light, even if I understand the desire to do it. (For the record, the game dragged on through the night and into the early morning, my early success rolling through Poland was halted and I grew irritable as my losses increased. So much for metagaming!)

Interestingly enough, a number of games seem to encourage it, including the Strat-O-Matic game company which publishes a baseball ratings guide

This article is all about metagaming. In it, I explore Strat-O-Matic’s pitcher fatigue system and compare the performance of a pitcher who has reached his point of weakness to his nominal performance when not fatigued. While I will utilize an equation or two here this is not intended to be anything more than a crude estimate.

This discussion will make more sense with an example so I will be referring to Randy Johnson’s 1995 Strat card throughout. Since I don’t have permission to include his card here, I’ll be sure to discuss only the relevant details.

Against right-handed batters, Johnson’s card includes two strikeouts that are changed to singles once he reaches his point of weakness. These occur in columns 4 and 6 and correspond to dice rolls of 7 and 9, respectively. The dice rolls are the same against lefties but occur in columns 5 and 6 respectively.

Now, I could spend a fair amount of time “reverse-engineering” Johnson’s card to show the probabilities associated with each result on his card and from these determine the probability he allows a walk, single, double, etc. I did a lot of that in preparation for this article to make sure the numbers I was calculating seemed reasonable. They did.

The mathematics of fatigue

Fortunately, trying to assess the effect fatigue has on his performance isn’t so complicated; we need only focus on the outs that get transformed to singles. To do so, we’ll need to consider how the probability of a right-handed batter getting a hit off Johnson increases when he is fatigued. This isn’t hard to do. The number of singles Johnson allows when fatigued that he would not have allowed otherwise is 6 + 4 = 10. This is due to the fact there are 6 chances of rolling a 7 and 4 chances of rolling a 9. This is from a total of 36 + 36 + 36 – 7 = 101 chances. Note that I am subtracting 7 in this case because Johnson allows a walk when a 7 is rolled in column 6 and walks don’t count as at bats and are thus ignored when calculating batting averages. At first glance, this would seem to indicate that Johnson allows opposing RH batters to bat 10/101 = .099 ≈ 100 points higher when fatigued. But this isn’t quite right.

Strat-O-Matic results are obtained from the pitcher’s card only half the time. In general, the formula looks like this:

BAnominal = [(BAb + BAp) / 2]

BAb is the batter’s batting average and BAp is batting average allowed by the pitcher. (Note: These are not raw averages. They have been adjusted to ensure players will duplicate their real-life statistics when facing the same level of competition).

Here is how things look when Johnson has reached his point of weakness and is said to be fatigued.

BAfatigued = [(BAb + BAp + 0.099) / 2] = (BAb + BAp) / 2 + .099 / 2 = BAnominal + 0.099 / 2

The last bit is important. It shows that Johnson will allow opposing batters to hits approximately 50 points higher when fatigued. I might have just said that a few paragraphs ago— it seems intuitive—  but I feel it’s important to be deliberate in these cases since our intuition is sometimes wrong. (Note: The same holds true for left-handed batters, whom Johnson faced far fewer times).

Did Strat-O-Matic get it right?

Does Randy Johnson allow opposing hitters to bat 50 points higher when he is tired? We don’t really know. According to the data (which includes both left- and right-handed batters), Johnson’s performance drop-off occurs during innings 4 thru 6, not innings 7 thru 9, but we don’t know if he reached his point of weakness in any of the 23 games he lasted into the seventh inning or further; we only know that his Strat endurance factor inning is the seventh inning. Of course, with the statistics available today, it would possible (though tedious) to calculate how many innings he pitched in real-life when he would have been considered fatigued by Strat-O-Matic rules but I won’t bother to do that here since the point isn’t to show whether or not the point of weakness rule is realistic or not. I happen to like it and suspect a big reason for it is to encourage realistic usage, which I  support.

Table 1: Randy Johnson’s 1995 Performance By Inning

Split G IP ERA PA AB H BB SO SO/W BA OBP SLG OPS
Innings 1-3 30 90 1.80 357 328 60 26 130 5.0 .183 .245 .265 .510
Innings 4-6 30 87.1 2.68 359 323 69 33 112 3.4 .214 .290 .337 .627
Innings 7-9 23 37 3.65 150 141 30 6 52 8.7 .213 .260 .312 .572

I did want to further explore how me might analyze the effects of fatigue on Johnson’s performance. We know that he will allow opposing batters to hit for a higher average but haven’t yet considered what that implies. I would like to know how many more runs he might be expected to allow under those circumstances. How much higher will his ERA be when fatigued?

The effect of fatigue on run-scoring

The table above provides an answer but it isn’t one I’d entirely trust. Most of Johnson’s stats are better in innings 7 thru 9 than they are in innings 4 thru 6 so it isn’t clear to me whether he was just extremely lucky to have allowed so few runs during those middle innings or simply unlucky in that regard in later innings. I’ll approach this question by tossing up a graph and a few equations that will make it seem like I’m doing real science when really all I’m doing is making a crude guess.

The graph below shows how a pitcher’s WHIP is related to runs scored during the 1995 American League season. Without delving into the details, I eliminated players with fewer than approximately 152 at bats and estimated runs scored using sabermetrics. Next, I plotted  the relationship between WHIP and runs scored (see below). It’s far from perfect but it provides at least a crude estimate. The linear regression equation in the white box on the right shows how changes in WHIP lead to more or less runs. For example, a 1.0 change in WHIP leads to 5.33 more or fewer runs.

Let’s try applying this equation to Johnson. Here are his statistics against right-handed batters in 1995:

Split PA  AB IP† R H BB BA WHIP R/G Estimated R/G
vs RHB 754 688 186.2 54* 145 59 .211 1.093 2.60 3.14

* The number of runs allowed vs RHB (54) and the number of runs allowed vs LHB (7) don’t add up to the 65 runs Johnson allowed. Thus, the R/G could be between 2.60 and 2.80.
† The number of innings pitched were estimated by multiplying the proportion of plate appearances versus RHB by the total number of innings pitched, 214.1.

They suggest that the linear regression equation, applied to Johnson, overestimates the number of runs he should allow by anywhere from about a third to a half of a run per game. While this isn’t the purpose of the equation, it’s important to keep in mind we’re pouring various quantities into beakers and test tubes, not performing real science.

So how do things change when Randy Johnson is fatigued? Since the only thing that changes is the number of hits (specifically singles) Johnson allows, that is all we’ll calculate: (.099 ÷ 2 × 688) ≈ 33.7 additional hits allowed. This produces a marginal WHIP rating of approximately 0.181. Since 0.181 × 5.3302 ≈ 0.96 runs per game, we conclude that Johnson’s ERA will be nearly a run higher when fatigued, an estimate that compares favorably with the actual data, which suggests a 0.97 rise in ERA from innings 4-6 to innings 7-9 (refer to Table 1 above).

Much Ado about nothing?

It obviously isn’t necessary to expend this sort of effort calculate a pitcher’s estimated performance when fatigued, but I wanted to provide a rough assessment of how fatigue effects even a pitcher of Johnson’s caliber and, as we observed, the effects are considerable. Beyond that, knowing how a pitcher performs when tired is really only useful when comparing him to relief pitchers who may replace him, and for this, it is probably enough to glace at their respective cards, noting that Johnson’s strikeouts will count as hits.