It's 4th and 1 from your opponent's 43-yard line. You're up 3 points, and there is 5:16 remaining in the 1st quarter. Should you go for it?
According to the 4th Down Calculator, the answer appears clear. Based on history, there is an estimated 74% chance of converting a 4th down in that scenario. Success yields 0.68 WPA; punting yields 0.61 WPA, and failure yields 0.55 WPA.
These odds tell you that on average, it is a good decision to go for it - just the same as on average you'll make money if you take a bet with those odds and that probability of winning. The "Expected Value" (EV) in this scenario is 3.62. This means that on average, you will gain .0362 WPA by going for it.
However, average doesn't always cut it. Because if it's not certain, you could still lose.
EV enthusiasts often object to that observation, but let's briefly consider an alternate scenario. Pretend that you come across a certain state lottery. For $1, you have a chance of winning $500,000,000 profit. And your chances of winning are 1 in 350,000,000. (Also pretend, for the sake of argument, that there's a new identical lottery every second, and there can't be multiple winners in a round.) Since 500 > 350, those are good odds. Say you can play only once. Should you buy a ticket? What if you could play multiple times, or buy multiple tickets? Should you spend your $5,000 in hard-earned savings on lottery tickets?
The answer is, of course, no. If you buy one ticket, you'll probably lose. If you spend $5,000 on one lottery, you are still more than 99.99% likely to lose. If you spend five MILLION dollars on lottery tickets, you are still roughly 99% likely to lose. In all those cases, you're very likely to exhaust all your bankroll before you can reach the expected value, unless you are already extremely rich. And this is all true even though the lottery game has an obviously positive EV.
So clearly, a positive EV isn't enough. What else do we have to factor in if we make a decision based on EV? This illustrates two key points about Expected Value:
• There has to be an expectation of multiple rounds, probably very many.
• You have to be able to afford the ideal bet.
To calculate the ideal bet, one tool to use is the Kelly Criterion. And in a normal bet, if you bet too big, you might lose enough of your bankroll that you can't bet again in a later round. If you are faced with a +EV scenario, such as a bet with good odds, the Kelly Criterion tells you how much you should bet as a percentage of your bankroll. The Kelly Criterion formula is (pb - q) / b. p and q are the odds of success and failure, respectively. b is the odds. The reason the Kelly Criterion works is because it is mathematically calibrated to maximize one's expected growth rate. It will eventually outperform every other strategy of how to bet when the odds are in your favor.
So how could this be used for football? Well, a football team wants to maximize their "football dominance", as in their chances of winning. Let's look again at the 4th-and-1 situation described above. Treating the punt as the default, risk-free outcome, the 4th-down calculation is in effect considering a bet, where you bet .06 WPA (failure) for a .07 WPA profit (success). The odds become 7/6.
The calculation becomes ((0.74 * 7/6) - 0.26) / (7/6) = 51.7% In other words, in a bet with those odds and chances of winning, you should feel comfortable betting 51.7% of your bankroll - or less, to be conservative. But NOT more, because it can exhaust the bankroll too quickly. In fact, if you bet more than the Kelly Criterion suggests, your average expected growth will be negative even when the odds are in your favor!
So if you're John Fox facing 4th and 1 from Baltimore's 43-yard line, and your Broncos are up 3 points, and there is 5:16 remaining in the 1st quarter, you should feel comfortable betting 51% of your bankroll. But what is a football coach's bankroll?
That's tougher to gauge. Maybe a bankroll is defined as how many times a coach can make (and fail) a controversial football decision before being fired or losing credibility. That could make sense, because a secure coach would feel more latitude to make risky decisions than one that is on the hot seat. After all, "no one ever got fired for punting on 4th down", as the saying goes.
But maybe the bankroll is something else. Let's look again at what Expected Value really means. It means that on average, your results will be in line with Expected Value. If you miss some early on, that's okay, because you'll catch up later. It will average out in the long run. However, the long run is cold comfort if you miss while on the verge of the playoffs at 10-6, and "catch up" during a pointless 5-11 season.
The real rational goal of a football coach is to maximize the chances of winning *each* particular football game. If that is what the coach is trying to maximize, then the bankroll could be described as how much influence a coach has over one particular game. They are trying to convert their influence into WPA. And while EV assumes the presence of a long run, in reality, a coach's influence over a particular game is rather limited. One or two failures might be sufficient to guarantee a loss.
This gives new insight into why a coach might *justifiably* want to punt the ball even if the EV suggests he should go for it. And it doesn't rely on the common explanations of momentum, or gut feeling, or unquantifiable "context". It shows that if a coach's goal is to use Expected Value to maximize the odds of winning *each game*, the coach's "bankroll" of "coaching decisions" might not be large enough to justify taking the bet. After all, if there might only be one +EV 4th-down scenario in the game, that would be betting 100% of the bankroll. Additionally, to be reasonably sure that it is a safe bet, the coach would have to feel comfortable that they face a high enough number of "rounds" to have a reasonable chance of reaching their EV.
Even beyond the bankroll, when we're talking about the context of one game, we're starting to run afoul of one of the main requirements in paying attention to EV: The need for multiple rounds.
Whipping out the old probability calculator, we can see that our 74% bet has a high bar to clear.
• To be 99% sure that you will eventually win a 74% bet, you'd have to face the scenario not once or twice, but four times. And even then, if you lost the first three times and won the fourth, you would still be "under water" in terms of EV.
• To be 99% sure that you will break even in terms of EV (this just means a net positive, not that you will average your EV of 3.62 WPA points), you'd have to face the scenario eight times.
• To be only 90% sure that you will eventually reach cumulative EV (that's 3.62 WPA points for each round), you'd have to face the scenario sixteen times.
From *our* perspective, as stats admirers and football fans, we are looking at it in the context of many coaches, many plays, many seasons - many rounds. Our "bankroll" is effectively infinite and there is no cost if we're wrong, so we're right that plays should always be called in accordance to EV. But when you're a coach looking at it in terms of 1-2 seasons of employment, or just one game's worth of plays, it really does change the equation in completely valid ways. The next time you see a coach choose to kick on 4th down even when the +EV says otherwise, he might actually be making a probabilistically valid decision.