It sounds like ticket design is the problem; an algorithm is needed that:
- produces a set of bingo cards or tic-tac-toe boards such that only one board per card is a winner
- produces cards that contain a number of almost winners
Simply randomly assigning numbers wouldn't work, and it seems the method they've developed for building "fun to play" cards contains weaknesses against statistical analysis.
Hmm... now I'll be stuck mulling over a good algorithm for this all day...
The solution is straightforward: generate truly random boards, evaluate them, and then use the set that fits your payout profile. The problem is that these boards will lack the enticing hooks that keep folks coming back. The complete solution is to also discard boards that are not enticing enough. The result will have fewer artificial patterns like the ones used in the article to determine winners.
That procedure is vulnerable to the same issues mentioned in the article.
Let's say you design a game that has outcomes Lose, Near-Miss, Win, and Invalid (tickets that must be suppressed, e.g. multiple wins). Then, imagine a rare, salient pattern, like 3 singletons in a baited-hook row. That might be an extremely rare occurrence in the overall lot of random boards, but it might still occur disproportionately in, or in the vast majority of, Win boards and Invalid boards.
If you then choose the Lose, Near-Miss, and Win cards randomly, in the desired proportion, from your truly randomly generated set, then the pattern will be statistically correlated -- potentially strongly -- with the Win cards. That's what the article describes.
A single confusing sentence in the article seems to have gotten a lot of people (including me, at first) thinking this had something to do with PRNG; rather, apalmblad's claim that this is a game design issue seems right.
Only if the "game design" purposely used a limited pool of numbers for the visible and hidden boards. If there was no guarantee that a number would appear 1.9 times, then seeing singleton numbers wouldn't be a predictor.
My guess is that the restricted number pool was used as a means of easily mapping a number (the number on the back of the card) onto a playing board. Anyone interested in the why and how of this should look at The Wizard of Odds[1]. If the number on the card was used to seed a PRNG which then produced a lot more data, there could be a more sophisticated board generator that doesn't need to take such compromising shortcuts.
So in a sense, the culprit is game design in that rules were created to allow for certain percentages of Loses, Near Wins, and Wins. But there is no reason to mix output control into game design. My suggestion is to make the game rules with no regard to controlling output. Instead, evaluate the generated boards and keep/drop them to control the output.
If there was no guarantee that a number would appear 1.9 times, then seeing singleton numbers wouldn't be a predictor.
Let's say there's no such demand, and the boards are generated truly randomly and neutrally. If visible-quality X is disproportionately correlated with invisible-quality "Win" then the game is already flawed; this may emerge naturally from the Win conditions and from the game design decision of what is shown in the baited-hook. If the output is controlled post-generation to increase the proportion of Win cards and Near-Win cards vs. Lose and Invalid cards, then the statistical correlation may be greatly increased.
Simplified example: Scratcher with two numbers 0-4: one bait, one hidden, pays if sum is 5 or 6. If everything is fair and truly random, the odds of winning are, 0 showing: 0%, 1 showing: 20%, 2-4 showing: 40%. Already a bad game, but now the game designers want to eliminate cards with sums of 7 or 8 because this confuses people, (not minding that they're changing overall odds of winning), so they block those Invalid cards from shipping without blocking anything else. That gives:
0 1 2 3 4
0 - - - - -
1 - - - - W
2 - - - W W
3 - - W W I
4 - W W I I
The new odds of a given ticket, given the visible number, are: 0: 0%, 1: 20%, 2: 40%, 3: 50%, 4: 67%. The point here is that now a 4-showing-card is >3x as good as a 1-showing card, when it used to be only 2x as good, and it might now have positive expected return.
Simply randomly assigning numbers wouldn't work, and it seems the method they've developed for building "fun to play" cards contains weaknesses against statistical analysis.
Hmm... now I'll be stuck mulling over a good algorithm for this all day...