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Tater10
08-20-2005, 10:22 AM
Here is a stupid game i invented when i began reading about game theory a couple years ago. I believe i have found optimal ev for this game, and would like to see if anyone else has the same results.

It's 1st and goal from the 10 yardline. Offense and defense have 2 dice each (a Red die for Run, a Purple die for Pass).

Here is how it works:
Offense and defense secretly select 1 die that he they are going to roll. They each roll at the same time (neither knows the color of the opponents die until the roll.)

This is just like football, you have 4 downs to get into the endzone.

How to gain yards and score:
Offense: Pass die; defense pass die: Incomplete pass, down is over.

Offense: Run die; defense pass die: Offense gains the yards on the die. (offense would gain 1-6 yards)

Offense: Pass die; defense run die: Offense gains 2x yards on the die. (offense would gain 2-12 yards)

Offense: Run die; defense run die: Offense gains 1 yard no matter what he rolls.

Caveat: Just like NFL, the endzones are 10 yards deep. So, if it's 4th and 2 and you throw a 12 yard pass, that pass is incomplete out the back of the endzone and the defense wins.

Obviously, if it's 4th and 1, 3rd and 2, 2nd and 3, offense would just run and there is no way for the defense to stop it, so the game is over when its 4th/1, 3rd/2, 2nd/3.

If the offense scores, he gets 1 unit. If the offense doesnt score, defense wins 1 unit. So, would you rather be offense or defense, and by how much? (you would be playing against someone who knew optimal strategy)

Extra Credit: It's 1st and 10 from the 20, and you can make a 1st down at the 10 yardline. Same rules except offense gets 2:1 from the defense. Would you rather be offense or defense, and by how much?

AaronBrown
08-20-2005, 01:06 PM
That looks doable, I'll take a look when I have a few mintues.

Here's one (with answer) from real football that's always bothered me.

Team A is down by 14 points and scores a touchdown with one minute to go. Obviously, any scenario in which Team B scores at all, or Team A fails to get another touchdown, means team B wins. So Team A should assume it will get another touchdown.

We all know that Team A always goes for one point in this situation. That has about a 98% chance of success in NFL football, versus 45% for two points. If it makes the one point, when it gets the next touchdown it will have a choice between going for 1, with 98% chance of overtime so a 49% chance of winning, or 2 with a 45% chance of winning. So going for 1 point on the first touchdown gives a 98%*49% = 48% chance of winning, assuming Team A gets another touchdown.

If Team A goes for two instead, it will make it 45% of the time. Then it will go for one at the next touchdown, for a 98% chance of victory. 55% of the time it will miss, then go for two on the second touchdown, for a 45% chance of overtime and a 22.5% chance of victory. 45%*98% + 55%*22.5% = 56% chance of winning.

Now I know specific game situations could change this, maybe the kicker has been erratic or the short-yardage running back is hurt. But these could work either way, so a lot more than half the time, two points is the right answer. But no one ever tries it. Why not?

Tater10
08-20-2005, 05:25 PM
I was reading the David Romer paper about going for it on 4th down, and how NFL teams are too conservative when I thought back to my football game. There was an article written in the 2+2 magazine here about going for it on 4th down, i think sklansky wrote it. Makes me think that sometime in the near future that a high profile college team will start going for it on 4th down and start going for 2 more often as your example. 'Moneyball II' will be in the near future.

TTChamp
08-21-2005, 02:03 AM
Another way off seeing this is that the chance of victory in regulation is the same if Team A goes for two on the first touchdown or the second touchdown, but there is a much higher chance of getting to overtime if Team A goes for 2 first.

I can't figure out why teams don't go for two on the first score more often. One explanation is that the typical football fan (or coach) is not very good at statistics.

I have two other sports situations where teams don't follow the best strategy:

1. A football team is down by 1 point with 90 seconds and the opposing team has 1st and goal from the 5 yard line. I think the best strategy is for the defense to let the other team score a touchdown and the best strategy for the offense is to go for 2 if they score a touchdown. Neither team usually takes the strategy though.

2. A basketball team is up by 3 with 20 seconds remaining. The other team has the ball. I think the best strtegy is to foul and not allow the other team a chance to shot a 3. Ocasionally a team will do this.

Has anyone ever seen or done an analysis of these situations.

PS Sorry for hijacking the thread.

AaronBrown
08-21-2005, 10:46 AM
Back to the original thread. Here's a start. On fourth down the optimal strategy from the 1 is for the offensive player to run with probability 1, and he wins with probability 1. Although it doesn't affect the optimal solution, the defense might as well play pass, because this could prevent the offensive win if the offensive player makes a mistake. I call this Offense Run 1 - Defense NA - Offense Win 1.

For the rest of the possible yardage points:

2 Offense Run 6/11 - Defense NA - Offense Win 5/11
3 Offense Run 5/9 - Defense NA - Offense Win 10/27
4 Offense Run 5/8 - Defense NA - Offense Win 5/16
5 Offense Run 2/3 - Defense NA - Offense Win 2/9
6 Offense Run 4/5 - Defense NA - Offence Win 2/15
7 Offense NA - Defense Run 0 - Offense Win 0
8 Offense NA - Defense Run 0 - Offense Win 0
9 Offense NA - Defense Run 0 - Offense Win 0
10 Offense NA - Defense Run 0 - Offense Win 0

A couple interesting observations. In each case, only one player's actions matter. If he does the optimal thing, the other player cannot affect the probability of winning. Also, the farther away the offensive player is, the more likely it is that he will run. This is because he can never win from beyond the 1 unless the defense guesses wrong. So he will pick Runs more often than Passes, because these give him less chance of winning, to force the defender to occasionally defend them.

Have you ever heard a football coach say "we like to run our bad plays more often than our good ones, to prevent the defense from gaining any advantage by defending the plays we do well"?

bobman0330
08-21-2005, 02:20 PM
OK, here's a general forumla for nth and X.

You need to start at 4th down.

For third down, for each situation, work out 4 probabilities: P(winning for (R,R))[A], P(winning for (R,P))[B], P(winning for (P,R))[C], P(winning for (P,P))[D] [offense's move listed first]. For (R,P) and (P,R), the winning chance will be an average of the 4th down winning chances from each die roll. For (P,P), it's the 4th down winning chance for the same yardage, and for (R,R) it's the 4th down winning chance for X-1.

In that case, equilibrium run percentage = (D-C)/(A-B-C+D). Pass percentage is one minus that.

PS, if Aaron forgot to adjust the win percentage in 4th and 2 for overthrowing, I'll be off slightly, but it's his fault. /images/graemlins/smile.gif

Yardage/Run %/Pass %/Win %
3 .5807 .4193 .6832
4 .5790 .4210 .5972
5 .6190 .3810 .5344
6 .6098 .3902 .4164
7 .7057 .2943 .3194
8 .6946 .3054 .1986
9 .7403 .2597 .1555
10 .8128 .1872 .0905

I'm a bit perturbed by the fluctuations in the running frequency. Anyone have an explanation or can anyone find my error?

Tater10
08-21-2005, 04:59 PM
Yeah, your numbers match up with mine if the offense were to win with a 12 yd pass on 4th and 2. I found the game to be better with a loss for the offense. (actually, i got +.00003 EV for the Offense if you were to split on a 12 yard pass on 4th and 2, but thats no fun)

4th down numbers (the D_run% is the same as the O_pass%):
Yds / O_run / O_pass /O_EV
6 / .80 / .20 / (.73333)
5 / .667 / .333 / (.55556)
4 / .625 / .375 / (.37500)
3 / .556 / .444 / (.25926)
2 / .500 / .500 / (.16667)

My 3rd down numbers:
Yds / O_run / O_pass / D_Run / D_Pass /O_EV
10 .8108 / .1892 / .1892 / .8108 / (.81945)
9 .7780 / .2220 / .2220 / .7780 / (.73071)
8 .7264 / .2736 / .2736 / .7264 / (.64769)
7 .7351 / .2649 / .3930 / .6070 / (.39843)
6 .6640 / .3360 / .4261 / .5739 / (.17520)
5 .6447 / .3553 / .4418 / .5582 / +.03893
4 .5800 / .4200 / .4769 / .5231 / +.18798
3 .5643 / .4357 / .4772 / .5228 / +.34163

and 'yes' to the earlier post by AB that if 1 player plays according to the percentages above, the other "plays" optimal as well.

Typed this fast, so sorry about grammar &amp; wording.

AaronBrown
08-21-2005, 06:01 PM
My bad. I didn't read carefully enough. I assumed that a 12 yard pass from the 2 would still be in the endzone, although I didn't assume that from the 1. Of course, that doesn't matter, because you never pass from the 1.

I think most people would take the offense, not knowing the numbers, and would play this game very badly. I think most people would pass too much. Throwing away game theory, I bet you win against most first time offensive players by playing pass every down. Not many guys would run more than twice out of four downs, especially if he's got relatively long yardage, and two runs give him only 1/6 chance of winning if you play pass defense all the time.

On offense, I'm guessing most first time defenders would start with Pass defense, so I run. If I'm wrong, or get a low roll, I stick with the run. If I'm right, or get a high roll, I switch to pass.

bobman0330
08-21-2005, 06:28 PM
Do you have a spreadsheet to do this for you?

If so, could you run 2nd and 1st downs and post the results?

Tater10
08-21-2005, 07:16 PM
This is a copy and paste from my excel spreadsheet. I hope it's legible.

Here is what i have for each down. You would want to be defense, only by .0029

Down / yards / O_Run / O_Pass / D_Run / D_Pass / O_EV

1 10 68.6% 31.4% 39.8% 60.2% \$(0.00290)
2 10 73.3% 26.7% 32.1% 67.9% \$(0.42981)
2 9 69.5% 30.5% 35.2% 64.8% \$(0.29851)
2 8 70.0% 30.0% 43.3% 56.7% \$(0.07825)
2 7 66.3% 33.7% 46.8% 53.2% \$0.13005
2 6 65.2% 34.8% 48.3% 51.7% \$0.32656
2 5 60.0% 40.0% 50.5% 49.5% \$0.46872
2 4 59.7% 40.3% 51.6% 48.4% \$0.60710
2 2 100.0% 0.0% 0.0% 100.0% \$1.00000
3 10 81.08% 18.92% 18.92% 81.08% \$(0.81945)
3 9 77.80% 22.20% 22.20% 77.80% \$(0.73071)
3 8 72.64% 27.36% 27.36% 72.64% \$(0.64769)
3 7 73.51% 26.49% 39.30% 60.70% \$(0.39843)
3 6 66.40% 33.60% 42.61% 57.39% \$(0.17520)
3 5 64.47% 35.53% 44.18% 55.82% \$0.03893
3 4 58.00% 42.00% 47.69% 52.31% \$0.18798
3 3 56.43% 43.57% 47.72% 52.28% \$0.34163
3 2 100.0% 0.0% 0.0% 100.0% \$1.00000
3 1 100.0% 0.0% 0.0% 100.0% \$1.00000
4 6 80.0% 20.0% 20.0% 80.0% \$(0.73333)
4 5 66.7% 33.3% 33.3% 66.7% \$(0.55556)
4 4 62.5% 37.5% 37.5% 62.5% \$(0.37500)
4 3 55.6% 44.4% 44.4% 55.6% \$(0.25926)
4 2 50.0% 50.0% 50.0% 50.0% \$(0.16667)
4 1 100.0% 0.0% 0.0% 100.0% \$1.00000

And, here is the strategy I have for 1st and 10 from the 20, with the offense getting 2:1 (still want to be defense by .0247)

Down YTG From O_Run O_Pass D_Run D_Pass Off EV
1 10 20 67.9% 32.1% 42.5% 57.5% \$(0.0247)
1 GL 10 68.6% 31.4% 39.8% 60.2% \$0.4957
1 GL 9 68.6% 31.4% 45.6% 54.4% \$0.7804
1 GL 8 66.3% 33.7% 49.1% 50.9% \$1.0455
1 GL 7 65.7% 34.3% 50.8% 49.2% \$1.2904
1 GL 6 61.7% 38.3% 52.7% 47.3% \$1.4706
1 GL 5 61.9% 38.1% 54.2% 45.8% \$1.6354
2 10 20 71.9% 28.1% 34.2% 65.8% \$(0.4521)
2 9 19 70.1% 29.9% 35.5% 64.5% \$(0.2985)
2 8 18 69.6% 30.4% 42.1% 57.9% \$(0.1042)
2 7 17 68.7% 31.3% 42.4% 57.6% \$0.1121
2 6 16 71.9% 28.1% 41.0% 59.0% \$0.2807
2 5 15 74.6% 25.4% 38.5% 61.5% \$0.4522
2 4 14 76.9% 23.1% 38.0% 62.0% \$0.6217
2 2 12 91.8% 8.2% 26.5% 73.5% \$0.9873
2 GL 10 73.3% 26.7% 32.1% 67.9% \$(0.1447)
2 GL 9 69.5% 30.5% 35.2% 64.8% \$0.0522
2 GL 8 70.0% 30.0% 43.3% 56.7% \$0.3826
2 GL 7 66.3% 33.7% 46.8% 53.2% \$0.6951
2 GL 6 65.2% 34.8% 48.3% 51.7% \$0.9898
2 GL 5 60.0% 40.0% 50.5% 49.5% \$1.2031
2 GL 4 59.7% 40.3% 51.6% 48.4% \$1.4106
3 10 20 81.2% 18.8% 18.8% 81.2% \$(0.8327)
3 9 19 77.4% 22.6% 22.6% 77.4% \$(0.7382)
3 8 18 72.3% 27.7% 27.7% 72.3% \$(0.6395)
3 7 17 73.9% 26.1% 36.0% 64.0% \$(0.4256)
3 6 16 72.7% 27.3% 36.4% 63.6% \$(0.2214)
3 5 15 72.8% 27.2% 36.5% 63.5% \$(0.0011)
3 4 14 70.0% 30.0% 38.1% 61.9% \$0.2046
3 3 13 68.3% 31.7% 38.9% 61.1% \$0.4174
3 2 12 84.2% 15.8% 54.5% 45.5% \$0.8017
3 1 11 60.3% 39.7% 28.6% 71.4% \$0.9413
3 GL 10 81.1% 18.9% 18.9% 81.1% \$(0.7292)
3 GL 9 77.8% 22.2% 22.2% 77.8% \$(0.5961)
3 GL 8 72.6% 27.4% 27.4% 72.6% \$(0.4715)
3 GL 7 73.5% 26.5% 39.3% 60.7% \$(0.0976)
3 GL 6 66.4% 33.6% 42.6% 57.4% \$0.2372
3 GL 5 64.5% 35.5% 44.2% 55.8% \$0.5584
3 GL 4 58.0% 42.0% 47.7% 52.3% \$0.7820
3 GL 3 56.4% 43.6% 47.7% 52.3% \$1.0124
4 6 16 85.8% 14.2% 14.2% 85.8% \$(0.7862)
4 5 15 74.8% 25.2% 25.2% 74.8% \$(0.5918)
4 4 14 69.3% 30.7% 30.7% 69.3% \$(0.3854)
4 3 13 62.5% 37.5% 37.5% 62.5% \$(0.2066)
4 2 12 57.3% 42.7% 42.7% 57.3% \$(0.0375)
4 1 11 78.8% 21.2% 72.0% 28.0% \$0.6705
4 GL 6 80.0% 20.0% 20.0% 80.0% \$(0.6000)
4 GL 5 66.7% 33.3% 33.3% 66.7% \$(0.3333)
4 GL 4 62.5% 37.5% 37.5% 62.5% \$(0.0625)
4 GL 3 55.6% 44.4% 44.4% 55.6% \$0.1111
4 GL 2 50.0% 50.0% 50.0% 50.0% \$0.2500

Tater10
08-21-2005, 07:20 PM
In the few times i have actually played this, i have done well because of what you said. People are pretty predictable.

Trantor
08-22-2005, 08:28 AM
[ QUOTE ]

T
If Team A goes for two instead, it will make it 45% of the time. Then it will go for one at the next touchdown, for a 98% chance of victory. 55% of the time it will miss, then go for two on the second touchdown, for a 45% chance of overtime and a 22.5% chance of victory. 45%*98% + 55%*22.5% = 56% chance of winning.

[/ QUOTE ]

If 2 points are made and the 1 point is then missed they still have overtime to win it so there is an additional .45 * .02 * .5 wins = 0.45%