The game “Higher or Lower” is a simple one in which playing cards are drawn randomly from a deck, and the player has to guess whether each subsequent card will be higher or lower in value than the previous.

What if "Higher or Lower" was played non-randomly, as a two-player game?

Let’s simplify the situation a little by removing the playing cards.

Player A must pick numbers from the range 1-50 without replacement. After the first number is announced by A, they pick the next in secret, and Player B must guess whether it is higher or lower than the first. B gains a point if they guess correctly, A gains a point if the guess is incorrect. This process is repeated until all 50 numbers have been exhausted.

The game is non-trivial because of one crucial feature: A's choices determine not only how easily B can guess the outcome, but also their range of options for the next pick. For example, if A's starting point is 2, then they might seek to thwart B by picking 1 as the next move with higher probability than all other numbers. However, if A does indeed pick 1, the next move is a free point to B as "higher" is the only option.

Is there a single optimum strategy for both players (and if so, what is it?), or must both players respond to the other’s strategies (and hence also, try to determine the opponent’s strategy)?

Note 1. Each player is competing against their own independent targets – I’m fairly sure that there is no strategy for A that will give a higher expected score than B’s. I’m asking about how A optimises their score in comparison with other hypothetical As, and the same for B.

Note 2. You can assume that both players have access to (pseudo-)random number generators, and as such are not vulnerable to being exploited via the human brain's notoriously bad performance in picking "randomly".

Note 3. While I have a few hunches, I do not know the solution to this problem. If it has already been studied I would be interested to know.

The game “Higher or Lower” is a simple one in which playing cards are drawn randomly from a deck, and the player has to guess whether each subsequent card will be higher or lower in value than the previous.

What if "Higher or Lower" was played non-randomly, as a two-player game?

Let’s simplify the situation a little by removing the playing cards.

Player A must pick numbers from the range 1-50 without replacement. After the first number (n1) is announced by A, they pick the next (n2) in secret, and Player B must guess whether n2 is higher or lower than n1. B gains a point if they guess correctly, A gains a point if the guess is incorrect. Next, A secretly picks another number (n3) from those remaining, and B must guess whether it is higher or lower than n2. This process is repeated until all 50 numbers have been exhausted, with B guessing whether n(i+1) is higher or lower than n(i) each time.

The game is non-trivial because of one crucial feature: A's choices determine not only how easily B can guess the outcome, but also their range of options for the next pick. For example, if A's starting point is 2, then they might seek to thwart B by picking 1 as the next move with higher probability than all other numbers. However, if A does indeed pick 1, the next move is a free point to B as "higher" is the only option.

Is there a single optimum strategy for both players (and if so, what is it?), or must both players respond to the other’s strategies (and hence also, try to determine the opponent’s strategy)?

Note 1. Each player is competing against their own independent targets – I’m fairly sure that there is no strategy for A that will give a higher expected score than B’s. I’m asking about how A optimises their score in comparison with other hypothetical As, and the same for B.

Note 2. You can assume that both players have access to (pseudo-)random number generators, and as such are not vulnerable to being exploited via the human brain's notoriously bad performance in picking "randomly". You can also assume that both players have excellent memories, and have perfect knowledge of which numbers have already been picked.

Note 3. While I have a few hunches, I do not know the solution to this problem. If it has already been studied I would be interested to know.