## Tuesday, December 7, 2010

### Fortune Cards

Cold has a guest post from Wukam up today about fortune cards.

TL;DR – Even though the cards, when you use them, will vendor for different prices, they all have the exact same statistical worth of X,where X is the potential value times the chance of getting that value.If mats are below X, you can buy absolutely risk free. If mats are above X, then you’re taking a risk.

Now, as a math and game theory geek who got into poker a few years back, and who then spent a while analyzing the variance of various tournament scenarios, deals and prop bets, I have a pretty good handle on this, and am poised to be the guy who takes the gold from a lot of unsuspecting statistically naive players.

But since I am a gnome and not a goblin, I will be posting the best explanations I can here of how to analyze this situation, and not be taken in.

I suspect that Wutan here is making a mistake primarily in wording, but this: "If mats are below X, you can buy absolutely risk free. If mats are above X, then you’re taking a risk.", is just plain wrong.

I've read and heard a million poker players say essentially the same thing. "Playing this game isn't a risk -- I'm a winning player."

Also wrong.

Every time you put money on the line in a game with chance involved, you are taking a risk, even when you will profit on average.

What's going on is that Wutan (and many poker players) are confusing positive expected value (a good bet) with a lack of risk. Even when the value of mats is less than the expected value of the fortune card and it is mathematically correct to gamble (assuming you have sufficient bankroll for your risk tolerance curve), there is still a great deal of risk.

In the example Wutan chooses, a craft has a 1/1000 chance of producing a 5kg vendor item, and a 999/1000 chance of producing nothing, or some unsellable trash. So, if the mats for a craft cost 4g, then you essentially make 1g per crafted card on average. A lot of people think this means that you can craft 1000 cards and be guaranteed a profit.

But it doesn't work like that. Crafting 1000 cards does not guarantee a 5k card. In fact, there is a ~36% chance that you will end up with absolutely nothing, having spent your 4k or whatever for mats. Of course there is also a chance that you will get 2, or 3 or even more 5kg cards, and make a huge profit. On *average* you will make 1k profit, but random numbers are random -- on any given crafting of 1000 cards, over 1/3 of the time you end up with nothing, so it would be stupid to do this with all your working capital, even though it is a big positive expected value. Even crafting 2000 cards leaves a 13.5% chance of getting back nothing (for minus 8k), and around a 27% chance of getting only 1 card, leaving you 3k in the hole for a 40% chance of taking a big loss.

This will only be anything like printing money for those who can stockpile so many mats that the chance of ending up a loser is negligible. But in Wutan's example that would be a *huge* amount of cards. Probably in the neighborhood of 100,000 craftings. And even then, you will still lose some of the time, just not very often.

Now, in the actual game, the variance won't be anywhere near that high, since there will be lots of cards that go for in between 0 and 5kg, so your chances of ending up with nothing after 1000 crafts will be infinitesimal. You will probably do worse than break-even fairly often on 1000 crafts, but when you do, you'll normally keep at least 1/2 of your investment and most of the time your losses will be in the 5-10% range.

Look at guys who play +EV Video poker for a good sense of what the payoffs on a +EV lottery look like. Generally, if your actual hits of royal/quads over some long stretch are significantly worse than expected, you will be losing over that stretch, because the chance at one is a fairly significant portion of your equity, and your EV is usually very very thin (max 1-2%).

The basic payoff structure and variance will probably be similar for these cards. The chance at rare cards like 5k/1k etc. will be a big portion of the value of an unknown card, so any run that is very low on those cards, is likely to be a run that loses, unless the EV is enormous by real-money gambling standards.

Now, the EV of this probably *will* be enormous if these cards are the main driver of demand for their mats. I judge this by the general profitability of other items with a random component such as ore prospecting and disenchanting. Generally, the cost of saronite tended by bottomed not by it's average value for prospecting and shuffling, but by the absolute baseline value of the most likely path: getting green gems, making rings/necks and vendoring them (or de-ing and getting all infinite dust). Now any given prospect could result in a good blue gem worth more, or a de could result in cosmic essence, worth more, but generally this was all profit. Generally goblins bought up saronite for this process only when we were almost certain to profit.

Once we start seeing what the percentages are for the various cards, we can do some statistical analysis to see when it makes sense to craft them given various risk tolerances and bankrolls. I will definitely be watching this market like a hawk, as there's a decent chance this will be hugely profitable. If I can figure out how to get blogspot to put up docs for download, I'll put up a spreadsheet in a future post that shows how to analyze this craft based on the chances of various cards, and your bankroll and risk tolerance.

With the big 5k reward possible, it's also quite possible that the unopened cards will, like scratch tickets, sell for significantly more than their expected value. And just as in real life, it will generally be some of the people with the least gold and the most difficulty making it, who are most willing to give it away in small probabilistic increments to those of us who understand expected value.

My next post will be about risk tolerance in general and how to talk about it mathematically.

1. I just geeked out on this post. Stats is what I do (seriously - they pay me for it), and I loved the read. You're going on my blog roll.

Cheers!

2. Chad has a point in Cold's thread. This may all just be wankery if people will radically overpay for these cards in the same way they do lottery tickets. We may well be able to make lots of gold crafting these for sale even though the expected value is less than the mat price.

Or they may be no profit opportunity at all, if there are enough innumerate or risk-philic scribes alone to push mat prices well above the expected value.

3. Yeah, Chad had a great point. I read this discussion across the blogosphere as a way for scribes to cash in on a shuffle - I think that's a bad idea - one that Blizz has certainly thought of.

In reality, these cards are worth far more sold for 20g & up to other players. Now that I see that the point isn't to shuffle, but to resale - it makes me sad that I don't have a high-level scribe. =(

4. Been making a killing on these things. Sales are amazing with the right tips on barking mysterious fortune cards.

5. Hey Gnome,

Great to see a post in the WoW world mentioning EV, (although I think that it will have gone over most peoples heads considering that the vast majority of poker players don't get it.)