THE BOOK--Playing The Percentages In Baseball

It might tell you something, but they take it for the wrong reason. The reason those coaches are punting is because they are risk averse, they prefer the safe choice. The Saints would choose the punt not because it’s a better percentage play for them, but because it’s a guarantee they get the ball.

In baseball some defenses would rather a player hit than bunt even though the hitting is a better percentage play because they are scared that the base they give up might cost them the game.

Something about this whole debate has bugged me. Burke uses expected value to boil the win probability of going for it down to 47%, and since 47% > 42%(the win probability if you punt) you should go for it, case closed.

This is certainly true in the long run, over many, many iterations. But the Falcons were making a decision in the present, not in the long run. The Falcons were not choosing between 42% or 47%, they were choosing between 42% and (57% or 18%). The 47% win probability doesn’t really exist in a single trial, so to base your decision on that number seems rather foolish.

Personally, it feels to me that the right way to approach the decision is to note that neither a 42% or 57% win probability are far removed from a coin flip, so why on Earth would you run the risk of dropping to an 18% win prob if the upside is still effectively a coin flip ?

I guess this is all a clumsy way to say that it feels like only the reward is being considered by those who advocate going for it, they are ignoring the risk. The decision to go for it really should be based on your risk appetite. If you are the underdog or the road team, or a key player has been hurt, basically if you have some reason to believe you are at a disadvantage then it seems your risk appetite should be high. You want to shorten the game, because small sample sizes are your friend. But if you have an advantage, you are the favorite or are at home, etc. you should be more risk averse. Large sample size is your friend. In the long run you should win, and you should avoid scenarios that effectively shorten the game.

TBW: you are so wrong, I don’t even know where to begin.

However, I do appreciate you making the point, because I’ve seen this point made a TON of times.  So, you speak for a huge group of people out there.

I’m hoping a Straight Arrow reader tries to come at it from a different angle, and straighten you out.

So, hold your thought, and let’s see how people can correct your thinking.

Maybe I don’t quite understand what win probability represents.

To rip off Taleb a bit, if there’s a $100 bill in the path of a steamroller and I have a 99% chance of getting to it before the steamroller, and a 1% of getting squashed then the expected value is $99. I feel like Burke’s argument amounts to my choice is between having $0 or $99, so obviously run out there and pick up the bill, it’s a no brainer. But in reality my choice is to choose to have $0 or (having $100 or being dead). The other possible result is so distasteful, at least for me, that I, and I suspect most people, would gladly forgo the chance at $100. In other words the reward simply doesn’t warrant the risk, despite the fact that the expected value is positive.

Am I digging my hole deeper ?

Death has a huge negative value, so that’s not a good analogy.

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All that matters is what is the chance you have of winning the game… by the end of the game.

Perhaps the best analogy is stealing a base: you add a small chance of winning if you make it, but kill yourself if you are thrown out.  Basically, your chance of winning goes up one step, but the chance of losing goes down two steps.

BUT, if the frequency of success is 66.7%, then you’ve reached breakeven.  If the frequency of success is 75%, then you TRY TO STEAL.  Yes, being CS really hurts you, and being successful helps you only half as much, but you are successful 3 out of 4 times.

The ONLY thing that matters is the chance of winning the game.

@TBW

The only thing that “exists in a single trial” is 1 whole win or 1 whole loss. The 42% chance of winning doesn’t “exist in a single trial” either.

So it’s not like choosing between $42 guaranteed vs. 75% chance at $57 and a 25% chance at $18. It’s like choosing between a 42% chance of $100 and a 47% chance of $100.

Yes, the second scenario requires two separate rolls of the dice—the first roll determines whether the odds of winning the final roll is 57% or 18%—but the overall odds of winning your $100 in the second scenario is 47%, and the first scenario still involves rolling the dice with only a 42% chance of winning $100.

Would you rather roll the dice with a 42% chance of winning $100 or roll the dice with a 47% chance of winning the same amount?

@TBW

And while it is true (in an abstract sense) that favorites prefer longer games, the idea doesn’t apply to this scenario. The remainder of this game had a fixed (and extremely short) duration.

The choice was between two endgames, one of which gave the team a 47% chance of winning and one of which gave the team a 42% chance of winning.

"Personally, it feels to me that the right way to approach the decision is to note that neither a 42% or 57% win probability are far removed from a coin flip, so why on Earth would you run the risk of dropping to an 18% win prob if the upside is still effectively a coin flip ?”

While it may “feel” like the difference between 42 and 57 is small, while the difference between 42 and 18 might “feel” large, that feeling isn’t helping you make good decisions.

In fact, the difference between 42 and 57 is 15, which is significant. The difference between 42 and 18 is 24, which is larger than 15, but not so horribly large that you should rule out the possibility of risking it.

What you’re feeling is loss aversion, which is a natural and normal thing for humans to feel, but is not useful in the context of a probability-based decision where you MUST roll the dice one way or the other.

Your hands are tied. There’s no scenario in which you’re guaranteed some intermediate outcome like 0.42 wins. You have to roll the dice, and the feeling that a 2-stage loss in which you first “lose” 24% of your win probability before losing the game is somehow “more painful” than any other loss is not a useful feeling.

re 6: I think you hit the point that I was stumbling on, the 42% is in itself an expected value. If the punt is 50 yards net, maybe the win prob is actually 50%, and if the punt nets just 25 yards maybe the win prob declines to 30%. The use of a single punt scenario obscured(for me anyway) the fact that the punt scenario was essentially a weighted average of all possible punt scenarios.

re 8: Since there is a clock in football, and since it was a regular season game, a tie is an option. In that sense you don’t have to keep rolling. Since different play selections presumably have differently shaped win probability distributions(passing has more big positive and negative plays than rushing), couldn’t one opt to try and minimize loss probability rather than maximize win probability and make different decisions as a result ? If you were literally playing not to lose you would presumably call running plays which would minimize the risk of a catastrophic play, and keep the play clock running thus shortening the game.

re 5: I get your point about stealing, but you don’t just say MLB baserunners are successful 75% of the time, so Jason Giambi should always have a green light every time he draws a walk. Context is everything, the quality of the pitcher’s move, the catcher’s arm, how fast the baserunner is, whether the baserunner has the flu, the score of the game, those are all critically important factors. The numbers Burke presented were very generic numbers, there is no reason to believe they applied in the particular situation. I would agree that they are strong evidence that the Falcons weren’t crazy to go for it, but the advantage that Burke claims exists by going for it could easily be wiped out by other factors.

Very few games end in ties.  More to the point, very few quarters end with no points scored, even ones where there was no score in the first two (very short) drives.  So, the probability of a tie is extremely low, to the point where it should be ignored at the time of the decision.  When it gets very late in OT, then the tie is a concern, but at this point P[Lose] = 1 - P[Win], so there’s no difference between “Not Losing” and “Winning”.

Right, exactly.  Brian had the process down correctly.  It’s then a question of coming up with the correct inputs for each of his assumptions.

Yes, we may expect Chase Utley to steal 75% under certain conditions, but it could very well be 50% under other conditions.

The point first, is to get the framework down.

Then, plug in average numbers to get a baseline.

Finally, put in more real numbers.

And, right, it shows they weren’t crazy.

TBW/2 - It seems you’re breaking things down to one extra step.

In the choice between 42% and 47%, the 47% is a weighted average of the 18% WP of failing to make it, and the 57% WP of succeeding, with the weight being the probability of making the first down. So that 47% is taking into account the risk as well as the reward.

Now you might quibble with the weighting: plug in a different percentage for making the first down and you may come to a different conclusion on whether to go for it or not.

Remember that whether a decision is correct or not is independent of whether it proves successful. Sometimes you make the right call but lose, or make the wrong call and win despite that.