The Universe of Discourse
Thu, 12 Dec 2013

Simplified Poker
Simplified Poker attempts to make poker as simple as possible, without losing the essential flavor of poker. (“But no simpler.”) Each player first pays an ante of value a into a pot and is then dealt a hand. The player makes the first move and the game proceeds according to one of the following 7 sequences of play:
Player Dealer Player Result
Pass Pass   Tie
Pass Bet Fold Dealer +a
Pass Bet See Show a+b
Bet Fold   Player +a
Bet See   Show a+b
Bet Raise Fold Dealer +a+b
Bet Raise See Show a+2b
The table should be self-explanatory to readers familiar with poker, but for others I will explain. The player moves first and has the option to bet a fixed amount b that her hand is best, or to pass and see what the dealer will do. If the player passes, the dealer gets the same choice; if both players pass the hand is dead and the players get their antes back. (Or equivalently the round is replayed with new hands.)

When faced with an opponent's bet, a player may “fold”, which means to drop out and forfeit the pot, or “see“, which means to match the bet by putting amount b in the pot. The dealer has an additional option when the player bets: he can “raise” means to increase the bet to 2b. After this the player can again see, at a cost of b, or fold, forfeiting the pot.

If a player folds, their opponent automatically wins the entire pot, for a net gain of at least a, and a+b in the case that the player folds after she bets. If neither player folds, the players expose their hands in a “showdown” (“show” in the table) and the better hand wins the entire pot, for a net gain to the winner of a+b, or a+2b if the bet was raised. If the two hands are identical, the pot is split and both players get back their money.

It remains to describe the deck and the hands. These are independent of the rules as described so far, and could even be a standard poker deck with standard poker hand rankings, but I believe the following is the simplest thing that retains any flavor of poker:

The deck contains an infinite number of cards of one suit, of which half are jacks (J) and half are kings (K). The king is high, the jack low. Two cards are dealt to each player. The first (the “hole card”) is face-down and, until the betting is complete, can be seen only by the player to whose hand it belongs. The second (the “up card”) is face-up and can be seen by both players.

The best hand is a pair of kings, followed by a pair of jacks. A hand without a pair is lowest.

(A possibly simpler game that might still satisfy the desiderata would have the two players share a single up card, but we can consider that another time.)

The reason I think this extremely simple game still contains the essence of poker is that it is not an entirely pure game of probability. There is scope for bluffing. Suppose that you are the player, going first, and the hands are as follows:

Opponent You
HoleUp UpHole
Here, your hand is worthless. The best you can hope for in a showdown is a tie. But you can bet anyway, pretending to have a pair of kings, and possibly trick the dealer into folding a pair of jacks. There may be a mathematically optimal strategy about how often one should bluff in this situation: you should bluff x% of the time, and then the dealer, not knowing if you are bluffing, should fold y% of the time and raise z% of the time.

But the mathematics does not tell the whole story, because you can win if you can persuade the other player that you hold a pair of kings. So the game is partly about psychology, and if you are good at persuading the other player to fold, you might win more than the mathematics says you should.

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Wed, 24 Jan 2007

Length of baseball games
In an earlier article, I asserted that the average length of a baseball game was very close to 9 innings. This is a good rule of thumb, but it is also something of a coincidence, and might not be true in every year.

The canonical game, of course, lasts 9 innings. However, if the score is tied at the end of 9 innings, the game can, and often does, run longer, because the game is extended to the end of the first complete inning in which one team is ahead. So some games run longer than 9 innings: games of 10 and 11 innings are quite common, and the major-league record is 25.

Counterbalancing this effect, however, are two factors. Most important is that when the home team is ahead after the first half of the ninth inning, the second half is not played, since it would be a waste of time. So nearly half of all games are only 8 1/2 innings long. This depresses the average considerably. Together with the games that are stopped early on account of rain or other environmental conditions, the contribution from the extra-inning tie games is almost exactly cancelled out, and the average ends up close to 9.

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Thu, 16 Nov 2006

Etch-a-Sketch blue-skying, corrected
In my last article I discussed a scheme for improving the Etch-a-Sketch which contained a serious mechanical error. I was discussing attaching gears to the two knobs of the Etch-a-Sketch to force them to turn at the exact same rate. Supposing that the distance between the knobs is 1 unit, I said, then we can gear the two knobs together by attaching a gear of radius 1/2 to each knob; the two gears will mesh, and the knobs will then turn at the same rate, in opposite directions. This was fine.

Then I went astray, and suggested adding an axle peg midway between the two knobs, and putting gears of radius 1/3 on the peg and on the two knobs. This won't work.

The one person who wrote to me to ask about the problem is a very bright person, but been seriously confused about how I was planning to set up the gears, so I evidently I didn't explain it very well. It needed a picture. So this time I'm going to try to get it right, with pictures. Here is an Etch-a-Sketch:

Recall that the distance between the centers of the two knobs (shown here in gray) is defined to be 1.

Here are some gears, which happen to have radii 1/3, 1/4, and 1/6:

The dark spots are the axle pins, which are at the centers of the gears.

Here's a picture of an Etch-a-Sketch with a radius-1/2 gear mounted on each knob:

Since the two gears are of equal size, the knobs are constrained to turn at the same rate, in opposite directions. This forces the Etch-a-Sketch to draw a line with slope -1, from northwest to southeast or vice-versa.

Here the knobs have been fitted with different-sized gears, one with radius 1/3 and the other with radius 2/3:

The left-hand knob is forced to turn exactly twice as fast as the right-hand knob, producing a line with slope -1/2. To get a line with slope -2, just reverse the positions of the two gears. I got this much right in the original article. (Although it didn't occur to me, before I saw the pictures, just how much of the screen would be occluded by the gears. Better make them transparent.)

Then I suggested that you could drill a little hole in between the two knobs, and use it to mount a third axle and a third gear. If all three gears are the same size, the two knobs are forced to turn at the same rate, this time in the same direction, and you get a line with slope 1, from southeast to northwest:

All fine, except that I said that since all three gears are the same size, they must each have radius 1/3. Wrong. They must have radius 1/4, as the diagram above should make clear: from the center of the left knob to the rightmost edge of its gear is one radius; the width of the middle gear is two radii, not one, and from the edge of the right gear to the center of its knob is the fourth radius. Since the total of four radii is 1, each radius must be 1/4. Oops.

This wrecks the rest of the details of my other article. Since we were already including gears of size 1/2 and 1/3, I reasoned, we can throw in a gear of size 1/6 and get some new behaviors from the 1/2 + 1/3 + 1/6 combination. The corresponding combination for 1/2 and 1/4 is 1/8:

And this gets us lines of slope 2, or, reversing the order of the gears, 1/2.

So what next? The calculations are a bit less obvious than they were back in the happy days when I thought that installing two gears of size p and q left space for one of size 1-(p+q). It's tempting to consider a radius-1/3 gear next, since it's the simplest size I haven't yet installed. But to mount it on the knobs along with a size-1/2 gear, we need to include a size-1/12 gear to go in between:

This produces a 2/3 or 3/2 gear ratio.

Once we have the size-1/12 gear, we can mount it with the size-1/4 and size-1/3 that we already had:

Well, you get the idea. I probably would not have gone on so long, but I was enjoying drawing the diagrams. I used linogram, which rocks. it's almost exactly the diagram-drawing program I've always wanted; expect an article about this next week sometime.

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Sun, 12 Nov 2006

I've always felt that the Etch-a-Sketch is a superb example of a toy that doesn't do as much as it could.

An Etch-a-Sketch is a drawing toy invented in 1959 by Arthur Granjean and marketed by the Ohio Art company since shortly afterward. It looks superficially like a flat-screen television with two knobs.

The underside of the screen (that is, the inside surface) is coated with a fine aluminum powder. Also under the screen is a hidden stylus. One knob moves the stylus horizontally, the other vertically. As the stylus moves, it scrapes the aluminum powder off the screen, leaving behind a black line. If you hold the Etch-a-Sketch upside-down and shake it, the powder again coats the screen, erasing the lines.

It is very easy to draw horizontal and vertical lines, but very difficult to draw diagonal lines. (Wikipedia says "Creating a straight diagonal line or smoothly curved line with an Etch A Sketch is notoriously difficult and a true test of coordination.") So although extremely complex drawings can be made with an Etch-a-Sketch:

(Etch-a-Sketch drawing of Albert Einstein by Nicole Falzone)

Most people can only produce something like this:

As a child, I had an Etch-a-Sketch. I did get some entertainment out of it by figuring out how it worked: if you manipulate the knobs to remove the aluminum powder from a large part of the screen, you can see the stylus inside. But for the most part I found it frustrating and disappointing. As an adult, I still find it so.

But it needn't be so. The most frustrating thing about the Etch-a-Sketch, I think, is that its potential has not yet begun to be unlocked.

Consider a forty-five degree line. To draw such a line, one must turn both the horizontal and the vertical knobs at the same time, at exactly the same rate. Suppose, for concreteness, that we're drawing a line from the upper left to the lower right. If you turn the horizontal knob a little too quickly, the diagonal line will bend rightward; if you turn it a little too slowly the diagonal line will bend downward. So in contrast to the mathematically exact vertical and horizontal lines that are easy to draw, it's next to impossible to draw a diagonal line that doesn't wiggle. And when you screw up, you can't fix the mistake without erasing the whole thing and starting over.

But the solution is obvious: If you can link the two knobs somehow, so that they can only turn simultaneously, you can easily draw a diagonal line. As a child, I experimented with rubber bands, trying to get one knob to drive the other. This wasn't successful. Clearly, a better solution is to use gears.

There are plenty of examples of toys that have good-quality cast-plastic gears. (Spirograph is one such.) The knobs on the Etch-a-Sketch could be geared together. If the gears are the same size, the knobs will rotate at the same rate, and the result will be a perfect 45° line.

If you gear the two knobs together directly, they will rotate in opposite directions, so that you can only draw lines with slope -1 (northwest to southeast), not with slope 1 (northeast to southwest). To fix this problem, we need to introduce more gears. There can be an axle peg sticking up from the case of the Etch-a-Sketch, in between the two knobs. Mounting three equal-sized gears on the two knobs and the axle peg gears will force the knobs to rotate in the same direction, at the same rate.

[ The remainder of this article contains a number of very dumb arithmetic errors. For example, you cannot fit three gears of size 1/3 on the knobs and pegs; you need to use three gears of size 1/4 instead. I will correct this on Monday, and provide an illustration to make it clearer what I mean. —MJD ]

[ Addendum 20061116: I have posted the correction, with illustrations. ]

Let's say that the distance between the centers of the two knobs is 1. We can get a line of slope -1 by mounting two gears, each with radius 1/2, on the two knobs; we can get a line of slope +1 by mounting three gears, each with radius 1/3, on the two knobs and on the axle peg. If we want to do both, we had better make the axle peg removable, or else it will interfere with the size-1/2 gears. This is no problem. It can mount into a socket on the front of the Etch-a-Sketch, and be pulled out when not needed.

But why have only one socket? We're including five gears already (two of size 1/2 and three of size 1/3) so we may as well put them to some more use. Throw in a size 1/6 gear, and add another socket for the axle peg, this time 1/3 of the way between the two knobs. Now you can mount a size 1/2 gear on the left knob, a size 1/6 gear on the axle peg, and a size 1/3 gear on the right knob. If the left knob turns at rate r, the middle gear turns at rate -3r and the right knob turns at rate 3r/2. This produces a line with slope 3/2, which is about a 56-degree angle.

Or put in another socket for the axle peg, 1/6 of the way between the knobs, and then mount size 1/2, size 1/3, and size 1/6 gears, in that order. The knobs are now producing a line with slope 3, a 72-degree angle. If you want a line with slope 1/3 (18°) instead, just reverse the order of the gears. (That is, exchange the large and the small ones.)

At this point adding a few more gears expands the repertoire significantly. Add a radius-2/3 gear and another radius-1/6 gear and you can mount [2/3, 1/3] to get lines with slope -1/2, [1/3, 2/3] to get slope -2, [2/3, 1/6, 1/6] to get slope 4, [1/6, 1/6, /23] to get slope 1/4.

Clearly, you can carry this onwards, limited only by the space for the axle holes and the expense of adding in more gears. Spirograph used to deliver fifteen or twenty plastic gears for a reasonable price, so it's clearly not implausible that Ohio Art could have done something like this.

Sometimes I even dare to think that they might have provided cams or elliptical gears. Properly designed cams could gear together the knobs to produce mathematically exact curved lines, squiggles, maybe even circles.

But no, as far as I can tell, it's never been done. Why not?

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