In the most recent article of my multi-part series on game theory and baseball, I applied mixed strategies to pitch selection. I considered a situation that I will use for several articles—a full count, with a pitcher deciding to throw a ball or a strike, and the batter simultaneously deciding whether to swing or take.

In the previous article, I constructed an approximate normal form set-up of payoffs under each situation, which I have duplicated below. The batter is equally well off when the outcome is strike/swing as when the outcome is ball/take, and the pitcher is equally well off when the outcome is strike/take as when the outcome is ball/swing.

**Table 1**

PitcherBatter | Swing | Take |

Strike (Fastball) | 1,-1 | -1,1 |

Ball (Curveball) | -1,1 | 1,-1 |

In this situation, we found the equilibrium by constructing players’ payoffs conditional on the other player’s strategies. We defined V(s) as the batter’s value of swinging and taking, respectively, conditional on the pitcher selecting a probability “p” of throwing a strike:

V(s) = (p)*(1) + (1-p)*(-1) = 2p – 1

V(t) = (p)*(-1) + (1-p)*(1) = 1 – 2p

The pitcher’s value of throwing a strike and a ball, respectively, conditional on the batter swinging with probability “q” were:

V(g) = (q)*(-1) + (1-q)*(1) = 1 – 2q

V(b) = (q)*(1) + (1-q)*(-1) = 2q – 1

The only equilibrium occurred when the batter set q=0.5 and the pitcher set p=0.5, leading them both to have no better response to the other’s strategy. The batter selected “swing” 50 percent of full count pitches, and the pitcher selected “strike” 50 per cent of full count pitchers.

### Great curveballs

Now, let’s tweak that scenario: Suppose that the pitcher has an amazing curveball, which is nearly unhittable out of the strike zone. We will re-interpret the strike/ball decision as a choice between throwing a fastball for a strike or a curveball for a ball. The difference between the new situation and the one above is that the payoff to the batter of swinging at a ball is even worse than -1: it’s now -1.5.

The question that we want to ask is not “what would happen if the payoff is 50 percent worse from strike/ball?” but rather is “what differences should we expect to see between pitchers with great out-pitches and pitchers with average out-pitches?” The answer may surprise you.

The normal form is as follows:

**Table 2**

PitcherBatter | Swing | Take |

Strike(Fastball) | -1,1 | 1,-1 |

Ball (Curveball) | 1.5,-1.5 | -1,1 |

Let’s re-solve the equilibrium. The batter’s payoff to swinging, conditional on the pitcher’s probability “p” of throwing a strike is:

V(s) = (p)*(1) + (1-p)*(-1.5) = 2.5p – 1.5

The value of taking, conditional on “p” is:

V(t) = (p)*(-1) + (1-p)*(1) = 1 – 2p

The batter will swing all the time if 2.5p – 1.5 > 1 – 2p, which is when p > (5/9). He will take all the time when p < (5/9). So the pitcher’s strategy that will make the batter indifferent between swinging and taking (thus being willing to randomize) is when p = (5/9).

The pitcher’s value of throwing a strike, conditional on a batter’s probability of swinging “q” is:

V(g) = (-1)*(q) + (1)*(1-q) = 1 – 2q

His value of throwing a ball is:

V(b) = (1.5)*(q) + (-1)*(1-q) = 2.5q – 1

The pitcher will prefer to throw a fastball for a strike when 1 – 2q > 2.5q – 1, which is when q < (4/9). The pitcher will prefer to throw a curveball for a ball when q > (4/9). The pitcher is indifferent when q = (4/9).

Therefore, the only equilibrium is when the batter swings 44 percent of the time and the pitcher throws fastballs 56 percent of the time. In other words, the pitcher with the fantastic out-pitch should throw it less often than the pitcher with the average out-pitch in full counts!

This is probably surprising. After all, pitchers have a tendency to rely more on their great pitches when they reach two strike counts. However, if batters were responding optimally, they would know how unhittable these pitches were and keep the bat on their shoulder more often. To entice the hitters to be willing to take a hack, pitchers should throw more fastballs. When they do throw curveballs, batters will be caught off guard.

The batters, on the other hand, should take more pitches, knowing that swinging too much in these counts will only encourage the pitchers to throw the hook. If they keep the bat on their shoulder more often, the pitcher is likely enough to give them a fastball when they do swing. Therefore, pitchers with great curveballs may actually get more called third strikes on fastballs than pitchers with mediocre curveballs.

Extrapolating this to other two-strike counts, the frequency of failed 0-2 and 1-2 pitches to land for strikes in major league baseball suggests that pitchers are probably over-relying on their great pitches, so much so that they are losing their potency.

### Discussion

It is not actually clear who is behaving sub-optimally in real baseball. It could be that batters are swinging at too much junk in an effort to save face, choosing to go back to the dugout knowing that they at least went down swinging. Of course, it could be that pitchers are afraid to get beaten when the count is in their favor, and instead lose their advantages by predictably throwing pitches out of the strike zone in two-strike counts.

Does all of this mean that batters and pitchers are foolish? Not necessarily—I made significant simplifications to formulate these set-ups. It is totally plausible that players may be strategizing incorrectly. Consider the usual reasons why you would trust an expert in other arenas. For instance, if I try to determine an optimal price for a big business in a competitive market, and come up with a price that is significantly different than what the business is charging, I am probably wrong. In a free market, a challenger start-up that profitably undercuts the market leader’s price will quickly begin to dominate the market. One who can wisely over-price the incumbent firm may be able to rapidly gain enough market share to hire its competitor’s employees away. The free market mechanism of “free entry” leads to an optimal price.

In fact, one would not need to use calculus to determine optimal pricing strategies. A business could simply be following a “rule of thumb” such as a 20 percent mark-up that is equivalent to an optimal mathematical derivation. For example, using calculus, the “rule of thumb” mark-up rule is equivalent to setting the derivative of the revenue function equal to the derivative of the cost function, under conditions of a certain constant elasticity demand curve. Effectively, business executives behave like economists without knowing it.

Why doesn’t this same line of reasoning apply to baseball? The answer is that there is no realistic free entry. I can do all the math and declare that Aroldis Chapman should be throwing fewer fastballs in an effort to blow hitters away when he does throw them, but I can’t enter the market. I don’t have a 103 mph fastball. If he’s throwing too many fastballs, there is no savvy new executive ready to step in and steal his customers. Limited entry of pitchers with fantastic fastballs means that the free market won’t solve this one. Aroldis Chapman throwing too many fastballs is still a better pitcher than me throwing the perfect mix of batting practice fastballs and hanging curves.

In the next article we will relax another assumption—we will allow hitters to take a look at the ball while it is on the way, and react accordingly. Let’s give the hitter a chance to take an educated guess, and then let’s give him a little more education.

**Tomorrow: If batters were Bayesians**

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