# 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

## 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?

• It looks like all prices are scaled by baseprice, so it should only depend on the probability distribution over patterns?

• We want to maximize the long-term growth.

• We want to limit risk of ruin.

• We want to limit

• We want to take into account income streams.

• We want to take into account occasional withdrawals.

• We want to account for imperfect execution when evaluating strategy.

Q. When should you sell the Turnips?

• What is the distribution of payoff?
• When should we sell, and when should we wait for more data?
• Should we sell all at once or spread the sell-off?
• What is the cost of flawed strategy execution?
• What is the cost of false assumptions?

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