# One-card draw poker

Introduction

Let’s play another simple card game: after a \$1 ante, you (the dealer) shuffle a standard 52-card poker deck, and deal one card face down to each of us.  After looking at our respective cards, we each have the option either to “stay” and keep our card, or to “switch,” discarding and drawing a new card from the top of the deck.  The player with the highest-ranked card (ace low through king high) wins the pot, where a tie splits the pot.  What is the optimal strategy and expected value of playing this game?

(Note that although all of the games discussed here are playable– and are even more interesting and complex– with three or more players, I am focusing on just two players here.)

One-card Guts

There are two interesting variations on these basic rules.  First, suppose that we declare our choice of whether to stay or switch simultaneously, as in the game Guts: each player takes a chip or bottle cap under the table, either places it in a closed fist (to stay) or not (to switch), then holds the closed fist over the table.  All players simultaneously open their hands palm-down to indicate their choice.

This is the simpler version of the game.  With two players, it can be shown that the optimal strategy for both players is to stay if they are dealt a nine or higher.  That the game is fair (i.e., the expected value is zero) is clear from symmetry; no one player is “preferred” or distinguished in any way by the rules of the game.

One-card draw poker

Now suppose instead that we play more like a typical draw poker game: after the initial deal, each player in turn decides whether to discard and draw a new card.  Remember that you are the dealer, so you get to make your decision after I have made mine.  Can you exploit this advantage?

This is the version of the game that motivated this post.  I came up with this while trying to find examples of games for a student, games that were “small” enough to be tractable to analyze and experiment with via simulation, but still with interesting strategic complexity.  It’s worth noting that the name “One-Card Poker” is also used to refer to a similar “stud” form of the game, where the card play is simpler (no discarding), but the betting rounds are as in normal poker, resulting in a game that is much more complex to analyze.

(One final note: use of an actual 52-card deck complicates the analysis slightly, in a mostly non-interesting way.  I find it convenient to present the game using an “infinite deck,” where the probability distribution of card ranks remaining in the deck does not change as cards are dealt.)

# Calories in, calories out revisited

“All models are wrong, but some are useful.” George E. P. Box

A couple of months ago, I wrote about my experience “counting calories,” particularly about the accuracy of a very simple model of daily weight changes as a recurrence relation, converting each day’s net calorie deficit (or excess) into a 3500 calorie-per-pound weight loss (or gain).  The resulting predictions agreed very closely with my actual weight loss… however, I raised some additional questions at the end of the post, and the post itself generated some interesting comments as well.  Since it’s New Year’s resolution time, I thought this would be an appropriate time to follow up on these.

The following figure shows my predicted weight loss (in blue) over a period of 136 days, compared with each day’s actual measured weight (in red).  The details of the predictive model are provided in the earlier post; briefly, the idea is simple: start with a “zero day” measured weight, then predict weight changes for all subsequent days using only daily calorie intake (from eating) and expenditure (based on weight and, in my case, running).

Predicted and actual measured weight over 136 days.

For side-by-side comparison, the following figure shows the corresponding estimated daily calorie intake over the same time period.

Daily calorie intake over the same 136 days.

The first 75 days (the focus of the earlier post) show reasonably consistent behavior: relatively aggressive calorie deficits that yield almost 2.5 pounds lost each week.  But what if I were not so consistent?  How well does this simple model handle more radical changes in diet?

As you can see from both figures, days 76 through 79 get messy.  During that long weekend I had to travel to give a talk, and thus I didn’t have ready access to a scale, hence the missing weight measurements; and I also had less convenient control over the food that I ate.  There is a banquet buffet lunch in there, some restaurant food, etc., where my best-effort calorie estimates are obviously much less accurate.

But although I certainly expected to have gained weight after returning from my trip, I was surprised at how much I had gained, much more than the predictive model could possibly account for.  However, over the next several weeks, when I returned to a more well-behaved diet (more on this later), my weight seemed to “calm down,” returning to reasonably close agreement with the simple “calories in, calories out” model.  It’s not clear to me what causes wild swings like this.  For example, there are also a couple of 4 am wake-up calls for early flights, late nights, and the general stress of travel during those four days.  Perhaps those departures from my normally routine lifestyle might also contribute to the fluctuation in some way?

Also, note the planned incremental increases in calorie intake over the last couple of months, and the resulting slowdown in the rate of weight loss.  I didn’t stop losing weight, I just started losing less weight each week as I approached my goal.  This ability to eat more while still losing weight may be counter-intuitive, but the math makes sense: it’s really hard to lose weight… but it’s much easier to maintain weight once you’re where you want to be.  (On the other hand, it’s an exercise for the reader to verify using the model that it’s also dangerously easy to gain weight, and to do so much more quickly than you lost it.)

Finally, this subject generated quite a bit of discussion about the initial question of whether “it’s really as simple as calories in, calories out.”  In particular, several commenters insisted that no, it is not that simple, that the human body “is not a bomb calorimeter,” but a much more complex machine where weight is influenced by many other factors, including genetic variation, gut flora, etc.

I don’t disagree with this.  In fact, while we’re at it, let’s point out several other limitations as well: this model treats calorie expenditure as a linear function of total body weight, instead of lean body mass, which is arguably a better fit (but is not nearly as convenient to actually measure).  It also treats calorie expenditure from running as a function of weight and distance, but not speed.

Which brings me to the quotation at the top of this post.  No, this simple recurrence relation does not reflect the full complexity of the biological processes occurring in the human body that contribute to weight loss or gain… but so what?  Don’t use a complicated model when a simple one will do.  In this case, most of that additional complexity is “rolled up” into the single coefficient $\alpha$ reflecting the individual’s “burn rate” based on gender, genetic variation, flora in the gut, etc.  Granted, that coefficient may be unknown ahead of time, but at worst it can be estimated using a procedure similar to that described in the original post.