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Auction theory typically (always?) begins by assuming that each bidder's valuation is a random variable. Now, it might seem reasonable (at least from a Bayesian perspective) for you to treat other people's valuations as random variables. After all, you don't know their valuations! However, what justification can there be for treating your own valuation as a random variable when deciding on how to bid? And are there any approaches which do not begin with this assumption?

Edit: Here's another way of posing the question. In the context of auction theory, people normally seem to define a strategy as a function mapping from my (random) valuation to my bid. But assuming that I know my valuation, why not simply define my strategy as my bid?

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    $\begingroup$ I think my answer already answered your edit. It is a Bayesian game, other people do not know my type. If they did, they would have no uncertainty about my valuation. If they are uncertain in equilibrium (which they are), their consistent beliefs demand that I can actually have multiple types. This is standard in Bayesian games. See also Bayes-Nash equilibrium. $\endgroup$ – Giskard Apr 16 '18 at 14:04
  • $\begingroup$ "why not simply define my strategy as my bid?" Do you mean "define my strategy as my valuation"? $\endgroup$ – Acccumulation Apr 16 '18 at 15:06
  • $\begingroup$ No, that is not what I meant (and is obviously not a sensible suggestion). $\endgroup$ – afreelunch Apr 16 '18 at 15:33
  • $\begingroup$ It's quite more sensible than "define my strategy as my bid". $\endgroup$ – Acccumulation Apr 16 '18 at 18:30
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    $\begingroup$ This appears to be mainly an epistemological issue: research papers are written from the point of view of an observer of a situation, not of a participant. Application manuals are a different story of course. If ones wants to study an auction from the point of view of a particular participant, then certainly this participant's bid will be conditional on its valuation that will be known to him (in some sense, not necessarily as a point value, but certainly an given input to its bid-decision function, that usually has more arguments that just own valuation. $\endgroup$ – Alecos Papadopoulos Apr 23 '18 at 20:02
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In most of the literature I have read on private value auctions you do actually know your own valuation in the auction, it is your private information. It has a distribution from the point of view of others, who can only guess at your valuation.

Another interpretation would be that your valuation is actually random before getting detailed information on the object you are bidding on. How much would you pay for this bottle of wine that I have here on my shelf? Oh, it depends on the wine? Then without knowing the exact type of wine and assuming it has some random distribution your valuation is also a random variable. Once you learn the exact type of the wine you learn your own 'type', which is modeled as getting a private signal about your valuation.

In case of noisy observation your valuation can actually be random. E.g. I tell you the type of wine but you have never had it before. Or in case of gas companies, they have an inexact estimate about the yield of a gas field. Their 'real' valuation depends on the 'real' yield, without knowing that they can only bid based on some expected values.

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  • $\begingroup$ What information one gets is itself a random variable, so what valuation one has is a random variable after one receives information. If one has no information, then valuation is not a random variable. $\endgroup$ – Acccumulation Apr 17 '18 at 5:10
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    $\begingroup$ @Acccumulation I decided to no longer argue with you. Perhaps you are the only sane person on the site and all other visitors of this question are wrong. Perhaps not. $\endgroup$ – Giskard Apr 17 '18 at 7:26
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You seem to not understand the concept of "random variable". The term "random variable" does not refer to a variable that is unknown, it refers to a variable that has a nondeterministic relationship with other variables. For instance, when you draw cards in poker, the result is a random variable. Just because you end up knowing that you ended up getting particular cards doesn't make them not random variables. You can still ask what the probability of getting those cards is. And since your opponents' actions will be based on that probability, you'd better be able to figure out what that probability is. If you're thinking "Well, I know that I have a pair of kings, so clearly the probability of having a pair of kings is 100%", you're probably not going to do well. To do Bayesian reasoning, you need to know what your prior probability of getting a pair of kings is.

For a more complicated example, suppose you have a population with mean $\mu$ and standard deviation $\sigma$. You then take a sample and calculate the sample mean. You then want to calculate the 95% confidence interval. This is an interval such that there is a 95% chance that your sampling procedure will result in a sample mean within the interval. This is often presented as "I have this sample mean, and there's a 95% chance that the true mean is within this measurement error". But this is suggesting that the true mean is some random variable with some probability distribution. But it's not a random variable, it's a fixed parameter. It's the sample mean that is the random variable, and it's the sample mean that has a probability distribution. Just because you've already taken a sample, and know what its sample mean is, does not mean it's not a random variable. There is a nondeterministic process that turns a population mean and std into a sample mean. The sample mean depends on the population mean, but does so nondeterministically, and so is a random variable. Similarly, you valuation is being modeled as coming from particular parameters, but doing so nondeterministically.

Presumably, your bid will be based on how much value you think the good will generate for you. This will in turn be based on what information you have received about the good. And what information you receive can be modeled as a random process. Basically, you have a good space G. You have some data space D. You have some valuation function V. Given a particular g in G, there is some distribution of D: p(D=d|G=g). Given a particular d, there is some valuation v: v = V(d). You want a strategy, given v, of choosing a bid. v is a function of d, which is a nondeterminstic function of g, thus v is a random variable depending on g. Given a particular v and a prior on the distribution of G, and the conditional distributions of d on g, you can use Bayesian reasoning to find the distribution of g. You can then find the bid that maximizes expected value (i.e. the value for each g, weighted by their conditional probabilities).

As for why the bid shouldn't just be your valuation: there's no reason to think that your valuation will be an unbiased estimator of the good's true value. Furthermore, the expected value of a bid has a complicated relationship between the bid and the good's true value. If you win the auction, then the value of your bid is the difference between the value and the bid, but the probability of winning the auction depends on your bid and the other participants'. Thus, even if you did have an unbiased estimator of the true value, you shouldn't use it as your bid; you need to take other participants' bids into account.

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  • $\begingroup$ Coming from the author of the other answer this will probably seem ungraceful but I think you really misrepresent the OP's claims. For example he never writes the bid should just be the valuation, he asks why the bid is not the entire strategy. $\endgroup$ – Giskard Apr 16 '18 at 15:36
  • $\begingroup$ @denesp "why the bid is not the entire strategy" makes no sense. Perhaps it is premature to take a guess what they meant, but calling it a "misrepresentation" to take a best guess does indeed seem a bit ungraceful. $\endgroup$ – Acccumulation Apr 16 '18 at 15:56
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    $\begingroup$ It makes sense to me. The whole premise of the question is that the state space does not offer more types ($v_i$ valuations) for a certain player $i$. If this were true then her strategy $b_i(v_i)$ may as well be denoted simply by $b_i$ as $v_i$ does not change, so there is nothing for $b_i$ to be a function of. $\endgroup$ – Giskard Apr 16 '18 at 17:20
  • $\begingroup$ By the way the OP also commented below your comment to clarify that is not what she meant. $\endgroup$ – Giskard Apr 16 '18 at 17:22
  • $\begingroup$ @denesp Your putative explanation for how it makes sense to you itself does not make sense. You introduce notation without explaining it, and you seem to think that random variables are the only type of variables. The only sense I make is that they are saying that if you curry the function, that makes it not a function; instead of b(v), you have $b_v$. Oh, look, now it's not written as a function! Yes, if you treat this as having a different strategy for each valuation, then a strategy isn't a function. Is that the point the OP is making? How is whether the valuation is random relevant here? $\endgroup$ – Acccumulation Apr 17 '18 at 5:08

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