I am self-studying the following question: there is a bag with unknown value (can be negative). Everyone bids a price and whoever bids the highest wins the bag. Design the optimal strategy. This is a quick open question and we can make reasonable assumptions as we want.
I decided to model it as a first-auction game. So suppose the value of the good is standard Normal $\theta \sim N(0, 1)$. Each bidder $i$ bids $y_i = \theta + e_i$ where $e_i$ is i.i.d. standard Normal. By using this, my expected payoff is $E(\theta - y_i | y_i \geq y_j, \forall j).$ However it seems hard to write down this expectation term. Can anyone help with this
Personally, I would surmise that each bidder is going to realize that the expected (i.e. average) value of the item is $(0)$. So, not only is there no apparent profit in making a positive bid, there is no apparent profit in bidding $(0)$.
So, I would assume that all bidders accept the above analysis, and all bidders believe that all other bidders accept the above analysis.
Therefore, the only hope for profit is to make a negative bid that has a small absolute value.
As you increase the size of the bid that you are contemplating, (i.e. as your bid approaches $0$ from below), then two things are happening:
You are increasing the probability that your bid will be the winning bid.
You are decreasing the profit that you will realize, if your bid is the winning bid.
If my competitor was a robot or android, then I would expect my competitor to bid something like $- 10^{-(100)}.$
In a real world setting, I would bid $(-1)$ and hope that all of the other bidders are too panicked to make a rational bid, like $-(0.5).$