Intereting Posts

Proving that a definition of e is unique
Moment of inertia about center of mass of a curve that is the arc of a circle.
How does the topology of the graphs' Riemann surface relate to its knot representation?
Lebesgue Spaces and Integration by parts
Holder continuity of power function
Solve the equation $2^x=1-x$
Smooth boundary condition implies exterior sphere condition
number of zeros of a complex polynomial
How to prove asymptotic limit of an incomplete Gamma function
Any other Caltrops?
Why does “the probability of a random natural number being prime” make no sense?
Proving the range of operator is closed
If $\gcd(m,n)=1$, find $\gcd(m+n,m^2-mn+n^2)$?
to prove $f(P^{-1}AP)=P^{-1}f(A)P$ for an $n\times{n}$ square matrix?
Maximum number of edges in a non-Hamiltonian graph

In this game, you are given empty balloons one by one, and for each balloon you are to inflate it with air until you are satisfied. If it does not burst, you gain

~~happiness~~points proportional to the volume of air in the balloon (say 1 point per ml). If it bursts, you gain nothing. If you attempt to inflate a balloon beyond 1000L, it will certainly burst. The balloons are~~inexhaustible~~infinite, and your goal is to maximize your cumulative gain. More precisely, you want to maximize the asymptotic behaviour of $g(n)$ as $n \to \infty$, where $g(n)$ is the cumulative gain after $n$ balloons. All you know about the balloons is that they are all made by the same factory process, and so you can assume that the maximum inflatable size is always drawn from the same fixed distribution with nonzero variance.$\def\pp{\mathbb{P}}$$\def\ee{\mathbb{E}}$$\def\lfrac#1#2{{\large\frac{#1}{#2}}}$Is there an optimal strategy, defined as a strategy that guarantees $\ee(\lfrac1n g(n)) \to c$ as $n \to \infty$ and also has asymptotically maximum $\lfrac{\ee(\lfrac1n g(n))-c}{\sqrt{Var(X)}}$ where $X$ is the distribution of the balloon’s maximum inflatable size, and $c$ is such that $c \cdot \pp(X \ge c)$ is maximum ($c$ is the optimal inflate size if we knew $X$)?

If we knew $X$, we could easily get an optimal strategy as follows. Simply always inflate to size $c$, which exists by the extreme value theorem because of the balloon size limit (which is one reason I imposed it). I can construct a strategy that guarantees $\lfrac1n g(n) \overset{a.s.}\to c$ as $n \to \infty$: With probability $\lfrac1{n}$ inflate until it bursts, otherwise use previous tries to estimate $c$ and inflate to that estimate. After $n$ tries we have on average roughly $\ln(n)$ samples, and so by CLT our sample distribution’s cdf will eventually converge to that of the correct distribution because it has finite variance (which is the other reason for the balloon size limit), and also we use the estimate with probability approaching $1$. The problem is to find a strategy that gives the fastest convergence (or prove that no optimal strategy exist). I don’t even know the asymptotic behaviour of this strategy that I’ve described.

- A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe
- Guess the number despite false answer
- Flip all to zero
- A variation of Nim game

I made up this generalized game after playing a simple Javascript variant some months ago, ~~but I can no longer find it~~. Thanks to *joriki*, this original variant is known as the Balloon Analogue Risk Task (BART), which can be seen as a variant where the distribution is discrete. The strategy I gave above works but can never be asymptotically as good as the strategy that only takes one sample and then always inflates to that size (as pointed out by *Keepthesemind*).

For each $k > 0$ there is a balloon distribution such that the optimal strategy cannot expect to gain more than $\lfrac1k$ of the actual possible gain (the sum of the maximum inflatable sizes of the balloons). This also means that there is no optimal strategy for the variant of the game where an adversary chooses the maximum inflatable size just before giving you each balloon.

- Game involving tiling a 1 by n board with 1 x 2 tiles?
- Good non-mathematician book on Game Theory
- Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?
- Deal or no deal: does one switch (to avoid a goat)?/ Should deal or no deal be 10 minutes shorter?
- Linear Programming with Matrix Game
- Pirate Game (modified)
- Understanding common knowledge in logic and game theory
- Try not to get the last piece, but with a twist!
- Flip all to zero
- Weighted War - Game of Mind and Probability

- How to evaluate $\int_{0}^{\pi}\ln (\tan x)\,dx$
- Probability question with interarrival times
- The inverse of the matrix $\{1/(i+j-1)\}$
- Injectivity of Homomorphism in Localization
- $ (\mathbb{Z} \times \mathbb{Z}) / \langle (a, b ) \rangle$ is isomorphic to $ \mathbb{Z} \times \mathbb{Z} _d $, where $ d = \gcd(a,b)$
- Double Integration with change of variables
- A conditional normal rv sequence, does the mean converges in probability
- Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
- Introductory text for calculus of variations
- How the dual LP solves the primal LP
- REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
- Compactness of Algebraic Curves over $\mathbb C^2$
- $X^n-Y^m$ is irreducible in $\Bbb{C}$ iff $\gcd(n,m)=1$
- To show a morphism of affine k-varieties which is surjective on closed points is surjective
- Localisation is isomorphic to a quotient of polynomial ring