With the information given, for any strategy you might pick, there exists an instance of the game that makes this strategy awful. For example, suppose $n=2$. Let $i$ be the index of the good you buy at day $1$, and suppose that $p_i^3=p_i^2+\epsilon$ and that for all $j\neq i$, $p_j^3=p_j^2+N$ for small $\epsilon$ and very large $N$.

If you want the notion of "optimal strategy" to make sense, you need to give more information. For example, you could ask that the daily price increments follow a given probability law on $\mathbb R_+$.

Assuming you have done so, let $X^d$ be the price increment vector between days $d$ and $d+1$, seen as a random variable in $(\mathbb R_+)^n$. If the $(X^d)_{1\leq i\leq d}$ are independant, then your strategy is optimal. Without independence, this is not true.

With the information given, for any strategy you might pick, there exists an instance of the game that makes this strategy awful. For example, suppose $n=2$. Let $i$ be the index of the good you buy at day $1$, and suppose that $p_i^3=p_i^2+\epsilon$ and that for all $j\neq i$, $p_j^3=p_j^2+N$ for small $\epsilon$ and very large $N$.

If you want the notion of "optimal strategy" to make sense, you need to give more information. For example, you could ask that the daily price increments follow a given probability law on $\mathbb R_+$.

Assuming you have done so, let $X^d$ be the price increment vector between days $d$ and $d+1$, seen as a random variable in $(\mathbb R_+)^n$. If the $(X^d)_{1\leq i\leq d}$ are independant, then your strategy is optimal. Without independence, this is not always true.

With the information given, for any strategy you might pick, there exists an instance of the game that makes this strategy awful. For example, suppose $n=2$. Let $i$ be the index of the good you buy at day $1$, and suppose that $p_i^3=p_i^2+\epsilon$ and that for all $j\neq i$, $p_j^3=p_j^2+N$ for small $\epsilon$ and very large $N$.

If you want the notion of "optimal strategy" to make sense, you need to give more information. For example, you could ask that the daily price increments follow a given probability law on $\mathbb R_+$.

With the information given, for any strategy you might pick, there exists an instance of the game that makes this strategy awful. For example, suppose $n=2$. Let $i$ be the index of the good you buy at day $1$, and suppose that $p_i^3=p_i^2+\epsilon$ and that for all $j\neq i$, $p_j^3=p_j^2+N$ for small $\epsilon$ and very large $N$.

If you want the notion of "optimal strategy" to make sense, you need to give more information. For example, you could ask that the daily price increments follow a given probability law on $\mathbb R_+$.

Assuming you have done so, let $X^d$ be the price increment vector between days $d$ and $d+1$, seen as a random variable in $(\mathbb R_+)^n$. If the $(X^d)_{1\leq i\leq d}$ are independant, then your strategy is optimal. Without independence, this is not true.