system
Given the following post, respond as if you're commenting on Reddit:
human
Who has the advantage in a quiz where two people have to guess a number (e.g. an amount or a prize). The first guesser or the second?

I am a radio host and sometimes conduct radio quizzes. As a tiebreaker between two contestants, I use a "guess how many"-question, like "How much money did the local government spend on traffic in 2015?" The first contestant gives an answer. The second one then answers. What is most fair? 
A: That the first one guess a number, and the second guess a number?
B: The first one guess a number, and the second says higher og lower?

My thinking is: In scenario B, number 2 has a 50/50 chance, but I am not sure if number 1 has the same 50/50 chance, because the success of his opponent is down to 50/50?

Another question: Is the fairness affected by the nature of the question, i.e. the difference between a random number and an actual number that might be, but unlikely is known by one of the contestants?
gpt
If both players are rational, then there should be no difference between your two modes of play (second player guesses or second player says higher/lower). Suppose player A says 50, and player B believes the answer is 60. It will always be strictly better for him to guess the minimal increment higher than the answer of A, say 51, rather than his actual guess. This way, if the true answer turns out to be 53, he would still win where this wouldn't have been the case had he answered his actual guess of 60.

So if your aim is to benefit the smarter player, you should keep the second guess. If your aim is to equalize the field, then go for higher/lower.

As for the second part of the quesiton: Even though the precise answer (the amount of money spent by the local gov't on traffic, for example) is likely to be unknown to players, smart players may be able to come up with a reasonably good order-of-magnitude estimate. This means that if a weak player is to guess first, there is a good chance for the smarter second player to score a nearly guaranteed win. For example if the first player guesses $100, the second player should immediately recognize this as far too low and guess "higher" (or $101). In reverse, if the smarter player goes first and produces a reasonable guess, then the second player might as well flip a coin to determine his guess of higher or lower.

When using random numbers, a similar effect is present, with the difference that there is a clear optimal strategy for both players. The first player should guess the average between the lower and upper limit of the random number (assuming symmetric distribution) and the second player can guess either higher or lower, it makes no difference. If the smarter player goes first, both players will always have a 50% chance of winning (assuming the higher/lower system is used), since the first player will guess the middle value. If the weaker player goes first, he may not guess the middle value, which would give an immediate advantage to the second player.

So to conclude: The best solution depends on what you want. Should it effectively boil down to a coinflip? Or should there be an advantage for smarter players? Using a random number has the best chance of ending up with 50/50 odds for both players since the optimal strategy is relatively easy to see, but it may become boring when the first player always guesses the middle value and the second player has no influence once this is done. Using an actual question is definitely more interesting, but the player with more knowledge / better estimation skills will benefit in this case, but primarily if he can play second.