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Consider a simplified version of eBay where everyone bids once on an item, nobody sees each-other's bid, and the highest bid wins. This is called a "First-price sealed-bid auction".

One day you find a trustworthy guy called Bill that promises that they can pay you \$200 for a Nexus One phone. You discover that Nexus One phones frequently sell for less than \$200 on eBay, and several of these phones are auctioned off every day. Bill hates to use eBay and won't ever use it no-matter what, and he doesn't really care what you bid for the phones, so long as he gets them. You realize that there is an opportunity to make some money here.

So what do you bid when you see one of these phones? Its a compromise, since if you bid \$200 you'll definitely win the phone, but you'll make no money. If you bid less, then you'll make more money if you win, but you'll win less frequently. If you bid too little then you'll never win and you'll make no money.

This is not a hard question to answer if you are reasonably smart and have a decent amount of information about past winning bids, since you'll be able to discover, given a bid $b1$, a function $f1(b1)$ that will tell you the probability of winning the auction, given whatever you bid is. Let's assume that you are smart enough and you do have enough information. Let's say your cut of the profit is $c1$, then, applying some high school math, your expected profit is:


So from that you are able to decide the optimal bid to maximize your own profit.

Then Bill comes to you and tells you that he may not always be able to pay you \$200 for the phone, sometimes he'll pay more, sometimes he'll pay less, but he will tell you what he will pay before you must place your bid. Fair enough you think, you've got your function $f1(b1)$, you can determine the optimal bid depending on whatever Bill is willing to pay.

Weeks go by and you are making good money off this relationship. One night you go out for a beer with Bill, and he drops a bombshell. He tells you that actually, he is playing the same game you are. He knows a guy, Jim, who is buying the phones from him. He won't tell you who Jim is (Bill isn't an idiot), but it turns out that Jim is paying even more for these Nexus Ones than Bill is! Worse still, it turns out that not only is Bill no idiot, he is at least as smart as you are. He has determined the probability of you winning your bid, and is choosing his profit margin to optimize his total profit, in the exact same way that you are (although his function, $f2(b2)$, won't be the same as your's because you are reducing his bid by your profit margin).

You go home drunk and don't think about it much, but the next day you have a real headache, and its not just all the beer you had with Bill last night. What exactly is the relationship between you and Bill? In one sense, you are on the same side. If your cut combined with Bill's cut is too big, then both of you will make less money. But in another sense, you are on opposite sides, the smaller his cut, the bigger your cut can be, and vice versa.

Given this relationship, and assuming that you can't deal directly with Jim, Bill can't deal directly with eBay, and Bill is at least as smart as you are, how do you maximize your profit?

It has been asked whether Bill or I get to choose our cut first. Actually we can each change our cut at any time. Note that we don't know each other's cut, although we will probably notice the effects if the cut is changed.

It has also been suggested that this is similar to the iterated prisoner's dilemma, however in that you have perfect knowledge, you know whether the other prisoner cooperated or defected immediately after each round. That is not the case here.

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The correct strategy is not to maximize profit, but to maximize profit/time, since you can spend the rest of your time making money some other way rather than fooling around on ebay. So the optimal solution is to tell your friends Tim, Tom, Joe, etc. that you will buy phones from them for some figure I cant be bothered to work out if they can find them on ebay.

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Maximizing profit per-phone is maximizing profit/time. Otherwise, I don't think you've answered the question. – sanity Jul 24 '10 at 19:26

I don't understand c1 in this. In the first case, I get 200-my bid for every phone I buy. So my profit is $200*f1(b1)*(200-b1). You are right I should be able to find the maximum of this.

In the second case, if Bill has been watching eBay (even though he won't buy there), he can figure out your function f(Bill's price,bid)=probability of winning. Then he can maximize his profit the same way to set the price he pays. Then you derive your bid from his price.

If Bill hasn't derived your function f (because he hates eBay so much he won't watch?), you have an advantage in information, so you should be able to do even better. But the situation is not well defined so it leaves the realm of mathematics.

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