Discussion of quizbowl game theory

[Mewto55555]

*At CO 2015 this year, we were up by 45 going into the last tossup against Seth's team. It was on physics. The lead-in was relatively easy and misplaced, but our team's two physicists didn't want to neg it, be wrong and have Seth power vulch. If they did so, and Seth vulched, and Seth's team 30'd the associated bonus, we would have lost. If they power and 30, we force overtime; any failure to 30 the bonus, we win. The dilemma is, game theoretically, is it better to wait until you are 60/95/100% sure and buzz, since you can always wait for power to expire and play overtime if necessary, or what? It's a nasty situation that isn't entirely obvious what to do, and if you want to reduce these types of situations, getting rid of powers sure helps that!

Let S be the percent probability of Seth powering, to the best of your knowledge. You buzz here your probability of being correct is bigger than max(100-S,50) -- so if this clue is that easy that you think Seth has a greater than 50% chance of knowing it, you should have gone in if you think you're at least 50% to be correct.

[Dominator]

Let S be the percent probability of Seth powering, to the best of your knowledge. You buzz here your probability of being correct is bigger than max(100-S,50) -- so if this clue is that easy that you think Seth has a greater than 50% chance of knowing it, you should have gone in if you think you're at least 50% to be correct.

This is a rather crude analysis, and for that matter, is one I'm sure Ike was capable of making himself. However, this is not the type of analysis most people can do in the second or so between bonus 19 and tossup 20, so it is not terribly helpful to a team closing out a game.

The bigger problem, though, is that you are ignoring a lot of relevant data. What is Seth's team's probability of a 30? There may be a small correction factor that the bonus category won't be the same as the tossup category. Also, what is Ike's team's priority? Are they trying to maximize the probability that they win the game or the tournament*? Are they concerned with controlling expectation or variance? You have decided, in your analysis, what his team values, but you might be incorrect.

The game theory of quizbowl is far from as simple as Max would have us believe. Regardless, that does not address the question of whether people want a game in which powers exist or not. There will be game theoretic analyses possible either way. Powers just complicate them.

*or maybe something else, like if notorious gambler Mike Sorice wagered $10000 on a CO exacta that the team was colluding to achieve

[Mewto55555]

Let S be the percent probability of Seth powering, to the best of your knowledge. You buzz here your probability of being correct is bigger than max(100-S,50) -- so if this clue is that easy that you think Seth has a greater than 50% chance of knowing it, you should have gone in if you think you're at least 50% to be correct.

This is a rather crude analysis, and for that matter, is one I'm sure Ike was capable of making himself. However, this is not the type of analysis most people can do in the second or so between bonus 19 and tossup 20, so it is not terribly helpful to a team closing out a game.

Right, which is why it might plausibly be useful to have some idea of the results of this analysis in advance of the game.

The bigger problem, though, is that you are ignoring a lot of relevant data. What is Seth's team's probability of a 30? There may be a small correction factor that the bonus category won't be the same as the tossup category. Also, what is Ike's team's priority? Are they trying to maximize the probability that they win the game or the tournament*? Are they concerned with controlling expectation or variance? You have decided, in your analysis, what his team values, but you might be incorrect.

The probability of a 30 is actually irrelevant to your strategy in this case -- in the universe where they don't 30 the bonus, you win the game regardless of what happens on the tossup. Additionally, I can't think of any circumstance where the win-the-tournament-strategy is different than the win-the-game-strategy (except if you want to throw the game, in which case the correct strategy is pretty obviously to neg and reveal what you think the correct answer to be in the process). The only factor like this that I think might have any bearing on the issue is that Ike might be sad if he looks like an idiot by negging tossup 20, but be fine with getting outplayed by Seth powering it, so he's willing to sacrifice a little win probability to avoid this outcome. I don't think that's how Ike thinks though.

I don't really understand what you mean here by expectation vs. variance. We're only concerned with the binary outcome of win/lose. Unless we're agents with really weird preferences, I think it's very reasonable to assume that we want to do whatever maximizes p(win).

The game theory of quizbowl is far from as simple as Max would have us believe. Regardless, that does not address the question of whether people want a game in which powers exist or not. There will be game theoretic analyses possible either way. Powers just complicate them.

I certainly don't think it needs to be as complicated as you seem to. All these "corrections" you think necessary are of an incredibly small magnitude compared to the errors in the estimates we have (like, I think I can differentiate between a clue Seth is 80% likely to know vs one he is 20% likely to know, but I doubt I can differentiate 50 vs. 60%), but I think the rough heuristic of "if Seth might know it but might not, I shouldn't be too afraid of a neg to buzz" is a pretty useful one, and runs counter to the way I see a lot of people play in these situations.

Obviously this has little bearing on whether there should be powers or not (you can get a similarly complicated situation when up 40 with no powers, for example).


EDIT: The one correction that actually does have a meaningful impact is whether each team is equally likely to win in a tiebreaker. In most cases, though, this should be close to 50% -- given that you just split the tossups 10-9 or 9-9 or something + your priors + Bayes' theorem, etc. If it's Big Three only and one team has a sizeable advantage there, though, then that matters.

[Dominator]

The probability of a 30 is actually irrelevant to your strategy in this case

Not really. The 30 decides whether a neg loses the game vs. takes it into overtime (in the event of a power), so if you knew that Billy Busse was superclutch on tiebreakers you'd not risk losing it in regulation.

I don't really understand what you mean here by expectation vs. variance. We're only concerned with the binary outcome of win/lose. Unless we're agents with really weird preferences, I think it's very reasonable to assume that we want to do whatever maximizes p(win).

Yes, winning games is the best strategy to winning tournaments, but you're talking about weighing strategies that risk losing, in which case it is necessary to decide how willing you are to lose.

I certainly don't think it needs to be as complicated as you seem to. All these "corrections" you think necessary are of an incredibly small magnitude compared to the errors in the estimates we have

Again, not necessarily. If Seth's team hasn't 30ed a bonus all day outside physics and you know that the physics bonus for the round is past you, then let Seth power it and <30 the bonus. That is probably not going to come into play here because Ike's team is unlikely to have that information and it is unlikely that such a thing would be true of such high-quality opponents. But you'd better believe that I coached my teams that way, and IMSA did a pretty good job of winning key matches against frequent opponents.

[Cody]

That doesn't matter; if they neg it and it's powered and 20d, they still win by 5. So, the analysis must proceed from the assumption that they'll thirty because in no other situation could you lose the game (i.e. in no other situation is there a point to analyze it). Now, sure, up 40 is a whole different ballgame.

[Ike]

I actually want to have a discussion about the game theory behind this instance, because I think it's incredibly interesting. I'm actually more on the same page as Dr. Prince rather than Cody or Max - I'm not really sure how you have to make the assumption that Seth's team is 30ing the bonus. I think assigning a probability value to the likelihood that they 30 the bonus and acting accordingly to that data is correct. If p = 0, then there's no need to use any strategy, but if p = 1, I think Max'x strategy might be right, I'd imagine that a mixed strategy of the two would be what I would use depending on what p is, especially if this is an iterated situation - and it is, since only two weeks later at VCU Open, we were up by 45 going into the last tossup! I'll also point out that I think Max's analysis fails to consider the probability that power is over at the given point in a question.

While on the topic of powers and game theory, I'll actually speak up and say I think it's cool to have a "game" element to quizbowl. I think one of my wiser moments was at BONGOS, being ahead by 15 going into the last tossup, and cringing upon realizing that the tossup was on orgo. I recognized some of the buzzwords, but eventually I heard the words Markovnikov's rule. Now I'm 90% sure I know what functional group to buzz in and say when I hear it, but I'm 99.99% sure Will Overman didn't put Markovnikov's rule in power for that clue. So instead of risking a neg, I just assumed power would be over, and I was correct. I think being able to make that type of decision-making is really, really, fun, and I actually laugh at players who decide to butcher their chances of winning by negging in that spot.

I think the most successful players have some sense of game theory behind them; it's rare that you see someone just knock every question out of the park with 100% knowledge. I'm not saying that every player is able to explain this game theory - Chris Ray for example, is a player who just strikes me as someone who knows when to take risks intuitively, but he may not be able to articulate it*. I think it's a topic we as quizbowlers choose not to discuss since we want to promote the intellectual aspects of the game, but it is interesting and useful.

* I say this as a compliment.

Additionally, I can't think of any circumstance where the win-the-tournament-strategy is different than the win-the-game-strategy (except if you want to throw the game, in which case the correct strategy is pretty obviously to neg and reveal what you think the correct answer to be in the process)

PS - For those who don't know, sometimes there is a compelling reason to want to throw a match at the ICT, none of which involve betting against your own team. Consider three purely hypothetical teams UVA-Evil, Chicago-Good, and Penn-Good. UVA-Evil's record right now is X-2. Penn-Good's record is undefeated, Chicago-Good's record X-1, and Penn-Good is going to play Chicago-Good in the last game. If UVA-Evil's p of winning is something crazy like .999, and they just happened to have run into variance, Penn-Good should consider throwing their last game against Chicago-Good and foregoing the advantaged final since under NAQT's current rules, if Penn-Good won their last game, UVA-Evil would do a playin game for an advantaged final. But if Penn-A-Good threw their last game, UVA-Evil would have no chance of clawing their way into victory from the depths of Hell.

[The King's Flight to the Scots]

Max's heuristic is useful precisely because it's simplified. Maybe it only captures 80% of the situation, but it takes 20% of the effort to calculate, which is vital in a fast-moving match.

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[Mewto55555]

I'm not really sure how you have to make the assumption that Seth's team is 30ing the bonus. I think assigning a probability value to the likelihood that they 30 the bonus and acting accordingly to that data is correct. If p = 0, then there's no need to use any strategy, but if p = 1, I think Max'x strategy might be right, I'd imagine that a mixed strategy of the two would be what I would use depending on what p is, especially if this is an iterated situation

I'm assuming that whether or not they 30 the bonus is independent of what happens on the tossup, so long as they hear it. That is, if you neg and they vulch, they have the same probability as if they'd gotten it without you buzzing.

In some sense, you can play the last TU/B cycle in reverse order. They read Seth a bonus. Then a tossup is read to both teams, scored normally. If Seth 20s, he's down by 25 going into the last tossup and can't win, so it doesn't matter what you do on the tossup. The tossup only matters when Seth would 30 this bonus.

[jonpin]

PS - For those who don't know, sometimes there is a compelling reason to want to throw a match at the ICT, none of which involve betting against your own team. Consider three purely hypothetical teams UVA-Evil, Chicago-Good, and Penn-Good. UVA-Evil's record right now is X-2. Penn-Good's record is undefeated, Maryland-Good's record X-1, and Penn-Good is going to play Chicago-Good in the last game. If UVA-Evil's p of winning is something crazy like .999, and they just happened to have run into variance, Penn-Good should consider throwing their last game against Chicago-Good and foregoing the advantaged final since under NAQT's current rules, if Penn-Good won their last game, UVA-Evil would do a playin game for an advantaged final. But if Penn-A-Good threw their last game, UVA-Evil would have no chance of clawing their way into victory from the depths of Hell.

I think I follow this, but I'm not sure if you mixed up which teams you wanted to declare Good vs Evil, as I think Maryland took the place of Chicago somewhere in there.

I'm not really sure how you have to make the assumption that Seth's team is 30ing the bonus. I think assigning a probability value to the likelihood that they 30 the bonus and acting accordingly to that data is correct. If p = 0, then there's no need to use any strategy, but if p = 1, I think Max'x strategy might be right, I'd imagine that a mixed strategy of the two would be what I would use depending on what p is, especially if this is an iterated situation

I'm assuming that whether or not they 30 the bonus is independent of what happens on the tossup, so long as they hear it. That is, if you neg and they vulch, they have the same probability as if they'd gotten it without you buzzing.

In some sense, you can play the last TU/B cycle in reverse order. They read Seth a bonus. Then a tossup is read to both teams, scored normally. If Seth 20s, he's down by 25 going into the last tossup and can't win, so it doesn't matter what you do on the tossup. The tossup only matters when Seth would 30 this bonus.

I've tried having this conversation insofar as it comes to football teams that are down 14 late in a game (say, under 5:00 to go) and then score a touchdown. Unless you get the ball back without conceding points (via onside kick or defensive stop) and score a second touchdown, nothing else matters, so we might as well take those as given if we're going to discuss the relative merits of certain strategies.
For what it's worth, in my opinion, the best strategy in that case is to go for two on the first touchdown, as it gives you a chance to win in regulation and a chance to salvage the tie if you miss the first time. This was my opinion even before they moved back the extra point, but even so I will be absolutely shocked if any NFL team ever tries that strategy.

[Ike]

I think I follow this, but I'm not sure if you mixed up which teams you wanted to declare Good vs Evil, as I think Maryland took the place of Chicago somewhere in there.

Oops, made a mistake, switched between Chicago and Maryland on accident. Fixed that. The point is, if a team is guaranteed to be in the final, they can throw their last game to influence who appears in the final. Even more hilariously, if a team is undefeated and everyone else has an X-2 record or worse, they can throw their last game with impunity (i.e., without giving up their advantage) to influence the rush for a spot in the final.

Ike

[theMoMA]

I split this discussion off (although I left bits of the game theory discussion in the other thread when they were interlaced with discussion of powers).