3
$\begingroup$

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.

$\endgroup$
3
  • 2
    $\begingroup$ 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. $\endgroup$
    – Bayesian
    Aug 6, 2021 at 8:18
  • $\begingroup$ so it would be (for valuation/resale price v and bid b) $\Sigma_{i=0}^b (v-i)\pi_i(p)$? $\endgroup$ Aug 6, 2021 at 11:35
  • $\begingroup$ 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) $\Sigma_{i=0}^b (v-i)(1-w(p))$? $\endgroup$ Aug 6, 2021 at 11:55

1 Answer 1

2
$\begingroup$

It seems like the answer to your question depends a bit on what assumptions you want to make regarding bidder decision-making and behavior, and which additional axioms of choice you want your bidders to satisfy. For example, you could assume that your bidders consider the problem as a single stage, such that their utility can be defined as a vector of payoffs that depends on whether they win the object and how much they pay. (In that case, you'd have to consider more than just p and 1-p, and take the integral of probabilities of winning given a bidding function.) Alternatively, you could make an assumption on how individuals evaluate compound lotteries similar to that of compound independence from Segal (1990), which would allow you to model the "win/lose" and "payoff given price paid and auction won" separately.

While there isn't too much to go off of, I'd recommend looking into these papers that examine payoff equivalence results under Dual Theory, as well as this paper that models different auctions under Dual Theory (and the updated version which seems to take the results in a slightly different direction, so both might be useful) and this paper which has a section describing how to apply non-expected utility theories to auctions. Other papers, like these, develop models of auctions where bidders have reference-dependent preferences (though not necessarily Dual Theory utility functionals) and might also be helpful in figuring out the best approach. (This one briefly mentions non-expected utility applications to auctions--while not particularly helpful on its own, it includes a few references that might be of interest.)

Sorry I don't have a more definitive answer, but hopefully some of these resources can at least be useful in developing your model!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.