Animal Crossing Turnip Markets

The goal is to use Bayesian estimation and Kelly strategy to optimally play the AC:NH turnip market.

It is interesting because it is a non-trivial model with some memory. An interesting question will be exploration vs. exploitation. To what extend do we want to take the current offer, versus waiting for more information.

Model

In the reverse engineered source code we find the following model:

Pattern state transition probabilities

$$ \begin{pmatrix} 0.20 & 0.30 & 0.15 & 0.35 \\ 0.50 & 0.05 & 0.20 & 0.25 \\ 0.25 & 0.45 & 0.05 & 0.25 \\ 0.45 & 0.25 & 0.15 & 0.15 \\ \end{pmatrix} $$

Based on the pattern, one of fo

Bayesian inferencing

Exploitation

In the first week we observe the following values, starting on Thursday AM:

$$ 57, 125, 64, 56, 124, 91 $$

Q. Given this information, what is the probability distribution of pattern?

$$ f(p) = \Prc{\mathtt{pattern} = p}{X_o = x} $$

This matches pattern zero with declen1 = 3, hilen1 = 4, hilen3 = 2.

Q. Assuming it is indeed that pattern, what is the probability distribution for baseprice?

$$ \begin{aligned} \Prc{\mathtt{baseprice} = p}{X_o = x} &= \frac{ \Pr{\mathtt{baseprice} = p ∩ X_o = x} }{ \Pr{X_o = x} } \end{aligned} $$

$$ = \Prc{\mathtt{baseprice} = p}{X_o = x} $$


$$ \begin{aligned} X_0 &∼ U(90, 110) \\\\ X_1 &∼ U(0.9, 1.4) \\\\ X_2 &= X_0 ⋅ X_1 \end{aligned} $$

$$ \Pr{\vec X = \vec x} = \Pr{X_0 = x_0} ⋅ \Pr{X_1 = x_1} ⋅ \Prc{X_2 = x_2}{X_0 = x_0 ∧ X_1 = x_1} $$

https://ermongroup.github.io/cs228-notes/

Given a set of vertices $V$ and a parent function $π: V → \powerset{V}$

$$ \Pr{\vec X = \vec x} = \prod_V^v \Prc{X_v = x_v}{\vec X_{π(v)} = \vec x_{π(v)}} $$

$$ p_{\vec X}\p{\vec x} = \prod_V^v p_v\p{x_v, \vec x_{π(v)}} $$


Questions

Q. How many Turnips should you buy on Sunday?

Q. When should you sell the Turnips?

Remco Bloemen
Math & Engineering
https://2π.com