My school is entering a college bowl with three rounds (with 18 questions each) of increasing difficulty. In the first round correct answers are worth 1 point, in the second round 5, and in the third round 10.
No points are lost for incorrect answers, but teams can announce a “challenge” before the question comes up that will double the points earned for the question, or subtract the same amount for a wrong answer.
Each team – there will be only three, of pretty much equal strength – gets a chance to answer before time is up. There is no buzzer.
Anyway, we have estimated, from looking at past competition questions, that we should be able to get 15 correct in the first round (83%), 12 in the second (67%) and 8 in the final round (44%).
Given this supposed success rate, when would be the best time to challenge? And how often in each round?
I may be reading this wrong but it looks like you should challenge on every question in the first and second round. Since your team should have a greater than 50% chance of getting any given question, that should increase your score. Somehow, that seems too simplistic. Are there any other rules such as a limit on the number of times you can challenge?
The 50% mark is not the breakpoint. The expected value when you don’t challenge is v * p, whereas the expected value is 2 * (v * p - v * (1 - p)) when you do challenge. So you want 2 * (v * p - v * (1-p)) > v * p, which simplifies to 2/3 > p. So, regardless of how many points a question is worth, you want to challenge when you have better that 2/3 odds of getting it right, and not challenge when it’s less.
Punoqllads, I’m not sure I see that. It looks to me like the challenge is a side bet, at even payout, in addition to the regular points for the question. And you should always accept a bet with even payout and better than even odds.
If a challenge applies to all teams, then on average, they make no difference. Your team is, on average, in the same position as every other team, so if it’s advantageous for your team to challenge, it’s advantageous for the other teams to challenge, too. But advantage is a zero-sum game, so the advantage has to be zero.
Another consideration: Is the final payout proportional to the score, or (more likely) only to the winning team? If the latter, then strategy for use of challenges changes dramatically. Most significantly, if you’re behind towards the end of the last round, it’s to your advantage to challenge even in the last round. For an extreme case, if you’re 19 points behind on the last question, then you should absolutely challenge. If you challenge, the other team gets it wrong, and you get it right (almost a 1 in 4 chance), then you’ll win. But if you don’t challenge, you’ll have no chance at all of winning.
I don’t think that’s what the OP meant. Freakstasy said:
Saying “the same amount” seems to indicate the same as the doubled points, as opposed to saying, for instance, “the original amount”.
Like you pointed out, though, all of this gets confuddled if any team announcing a challenge will alter the points scored by all the teams for that question. In that case, challenging only makes sense at the end, when you are in a position such that, if all remaining questions are unchallenged, you would not win even if you get all of them right and your opponents get all of them wrong.