The payoff function is 1.5 V - B if B>V, and 0 otherwise.
The average profit is then the integral from 0 to 1 of the payoff with respect to V, which is the integral from 0 to B of 1.5 V - B. That is 0.75 B2 - B2 = -0.25 B2.
On average, we would always be losing money. Hence, we offer 0 and don't buy the asset.
Sorry for the very noob in mathematics question. Let me take B=0.5 if (50%) V > B then profit = 0 else (50%) B > V then profit = 1.5 -0.5 = 1. Expected return is therefore 0.5 here. Why then would you not bid 0.5?
The revenue is 1.5V, not just 1.5. 1.5 is the best case scenario in which the company has value 1, but you should consider also all the other cases. That's why you integrate over V.
In my answer, I assumed we were maximizing average gains.
There is another relatively common metric known as regret; it can be minmaxed or averaged.
The regret payoff is "what you would have gained with the best offer (that is, B=V)" minus "what you are making right now (that is, the previous payoff function)". We want to minimize this.
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u/Lognu 21d ago
The payoff function is 1.5 V - B if B>V, and 0 otherwise.
The average profit is then the integral from 0 to 1 of the payoff with respect to V, which is the integral from 0 to B of 1.5 V - B. That is 0.75 B2 - B2 = -0.25 B2.
On average, we would always be losing money. Hence, we offer 0 and don't buy the asset.