# What would be the Dual Expected Utility function for an English Auction?

So the DEU function is $$V(p)= \Sigma \,x_i\pi_i(p),$$ and since an auction only has two outcomes for a bidder, failure (with probability $$p$$) or success (probability $$1-p$$) the function becomes $$V(p)= x_1w(p) + x_2(1-w(p)).$$

But I'm not completely sure how to characterize the $$x$$'s. DEU being rank dependent, would the $$x_1=0$$ since it is a failure outcome and bidder doesn't get the item? If so, how would $$x_2$$ be described?

Thanks for any help.

• There are more than two outcomes. One is losing the auction (not getting the good and not paying), and there is a different outcome for winning and paying a price p for each p lower than the bid. Aug 6, 2021 at 8:18
• so it would be (for valuation/resale price v and bid b) $\Sigma_{i=0}^b (v-i)\pi_i(p)$? Aug 6, 2021 at 11:35
• On second thought, wouldn't the winning bids be grouped in under the $x_2$, as that term has the probability weight of success, thus making that term (for bid b and valuation v) $\Sigma_{i=0}^b (v-i)(1-w(p))$? Aug 6, 2021 at 11:55