When you think about it, quizbowl is like a duel.

Okay, maybe not all duels. But some.

Consider this scenario. Two gunfighters, Pat and Quinn, are having a duel. Each has exactly one bullet in the chamber. They start facing each other from far away and slowly walk nearer. At any point they can fire their shot, but if they miss, it's a sure win for the other side -- the opponent can save their shot until it's in point-blank range. In this scenario, when should the fighters take their shots?

When the duel starts, both Pat and Quinn have a small probability of hitting their target. But as time goes on, that probability steadily goes up as their target gets closer, all the way until the theoretical end of the duel where their barrels are literally against their opponents' chests. At that point, it's a sure thing -- they'll hit their shots 100 percent of the time.

I'm a mathematician by training, so I'm tempted to define variables here. Let's use p and q to represent Pat's and Quinn's probabilities of hitting a shot. Remember that p and q are changing, starting off near 0 and ending up at 1. Now we can use these variables to decide when to fire.

Let's start from Pat's point of view. Pat knows that if she shoots first, she has a probability p of hitting her shot and winning the duel. On the other hand, if she waits for Quinn to shoot first, she'll lose if Quinn hits and win if Quinn misses. Algebraically, by waiting for Quinn to shoot, Pat loses with probability q and wins with probability 1-q. So is it better to shoot first or be shot at first? Let's go to the numbers. By shooting first, Pat wins with probability p, and by waiting to shoot, she wins with probability 1-q. At the beginning, since p and q are close to 0, the chances of winning by waiting are better than by shooting. But by the end, p and q are both 1, so Pat would definitely be better off shooting than not. There must be some point in the middle where it changes, where Pat realizes that she is now suddenly better off shooting, and she goes ahead and fires. That change happens as soon as p is no longer smaller than 1-q, when p=1-q (or, alternatively, when p+q=1).

What is Quinn thinking during this whole process? The situation basically looks the same for him as it did for Pat, just with the ps and qs switched. Quinn has a probability q of winning by shooting and 1-p of winning by waiting. Quinn will shoot just as soon as q is not smaller than 1-p. That happens when q=1-p (or, alternatively, when p+q=1).

The astute observer may notice that, according to the numbers, Pat and Quinn are choosing to shoot at the exact same time. We can't say who is going to win, since that depends on which shots hit, but we can see when those shots should happen.

The skeptical reader may not see yet how this duel relates to quizbowl. This duel is a very close analogy to how a tossup question plays out between two players. At the beginning of the tossup, both players have essentially no idea of what the answer is. They need to wait, and while more clues are being read, they get a better chance of what the answer is. Maybe they hear a familiar clue, or they can use context to think of some reasonable guesses. The more of the question that gets read, the higher their probabilities of knowing the right answer, all the way until the giveaway clue. So, at what point should a player fire her shot buzz in? In this sense, the duel we've been looking at is a perfect analogy for quizbowl. And the conclusions we've reached about the duel offer some insights into how quizbowl ought to, in the mathematical sense, play out.

The first is that buzzer races -- when both players ring in at the same time -- should be common. In fact, they're the optimal state. If there's not a buzzer race on a tossup, then either the buzzer is being too aggressive or the opponent is too timid in not taking a calculated risk.

The second lesson is about how to play as the underdog. The team with the lower correctness probability should have a lower threshold for buzzing -- they should be ringing in less sure than their opponents. In a hypothetical world where Quinn is twice as good as Pat, Pat should be ringing in when she is one-third sure of the answer. At that moment, p is equal to 1/3, q is twice as big, and p+q is equal to 1. Sure, Pat may miss the question (in fact, that's likely to happen), but nonetheless, choosing to buzz at that point gives Pat the best chance of getting the points. In order the play the game optimally, underdogs need to be willing to gamble.

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