Game Theory and Chinese Auctions

Factual Questions
1

I’m not sure if the term applies to only one type of auction. But the type I’m thinking of has an unlimited number of tickets all sold at the same price, which can be placed (in any number) in the pool towards any particular prize. But the prizes themselves have varying values, in some cases explicitly so. (In some cases the prize is itself $X cash, but even where it’s not, there is frequently an option for an $X cash value alternative. And in some cases the differing values are pretty obvious.) So there’s an interplay of prize value and likelihood of winning, the former of which would undoubtedly influence the latter to some unknown extent.

To give a very simple example. Suppose there were 1,000 tickets sold at $10 each. There are exactly 2 prizes, one of which is $2,000 in cash and the other $4,000 in cash. You have one ticket. Now the first thought is that for the same $10 you can get a chance to win $4K, so why put in for a chance to win $2K when you can get a chance to win $4K for the same price?

But suppose every single other person in that auction made that exact calculation? In that case, you would be guaranteed $2K if you put in for the $2K pot (since you would be the only ticket in that pot) while your chances would be 1/1,000 if you put in for the $4K pot. So perhaps you should go for the $2K prize.

OTOH, some unknown number of other people are undoubtedly making that exact same calculation, and will put in for the $2K prize for that exact reason. If 50% of the participants made that calculation, then you’re back to the same odds of winning in either pool, so you may as well go for the higher prize. And even if less than 50% make that calculation, you would get the same expected value from either pot if 1/3 of the players put in for the lower value. But it might be that very few people think that way.

So you’re essentially making an assessment on how many people will make one calculation versus the other (as are some of the other participants, themselves).

Of course, it’s more complicated than that, because some participants don’t think of these things rationally, and in addition the cash value of the various prizes is not always so explicit and known. (In addition there’s the fact that the utility of money is not the same at various values, and this itself varies by person.) But the basic dynamic is there.

So the question is basically: is there an optimum strategy in this type of situation, and/or some rules of thumb that govern this interplay of factors?

[Not sure if this belongs in GQ or IMHO - I’m wondering if there are studies of this, as well as asking for thoughts.]

2

Define “rationally”. If we’re looking just at expected value for your money, then the most rational move is to not buy any tickets in the first place. Anyone who is buying tickets, then, must have some other motivation, and we need to know what that motivation is. It might, for instance, be because they find gambling to be fun in itself, in which case they might get more fun from going for the big prize, regardless of the odds. It might be that their only motivation for buying tickets is to support the nonprofit that’s organizing the event, in which case they might not care about winning at all, and could put their tickets in either bucket, or even in no bucket at all. Or their motivation might be some mix of expected value, fun from gambling, and support of the nonprofit, in which case we’d want to know what that mix was.

3

I don’t think you can say that. It depends on the conditions of the auction. If you’re the only bidder, the obviously it makes sense to bid on every item. If the total payout is higher than the total cost of all tickets, there may be a positive expected value. If I know something about the behavior of the other bidders (maybe they’re all Star Trek fans and one of the items is Kirk’s chair), I may be able to use that knowledge. Of course generally in problems like this we assume only that the other bidders are acting “rationally”; that is, trying to maximize the expected value of their bids.

I’ve found a few papers, although several of them mentions that Chinese auctions have not been studied much:
https://www.sciencedirect.com/science/article/pii/S1517758015000077
http://www.gtcenter.org/Archive/Conf07/Downloads/Conf/Matros478.pdf

4

All kinds of things motivate people to buy raffle tickets - charity is probably the greatest motivator of all. UK National lottery tickets are sold on the way that some (28%) goes to charity.

I used to run a raffle in which we had a number 5x5 boards with 25 numbers. Players could buy a number of their choosing, so long as it was available, for £1 - the winning number was only chosen when the board was sold out. The prize was a choice of whisky/gin/etc which cost us around £8 and a 'profit to the charity of £17. We could easily sell up to 20 boards in an afternoon.

5

I think you need to distinguish between the motivation for buying a ticket to begin with and the goals/strategy for ticket placement once in the auction.

IME most people (virtually all, at least overtly) are primarily motivated to participate based on support for the charity. But once in the game, people try to maximize their expected value or at least utility. Just not necessarily in the most rational manner.

6

Leaving aside whether buying a ticket in the first place is a good investment, I think this might be the answer:
an X% chance of winning the 4K prize is twice as valuable as an X% chance of winning the 2K prize, or, equivalently, an X% chance of winning the 4K prize is exactly as valuable as a 2X% chance of winning the 2K prize.
So, if you had a single ticket to put down, and you could see how many tickets were in each bucket, you’d put the ticket in the 4K bucket if there were less than twice as many tickets in there as in the 2K bucket. If there were more than twice as many, you’d put it in the 2K bucket. If there were exactly twice as many, you wouldn’t care, so flip a coin. But you always maximize your return by moving the balance to exactly twice as many tickets in the 4K bucket as in the 2K.

Now, suppose you had three tickets, and could see that there were exactly twice as many tickets in the 4K bucket. Your optimum move is clearly to put two tickets in the 4K and one in the 2K.

But we don’t know how many tickets are in each bucket. However, we’re going to – for the sake of argument – assume everyone else is also rational and is making the exact same calculation as we are. So the only robust strategy is that everyone is always trying to move to that 2:1 balance, so it’s clear the most likely configuration is that the balance is close to 2:1. Therefore we should, always put twice as many tickets in the 4K box as the 2K box. If we only have one, flip a three-sided coin— or, if you don’t have one, roll a six-sided die. On a 1,2,3,4, put the one ticket in the 4K bucket and on a 5,6 in the 2K bucket.

I think this same analysis holds for more than two prizes and buckets, but I haven’t completely thought through situations with a relatively small number of tickets, so it might break down then.

7

I don’t understand this at all.

Even if we grant that everyone else is rational and making the same calculation and always trying to move close to that 2:1 balance, the thing is that no one else has any idea which bet moves the balance closer to a 2:1 ratio.

Based on your assumptions you would assume that if people knew that the lower prize currently had less than 1/3 of the tickets, that they would put in for that prize until the balance got to 1/3. But no one has any idea whether the balance is currently lower than 1/3, or how many tickets on that prize would make it 1/3. So assuming everyone is rational doesn’t tell you anything about what the current ratio is (or what it’s likely to end up at).

I suspect that you’re envisioning a situation where everyone is dividing up multiple tickets between the two prizes, and your assumption amounts to assuming that all these people allocate 2/3 of their tickets to the higher prize. But I don’t think the assumption that everyone has multiple tickets that they allocate in accordance with the prize value is valid. And once there are some people who only put in one ticket or who don’t distribute evenly, then the ratio is skewed, and there’s no reason to assume that it ever becomes unskewed based on what you describe, since no one knows that it’s skewed or in which direction.

I don’t get this either. At the point where there are twice as many tickets in the $4K box, then you would be ambivalent as to which box you choose, and could put all your tickets in either. (Unless you’re saying that your own tickets will skew the balance.)

8

There’s a simple statistics formula for situations like this: % likelihood of winning * payout = “value” .You’re going to have to guess at the likelihood, (assuming you can’t observe where people are placing their bets for some time period).

9

Some people might be playing irrationally, but we can’t know how they’re irrational, and it might be in either direction, so the best we can do is assume that they’re all playing rationally. Rational play is to allocate their tickets 2/3 to the big prize, and 1/3 to the small one. So if we assume that they’re doing that, then we should do that, too.