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# Game Theory: Going for Two with the Lead

When is it right to try a two-point conversion while you're ahead?

Two years ago, I wrote an article on when to go for two when trailing. There's a lot of math involved there (and this one will have a bit too), but a brief summary on late-game strategy:

• If you need a two-point conversion, it's often best to do it early - sometimes there are alternate paths to success if the two-point try fails early, but postponing the conversion puts all your eggs in one basket.
• If down 8, 11, 15, 18, 19, 22, 23, or 24 before the touchdown, you should always go for two.
• If down 6, 9, 13, or 20, you should always kick.
• Down 16, you should only kick if you think your chances of another TD and a FG are very high and your chances in OT are very low. Almost all realistic assumptions give you a better chance by going for two.
• Down 14 or 21, you should usually go for two - success means you avoid OT entirely, while failure is recoverable. This also applies down 17, though the awkward strategic situations down 11 if you fail make it a little less valuable there. Only a team that is much more likely to win in OT than to make a two-point conversion should kick and play for OT.
• Down 7 or 10, go for two if you are more likely to succeed at that than win in OT, but err on the side of kicking if there is significant time left down 10 (or more than about 10 seconds left down 7). Note that it's much better to go for two on the first touchdown while down 14 than it is to do it on the second after kicking the first time.
• Down 12, you should go for two late in the game but probably not early. Going for two is harmless if the opponent doesn't score again, necessary at some point if they do, but potentially risky if they score multiple times.

Now, let's look at scenarios where the touchdown has already put you ahead (score is listed by the pre-touchdown margin). Some simplifying assumptions which are applied unless otherwise stated:

• P(2) (probability of success), P(O2) (opponent's probability of making a two-point conversion), and P(WOT) (probability of winning if the game goes to OT) are independent constants within a game.
• You don't score again (if you do, you're likely to run enough time off the clock that it's moot, since this is targeting endgame strategy).
• The opponent has enough time for as many scores as are needed to tie but probably not any more (alternative strategies open up quite a bit if they have time for more scores, and nothing matters if they don't have time for enough to tie).
• Kicks are assumed automatic.
##### Down 5

Let's start with an obvious one. If you kick, you're up two and lose if the opponent kicks a field goal. Fail the two-point try and the situation is effectively the same, but if you succeed a field goal only ties the game. There's no practical downside to going for it and the payoff is big. Go for 2.

##### Down 4

If you kick, a field goal for the opponent ties the game. If you go for two, either 1) you miss and a FG wins the game for the opponent or 2) you make it and they need a touchdown. In mathematical terms:

Kick: Your chances of winning are P(opp doesn't score) + P(opp FG)*P(WOT).
Success: P(opp doesn't score) + P(opp FG)
Failure: P(opp doesn't score)

Superficially, this suggests that (as in the down 7 case) the break-even point is when you are just as likely to make the 2 as to win in overtime. However, the probability of the opponent scoring is not a constant across these three cases. A team that needs a touchdown isn't going to kick a field goal and will instead take a riskier strategy to increase their chances of a touchdown, while a team that wins with a field goal is likely to play it safe once in comfortable field goal range. (If a field goal ties, they'll likely fall somewhere between these extremes, though most teams tend to err on the side of forcing overtime if they are in a "FG ties, TD wins" situation.) In effect, if you go for two, the opponent knows what they need and adjusts accordingly. Thus, the break-even point is significantly lower and you should only go for two if you really don't like your chances in overtime.

##### Down 3

Another easy one: missing the two-point try leaves you up only a field goal, kicking or success means a touchdown beats you but a field goal is useless. If the opponent scores a touchdown, a failed two now means you would have to score a touchdown to answer, while success may not be enough to make a field goal win the game (the opponent would likely go for two as well, attempting to pad the lead back to 3). Kicking it is the right call.

##### Down 2

The opponent needs a touchdown no matter what. If you kick, the opponent will likely go for two to try to make it a 3-point lead. If you succeed at a two-point try, the opponent will likely kick to take the lead, whereas if you fail they will almost always kick to go up 3 (they could, as noted in the "down 4" heading, go for 2 to try to make it 2 or 4, but that's usually poor strategy). Either way, you will end up down either 1 or 3, but you get to decide whether it is your two-point try or your opponent's that determines which. The break-even point is where you are equally likely to succeed in your own two-point try or stop the opponent's; if you are more likely to make it, go for 2, but if you are more likely to stop them, kick and make them do it.

##### Down 1

If you kick, your opponent (after scoring a touchdown) can kick for the win. If you go for 2 and fail, you're in the same position. If you succeed, they have to kick to tie (or go for two for the win). The correct decision is obvious: go for 2.

##### Tied

Assuming the opponent matches with a touchdown of their own, there is another question to consider: are your opponents risk-averse or will they go for 2 on that touchdown? If you kick, your opponent can kick to force OT but might put everything on a two-point conversion instead; if you go for two, your opponent is forced to match to tie if you succeed but will just kick for the win if you fail. The scenarios:

Kick: Your chances of winning (assuming an opposing TD) are either P(WOT) if they always kick or the minimum of P(WOT) and 1-P(O2) if they are smart (assuming their answering score is very late).
Fail on the two: If the opponents score a touchdown, you lose.
Succeed on the two: (1-P(O2)) + P(O2)*P(WOT)

The graphs below illustrate the break-even points:

Interestingly, the two graphs show somewhat opposite behavior. If your opponent follows the conventional strategy of just forcing overtime, the better your chances are in OT, the less likely you should go for two (since kicking guarantees OT). Similarly, the more likely they are to make a two-point conversion if forced to attempt one, the less you have to gain by forcing them to do so, so kicking becomes more advantageous. But if the opponent considers going for two themselves, you should be more likely to go for it the more heavily favored in overtime you are (the opponent won't take their chances with overtime anyway) and more likely to go for it if the opponent is good at two-point tries (they will go for it anyway, so your best chance is to preemptively match their two-point conversion).

Even under the assumptions that your opponent will play it smart if you kick, you're still better off kicking in most cases unless your opponent is extremely good at two-point conversions. This is another instance of the principle that early risks help the side whose strategy is most affected by the new information, but here the opponent is the one who benefits.

##### Up 1

This is a situation a few people have mentioned as a possible situation to go for 2, the theory being that success puts you up 9 and makes it a two-score game. While true, failure means the opponent can tie just by kicking and has a chance to win the game in regulation by going for two. Worse, if you do it early enough, the opponent knows they need two scores and will adjust their strategy to maximize their chances of getting them. Down 7 or 9, they know how many scores they need; down 8, they don't.

If you kick, your chances of winning are (1-P(O2)) + P(O2)*P(WOT). If you go for two, assuming the opponent has no time for a second score, your chances of winning are P(2) + (1-P(2))*min(P(WOT), 1-P(O2)) if the opponent follows the smart strategy down 7 (if they follow the conventional strategy, the second term is just (1-P(2))*P(WOT)).

Assuming conventional strategy, the break-even point is where your chances of success on a two-point try and stopping your opponent on one are equal, as in the down-2 case. However, if your opponent follows the optimal strategy, you also need P(2) > P(WOT) to make going for two correct because the opponent may decide to go for two and win the game outright if you fail. Note that in both cases, this is slightly skewed toward going for it, because it doesn't include the possibility of two scores by the opponent, which is much more likely if they know they need two scores. Thus, if it's close, kick.

##### Up 2

Kicking guarantees that the opponents need two scores. The benefit of going for two is that a TD+1 and FG only ties the game, but failure leaves the possibility of a TD+2 to tie by itself. Better to play it safe and kick.

##### Up 3

This is essentially the same as the tied scenario (with the opponent needing an extra FG). The biggest change is that success puts you up 11, which is a strategically awkward position for your opponent - on 4th down in field goal range, should they kick and possibly find out later that they needed two touchdowns, or go for the touchdown first so they will know if a field goal will help next time? While that's useful, I don't think it's enough to move the break-even point very far, although it's enough to make going for 2 slightly more attractive.

##### Up 4

A kick leaves your opponent needing a TD+2 and FG to tie. Failure on the 2 makes it a TD+1 and FG, while success means two touchdowns are needed to win. The awkwardness of being down 11 makes me lean toward kicking here. The main difference between this and up 1 is that success on the two-point try doesn't force the opponent to make an extra score; it just forces both scores to be touchdowns, which is a much smaller advantage.

##### Up 5

Failure leaves the opponent the opportunity to tie with a TD+2 and FG; a kick or successful two-point try makes it two touchdowns to win. Since there's little benefit to success on the two and a potentially steep penalty for failure, kicking is correct.

##### Up 6

Kick or fail the two-point try and two touchdowns will beat you; success on the two-point try makes it two touchdowns to tie (or TD+2 and TD+1 to win). Here there's no harm to failure and plenty to gain from making the two, so go for it.

##### Up 7

Failure makes it two touchdowns to win. Kicking makes it two touchdowns to tie or TD+2 and TD+1 to win. Success makes it TD+2 and TD+1 to tie or TD+2, TD+2 to win. If the opponent follows conventional strategy, this breaks down exactly the same as the tied scenario, but if they're smart:

Kicking wins the game (assuming two touchdowns by the opponent) with probability either P(WOT) or (1-P(O2))*((1-P(O2)) + P(O2)*P(WOT)), whichever is less. (The opponent will likely go for two on their first touchdown.)
Going for it wins with probability P(2) * (1-P(O2) + P(O2)*min(P(WOT), 1-P(O2))); the opponent will certainly go for two the first time and might the second time (to go for an outright win).

The resulting graph:

The graph follows three ranges: at the far left, it matches the strategy while tied with the opponent following conventional strategy. The flat part represents where the opponent should go for 2 down 14 but not down 7 and the break-even point is where you are as likely to make your two-point try as to stop theirs. On the right, the opponent should go for two twice if down 15 to try to win the game outright.

Going for two against a smart opponent is better than doing so against one following conventional strategy (since they have better options than to just kick and play for OT if you kick), but it's still usually the wrong move.

##### Up 8

If you kick, the opponent needs a TD+2 and TD+1. If you succeed at a two-point try, they need to match it twice; failure allows a tie with TD+1 x2. Assuming conventional strategy from the opponent:

Kick wins with probability (1-P(O2)) + P(O2)*P(WOT)
Going for two wins with probability P(2)*((1-P(O2))^2 + P(O2)^2*P(WOT)) + (1-P(2))*P(WOT)
Break-even: P(2) = 1 / (1 + P(O2))

However, with optimal strategy the opponent can do a little better in both cases (possibly winning outright if down 15 by going for two twice, or increasing their chances significantly by going for two the first time down 14). The resulting chart:

Here there's very little to gain unless the opponent is excellent at two-point conversions and you are a massive favorite in overtime. (If the opponent follows conventional strategy, the flat line at the far left extends all the way across; your chance of winning in OT is completely irrelevant since the opponent will never win in regulation.)

##### Up 9

Kicking makes it 2x TD+2 to tie. Making the two forces them to get three scores, but failure allows TD+2 and TD+1 to tie. Assuming exactly two scores, the break-even line works out identical to that of the up-1 scenario, since an additional TD+2 is needed in all cases; however, I'd err even more on the side of kicking than I do up 1 because it's a smaller adjustment going from needing two scores to three (with a reasonable amount of time left) than going from needing one score to two.

##### Up 10

Kicking forces three scores to tie. Making the two does require the opponent to either score three touchdowns or get a TD+2, TD+1 and FG, but failure allows the possibility of tying with two scores (2x TD+2). Just kick it to get up three scores.

##### Up 11

This works out identically to up 8, except with an extra field goal needed. No matter what the outcome, the opponent needs three scores, and a FG may be one of them if they make sufficiently many two-point tries. Three touchdowns beats you regardless.

##### Up 12

This is three scores regardless. Kicking makes it TD+2, TD+2, FG to tie; a failed two makes it TD+2, TD+1, FG; success makes it three touchdowns to win. If we assume two touchdowns and a field goal, the decision comes out identical to up 9 (and therefore up 1), but the gain from making the two-point conversion is smaller since it does not force an additional score but instead changes a field goal to a touchdown. If it's at all a close decision, kick.

##### Up 13

If you kick, you always lose to three touchdowns but the opponent cannot tie with two TD+2s and a FG. Making a two gives you a chance to win opposite three touchdowns, but failure allows TD+2, TD+2, FG to tie. For typical numbers, assuming smart strategy by the opponent, success on the two-point try gives you about a 35-50% chance of surviving three touchdowns (possibly more if the opponent is an underdog but plays for OT anyway), while failure gives the opponent about a 7-15% chance of beating you given two touchdowns and a field goal.

The calculations here are quite complex since you have to consider whether the opponent would settle for a field goal down 19 or 11 (before getting both of the TD+2s they need) if you fail, whether three touchdowns is more likely down 20 or 21 than down 19, how much more likely a field goal is than a third touchdown, and several additional factors. However, unless your opponent is very good at two-point conversions and you are very poor at them, the risk of the miss, TD+2, TD+2, FG sequence is probably not enough to offset the possibility of surviving three touchdowns. Go for two.

##### Up 14

If you go for two and fail, three touchdowns always beats you. If you kick, the opponent can automatically force OT (assuming three touchdowns) or can go for the win with a successful two-point conversion or failure followed by two successes (or fail, succeed, kick for OT). If you go for two and succeed, the opponent has to go for 2 the first time; if they fail, they have to make the next two to force OT, but if they succeed they can kick twice for OT or go for two again to try to win in regulation.

The equations here, assuming the opponent scores three touchdowns and follows optimum strategy (if they always play conservative, the rule is the same as if you are tied and assume the opponent will play conservative):

Kicking wins with probability P(WOT), (1-P(O2))*(1-P(O2) + P(O2)*P(WOT)), or (1-P(O2))*(1-P(O2)^2), whichever is lowest.
Going for two wins with probability P(2) * ((1-P(O2))*(1-P(O2)^2 + P(O2)^2*P(WOT)) + P(O2)*min(P(WOT), (1-P(O2))*(1-P(O2) + P(O2)*P(WOT)))).

The chart:

The fact that failure is an automatic loss (assuming the opponent scores three touchdowns) makes kicking usually the correct play, but when the opponent is good enough at two-point conversions to go for best two-out-of-three instead of just kicking for overtime if you kick, it can be worth the risk. That said, for typical two-point conversion rates, it's almost always right to kick.

##### Up 15

If you kick, the opponent needs one two-point conversion (with a backup plan of making the next two after failing the first) to tie. If you go for two and fail, they can tie with all kicks; if you succeed, they need two successes to tie (with no second chance). If the opponent follows the conventional strategy of attempting only as many two-point conversions as they absolutely have to, this case simplifies to a question of whether you are more likely to make your own two-point attempt or stop the opponent (going for it either forces them to try an extra one if you succeed or permits them to skip it if you fail). If the opponent follows the smart strategy, it gets messy; I'll spare you the full set of equations here, because it's significantly messier even then the up-14 case. But the graph is as follows:

Since the opponent has to attempt a two-point conversion unless you fail, it is often better to kick. When the opponent follows the smart strategy, the advantage of kicking is increased, as it gives the opponent fewer chances to try to win in regulation.

##### Up 16

If you kick, the opponent needs two two-point conversions. A successful conversion of your own bumps that number up to three, but failure reduces it to one and allows them a second chance (by making two in a row after missing the first). You should only consider going for two if you are extremely likely to make it, your opponent is extremely likely to make theirs, you are a huge favorite in overtime, and you are worried about the possibility of the opponent making three conversions in a row to win in regulation.

##### Up 17

As in the up-1 and up-9 cases, a successful two-point try forces an extra score at the expense of potentially removing one forced two-point try from the opponent. The break-even point is the same, although the extra information of knowing they need an extra score once again skews things in favor of kicking if it's a close decision.

Note that when leading, it's generally rarer to attempt a two-point conversion than when trailing. This is because the extra information about what they need is of more use to your opponent than to you. While trailing, on the other hand, you are usually the one to benefit from that information and thus it pays off to go for two more often. A summary:

• Down 5 or 1 before scoring, or up 6 or 13, go for 2.
• Down 2 or up 1, 9, or 17, go for 2 if you're more likely to succeed than to stop the opponent on a two-point try (but err on the side of kicking if you're already ahead).
• Tied or up 7, 8, 14, or 15, go for 2 only if you and the opponent are both very likely to succeed on two-point attempts. Up 7 or 15, you should be less likely to go for 2 if you're likely to win in overtime; tied or up 8 or 14, you should be least likely to go for 2 if overtime does not particularly favor either team.
• Down 3 or 4 or with any lead not listed here, kick unless the circumstances are extreme.