The way you describe this object (I will call it a coin) it exists in some finite quantity with no reasonable substitute and no new coins can be made. Suppose before you go on the proposed endeavor the world starts with a Pareto efficient allocation of these coins. Additionally, assume the act of destroying a coin has no impact on how much other owning these coins value their collection. This implies a few things:
- Each individual who has a coin values the last coin in their collection more than anyone who does not have a coin, otherwise it would be Pareto-improving for someone with a coin to sell it to someone with a higher valuation.
- The market-clearing price of coins is the lowest valuation of anyone who owns a coin.
- Every individual values their first or an additional coin at something equal to or less than the market clearing price, otherwise they would purchase the coin.
- You will never pay less than the market-clearing price for a coin, otherwise someone with a higher valuation would have bought it already.
These facts imply that you can never make money in your endeavor. When you buy a coin you will pay a price equal to the lowest valuation of anyone that already has a coin. But after destroying the coin you haven’t changed the price because the marginal buyer is still the same and has the same valuation. You can repeat the process and pay a higher price for the next coin, but you still haven’t changed the valuation of the marginal buyer, so the market-clearing price will not change.
This conclusion relies on two assumptions, which may not be true in the real world. If the initial allocation isn’t Pareto efficient, you could make more money simply acting as a broker, buying coins from people with low valuations and selling them to people with higher valuations (since the coins are clearly less valuable after you destroy them). This could even pay for your travel.
It’s more complicated if valuations are impacted by your act of destroying the coins. In this case, you could make money, but it would be difficult to determine ahead of time. As a buyer/destroyer of coins you can observe the total number of coins, the number of people who own them, and the market-clearing price of coins. That price tells you nothing of how the preferences of people with valuations higher than the market-clearing price change with each coin you destroy. For example, if you destroy a coin and the market-clearing price goes up by $100 it may be that everyone increased their valuations or only the marginal buyers increased their valuation. Predicting how prices would change after each coin destruction given what you observe would require assuming the highest-valuation buyers behave similarly to the marginal buyers, which can be a stretch.