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Subject: 4th and long odds rss

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James Fung
United States
San Diego
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I had some time to kill on the bus yesterday, so I wrote an AI for Finger Ball, taking into consideration both the point advantage/deficit and the current quarter. I'll leave the details of how the AI was trained for another time. What I wanted to talk about was one of the counterintuitive findings (at least for me) of the AI: if you're losing, you should play 3 fingers less often.

Long odds. The reason, I believe, is this: being in the lead is a strong position. A team up by +1 or +4 points in each quarter has the following probability of winning:

1st 2nd 3rd 4th
+1 56% 58% 62% 70%
+4 68% 72% 77% 86%

(Note: the above numbers were calculated ignoring ties, i.e. in the games where one team was up by 4 points in the 4th quarter, in the games that did not end in a tie, the leading team won 86% of those games.)

This itself is counterintuitive because, if a football game reaches the 4th quarter with only a gap of 1 or 2 points, it's considered a pretty close game that could go either way. You're unlikely to place 7-to-3 odds against the trailing team (or 6-to-1 for a team trailing by 4). And this is because Finger Ball differs from football in one fundamental way: quarters can be really short. A couple tied plays ends a whole half. Meanwhile, in real football, a team can't sit on the laurels of a 1st quarter field goal for the whole game; they still have to make each drive count.

Defensive play. As has been posted before, the Nash equilibrium based on points is 3/13ths one finger, 7/13ths two fingers, 3/13ths three fingers, or about a 40% chance of ending the quarter, about a 1-in-6 chance of finishing two quarters with no further scoring. For this reason, the leading team can afford to play conservatively: play 3 fingers more often, generally at the expense of choosing 1 finger, though 2 fingers is still chosen almost half the time. The reason for this is, by choosing primarily 2 or 3 fingers, the winning team will rarely give up a touchdown that would put it at severe disadvantage.

Don't go for broke. Now for the losing team, especially one down by 4 or more points, one might think you should pull 3 fingers to get that touchdown and flip the tables. However, a smart opponent is aware of the situation and will play 3 fingers 40+% of the time in the 4th quarter, hoping to catch you and end the game. Instead, a losing team is actually better off playing mostly 1 or 2 fingers, aiming for field goals to narrow the gap.

To err is human. All of the above were calculated assuming computer players executing mixed strategies, ignoring what plays were made before, and basing the strategy solely off of the point different and current quarter. Humans, however, rarely are this random. I find the humans don't play 2 fingers often enough (they should be playing it about half the time). Humans don't like repeating the same play too often, thinking it isn't random enough. When they lose one or more plays in a row, humans tend to get flustered and go with their gut rather than thinking clearly.

Thus, based on the above, I give this advice:
- Don't pick 1 finger too often. The main attraction of playing 1 finger is scoring a field goal against an opponent who plays 2 fingers about half the time.
- If you're winning by 4 or more points, go for 2 or 3 fingers. The losing player may be desperate for a touchdown, and scoring an extra field goal or ending the quarter are both good results.
- If you're losing, going for a touchdown might cause the game to end instead. Rather, try for a field goal with 1 or 2 fingers. However, remember the first point above.
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