## Game Theory: Going for Two, Part 1

Dave Reginek

Coaches generally rely on "the card", but what should their decisions really be?

When to go for two and when to kick is sometimes a hotly debated topic, but the prevailing wisdom is codified in what is simply known as "the card". Here is one early version of it, organized by the score after the touchdown (but before the conversion attempt). I don't know if this is the exact version still used today, but today's is still recognizably similar. Statistically, though, the prevailing strategy leaves much to be desired. Because there are a lot of scenarios to cover, I'm splitting this into two articles: this one focuses on decision-making when behind, the second will look at when to go for two when you're ahead.

In the first Game Theory article, I said that risks generally favor the underdog because of the increased variance and also favor the team that can adapt their strategy based on the result. The first is such a minimal effect here (going for two on every score probably wouldn't even raise the standard deviation by half a point - you're adding approximately 1 to the variance each time, but the standard deviation is the square root of that) that it can be safely ignored, but the second is rather important, especially late.

Early in the game, when the endgame situation is still unclear, both teams have a chance to adapt their strategy based on the outcome of a two-point try, but the one that benefits more is the one that needs to score more often, because in the case where that one point matters they will have more decisions to make that are aided by that information. Even so, the gain is minimal early on. In the first half, I would lean toward going for two only when one of these two conditions applies:

1. You have a very high chance of succeeding (either due to an overpowering offense or the element of surprise - Oregon, for instance, tends to line up in the swinging gate on conversion attempts and will frequently go for it on the first touchdown if they believe they've caught the opponent napping), or
2. Succeeding on a two-point try will get you to a simple margin (0, 3, or some multiple of 7) in either direction. This can backfire, but often you will end up in a scenario where it would have been necessary to go for two eventually anyway, or in making it you may force your opponent to match later (for instance, if you go for two to get to a 7-point lead and the opponent cuts it to 1 with two FGs, another TD will force them to go for two to match).

Later on, though, the endgame scenarios are a little clearer and more amenable to direct analysis. When trailing, you need to consider two probabilities primarily: your chance of making the two-point conversion (P(2)), and your chance of winning should it go to OT (P(WOT)). Sometimes, the probability of getting a FG versus a TD (P(FG) and P(TD)), either for you or your opponent, is also relevant. When you're ahead, your opponent's two-point conversion rate (P(O2)) may also matter. We will make the following simplifying assumptions:

• All of the parameters above are constants for a particular game and independent of each other.
• The team leading after this score is assumed not to score again unless otherwise specified; if they do score, apart from widening the lead they'll likely take a significant chunk of time off the clock, making a comeback very unlikely.
• There is enough time for the minimum number of scores to tie, but unless otherwise specified the probability of any scores beyond the minimum is negligible.
• If you are ahead after this score, we'll run two sets of calculations, one assuming a strategy based on conventional wisdom for the opponent and a second based on optimal strategy where that differs.
• Kicking will be assumed automatic unless otherwise specified; even awful D-1 kickers can usually make at least 95% of PATs. (Case in point: 2005's kicking game was a disaster of epic proportions, going 5 for 22 on FGs and making none beyond 35 yards, but the kickers still went 43 for 44 on PATs.)

All scenarios here are listed based on the score before the touchdown for which you are making the decision (that is, "down 24" means you will be down 17 if you kick or 16 or 18 if you go for two).

##### Down by 24

Tying the game with three scores requires all of them to be TD+2. Unless your chances of making a two-point conversion are truly dismal to the point that getting an additional FG beyond the required three touchdowns is more likely than making the first two-point conversion, going for two is obviously correct. (If you make the first and miss the second, you're no worse off than kicking both, so only missing the first hurts your chances in the four-score scenario.)

##### Down by 23

Tying the game with three scores requires two TD+2 and a TD+1. (Winning outright requires three TD+2, but you can make that decision later.) If you fail to get the required two-point conversions, you need a fourth score, and you may as well find out early. Your chances of getting the fourth score are likely to be poor regardless, but they'll be somewhat less dire if you plan for it early. As with down-24, unless your two-point conversions are nearly certain to fail such that going for four scores is actually more likely than making the first two-point conversion, go for two.

##### Down by 22

There are two ways to tie the game with three scores: a TD+2 and two TD+1, or two TD+2 and a TD+0 (failed try). The latter scenario makes going for two the first time much better than putting it off; in the down-23 scenario, the only gain to going for two early is that you know sooner if you need to resort to truly desperate measures, but here doing so gives you a backup plan if you fail.

You can make up for a failed two-point conversion here by going for two on the next two, but if you kick the first one you no longer have an opportunity to recover if you fail when you ultimately do go for two. Assuming you play for the tie (about which, see the down 14 section) and that you get the necessary three touchdowns, your probability of tying if you postpone the two-point try is P(2), but if you go for it the first time it's P(2) + (1-P(2))*P(2)^2, which is a clear win no matter what the value of P(2) is (unless it's zero). Kicking here would be even more wrong than doing so down 23 or 24.

##### Down by 21

The obvious answer is that three TD+1 gets you to overtime, and therefore there's no reason to go for two. The obvious answer is wrong. The analysis below assumes you get three TDs, because nothing matters if you don't.

If you kick all three, your probability of winning is P(WOT). If you go for two on the first, there are two scenarios to consider:

• You make it. Kick the next two and you win. Probability of winning this way = P(2).
• You miss. Go for two again on the next one to make up for it, and on the third you have the choice of playing for OT by kicking or going for the win outright with a third two-point try. Probability of winning this way = (1-P(2))*P(2)*max(P(2), P(WOT)).

The break-even point for these strategies is when P(WOT) = P(2) + (1-P(2))*P(2)*max(P(2), P(WOT)). If the left side is greater, kick and play for overtime. If the right side is greater, you should go for two. The graph below shows the break-even line; note that your chance of winning in OT must (except at the extreme low end) be significantly better than your chance of success on a two-point conversion to come out ahead by kicking. If OT is a coin flip, you only need about a 38% chance of success on two-pointers to make going for it the correct decision; if your two-point tries are coin flips, you would need to be a 2-to-1 favorite in OT for kicking to work out as well.

If you decide that you will go for two on any of the three scores (assuming that there is no point in worrying about what the opponent does if they score because you won't have time to catch up), it's important to go for two on the first touchdown. Waiting until the third allows no recovery from failure (the second term on the right side above disappears), and waiting until the second leaves you no option of playing for an outright win by going "best two out of three" on the two-point tries.

##### Down 20

This time, the obvious answer is correct. Three TD+1 is enough to win, two TD+2 doesn't even buy you the opportunity to tie with a FG. Unless your kicker is either severely injured or outrageously incompetent by D-1 standards, going for two is a big risk for an almost certainly meaningless reward.

##### Down 19

Three touchdowns win (unless the PATs result in 0-0-1 or 0-0-0), but two TD+2 gets you the chance to tie with a FG as well. Going for two is clearly correct; fail and two TD+1 still win, but if you succeed you can get to OT with a TD+2 and FG. An extra score from the opponent doesn't add any risk to going for two; if they add a FG (making you effectively down 22 now), going for two first is still correct.

##### Down 18

Again, three touchdown wins (unless all three PATs come up empty), but a two-point conversion gives you a chance to make it with a FG. If you kick, you still have the opportunity to go for two on the second TD, but you won't know until that one whether the third score has to be another touchdown or whether a field goal will be enough. The less time left when you find out, the more effectively the defense can adjust (playing prevent is only a reasonable strategy when there is a very small amount of time left) and the less effectively you can, so it is best to find out early.

Down 8 and down 11 are the most awkward no-man's-land situations for offensive strategy - in the first, you don't know if you need one score or two, and in the second you don't know if you need two touchdowns or if one of them being a field goal is enough. (The same argument could be applied to 15 or 18, but you're in the position of needing multiple touchdowns at that point anyway, so if you don't have time to find out you're already in deep trouble.) When given a choice between kicking to reach one of those scores or going for two, I would always go for two, and so it is here.

##### Down 17

This is a surprisingly interesting case. The obvious route to overtime is two TD+1 and a FG, but there may be a better way. If you kick and we assume that you score two TDs and a third score which could be a FG or TD, your odds of winning are P(TD) + P(FG)*P(WOT). If you go for two, there are three scenarios to consider:

• Make it and it doesn't matter what the third score is anymore, you win in regulation either way. Probability: P(2) * (P(TD) + P(FG))
• Miss, go for two on the second TD and succeed, and you're in the same position as if you'd kicked both. Win probability: (1-P(2)) * P(2) * (P(TD) + P(FG)*P(WOT))
• Miss, go for two and miss again, a third TD wins but a FG no longer helps. Win probability: (1-P(2))^2 * P(TD)

If you go for two, the total probability of winning is P(TD) + P(FG) * (P(2) + (1-P(2))*P(2)*P(WOT)). The P(TD) cancels out on both options, as does P(FG), leaving the same break-even line as in the down 21 case. However, if you go for two and miss, you're down 11, which is an awkward position strategically. If your next drive stalls out in field goal range, do you take the three points and gamble that you'll get the two-pointer next time, or do you press on hoping to score a touchdown and find out in advance whether a field goal is good enough? I'm not sure how much this uncertainty should count against going for two in this situation - probably not as much as it does in most coaches' minds, but it's not totally negligible like it is for down 21. Thus, if I think I'm close to the line I'd probably kick even if I suspect I'm on the "go for it" side of the line.

##### Down 16

Two TD+2 is the only way to tie with just two scores. If there's sufficient time to get three, two TD+1 and a FG (or TD+2, TD+0, FG) wins; TD+0, TD+1, FG ties; two TD+0 requires a third touchdown. Assuming only two scores, you must go for two. If you wish to consider the three-score possibilities, kicking succeeds if and only if you get a third score of any sort (P(FG) + P(TD)). If you go for it:

• Make it: TD+2 ties, third score wins. Win probability: P(2) * (P(FG) + P(TD) + (1-P(TD)-P(FG))*P(2)*P(WOT))
• Miss: A third TD wins regardless, but with a TD and FG, you can either play for OT or go for 2 and the win in regulation. Win probability: (1-P(2)) * (P(TD) + P(FG)*max(P(2), P(WOT)))

The break-even line here (after canceling P(TD) from both sides) is P(FG) = P(2)^2*P(WOT)*(1-P(FG)-P(TD)) + P(2)*P(FG) + (1-P(2))*P(FG)*max(P(2), P(WOT)). If the left side is greater, kick; if the right side is greater, go for two. This is a pretty messy equation, but the chart below shows example lines for cases where the probability of a third score is 5%, 10%, or 20%, either always a FG or split evenly between FG and TD cases. When the probability of a FG for your third score is high, kicking becomes a better option, but even at 20% to get a FG for a third score, a team with normal two-point success rates and a 50-50 shot at winning in OT should go for two.

##### Down 15

To tie, you need a TD+2 and a TD+1. Because your strategy will be very different if you need three scores than it is if you need only two, it is best to find out early by going for two now. Unlike down 16, the three-score case doesn't put pressure on you to consider kicking - if you fail the two here, a TD+1 and FG still wins outright.

##### Down 14

As with the down-21 case, the obvious answer (kick twice and go to OT) is wrong. If you kick both, your win probability is P(WOT). If you go for two the first time, either you make it and kick the second to win (P(2)) or miss and go for two again to try to force overtime ((1-P(2))*P(2)*P(WOT)). This produces the same decision chart as down 21.

##### Down 13

Going for two gains nothing unless the opponent scores again (or you kick two FGs instead of getting a TD). Unless your kicker is abysmal, kick - if P(2) and P(WOT) are both 50%, you only gain by going for two if your kicker is worse than 75% on PATs. If your opponent scores a FG to make it effectively 16, you'll wish you had gone for two, but the odds of that (especially late in the game) aren't high enough to overcome the advantage of kicking when they don't score.

##### Down 12

As with 13, going for two here has no immediate gains - but it doesn't have any immediate losses either (a TD+1 still wins even if you fail) and may provide a benefit if the opponent gets a FG (instead of being down 8 and not knowing whether you need one score or two, you're down 7 or 9 and know). Early on, when the endgame situation is unknown, it's probably still right to kick, but late in the game going for two is probably correct.

##### Down 11

If you go for two and succeed, a FG ties. Kick or fail the two and you need a touchdown. Going for two is clearly correct. Sure, if your opponent kicks a FG you may find yourself chasing the "lost" point later if you fail; on the other hand, a TD+1 would now win outright if you succeeded. (The analysis works out the same as in the down 14 situation.) A more interesting question is what to do on 4th down in FG range, but that's a topic for another day.

##### Down 10

If you go for two, a FG for your second score either wins or is useless. Kick, and a FG ties the game. A second TD wins regardless. If you kick, your win probability is P(TD) + P(FG)*P(WOT). Go for two:

• Make it and a FG wins. Probability: P(2) * (P(TD)+P(FG))
• Miss and you need a TD. Probability: (1-P(2)) * P(TD)

Here (after canceling terms), the break-even line is P(WOT) = P(2). However, if the opponent kicks a FG at some point, the go-for-2 case becomes all risk, no reward (see the down 13 scenario). If there's enough time that conceding a FG is not necessarily "game over", you probably want to err on the side of kicking if it's a close decision.

##### Down 9

A FG wins if you kick or make the two; if you fail, a FG only ties. Kicking is clearly a better choice, though as with 13, the opponent scoring again (a touchdown this time) might make you wish you had gone for two.

##### Down 8

A two-point conversion ties the game; kick and you have to score again. I don't really have to explain this one, do I?

##### Down 7

Kicking ties the game; a two either puts you ahead or leaves you needing another score. With no time left (or going second in OT), the comparison is straightforward (as with down 10, P(WOT) = P(2) is the break-even line).

With a small amount of time, the possibility of another score complicates things. Your probability of winning with a two-point try becomes P(2)*(1-P(opponent scores)) + (1-P(2))*P(you score again), while a kick becomes (1-P(opponent scores))*P(WOT), assuming the opponent will either score or run out the clock. Additionally, P(opponent scores) is not necessarily the same in those two scenarios; a team that is tied might sit on the ball and wait for OT, but if you take the lead that's not an option anymore. If the probability of the opponent scoring is small and not too different between the two cases, you can probably safely ignore the correction and stick with the simple P(WOT) vs. P(2) comparison, but if your opponent has a good quick-strike offense and it's a close decision without the correction, you may want to kick and try to bait them into playing for OT instead of forcing them to rely on that strength. (If nothing else, if you force a fourth down they might consider punting, which they can't do if you are leading.)

##### Down 6

A kick or made two puts you ahead; failure leaves it tied. Going for it only gains if you manage to give up a safety and costs you big if you fail, so kicking is obviously correct.

That's all for Part 1; Part 2 will cover scenarios from down 5 to up 17. To summarize so far:

• Always go for two when (prior to the touchdown) you were down 8, 11, 15, 18, 19, 22, 23, or 24.
• Always kick when you were down 6, 9, 13, or 20.
• Almost always go for two down 16; only consider kicking if you think your chances of getting a third score (especially a FG) without the opponent scoring are high and your probability of winning in overtime (plus 2 two-point tries to get there) is very low.
• Down 14, 17, or 21, you should go for two unless you are significantly more likely to win in OT than to succeed on the first two-point try; you can recover from failure and avoid OT entirely with success. Be somewhat more conservative down 17 than 14 or 21, however.
• Down 7 or 10, you should go for two if your chances of making the two are better than your chances of winning once the game gets to OT. With a moderate amount of time left down 10 or a short amount of time against a quick-strike offense down 7, though, err on the side of kicking.
• Down 12, kick early but go for two late.

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