The very concept of bunting has become anathema in the minds of many performance analysts these days. The argument is summed up as follows: "If you play for one run, you will only score one run." Play for the big inning and forget the sacrifice bunt.

Of course, sometimes you only need one run - as in a tie game in the bottom of the ninth. But sometimes you need two. And sometimes you're just looking for a little insurance to pad a lead. In these situations, the statistically-oriented break out their expected-runs matrix (XRM), win-expectancy charts, and TI-83s and squeal until they're blue in the face.

I'd like to propose a more sophisticated framework for studying the in-game decision-making. Part of me is interested in the search for baseball truth, but mostly I just want to see a manager typing away at a laptop before he calls for a bunt or stolen base. The easiest way to illustrate the point is via example (proof by example? I never claimed to be a mathematician). Maybe when I'm unemployed and/or washed-up as a scientist, which shouldn't be too long from now, I'll write a computer program to generalize the issue. Until then, I have neither the chops nor the time for such an effort.

* * * * * * * * * * * * *

You're Mike Scioscia (keep reading, I promise it gets better), it's the bottom of the ninth and you're trailing by one run at home. Vlad Guerrero works a walk and Garret Anderson singles up the middle, and now you have runners on first and second with Darin Erstad coming to the plate. Most managers would be frozen with horror at the the thought of Rex Hudler attempting to analyze their moves, but you're Mickey Scioscia, former manager of the year, and you wonder: do I let Darin Erstad swing away and risk the double play? Or do I recognize Erstad's bat-handling abilites and call for the sacrifice bunt?

You have been given a fortune (the expected runs of this situation) to invest. Since you understand the intricacies of Moneyball, you pull out your Dell Inspiron 1150 and, after Bud Black points out the power button, make a little Excel Spreadsheet:

Initial situation:     Home team, down by 1, runners on 1st and 2nd, ninth inning, 0 outs
Initial Fortune:       1.473
Initial Utility:       0.542


Investment A: Bunt the runners over
Outcome     Probability   New Situation           Return   
Sac Bunt        90.0%     -1, 2nd/3rd, 1 out      1.435    
Failed Bunt      7.0%     -1, 1st/2nd, 1 out      0.939    
Error            3.0%     -1, 1st/2nd/3rd, 0 out  2.343    
                            Expected Return    1.428    

Investment B: Swing away
Outcome     Probability   New Situation           Return   
BB              7.96%     -1, 1st/2nd/3rd, 0 out  2.343    
1B             19.47%     -1, 1st/2nd/3rd, 0 out  2.343    
2B              4.96%      0, 2nd/3rd, 0 out      3.081    
3B              0.53%     +1, 3rd, 0 out          3.570    
HR              0.89%     +2, 0, 0 out            3.520    
K              16.60%     -1, 1st/2nd, 1 out      0.939    
Out            34.60%     -1, 1st/2nd, 1 out      0.939    
Sac Fly         5.00%     -1, 1st/3rd, 1 out      1.207    
Productive Out  3.00%     -1, 2nd/3rd, 1 out      1.435    
GiDP            7.00%     -1, 3rd, 2 out          0.373    
                            Expected Return    1.456    

Return  = Change in Expected Runs + Runs Scored in new situation

Note: Erstad stats are for 2005.  Sac Bunt, Failed Bunt, Error, Sac Fly, Productive Out, and GiDP
estimated

"Oh yeah?" you say.

"Yeah. When you benched me last week, Adam Kennedy loaned me his copy of An Elementary Introduction to Mathematical Finance."

"Hey, is that the book by Professor Sheldon Ross published by the Cambridge University Press?"

"Second Edition," says Steve Finley, smiling. "Anyway, I was reading the section on Utility Functions - chapter 9 - and I think it will help you out here. The spreadheet you made is an attempt to maximize the return on your investment, where, in this case, the investment is the to-bunt-or-not-to-bunt strategy and the capital is your current expected runs."

"Well, yeah, that's what Ken Macha was telling me at dinner last night - the A's decide on strategy by trying to maximize the return on their investments."

"Not quite, Coach. What you probably heard is that the A's try to maximize the utility of their investments."

"What's the difference?"

"Well, the idea is not to maximize the return you receive on the investment, but to maximize the utility - i.e., the chance of winnning - of your investment. A Utility Function (UF) is an economic concept that is often used in choosing investments. Let me give you a quick example: you have $1000 and wish to invest it. There are two choices - keep your $1000 dollars under the mattress, or play Uncle Sal's riverboat gambling game. In the gambling game, you can win $10,000. The catch? You only have a 10% chance of winning. If you are risk-neutral, both choices are the same, since the expected return on your investment is $1000."

"Hmmm...I would still rather keep the $1000," you say.

"That's what most people would do, because they are naturally risk-averse. Now I propose another game in which you have a 1% chance of winning $2,000,000. In this case, the expected return on the gambling game is $20,000 (1% * $2MM) - far more than the $1000 you started with - but I suspect you would not take the bet. This is classic risk-averse behavior. A UF is a way of codifying, based on your personal financial position, of how much risk you are willing to assume. A UF tells you at what odds/payoff you are likely to change your mind and take the gamble instead of stashing the $1000 under the mattress."

"Okay," you say, intrigued. Finley never shows this kind of acumen at the plate, but what he is saying is interesting. "Go on."

"How does this apply to baseball? The UF in baseball is very easy to define because there is one desired outcome: to win the ballgame. The utility of a particular strategy is the how it changes your probability of winning the game, or win-expectancy (WX). I was surfing around SBNation and I've seen plenty of WX charts, but in case you haven't, check out Lookout Landing for examples. The problem with the XRM is that sometimes they will sucker you into playing Uncle Sal's gambling game."

Finley sidles next to you on the dugout bench and starts tapping away at your Dell, occassionally using the hi-speed wireless internet connection that Arte Moreno installed at the behest of PartyPoker.com obsessed Frankie Rodriguez.

"What's that website you're using?" you ask Finley.

"Oh, it's the Win Expectancy Finder by Chrisopher Shea and Phil Birnbaum. I'm going to add a column to your spreadsheet that shows the change in win-expectancy - that is, the utility - of each move and the overall expected utility from each strategy. Here we go:"

Initial situation:     Home team, down by 1, runners on 1st and 2nd, ninth inning, 0 outs
Initial Fortune:    1.473
Initial Utility:    0.542

Investment A: Bunt the runners over
Outcome     Probability   New Situation           Return   Utility
Sac Bunt        90.0%     -1, 2nd/3rd, 1 out      1.435    0.563
Failed Bunt      7.0%     -1, 1st/2nd, 1 out      0.939    0.390
Error            3.0%     -1, 1st/2nd/3rd, 0 out  2.343    0.614
                            Expected Return    1.428    0.552

Investment B: Swing away
Outcome     Probability   New Situation           Return   Utility
BB              7.96%     -1, 1st/2nd/3rd, 0 out  2.343    0.614
1B             19.47%     -1, 1st/2nd/3rd, 0 out  2.343    0.614
2B              4.96%      0, 2nd/3rd, 0 out      3.081    0.881
3B              0.53%     +1, 3rd, 0 out          3.570    1.000
HR              0.89%     +2, 0, 0 out            3.520    1.000
K              16.60%     -1, 1st/2nd, 1 out      0.939    0.390
Out            34.60%     -1, 1st/2nd, 1 out      0.939    0.390
Sac Fly         5.00%     -1, 1st/3rd, 1 out      1.207    0.507
Productive Out  3.00%     -1, 2nd/3rd, 1 out      1.435    0.563
GiDP            7.00%     -1, 3rd, 2 out          0.373    0.150
                            Expected Return    1.456    0.479

Return  = Change in Expected Runs + Runs Scored in new situation
Utility = Win Probability of new situation

"See coach," says Finley, "Even though bunting the runners over gives you fewer Expected Runs than letting Ersty swing away, it actually decreases the expected utility. A bunt gives you a 55.2% chance of winning and letting Erstad swing away gives you only a 47.9% chance."

"Why is that?"

"It's simple: at this stage in the game, one run is worth quite a lot, or in the parlance of economics, it has a lot of utility. If this were say, the seventh inning, the difference would not be so dramatic. Let me show you:"

Initial situation:     Home team, down by 1, runners on 1st and 2nd, seventh inning, 0 outs

Initial Fortune:    1.473
Initial Utility: 0.544

Investment A: Bunt the runners over
Outcome     Probability   New Situation           Return   Utility
Sac Bunt        90.0%     -1, 2nd/3rd, 1 out      1.435    0.509
Failed Bunt      7.0%     -1, 1st/2nd, 1 out      0.939    0.444
Error            3.0%     -1, 1st/2nd/3rd, 0 out  2.343    0.711
                            Expected Return    1.428    0.511

Investment B: Swing away
Outcome     Probability   New Situation           Return   Utility
BB              7.96%     -1, 1st/2nd/3rd, 0 out  2.343    0.711
1B             19.47%     -1, 1st/2nd/3rd, 0 out  2.343    0.711
2B              4.96%      0, 2nd/3rd, 0 out      3.081    0.745
3B              0.53%     +1, 3rd, 0 out          3.570    0.848
HR              0.89%     +2, 0, 0 out            3.520    0.916
K              16.60%     -1, 1st/2nd, 1 out      0.939    0.444
Out            34.60%     -1, 1st/2nd, 1 out      0.939    0.444
Sac Fly         5.00%     -1, 1st/3rd, 1 out      1.207    0.515
Productive Out  3.00%     -1, 2nd/3rd, 1 out      1.435    0.509
GiDP            7.00%     -1, 3rd, 2 out          0.373    0.287
                            Expected Return    1.456    0.533

Return  = Change in Expected Runs + Runs Scored in new situation

Utility = Win Probability of new situation

"Wow, Steve. I think I'm starting to get this. Thanks a lot. How can I return the favor?"

"Start me tomorrow night?"

"Yeah, right."

* * * * * * * * * * * * *

Sometimes you want the big inning. Sometimes, you only need one run. And sometimes you need two. When combined with player data, Win-Expectancy is a way to quantify what strategy will be the most effective. Managers can manage with the confidence with which Warren Buffet invests. It is often said that baseball is economics, with outs and runs being the currency. Applying an elementary economic concept like the utility function is a good place to start.

Notes: The estimates made will affect the numbers but I do not believe they affect the conclusion. There are limitations here, based on right/left splits, quality of pitcher, etc. As with all statistical reasoning, there is a delicate balance between the specificity of the situation and the size of the sample. Still, the concepts behind this study remain the same. I encourage others to play with different situations that call for stolen bases, bunts, etc.