# Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have regained interest in this question recently and so I decided to post on here (too late for migration. If I should do anything else to move the question here please let me know).

The original problem is as follows:

Alice and Bob play a game. The game terminates when Alice reaches $$a$$ points and Bob reaches $$1$$ point. Alice wins if she ever reaches $$a$$ points ($$a$$ is a parameter of the game, and therefore fixed throughout the game). In each round of the game, Alice picks a real number $$x$$. Then a coin is flipped, Alice is awarded $$x$$ points if the coin comes up heads, and Bob is awarded $$x$$ points if the coin comes up tails. Your task is to find the probability that Alice wins for each $$a$$ given she is using the best strategy.

The formalization of the problem is not quite trivial, so I will offer the following to ensure that there are no ambiguities. First, note that at the beginning of any round, we can normalize the game so that Bob needs exactly $$1$$ more point. We will define a strategy $$s:\mathbb{R^+}\to\mathbb{R^+}$$ as a function that takes as an input the number of points Alice needs, and outputs her choice $$x$$. Define $$p_s(a)$$ to be the probability that Alice wins the game needing $$a$$ points and playing strategy $$s$$. Let $$S$$ be the set of all strategies. Find $$p, p(a)=\sup_{s \in S}p_s(a).$$

The following is an exhaustive list of the progress on this question that I am aware of:

\begin{align} p(a) &\leq \frac 12 + 2^{-a} \tag{1} \\ p(a) &\leq \frac 12 + \frac 1{2a} \tag{2} \\ p(a) &= 1, a \leq 1 \tag{3} \\ \end{align}

Furthermore, $$(1)$$ is tight over the naturals, and $$(2)$$ is tight over $$a=\frac{1}{1-2^{-n}}$$. Proofs and the strategies that give the tight bounds can all be found in the linked MSE question.

If I should do anything else to make the migration of content easier please let me know (resubmit answers, etc.)

• @PatDevlin what do you mean negative? Alice and Bob each have a strictly positive score thought the game – DreamConspiracy May 11 at 0:25
• (Sorry! Posted accidentally before finishing typing [or proofreading].) – Pat Devlin May 11 at 0:32