# Auction models for tickets

Consider a typical event for which tickets are sold. The event only occurs once, and for simplicity we can assume that only one kind of ticket is provided, i.e, no VIP seats etc. $N$ tickets are available for purchase, and the $N$ highest bids win the tickets.

Does it exist any clever method for auctioning this type of good?

In a classical auction, the highest bid at any time is shown and participants can increase their bid to be the winning participant. But when auctioning this type of good that does not make any sense, since there will be multiple winning bids.

Should we perhaps display the currently lowest bid that is needed to win a ticket?

What if we think that it is desirable that all winners pay the same amount, how does this effect things?

I have tried to find existing research about this, but to no avail. Maybe there exist some other type of good with similar characteristics to tickets (oil/electricity?) that is auctioned in a similar way? Pointers to existing literature would be greatly appreciated.

• What do you mean by "clever"? Profit maximizing for the seller? – Herr K. Dec 4 '17 at 23:15
• @HerrK.: That was a bad wording on my part, sorry. Profit maximising and also practical to implement. If for example auctioning one ticket after another in a sequential fashion could be argued to be profit maximising, it still wouldn't be feasible. Each individual auction would have to be extremely short, or the whole process would take a ridiculous amount of time. – Filip Nilsson Dec 4 '17 at 23:45
• I also found a newly published document when doing some new research right after posting this question: faculty.chicagobooth.edu/eric.budish/research/… – Filip Nilsson Dec 4 '17 at 23:46

First, you should note that there is a Revenue Equivalence Theorem that says under a set of conditions, the seller's revenue from using different forms of auctions will be the same. This same result holds also in the multi-object case.

Thus, if you have $n$ identical objects, you can sell them using a ($n+1$)th price sealed bid auction, where

• each bidder submits a bid,
• the highest $n$ bidders each gets $1$ object, and
• they all pay a price equal to the ($n+1$)th highest bid.

In such an auction, it is a weakly dominant strategy for every bidder to bid their true valuation of the object. The reasoning is similar to the second price auction.

• But the nth bidder wasn't willing to pay the (n - 1)th highest bid. Could it be the (n + 1)th highest bid? That would be consistent with a second-price auction for a single item, as per your last link. – 410 gone Dec 5 '17 at 0:42
• @EnergyNumbers: Oh yes, thank for catching that! – Herr K. Dec 5 '17 at 0:58
• Things get a lot more complicated if some of the bidders want to buy more than one ticket because then you have to worry about demand reduction. – Ubiquitous Dec 5 '17 at 9:20
• @Ubiquitous: Can we not treat a bidder who demand multiple units as multiple bidders who each demand one unit (at potentially different prices)? – Herr K. Dec 5 '17 at 19:11
• @HerrK No. Quick example. Suppose there are three units and I am willing to pay 10 for each of two units and you are willing to pay 10 for each of two units. If we both bid sincerely then the price is 10 and I get zero surplus. But if I bid as though I only wanted one unit and set my second bid equal to zero then the price is zero and I get a surplus of 10! This is known as demand reduction and is a more general phenomenon in multi-unit auctions with multi-unit demand. – Ubiquitous Dec 5 '17 at 20:34