<p>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 <a href="https://www.jstor.org/stable/2938207" rel="nofollow noreferrer">Segal (1990)</a>, which would allow you to model the &quot;win/lose&quot; and &quot;payoff given price paid and auction won&quot; separately.</p>
<p>While there isn't too much to go off of, I'd recommend looking into these papers that examine <a href="https://www.sciencedirect.com/science/article/pii/S016517650200054X" rel="nofollow noreferrer">payoff equivalence results under Dual Theory</a>, as well as this <a href="https://economics.smu.edu.sg/sites/economics.smu.edu.sg/files/economics/pdf/Seminar/2020/20201105.pdf" rel="nofollow noreferrer">paper that models different auctions under Dual Theory</a> (<a href="https://www.econ2.uni-bonn.de/moldovanu/revision18.pdf" rel="nofollow noreferrer">and the updated version</a>  which seems to take the results in a slightly different direction, so both might be useful) and <a href="https://www.econ2.jhu.edu/people/karni/ambiguity%20and%20nonexpected%20utility.pdf" rel="nofollow noreferrer">this paper</a> which has a section describing how to apply non-expected utility theories to auctions. <a href="https://www.sciencedirect.com/science/article/pii/0165176586900716?via%3Dihub" rel="nofollow noreferrer">Other papers</a>, <a href="https://tspace.library.utoronto.ca/bitstream/1807/4778/2/450.pdf" rel="nofollow noreferrer">like these</a>, 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. (<a href="https://core.ac.uk/download/pdf/345074064.pdf" rel="nofollow noreferrer">This one</a> 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.)</p>
<p>Sorry I don't have a more definitive answer, but hopefully some of these resources can at least be useful in developing your model!</p>

CC BY-SA 4.0
2021-08-07T14:16:24.607Z
2021-08-07T14:16:24.607Z
AndrewC
<p>So the DEU function is <span class="math-container">$$V(p)= \Sigma \,x_i\pi_i(p),$$</span> and since an auction only has two outcomes for a bidder, failure (with probability <span class="math-container">$p$</span>) or success (probability <span class="math-container">$1-p$</span>) the function becomes <span class="math-container">$$V(p)= x_1w(p) + x_2(1-w(p)).$$</span></p>
<p>But I'm not completely sure how to characterize the <span class="math-container">$x$</span>'s. DEU being rank dependent, would the <span class="math-container">$x_1=0$</span> since it is a failure outcome and bidder doesn't get the item? If so, how would <span class="math-container">$x_2$</span> be described?</p>
<p>Thanks for any help.</p>

CC BY-SA 4.0
2021-08-06T08:18:39.483Z
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.
Bayesian
CC BY-SA 4.0
2021-08-06T11:35:27.463Z
so it would be (for valuation/resale price v and bid b) $\Sigma_{i=0}^b (v-i)\pi_i(p)$?
OldDogNewTrick
CC BY-SA 4.0
2021-08-06T11:55:18.820Z
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) $\Sigma_{i=0}^b (v-i)(1-w(p))$?
OldDogNewTrick
<auctions><expected-utility>
What would be the Dual Expected Utility function for an English Auction?
OldDogNewTrick
# What would be the Dual Expected Utility function for an English Auction?

(Asked by: username_0 on 2021-08-06)

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.

**Tags:** auctions, expected-utility
> **Comments:**
> **username_1**: 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.  
> **username_0**: so it would be (for valuation/resale price v and bid b) $\Sigma_{i=0}^b (v-i)\pi_i(p)$?  
> **username_0**: 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) $\Sigma_{i=0}^b (v-i)(1-w(p))$?  

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## 1 answers

### Answer by username_2 on 2021-08-07. Score: 2

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)](https://www.jstor.org/stable/2938207), 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](https://www.sciencedirect.com/science/article/pii/S016517650200054X), as well as this [paper that models different auctions under Dual Theory](https://economics.smu.edu.sg/sites/economics.smu.edu.sg/files/economics/pdf/Seminar/2020/20201105.pdf) ([and the updated version](https://www.econ2.uni-bonn.de/moldovanu/revision18.pdf) which seems to take the results in a slightly different direction, so both might be useful) and [this paper](https://www.econ2.jhu.edu/people/karni/ambiguity%20and%20nonexpected%20utility.pdf) which has a section describing how to apply non-expected utility theories to auctions. [Other papers](https://www.sciencedirect.com/science/article/pii/0165176586900716?via%3Dihub), [like these](https://tspace.library.utoronto.ca/bitstream/1807/4778/2/450.pdf), 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](https://core.ac.uk/download/pdf/345074064.pdf) 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!


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