# Game theory and professional mixed-strategy models.

Game theory is a branch of mathematics involving the creation and study of models of situations in which outcomes are interdependent on choices made by two or more actors. A game model requires the following elements:

1. Players. These are assumed to be rational actors weighing costs and benefits as they pursue their own goals. There must be two or more players.

2. Rules of the game. These define the limits of action - what can and cannot be done in the game.

3. Strategies. These are the choices that the players can make within the rules of the game. A strategy is a complete set of choices from beginning to end of the game. For example, if a player can make three different decisions, and for each decision there are two alternatives, he has [2.sup.3] = 8 different strategies for the whole game.

4. Payoffs. These are the outcomes that accrue to players depending on the choice of strategies they and their opponents make. Payoffs may he either ordinal or cardinal.

5. Solutions. A solution is the set of payoffs arising from the strategies that rational players would choose under the rules of the game. Sometimes there are multiple solutions. Indeed, sometimes there are multiple solution concepts, that is, more than one line of reasoning that rational actors might employ.

Game theory emerged as a distinct intellectual enterprise in 1944 with the publication of the Theory of Games and Economic Behavior, by John yon Neumann and Oskar Morgenstern. Its maturity was signalled fifty years later by the award of the 1994 Nobel Prize for economics to three eminent scholars in the field. Game theory is now widely used in economics and political science, and to a lesser extent sociology, psychology, and the other social sciences. Good introductions for nontechnical readers have been published by Dixit and Nalebuff (1991), Davis (1983), and Hamburger (1979).

Curiously, game theory has been little applied to the analysis of athletic events, even though these competitive contests have all the characteristics of games as described above. This paper is intended to show how game theory can contribute to the understanding of sport behavior.(1)

Baseball is well known for the quantity and quality of statistical data associated with the game; and at least one major work (Cook, 1966) has used these data to propound mathematical rules of strategy. However, Cook's approach was not game-theoretical because it did not focus on the interplay of strategic choices made by the offence and defence. Cook looked at offence and defence in isolation, calculating the productivity of various strategies over a long run of games. This amounts to conceiving baseball as a "game against nature," in which players make optimizing choices against random events. But a baseball game is a contest between two teams of opponents, each of which has independent choices to make in the attempt to defeat the other. As such, it is an ideal subject for game-theoretical analysis.

The particular solution concept employed in this paper is known as "solution in mixed strategies." It derives from the minimax theorem, proved by John von Neumann in 1926, which was a turning point in the development of game theory. Von Neumann showed that every two-person, zero-sum game with a finite number of strategies has a solution. When that solution is a pair of pure strategies, it is known as a saddlepoint. Sometimes, however, players do not have a single best strategy, in which case their rational choice is a random mixture of pure strategies in certain calculable proportions (Davis, 1983; Hamburger, 1979; Poundstone, 1992; Rapoport, 1966; Williams, 1982). In this article, the solution of the various models always represents the proportions of at-bats in professional baseball taken by left- and right-handed batters and pitched by left- and right-handed pitchers.

Pitchers vs. Batters

Although baseball is a complex team sport, the focus of attention is the duel between pitcher and batter. This central contest can readily be modelled as a two-person zero-sum game with a binary choice of strategies for each player. The pitcher can throw with either the right or the left hand, and the batter can hit from either the right or the left side. There are four possible pitching-batting combinations, as shown in Table 1.
```Table 1

Pitching-Batting Combinations

Pitcher
Throws Left Throws Right

Hits Left L vs. L L vs. R
Batter
Hits Right R vs. L R vs. R
```

Table 1 does not apply to a single encounter. Pitchers have only one good throwing arm, and batters are not allowed to switch sides during their turn at the plate. However, the model does apply in a sense to managers assembling a lineup. All batting orders contain a mixture of right- and left-handed hitters, and all pitching rosters contain a mixture of right- and left-handed pitchers. During the season a pitcher can expect to face batters standing on both sides of the plate, and a batter can expect to face pitchers throwing from either side of the mound.

However, we would not expect the model to predict the composition of any particular team. Professional baseball rosters are relatively small, containing about 10 pitchers and 15 other players for a total of 25. Pitchers, since all they do is pitch, might be balanced between left and right according to some mathematical principle; but hitters (except for designated hitters in the American League) have to play a defensive position and be able to run the bases. Moreover, because players under long-term contracts are not freely available to be hired and fired, managers work with a limited pool of candidates in any particular year. Hence, any roster will have to balance a number of athletic and economic considerations and cannot be predicted solely from a model of batting success.

The model, however, can be tested at a higher level of aggregation, in which there is a sort of statistical contest being waged by hitters as a group against pitchers as a group. In 1995, not counting at-bats by National League pitchers, 583 batters faced 550 pitchers in 133,621 at-bats (James, 1995). We could reasonably expect a game-theory model to be predictive at this level. Such an approach resembles the work of John Maynard Smith (1982), Richard Dawkins (1989), Robert Axelrod (1984), and others in modelling population polymorphism using game matrices with solutions in mixed strategies.

Even the most casual baseball fan knows that right-handed batters tend to do better against left-handed pitchers, while left-handed batters do better against right-handed pitchers. We have all seen games held up while one manager sends in a pinch hitter to get the batter's opposite-side advantage, and the other manager responds by changing pitchers to restore the hurler's same-side advantage. The reason for the batter's opposite-side advantage has to do with the behavior of breaking-ball pitches. A natural curve ball, i.e., one thrown with outward rotation of the pitcher's arm, breaks to the opposite side - to the left for a right-handed pitcher and to the right for a left-handed pitcher. A curve ball also drops and acts as a change of pace because it is slower than a fastball. All these factors fool the batter to some extent, but a curve ball moving toward you is easier to hit than one moving away from you. Robert K. Adair (1994) explains why:

Batting against the curve ball, the batter tends to swing too quickly at the relatively slow pitch, and he tends to underestimate the in-out curve deviation. These errors tend to add up for out-curves but cancel for in-curves. Hence, the in-curves may be a little easier to hit, accounting for the small advantage batters have when they face a pitcher throwing from the opposite side. (p. 96)

Data published by STATS, Inc., can be used to develop and test a model of the contest between pitchers and hitters. First I use the so-called "lefty-righty" statistics for the 1995 season, as published by James (1995) and available on diskette for further analysis. For a second test, I use slightly different data published in aggregated form by Adair (1994) for the 1984-87 seasons.

Three indicators of batting success are readily available in the STATS, Inc. data - batting average, on-base percentage, and slugging percentage. Batting average is the best known and most widely quoted (Runquist, 1995), and has been found in an econometric analysis to be the single most important variable in explaining salary variation (Akers & Buttross, 1988). It takes account of bases on balls, sacrifice flies and hits, and being hit by a pitch by removing these outcomes from the total of "at-bats" in the denominator. This quite sensibly gives these lesser forms of offence some weight, but not as much weight as is given to base hits, which also appear in the numerator of the fraction. A sacrifice hit or fly moves a runner forward one base but also produces an out, so it is not as good as a single. Nor, in most cases, is a base on balls. For example, three walks in a row will load the bases, but three singles in a row will score one or even two runs. On-base percentage overvalues walks by giving them the same value as hits in the numerator.

Admittedly, batting average does not do justice to extra-base hits because it counts them the same as singles. However, slugging percentage overvalues extra-base hits by counting all bases equally. To see why that is not justified, compare a bases-empty home run to four singles in a row. Both are worth four in the total bases count, which is the numerator of slugging percentage; but four singles in a row will produce two or even three runs, whereas a bases-empty homer produces only one run.

Because each of the three main batting productivity indicators has obvious imperfections, I calculated separate versions of the model with all three, plus a fourth using another indicator of my own construction. Batting average gave the closest fit to the actual data, so the rest of the paper relies on it as a payoff indicator. Results from using other indicators are presented in note 2.(2)

Table 2 summarizes major-league batting averages for the 1995 season. I have removed the at-bats of pitchers in the National League because pitchers do not bat at all in the American League and are not chosen in any case because of their batting skill.
```Table 2

Major League Batting Results, 1995

Batter Pitcher At-Bats Hits Batting Average

(1) L L 8,360 2,144 .2565
(2) L R 30,479 8,559 .2808
(3) L L&R 38,839 10,703 .2756
(4) R L 20,609 5,713 .2772
(5) R R 48,361 12,746 .2636
(6) R L&R 68,970 18,459 .2676
(7) S L 7,165 1,927 .2689
(8) S R 18,647 5,030 .2697
(9) S L&R 25,812 6,957 .2696

(10) 133,621 36,119 .2703
```

I begin with the simplifying procedure of counting switch-hitters as left-handed batters when they face right-handed pitchers and right-handed batters when they face left-handed pitchers. This means combining lines (8) and (2) and lines (7) and (4) from Table 2 in order to produce Table 3. The purpose is to enable construction of a 2 x 2 model, which is the easiest to solve and interpret. However, there is clearly some distortion in this procedure; being a switch-hitter is not the same as always batting from one side, because the switch-hitter chooses his batting side after seeing which pitcher he will face. This introduces a sequential element into the game and violates the assumption that the players are making simultaneous, independent choices. Hence, I will later treat switch-hitters as a separate category in a 3 x 2 model.
```Table 3

Pitcher-Batter Game, 1995 (Switch-Hitters Merged)

Pitch
Left Right

Left .2565, -.2565 .2766, -.2766
Bat
Right .2751, -.2751 .2636, -.2636
```

The solution to zero-sum games is based on the concept of security level. Each player is assumed to act so as to guarantee for himself through unilateral action the best of the worst that could happen to him (i.e., play "maximin," maximizing the minimum). But since in a zero-sum game your opponent's payoff is the negative of yours, you minimize his maximum ("minimax") when you maximize your own minimum.

If each player has a single strategy for which maximin equals minimax, the resultant strategy pair is the saddlepoint of the game or solution in pure strategies; but inspection of Table 3 shows there is no saddlepoint for the 1995 Pitcher-Batter Game. Neither side has a dominant strategy; it all depends on what the other side is doing. If the pitcher is left-handed, you will do better on average by sending out a right-handed batter, and vice versa. Hence we must look for a solution in mixed strategies, which means that each player should play a random combination of strategies in prescribed proportions. The problem is to compute the proportions. This means finding the prescribed probability (p) of batting left and (1-p) of batting right, and the prescribed probability (q) of pitching left and (1-q) of pitching right.

By looking for the point where maximin = minimax, each player wants to make himself invulnerable against the other's choice. You reach your security level, i.e., make yourself invulnerable, when nothing your opponent does can hurt you. Each player achieves this by selecting probabilities such that his opponent's expected payoff is the same no matter which strategy the opponent selects. That is, batters want to mix their likelihood of batting right and left in such proportions that it does not matter whether the pitcher throws right or left. Similarly, pitchers want to mix their probabilities of throwing right and left in such proportions that it does not matter whether hitters bat right or left.

Algebraically, this implies the following set of equations for calculating (p) and (1-p) in the 1995 Pitcher-Batter Game:

Let EV = Expected Value

EV(Pitch L) = EV(L vs. L) + EV(L vs. R)

EV(Pitch L) = -.2565p + (-.2751)(1-p)

EV(Pitch R) = EV(R vs. L) + EV(R vs. R)

EV(Pitch R) = -.2766p + (-.2636) (1-p)

EV(Pitch R) = EV(Pitch L)

-.2565p + (-.2751)(1-p) = -.2766p + (-.2636) (1 - p)

p = .364

1-p = .636

If there had been a manager in charge of all hitters for 1995, he should have chosen lineups such that players batted left 36.4% and right 63.6% of the time. With this combination, it would not have mattered whether his roster of batters faced a right- or left-hand pitcher; the expected payoff would have been the same. This is the best that such a hypothetical manager could have done. Had he fielded more left-handed batters than the optimum, the defence could have done better by pitching left-handed as a pure strategy; and similarly if he had played more right-handed batters, his hitters could have been victimized by right-handed pitchers.

The calculation of (q) and (1-q) proceeds along similar lines:

EV(Bat L) = EV(L vs. L) + EV(L vs. R)

EV(Bat L) = .2565q + .2766(1-q)

EV(Bat R) = EV(R vs. L) + EV(R vs. R)

EV(Bat R) = .2751q + .2636(1-q)

EV(Bat L) = EV(Bat R)

.2565q + .2766(1-q) = .2751 + .2636(1-q)

q = .411

1-q = .589

A hypothetical manager in charge of all pitchers in 1995 should have had 41.1% of at-bats pitched by left-handers and 58.9% by right-handers. The offence could have penalized any departure from this policy by converting to the pure strategy of always batting right or left, in coordination with the shortage of pitchers (i.e., if there are not enough left-handed pitchers, always bat left, and so on).

A second test of the model can be performed upon the data published by Adair (1994) for the 1984-87 seasons and presented below in Table 4. His data include only those players who had over 250 at-bats.
```Table 4

Pitcher-Batter Game, 1984-87 (Switch-Hitters Merged)

Pitch
Left Right

Left .2568, -.2568 .2797, -.2797
Bat
Right .2782, -.2782 .2649, -.2649
```

The solutions in this second model are astonishingly close to those in the first: p = 36.7% for 1984-87, compared to 36.4% for 1995; and q = 40.9% for 1984-87, compared to 41.1% for 1995. Because the two sets of numbers are so close, I will simply refer to the 1995 model in the rest of the paper.

The empirical test of the model is to compare its predictions to the actual proportions calculated from Table 2. The comparison is presented in Table 5.
```Table 5

Comparison of Actual and Predicted Values from the Pitcher-Batter
Game, 1995 (Switch-Hitters Merged)

Bat Pitch Actual % Predicted %

(1) L 43.0 36.4
(2) R 57.0 63.6
(3) L 27.0 41.1
(4) R 73.0 58.9
```

The model underpredicts the percentage of left-handed at-bats moderately (36.4% vs. an actual figure of 43.0%) and overpredicts the percentage of left-handed pitching turns substantially (41.1% vs. an actual figure of 27.0%).(2) An attempt to explain these discrepancies leads us into the realm of human biology. According to Coren (1992), about 14% of males are born left-handed. Although left-handed people experience higher mortality, that is not a significant factor at the age at which professional baseball is played. Hence the pool of potential professional players is about 14% left-handed. Experience shows that it is exceedingly rare to be able to throw effectively with the off hand, and the rare exceptions prove the rule. When Greg Harris of the Montreal Expos pitched both right- and left-handed in the same inning on September 28, 1995, he was the first pitcher to do so in a major-league game since 1888 (McKnight, 1995). But Harris is normally a right-handed pitcher, and the Expos' manager let him carry out the stunt for the recognition it brought (Gallagher, 1995).

This suggests that baseball teams would hire more left-handed pitchers if they could find them, but there are just not enough available. Indeed, there is considerable anecdotal evidence that left-handed pitchers are always in demand and can command premium salaries (Shaughnessy, 1989; Adams, 1990). Scouts go deeper into the talent pool for left-handed pitchers than they do for right-handers; but at a certain point the trade-off will become unfavorable because the advantage of having another left-handed pitcher will be outweighed by his lower native ability.

In contrast, experience shows that many players can learn to bat effectively from the opposite side. In fact, in the 1995 season, 53.6% of those who batted left threw right (James, 1995), even though their batting average was not as high as those who were natural left-handers. Although data on this precise question are lacking, it may well be that a significant proportion of these right-throwing, left-batting players are only weakly right-handed, or have some particular combination of dominant hand, eye, and foot that enables them to bat differently than they throw (Coren, 1992). Coren has found that some sports, e.g., tennis, favor athletes who are strongly one-sided, while others, e.g., hockey and basketball, flavor those who are less laterally oriented (Porac & Coren, 1981). Thus it is not unlikely that lateral orientation also plays a role in choice of strategies within particular sports.

Training is also a factor because many fathers and coaches are aware that there is an advantage to batting from the left. For one thing, the hitter starts out one or two steps closer to first base. Over the course of a season, this can mean several more bunts or ground balls turned into infield hits. Also, as long as there is a shortage of left-handed pitchers, there is an advantage to batting from the left because the batter faces a surplus of right-handed pitchers throwing curve balls that break towards rather than away from him. For these two reasons - and perhaps others, such as closer right-field fences, which turn harmless fly balls into home runs (Cook, 1966) - left-handed batters do better overall than right-handed batters: .2756 against .2676 in the data from Table 2. Natural left-handers, i.e., those who both throw and bat left, do best of all (.2827), while converted left-handers, i.e., those who bat left but throw right, have an average of .2687, still slightly above right-handed batters. Baseball teachers, being aware of these facts, are likely to encourage right-handed youngsters to convert to left-handed batting if they show any aptitude for it. In contrast, players who throw left-handed rarely are taught to bat from the right side. There are a few prominent exceptions, such as Ricky Henderson; but in 1995 only 1.8% of players threw left and batted right (James, 1995).

If biological realities explain the shortage of left-handed pitchers, how do we explain the actual surplus of left-handed batters? Here the logic of game theory comes into play. There is no penalty for having too many left-handed batters because the defensive side does not have enough left-handed pitchers at its disposal. Indeed, given the shortage of left-handed and the surplus of right-handed pitchers, the batting side would actually do better to abandon its mixed strategy and to play nothing but left-handed batters. The offensive value of the game at equilibrium, when both sides play their optimum strategy mixtures, is a batting average of .2683. Given the actual proportion of only 27.0% left-handed pitchers, the batting side could obtain an average of .2719 by batting left all the time. In fact, according to Table 2, batters achieved a collective average of .2703 by batting left 43.0% of the time and right 57.0% of the time.

In a sense, then, the question is, given the shortage of left-handed pitchers, why are there any right-handed batters at all? One answer may be that the defensive positions of second base, shortstop, and third base favor right-handed throwing for obvious reasons. The reasons are not so obvious for the catcher's position, but there too right-handers are almost universal. Because right-handed throwers are needed for many defensive positions, right-handed batting often comes along as part of the package.

Another reason is that the mixed-strategy game model presented here does not take account of individual variation. It treats all batters as if each one achieved the same success measured by batting average, but in reality there is a great range of abilities. Even though left-handed batters do better on average against right-handed pitching than right-handed batters do, a manager will not turn down a Frank Thomas, Albert Belle, or Mark McGwire because they bat right-handed.

An Evolutionary Perspective

While it may be obvious to managers that they should have both right- and left-handed batters and pitchers, the "correct" proportions are certainly not self-evident; and we would hardly expect baseball managers to solve the model in a formal sense as is done here. Moreover, research with experimental games has shown that players do not easily hit upon the correct mixture of strategies under laboratory conditions. Even in a symmetrical game with one player acting as tacit instructor playing the optimal mixture, the other player has no incentive to imitate the instructor. The fact that the instructor is playing optimally equalizes the second player's outcomes so that he gets the same average payoff no matter which strategy he chooses. Under these conditions, there is no reward for success and no penalty for failure, and hence no incentive for learning (Rapoport, Guyer, & Gordon, 1976).

However, it seems that an evolutionary mechanism might be posited for baseball, given a sufficiently long period of time. Assume that at the beginning of baseball in the middle of the nineteenth century, players were represented in the game in the same proportions as in the general male population, i.e., 14-86 left- to right-handed. In fact, at that time the proportion of left-handers may have been even lower because parents and schools often attempted to convert left-handed children to favor their right hand. With only 14% or fewer left-handed pitchers, left-handed batters would have feasted upon the mainly right-handed pitching they faced. It would have become obvious quite quickly that left-handed batters were an asset. But as left-handed batters proliferated, it would also have become obvious that left-handed pitchers were more adept at getting them out.

This line of thought yields two empirically testable predictions: (a) the proportions of left-handed players, both batters and pitchers, should tend to rise over time until reaching their respective optimums; and (b) the increase in left-handed batters should start before the increase in left-handed pitchers and should continue to lead until reaching equilibrium.

These hypotheses were tested against two samples from The Baseball Encyclopedia, which contains career records of all men who have ever played big-league baseball (Reichler, 1979). For each case in the sample of 174 offensive players, I noted his batting side and the year in which he entered major-league baseball (ranging from 1876 to 1974). If there ever was a time when the percentage of left-handed batters approximated the general population percentage, it must lie in the distant past before 1876. Of the 26 players who started before 1900, 38.5% batted left or switched; and of the 93 who started before 1940, 36.6% batted left or switched. More recently, there has been a slight decline in pure left-handed batters but a large increase in switch-hitters, who of course bat left most of the time. Of the position players listed in the 1996 Major League Handbook, 28.4% batted left and 16.4% were switch-hitters.

The story is quite different, however, in the sample of 222 pitchers. Of the 24 who started before 1900, only 2 (8.3%) were left-handed. In comparison, 26.8% of those who started after 1900 were !eft-handed [[Chi].sup.2](1, N = 228) = 3.3, .10 [greater than] p [greater than] .05; and for the 1995 season, 29.8% of all pitchers were left-handed (James, 1995). The evidence seems to confirm the prediction of the evolutionary model that the growth in left-handed batting should lead the growth in left-handed pitching. In approximate terms, left-handed batting reached its current level well before 1900, whereas left-handed pitching did not start to catch up until after the turn of the century.

Switch-Hitters

Up to this point, we have been using the simplified 2 X 2 model, which merges switch-hitters with others; now we must try to isolate that important group of batters, who took 19.3% of at-bats during the 1995 season. The 3 X 2 model including switch-hitters as a separate group is shown below in Table 6.

There are at least three surprises buried in this matrix. First, it is mildly surprising that switch-hitting is not a dominant strategy for batters. A switch-hitter always takes the advantageous side, batting left-handed against a right-handed pitcher and right-handed against a left-handed pitcher. Therefore, one might think that switch-hitters' results should be better than those of batters who are always stuck on the same side of the plate. However, this expectation is not confirmed. Switch-hitters do better than right-handed batters do against right-handed pitchers, but they are less successful than left-handed batters are against right-handed pitchers. A similar pattern exists against left-handed pitching, as can be verified from Table 6. Against both types of pitching, switch-hitters average .2696, a little better than right-handed hitters (.2676) but not as good as left-handed hitters (.2756).
```Table 6

Payoff Matrix for Pitcher-Batter Game, 1995

(Including Switch-Hitting as a Separate Strategy)

Pitcher
Throws Left Throws Right

Hits Left .2565, -.2565 .2808, -.2808
Batter Hits Right .2772, -.2772 .2636, -.2636
Switches .2689, -.2689 .2697, -.2697
```

A second surprise is that, far from being a dominant strategy for batters, switch-hitting is actually a dominated strategy - dominated by the optimal mixture of pure right- and left-handed batting. According to game-theoretic reasoning, a dominated strategy - one which always yields less than some available alternative, whether a pure strategy or a mixture of strategies - will never be chosen (Hamburger, 1979).

This is illustrated visually in the graphic representation in Figure 1. Each of the three batting strategies is depicted by the straight line connecting its payoff against right-handed pitching on Axis 1 to its payoff against left-handed pitching on Axis 2. The fact that the lines R and L intersect above line S shows that the optimum mixture of R and L dominates S. In general, the solution of an M X 2 game always entails a mixture of only two strategies for the player with M alternatives, and the critical pair is identified by the lowest point on the upper boundary of the set of lines (Williams, 1982).

Using the payoffs in Table 6, the mixed strategies for batting at equilibrium are 35.9% pure left-handed, 64.1% pure right-handed, and 0% switch-hitting; for pitching they are 45.4% left-handed and 54.6% right-handed. At equilibrium, the value of the mixture of left- and right-handed batting is .2697, higher than the value of .2693 that switch-hitters would earn against the optimal pitching mixture. Switch-hitting is not dominated by much, but it is dominated.

Table 7 compares the predictions of the 3 X 2 model against the actual data for 1995:
```Table 7

Comparison of Actual and Predicted Values 3 X 2 Pitcher-Batter Game,
1995

Bat Pitch Actual % Predicted %

(1) L 29.1 35.9
(2) R 51.6 64.1
(3) S 19.3 00.0
(4) L 27.0 45.4
(5) R 73.0 54.6
```

The 3 X 2 model is less successful than the 2 X 2 model on two counts. First, even though the 3 X 2 model was designed to take account of switch-hitting, it predicts that there should be no switch-hitters at all. Second, the 3 X 2 model's prediction of the pitching proportions lies even farther from the actual figures than does the 2 X 2 model's prediction.

Although far off the mark empirically, the 3 X 2 model does lead to a fruitful examination of the phenomenon of switch-hitting. Coaches recommend switch hitting to a certain kind of player - the "Charlie Hustle" type, exemplified by Pete Rose, who fields well, runs the bases, bunts, and does whatever else it takes to help the team. Mickey Mantle and Eddie Murray are well-known exceptions, but in general there have been few long-ball switch-hitters. They are generally contact hitters whose short, compact swings produce line-drive and ground-ball singles rather than home runs and extra-base hits (Weiskopf, 1985). Almost all the great natural hitters from Babe Ruth and Ty Cobb onwards have batted from only one side of the plate. It is noteworthy that in 1995, of the 25 active players with the highest career slugging percentages, none were switch-hitters (James, 1995).

This suggests that switch-hitters probably have less natural hitting ability on average but make up for it with greater practice and effort. For them, learning to bat from the left (almost all are natural right-handers; only 6.2% of switch-hitters threw left in 1995) is an extra advantage that brings them, as a group, almost up to the batting average of the whole population of players. From there they can build a career with base-running and defensive play.

Conclusions

Although neither the 2 X 2 nor the 3 X 2 model is entirely accurate in a predictive sense, the attempt to use game theory to model the duel between pitchers and batters has led to a number of interesting hypotheses at least tentatively supported by empirical data. The most important are the following:

1. Due to biological constraints, there is a chronic shortage of left-handed pitchers.

2. The shortage of left-handed pitchers makes profitable a surplus of left-handed at-bats. Thus players who naturally throw right-handed take up left-handed batting or switch-hitting.

3. Left-handed pitching increased as a response to the success of left-handed batting in the early decades of major-league baseball.

4. The recent rise of switch-hitting may be a response to the increase of left-handed pitching.

Baseball is a game in which actors make strategic choices as permitted by rules, and the outcomes are interdependent on the choices of both sides. This analysis of baseball shows that a mixed-strategy model can be used with some success to predict the proportions of strategies chosen at the population level, and the errors in the predictions can be plausibly linked to known factors of human biology. Moreover, the attempt to explain the discrepancies between predicted and actual data values point to known facts about switch-hitting and generate new and non-obvious hypotheses about the evolution of professional baseball - hypotheses that can also be supported by data. In view of these promising results, researchers may wish to consider further applications of game theory to baseball and other sports.

Note

1 After this article was in press, I received a copy of Goldstein and Young (1996). Their paper also applies game theory to baseball and reaches conclusions similar, though not identical, to mine.

2 This note summarizes the results derived from using other indicators of batting productivity and compares them to the actual data as well as the predictions derived from using batting average. (On-base percentage and slugging percentage follow standard definitions. For "total batter offence," the denominator is plate appearances and the numerator is total bases + bases on balls + hit by pitcher + sacrifice flies + sacrifice hits.)
```Indicator predicts % Bat Left % Pitch Left

On-base percentage 39.9 53.7
Slugging percentage 28.4 12.8
Total batter offence 20.5 47.5
Batting average 36.4 41.1

Actual 43.0 27.0
```

On-base percentage gives the best prediction of batting but is far off the mark on pitching. Slugging percentage does almost as well on pitching as batting average does but is less accurate in predicting batting. Overall, batting average performs the best.

References

Adair, R. K. (1994). The physics of baseball. New York: HarperCollins.

Adams, C. (1990). Why left-handed pitchers are prized in the big leagues. Baseball Digest, 49, 36-38.

Akers, M.D., & Buttross, T. E. (1988). An actuarial analysis of the production function of Major League Baseball. Journal of Sport Behavior, 11, 99-112.

Axelrod, R. (1984). The evolution of cooperation. New York: Basic Books.

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