How to win big at Chinese AuctionsPosted: June 26, 2011
(Disclaimer: I wrote my Senior Thesis at Princeton about eBay’s auction mechanism, so I’m kinda obsessed with auctions)
So, there are lots of different types of auctions out there, and lots of different auction houses. There’s Christie’s, a major art auctioneer; eBay, the #1 online auction site; English auctions, Dutch auctions – the list goes on and on. As one would then expect, the academic literature on auctions is huge. To attempt to summarize it here would be impossible. For the barest of an overview, check out Wikipedia’s page. I guess I can provide a couple of brief sentences: in the first-price auction (where you pay what you bid if you win), the Nash equilibrium symmetric strategy is to bid what you expect the guy with the next-highest bid values the object. In the second-price auction (where you pay what the next highest bid), you should always bid how you value the object, no matter what anybody else does.
Surprisingly, there is virtually no literature on “Chinese auctions,” which is very surprising since this type of mechanism is not at all uncommon. The Wikipedia page says that Chinese auctions are “typically featured at charity, church festival and numerous other events.” I know that every year the local Mikvah (Jewish ritual bathhouse) holds a Chinese auction to raise money (my family has actually done quite well with winning stuff, so I don’t know how they would make money if not that the items for auction are donated, but that’s another story). Yet when doing a Google Scholar search, only one relevant article (which you can’t even access, and has only been cited twice) is listed. Compare that to, say, second-price auctions, which generates hundreds, if not thousands, of relevant hits. Not 100% sure why that is.
Anyway, I guess I should describe just how a Chinese auction works. Basically, an item/good is up for sale (say, two tickets to that thing you love), which different people can value differently. I suppose we can make things simple here by having each person value the object independently (not depending on how others do), with a uniform distribution over [0,M]. Each person (i) of the total of N people buys a certain number of tickets (), and a ticket is chosen at random from those bought, selecting the winner. Thus the probability that person i wins the item is , i.e. the number of tickets they bought out of the total number of tickets sold. Again for simplicity, we assume that , and the cost is $1/ticket.
Let’s assume for now that if you like the item, you’re going to buy more tickets. Makes sense – you’re willing to invest more to ensure that you win. Let’s just check to see that we can find a Nash equilibrium with people following a strategy like this one.
At equilibrium, no person wants to buy any more or any less tickets – they are best off by buying exactly based on how much they like the good (). So, they cannot improve their expected payoffs by buying a different number of tickets. That is, for all i,
This is way too complicated to try to analyze directly. So let’s make things simpler. Say there are K other tickets in the pot. Then one will want to buy tickets so that
Thus, when the number of other tickets in the pot is known to be K, it is best to put in exactly tickets. Of course, this value could be negative – in which case, it would be best to buy no tickets at all! This is because the cost of any ticket is greater than the return one can expect from getting it, and one might as well sit out the auction. Notice that this gives us the characteristic we wanted – people who want the good more will buy more tickets.
Now, let’s try the case where each person knows exactly how much everyone else likes the good. If there are T total tickets, where , then we can rewrite the optimal number of tickets that person i purchases as
Thus we immediately see that only those people who value the object more than the total value of tickets in the pot will actually submit any tickets at all. We’ll assume that’s the case for simplicity’s sake; otherwise, we can just ignore those people who don’t want to get anything.
Summing the i equations of the above form, we get
From there, it is easy to plug in to the equations for to solve for .
Example: Suppose there are two people going for an iPad (I’m still a PC person, but whatever). The first values it at $1000, while the second at $500. By the arguments above, we plug in , , and , and get
One more note: when everyone’s valuations are known, it is still a dominant strategy to submit your valuation as your bid in the second-price auction. This yields a revenue (on average) of , where is the highest appraisal of the item. Yet here, the ticket sales are (at most) just (N-1)/N times the harmonic mean of how much everyone values the item. Indeed, when we take a second look at how many tickets each person is willing to buy, we see that the total number of tickets must be less than how much at least two people value the item, which is less than the amount paid by the winner in the second price auction. Thus it appears that Chinese auctions yield lower revenues to the auctioneer than second-price auctions. Perhaps this is not the best way to run a charity auction. Something to keep in mind; maybe I’ll look more into it at a later point.
Marli will write another post later this week.
(Final disclaimer: the arguments in this post constitute a sketch of an argument, not a rigorous proof. As such, the results here should be considered tentative, and this post should not be construed to be the final word on the subject.)