Probability related finance question: Need a more formal solution

Question

You are offered a contract on a piece of land which is worth $1,000,000$ USD $70\%$ of the time, $500,000$ USD $20\%$ percent of the time, and $150,000$ USD $10\%$ of the time. We're trying to max profit.

The contract says you can pay $x$ dollars for someone to determine the land's value from which you can decide whether or not to pay $300,000$ USD for the land. What is $x$? i.e., How much is this contract worth?

$700,000 + 100,000 + 15,000 = 815,000$ is the contract's worth.

So if we just blindly buy, we net ourselves $515$k.

I originally was going to say that the max we pay someone to value the land was $x < 515$k (right?) but that doesn't make sense. We'd only pay that max if we had to hire someone to determine the value. We'd still blindly buy the land all day since its Expected Value is greater than $300$k.

Out of curiosity: what is the max we'd pay someone to value the land?

Anyway, to think about this another way:

If we pay someone $x$ to see the value of the land, we don't pay the $150,000$ USD $10\%$ of the time and we still purchase the land if the land is worth more than $300,000$.

Quickest way to think of it is to say "valuing saves us $150,000$ USD $10\%$ of the time and does nothing the other $90\%$, so it is worth $15,000$ USD".

Is there a more formal way to think of this problem?

Answer 1

There is no unique arbitrage-free solution to the pricing problem with $3$ outcomes, so you will need to impose more assumptions to get a numerical value for the land.

Answer 2

To get a solution, you'll need to specify an objective that you are trying to maximize.

If you are just maximizing the expected profit, then your $\$15,000$ answer is correct.

Typically, however, in finance we assume people are risk-averse. A standard objective function in that case is $E[\text{profit}] - \lambda \cdot Var[\text{profit}]$ where $\lambda$ is the level of risk-aversion that you choose. In that case you'd be willing to pay more than $\$15,000$ for the information in order to avoid the risk of losing $\$150,000$.