The Mac version of Bonnie’s Bookstore is out – you can try & buy it on my own site or on the site of my publisher, PopCap.

I’m quite curious to see if it generates much sales. I’ve seen some small game makers report 40-60% of their sales from the Mac market, but that’s often off a very small sales base, and a single mention on apple.com will cause Mac sales to dwarf Windows sales.

While the initial ‘get it up and running’ port to Mac took only a few days (3?), I ended up investing far more time into the Mac version than I’d planned – maybe 15% of so of the overall ‘time budget’ for the game. I’m skeptical that it will generate 15% of the total sales, but we’ll see.

How did a quick port get bogged down, and end up not coming out until 3 months after the Windows version? Long story that I won’t fully detail here, but it was, in part, a result of trying to do a really clean port, that’s very ‘Mac-like’, fulfilling all kinds of tricky requirements, as well as a rather stop ‘n start development and testing process, that was not nearly as efficient as a one shot, “port, test, and publish” approach might have been.

See here.

Well, let me warn you now that you’ll find it just as hard to stop playing Bonnie’s Bookstore. So better be prepared to spend a lot of time on this little gem from developer New Crayon and publisher PopCap Games.

So check out Bonnie’s Bookstore today. Just don’t blame me if you get addicted.

This review is written by the same author who wrote a piece a few weeks back on the making of BB.

There’s a 3 page article in this month’s (May) Computer Gaming World about Bonnie’s Bookstore. It’s nominally a review (and very positive), but it’s also about the Casual Games industry in general, and about my personal transition from mainstream commercial PC game developer to Casual Game developer, with a sidebar on PopCap (my publisher) and their history.

No link unfortunately – you have to stop by your local magazine rack to check this one out…

A follow up to yesterday’s quasi-riddle, on Tycoon Tommy, who despite apparently high average returns, lost money over a 20 year period.

As noted by the commenters, Tommy fell victim to a mathematical illusion. Here’s the full set of returns I postulated when setting up the problem:

Specifically, while the arithmetic average of Tommy’s returns was high (10.5%), the geometric average was slightly negative (-.04%, if my math is right).

In a nutshell, taking an initial sum, then adding 20% to it, then subtracting 20% from THAT result, does not result in your original value, but rather less (i.e. 100 * (1+.20) * (1-.20) = 96). It doesn’t even matter whether the negative result comes before or after the positive result.

The geometric mean is basically the Nth root of the total growth. So, for the simple +20%, -20%, above, with a final result of 96, the geometric mean can be computed as:

((final total/initial total) ^ (1/num periods)) – 1
or
((96/100) ^ (1/2)) – 1 = -2.02%

If there is no deviation in the annual results, the arithmetic mean will be the same as the geometric mean (i.e. as in Carl’s steady 5% return).

The wider the deviation in annual results, the greater the difference between the geometric mean and arithmetic mean. Even when all the returns are positive. For instance, two years of +10%, then +90%, gives you (100 * 1.1 * 1.9) = 209, with a geometric mean of 44.6%, which is noticeable less than two years of +50%, then +50% (100 * 1.5 * 1.5) = 225, geometric mean, of course, of 50%.

You’ll note in the table above a third column, showing the results of a 30/70 blend – with 30% of the money invested in Tommy’s fund and 70% in Carl’s fund, and with the money rebalanced each year to restore that 30/70 balance. Interestingly, the final result here is noticeably better than either of the 100% approaches. This goes against many people’s expectations that you should invest as much as possible in whatever has the highest return.

In fact, it is usually possible to get a better result by splitting money between multiple investment options, then periodically rebalancing the returns. A full explanation of this phenomenon, and how to apply it in the real world, is beyond the scope of this article, but, if interested, I’d strongly encourage you to do on-line searches for the terms ‘Efficient Frontier’ and ‘Mean Variance Optimization’, and/or read up on this site. You actually get a double bonus by this strategy – higher long term returns (geometric returns, the ones that count), and lower volatility/deviation.

Continuing the theme from last week, a riddle disguised as a parable…

Tycoon Tommy wasn’t really a tycoon, but sometimes he thought he was. On his 21st birthday, he received a $10,000 inheritance, as did his twin brother Conservative Carl.

Carl lived up to his name, and invested the money in a bond paying 5% interest. This bond had a special feature guaranteeing that the interest could be reinvested each year at the same 5% rate.

Tommy had an inside line on a hot off-shore fund. The fund was VERY off-shore, in fact. It didn’t conform to typical reporting requirements, and really, Tommy’s investment would normally have been too small for the fund, but Tommy called in a few connections and was able to invest.

In the years after, Tommy didn’t get a lot of details on his fund’s performance. Every year, he received a short note stating that his fund was up or down by such and such percent for that year, and that’s all the detail he got. After a few years, he wasn’t even sure how much his fund investment was worth exactly. Still, Tommy waited anxiously each year to see how the fund had done that year, and by and large, he wasn’t disappointed. The first year, the fund generated a 14% return for all investors (including Tommy). Year 2 was +41%! Year 3 was a meager +3%, and year 4 was a bear year, minus 36%. Still, such things, one supposes, come with such a volatile, exotic investment, and the next year, year 5, Tommy was exhuberant when his fund had a +114% year. Doubled in a single year! Sometimes when he saw his brother, Conservative Carl, Tommy would rib him a bit for his foolishly conservative investment, with it’s 5% annual returns.

The years went by, and overall, Tommy was pleased. After 20 years, the fund had had only 6 down years, and 14 up years (including the +114% year, and a +102% year as well). Tommy was no math genius, but he added up all those annual return percentages, divided by 20, and was delighted to see that the average was 10.5%, more than twice as high as his brother’s slow and steady 5%.

Coincidentally, after year 20, Carl cashed out his investment, which, with 5% compounding had grown to $26,532. Carl bought a nice, but smallish, boat and invited Tommy out for the weekend.

Tommy thought, “Hmm, my fund has averaged a 10.5% return, against Carl’s 5%. With that 5%, his money has grown to $26,532. I probably have at least twice that much in my fund by now! I’ve heard a little about the miracle of compound interest – maybe that extra return has magnified my investment to $100K or more! Tommy will be green with envy when he sees the size of my boat…”

Tommy contacted his fund manager and requested a full redemption. A few days later, Tommy received a check in the mail from the fund for $9,929, and an attached statement confirming that his fund value had, despite intervening fluctuations, ended up slightly DOWN from where it started 20 years earlier. “Fraud! Theft!”, screamed Tommy. He stormed down to the local SEC office and demanded that the SEC take action against his off-shore fund.

After analyzing the statements, the SEC agent responded, “I’m sorry, there’s nothing we can do.”

It was as Tommy feared, “You mean you can’t go after them because they’re off-shore?”

“No.”, said the SEC agent, “We can’t go after them because no fraud or theft has been committed. This report matches up exactly with the rather terse annual statements you’ve been getting for years. There’s no funny business fees, penalties, etc, involved. The fund has been entirely honest with you throughout, and your final balance of $9,929 is correct.”

For the reader: How can this be so?

Thanks to all who commented on the riddles yesterday. The first riddle had the highest number of responses to any post I’ve made to date. Here’s my proposed answer to the riddles (paradoxes?) posted yesterday, here and here.

Starting with Riddle 2-A. In brief:

Suppose we are presented with the opportunity to open our wallets. Whoever has more money has to give it to the other guy.

A simple analysis suggests that you have a 50/50 chance of winning, and if you do, you’ll gain more money than when you lose.

I think this is the easiest to address:

Clearly, no new money is being generated in the interaction, so there can be no overall positive expectation for playing the game (assuming equal information between the players).

If in fact, this game DID generate positive total expecations, then I would encourage all married individuals to go right now to their spouses and play this game – the two of you will come out ahead, in the aggregate.

Or not. Anyways, I could run a simulation to address this problem, but I think it’s simple enough that it’s unnecessary. I won’t present math to try to analyze WHY this apparent paradox exists, but follow the Wikipedia link at the end to those who do.

2-B is basically the same as 2-A, except that rather than their being an unknown difference between the two player’s wallets (envelopes), it’s known that one player has twice the other player’s money. I’ll skip it, because I think I address the case better in analyzing Riddle 1, below.

Riddle 1. In brief:

An eccentric billionaire invites to participate in a challenge.

He has taken a number of cards, and had a number printed on each of them. On the reverse side of the card, there is another number – twice the value of whatever is on the front.

The cards are put in a bag. You draw one, and see that it shows 53. By prior agreement, he will pay you that sum (in dollars), or allow you to pay a 10% premium (i.e. $5.30) to flip the card over and accept whatever sum is on the back, instead.

Do you pay to flip the card?

The key here is that, although it APPEARS that the amount of money on the other side of the card has a 50/50 chance of being twice or half what’s on the side you see, it’s not quite that simple. In fact, the cases where the amount on the other side is higher are disproportionately drawn from the set of card face values with low values on them. Confusing? Sure, but it’s easy to simulate:

Here’s the code to test this. It sets up a simulation where 10 cards receive a random value from 1 to 100, and the back sides receive double that amount. Then I draw a card and record the results. I repeat the experiment 1,000,000 times (with a new, random set of cards each time), and here are the results:

Flip mean: 68.35
No-Flip mean: 75.53
Flip advantage (penalty) -9.51%
Flip winners percent: 50.34%
Flip winners mean (starting): 50.51
No-Flip winners mean (starting): 100.89

By flipping, you end up with a 9.51% overall penalty – very close to the 10% penalty you’d expect (i.e. the cost of the flip.)

Note that the mean result from not flipping is a bit over 75. That’s the average of the front side (1..100, average = 50.5), and the back (2..200, average = 101), total average = (50.5+100)/2 = 75.25.

Note that on the 4th line, we see that in this simulation, the odds of ‘flip’ improving your position on any ONE bet were 50.34% – close to the expected value of 50%.

However, the next two lines show why the whole paradox exists: The occasions when ‘flip’ works are those where the starting value of the card (i.e. the number you see) are relatively low (mean 50.51), and the no-flip works on higher relative values (mean 100.89). So even though, on any SINGLE observation, flipping should work half the time, the problem is that those observations cluster into the group where the initial observed value is low.

I also did the same test altering one aspect of the simulation. What if, rather than the ‘flip’ option giving you whatever is ALREADY on the other side of the card, it gives you a random (50/50) chance at double or half your money. Here’s the results:

Flip mean: 87.41
No-Flip mean: 75.53
Flip advantage (penalty) 15.72%
Flip winners percent: 50.47%
Flip winners mean (starting): 75.54
No-Flip winners mean (starting): 75.52

Flipping here improves your outcome by about 15%, as expected. The key difference? The starting values of the flip winners and the no-flip winners are effectively the same (~75). So flipping is not paying off only in the low value cases, but rather, in an even spectrum across all cases.

Some might argue that in the original problem, the eccentric billionaire’s range of possible numbers for the cards was unstated, and in my simulation, I’ve arbitrarily chosen 1-100. Fair enough. Here’s the results on two other ranges.
(All using the ‘preset deck’ scenario)

Card Range 1-20,000:
Flip mean: 11636.22
No-Flip mean: 12858.08
Flip advantage (penalty) -9.50%
Flip winners percent: 50.36%
Flip winners mean (starting): 8596.28
No-Flip winners mean (starting): 17180.83

Card Range 1524-7621 (i.e. a fairly arbitrary low and high end)
Flip mean: 6006.02
No-Flip mean: 6637.75
Flip advantage (penalty) -9.52%
Flip winners percent: 50.35%
Flip winners mean (starting): 4437.15
No-Flip winners mean (starting): 8869.06

Note that I couldn’t easily test with max values above 32,768, because of limitations of my random number generator. But if you still have doubts, download the code, put in a better random number generator, and test from 1 – 1 billion. The results will be effectively the same.

Finally, I know this is only a simulation, and doesn’t include a good mathematical explanation of WHY this works in such a counter-intuitive manner. The math of this problem is beyond me, but I’d refer you to the Wikipedia article on this subject, with many links to mathematical analyses at the bottom.

OK, I posted this inside the thread comments to the previous riddle, but I wanted to call it out with it’s own post.

Riddle 2-A (taken from here):

Suppose we are presented with the opportunity to open our wallets. Whoever has more money has to give it to the other guy.

A simple analysis suggests that you have a 50/50 chance of winning, and if you do, you’ll gain more money than when you lose. So you should take the bet. But the same analysis suggests that I too should take the bet, and it’s a zero-sum game, so it can’t be advantageous to both of us!

Riddle 2-B (slightly rephrased)

I have two envelopes with checks (made out to ‘Cash’) in them. Andy and Bob randomly draw the envelopes, Andy ends up with envelope A and Bob with B. Andy is offered the opportunity, sight unseen, to make the same deal as in Riddle 2-A (i.e. if his envelope has a smaller check, he gets both envelopes, otherwise he loses his envelope). Should he do it?

Is there a fundamental difference between 2-A and 2-B above?

How about between these two and Riddle 1 from the previous post? (other than the concept of paying a premium to make the switch in the previous riddle…)

I posted this on a forum I frequent, but so far haven’t gotten any answers I’m satisfied with.

—–

An eccentric multi-billionaire with an interest in psychology and probability invites you to participate in a challenge.

He has taken a number of cards, and had a number printed on each of them. On the reverse side of the card, there is another number – twice the value of whatever is on the front.

The cards are put in a bag, and you are invited to draw one. You reach into the bag, feel around a bit, and pull out a card. You see that it shows 53. He will pay you that sum (in dollars), or allow you to pay a 10% premium (i.e. $5.30) to flip the card over and accept whatever sum is on the back, instead.

Do you pay to flip the card? Why/Why not?

Further info:
* All rules are explained/guaranteed to you up front – the ‘flip’ offer was not in any way conditional upon whatever card/value you initially pulled, but rather was explained to you before you pulled the card.
* You have no particular idea as to the range of values printed on the cards. The experimenter is a billionaire with a rather cavalier attitude towards money. You will only be allowed to do this experiment once, so any knowledge you gain won’t be applicable to future tries.
* You can’t peek into the bag, feel the type on the back of the card, or any other ‘cheating’ solution like that. Looking at the card in front of you, you have no idea whether it was originally the ‘front’ or the ‘back’.
* The fact that the value you pulled ended in an odd digit is not in any way significant. The billionaire tells you that, subject to the range of the experiment that he had in mind, he had values randomly generated, down to fractions of a penny, then took the original value, and the doubled value, and rounded them to the nearest whole dollar (i.e. the ‘unrounded’ number on the reverse side could be anything from 26.5 to 26.999999, or 106 to 107.999999)
* The billionaire is honest, and, after the experiment, will allow you to fully examine everything (the cards, his number generator and it’s output, etc)

In answering why you did or didn’t pay, don’t rely on something simplistic like “I’m a gambler”, or “I’m risk averse”. Try to figure out mathematically/logically why one or the other choice yields the best expectation.

[If you're reading this site via RSS feed or aggregator, be sure to click through to read proposed responses...]

A new site has been set up to compile a ‘blended average’ top 10 - apparently taking top 10 lists from 15 different portals and creating a composite. They don’t go into much detail on their methodology, but it’s still a nice tool to have, supplementing James Smith’s excellent site focusing on Real Arcade sales.

DFC Intelligence just released a research report titled Will Online Games Decide the Upcoming Video Game Console War?

Their answer was no (more or less).

They took the position that, in spite of Sony’s public posturing, Sony wouldn’t emphasize their on-line service (competing against X-Box Live) very heavily for the coming generation. Particularly striking was this:

However, even with our strong growth forecasts we estimate that less than 25% of the new console systems get connected to an online service by 2011.

For me, this is interesting in thinking about X-Box Live Arcade – the service on X-Box that sells downloadable casual games. Some in the casual industry have been touting it as a potentially huge market in the next 2-3 years. Upon X-Box 360’s launch, with a relatively weak lineup of AAA titles, there was a lot of attention on the Arcade lineup of simple 2D and 3D casual games – Geometry Wars in particular got a lot of attention, and conversion rates were supposedly pretty high.

Like DFC, I’m not entirely convinced that X-Box Live Arcade, and it’s equivalents on the Sony and Nintendo platforms, will eclipse the PC casual/downloadable market in the next couple years. Early adopters of consoles tend to be much more technical – more willing to hook consoles up to the internet and try out odd download services. I think connection rates, download rates, and conversion rates will all fall off significantly as the next-gen installed base goes up.

Maybe it’s just my personal biases. I’ve got all 3 previous gen consoles in my home office, hooked to a nice TV, with a broadband connection 3 feet away. But I’ve never connected any of the consoles. I can get the casual games I want easier through the PC. And so far, the temptation to try sports games on-line against trash-talking twelve-year olds hasn’t tickled my fancy.

If the portals are correct and the biggest market for casual games is 30-50 year old women, then I don’t see that consumer rushing out to buy $500 consoles and hook them to the internet.

Long term – 5 or 10 years from now? Maybe. Would I turn down good opportunities on X-Box Live Arcade in the short term? No. But it’s not going to be a focus of mine in the near term.

OK, I’m a computer programmer – a math geek. I can do fairly complex problems in my head. So maybe I’m not typical of, say, the average customer at Burger King. Still…

I stop at BK for lunch today. They’ve got 4 chicken tenders for $1, or 8 for $2.80. The former is listed in the ‘dollar menu’ section, the latter about 5 items down in the regular section. Neither is on sale – they’re both regular ala carte prices.

Is this a symptom of the inability of BK managers to add 1 + 1, or do they think their typical customer can’t add 1 + 1?

When I was in college, the popular local diner had a ’special’, which was a combo meal of a cheeseburger, fries and soda, for $3.40. However, if you actually added up the ala carte prices of those same three items, they totalled $3.20 (plus or minus a nickel or so – I can’t remember exactly.) So, we’d have this ordering sequence:

Me: “I’d like a cheeseburger, fries, and a diet coke”
Server: “OK, one special coming up”
Me: “No, I don’t want the special – I want a cheeseburger, fries, and a diet coke”
Server: “That’s a special…”
Me: “I don’t want the special, I want…”
(etc..)