Na 2004, 2006 en 2008 was het dan afgelopen weekend tijd voor de vierde editie van de biertest.

De opzet is simpel: een aantal verschillende bieren wordt blind getest op de criteria uiterlijk, smaak, geur en nasmaak, en er wordt en eindoordeel gegeven. Bovendien moeten de deelnemers raden om wel bier het gaat.

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Finally, almost 6 months after last working on this, I wrote a paper summarizing my research on the game of thirties. Abstract:

Abstract

In this article, we discuss the game of 30s, a fairly simple game commonly played as a drinking game. We discuss the rules, various goals one may have in the game, and the results of a calculation of the optimal strategies to achieve these goals.

First, the rules of the game are introduced, and some strategies and goals one may have in the game are discussed. Next, a calculation for the so-called second stage of the game is executed, and an implementation for the simulator of the first stage is discussed. Next, we show how to use its output, and analyze some of its results. Finally, we claim that by some formal definition, the game can be called a ‘game of chance’.

All relevant files can be found here:

• The paper describing my research
• A handout (suitable for taking along when you play the game)
• Source code of the simulator
• Windows command-line version of the simulator
• Simulator with advanced knowledge, used for the game of chance/game of luck part of the paper

Note that the earier handout and blog posts contained a (I now think) erroneous calculation of the expected value of the second stage of the game. Happy reading!

Some three years after being spotted on German television at Rock am Ring – and of course, some months since having been on the BBC at a snooker match – , finally I was spotted again, this time on the Fabchannel recording of last years Wir Sind Helden concert in Paradiso:

After the 2004 edition and the 2006 edition, last weekend again it was time for the Great Exciting Beer Test.

The set-up is quite simple: 18 beers are served in anonymous glasses — this year by Hilde, thanks very much! The participants then grade on a scale from 1 to 10 for looks, smell, taste, aftertaste and general opinion, and try to guess which beer is which. Of cource the combination guarantees some fun results when people fail to recognize malt beer, or when they give low grades to their predeclared “favorite beers”. The beers tested this year, in alphabetical order:

Alfa, Amstel, Amstel Malt, Bavaria, Brand, Carlsberg, Chimay Trappist, Dommelsch, Duvel, Export, Grolsch, Heineken, Hertog Jan Grand Prestige, Krusovice, Mongozo, Palm, Schutters, Warsteiner

The best beer

Let’s first look at the most important result: the best-graded beer. The following table summarizes this:

More results later; for now, enjoy the raw data and this graph:

For the class in elliptic curves that I am currently following, it came in handy to have a script to find small integer solutions to equations such as Y^2=X(X+1)(X-1). For this, I wrote a small Perl script that generates code to check all small values for all variables in the equation, and then runs the code. For instance, consider

meilof@meilof-laptop:~$ perl solfinder.pl "y*y==x*(x*x+4*x-6)"
(y,x)=-18,6
(y,x)=-3,-1
(y,x)=0,0
(y,x)=3,-1
(y,x)=18,6

This works for an arbitrary number of variables (as long as they are single characters), and any boolean expression in Perl syntax. It looks for solutions in which all variables have values with absolute value smaller than 50. Internally, this generates and then runs the following Perl code:

for ($y = -50; $y <= 50; $y++) {
  for ($x = -50; $x <= 50; $x++) {
  if($y*$y==$x*($x*$x+4*$x-6)) { print "(y,x)=$y,$x\n" }
  }
}

Grab the script here.

Note that the original version of this article contains an erroneous calculation. This is the version as of 28 februari, 19:16.

As we have seen, different strategies in the game of thirties give us different expected results, even if the game is largely determined by luck. This raises the question: to what extent is the game a game of chance?

Obviously, to answer this question one needs to have a rigurous definition of what exactly is a game of chance. Some research in this area, particularly to be able to advise courts in The Netherlands, has been done by Dr. Ben van der Genugten. The method is explained in this article in the Dutch state newspaper:

The idea is that one defines the learning effect LE in a game to be the expected value of the optimal strategy minus the expected value of a strategy that may be employed by a “beginning player”, and the random effect RE to be the expected value of a player with advance knowledge (i.e., he knows what he is going to throw, should he throw again) minis the expected value of a player employing the optimal strategy. Then, the relative skill of a game is defined S=LE/(LE+RE).

On the grounds of Dutch jurisprudence, the boundary value has been set at 0.20; for example, roulette has value 0, black jack has value 0.049, and some simple forms of poker (the full game is too difficult to analyze) have values around 0.40.

To be able to calculate the random effect, I wrote a small simulator that, for random dice throws, calculates the optimal strategy. This suggests that with prior knowledge, the game has an expected value of about 32.73. As we know, the expected value given an optimal strategy is 30.1520.

So what, in this case, is the relative skill? For this, we need to define what a beginning player would do. Experience suggests that the strategy “put all fives and sixes away, and the last dice if it is 4 or higher” may be reasonable. In this case, one gets expected value 29.7232. This would suggest that the relative skill is 0.14, so the game just about a game of chance. This would seem to be reasonable.

There are some pitfalls, though. First of all: is this choice of strategy reasonable? As we know, the heuristic strategy is pretty easy to remember, and has an expected value which is just about equal to the expected value of the optimal strategy. In this case, the relative skill is only 0.0024.

Also, it is a bit difficult to precisely specify a random execution of the game, since the number of different throws may vary. In an earlier version of the simulation, I actually did this wrong. Calculating exactly (i.e., not simulating) the complete expected value with prior knowledge would probably be a large computation..

While playing a bit of the game of 30s Tuesday, I learned that the second stage of the game is actually a bit different from what I previously described. In fact, it works as follows: say one has thrown 32. Then one starts throwing again, putting away 2s while new ones are thrown. But now, the previous score (in this case, 2*6 thrown in the second stage+2) gets doubled, and one starts throwing 3s. This can go on (doubling the score each time) until one gets at 6s; in this case, if 6*6s are thrown, one just continues throwing 6s — presumably, this is already bad enough for the opponent.

Unfortunately, this makes the calculation of the expected value for the second stage game more difficult for two reasons. First, the expected value is no longer linearly dependent on the value of the dice: the lower the score on the dice, the more impact going from one dice value to a higher one is. This is not a real problem since it only increases the number of equations by, say, 6. The real problem is the doubling, which makes the score dependent in a non-linear fashion on the score obtained earlier.

I’ll spare you the analysis for now, but here’s the Maxima code I used to define P(x,n), which is the expected value when throwing xs when n points have already been scored. I do this by ignoring terms for very large n. This seems justifyable given that, letting n tend to infinity, the calculated values converge.

d(x,n):=if(x=0) then 1 else if n=0 then (5/6)^x else
   sum((1/6)^i*(5/6)^(x-i)*binomial(x,i)*d(x-i,n-i),i,1,n);
d(6,0)+d(6,1)+d(6,2)+d(6,3)+d(6,4)+d(6,5)+d(6,6);
a:sum(d(6,i)*6*i,i,0,6);
b:d(6,6);
P6(x):=if x>10000 then 0 else x+a+b*(36+x+P6(72+2*x));
P(y,x):=if y=6 then P6(x) else
   x+sum(d(6,i)*y*i,i,0,6)+b*(6*y+x+P(y+1,12*y+2*x));

This gives the following results:

Note that these values are considerably higher than the 2.88d obtained using the simpler variation of the endgame.

Small update: a reversed version of the source code is available. It now has command-line support, so that recompilation if one wants to use another strategy is no longer necessary. A Linux binary (static, gzipped) is available as well. Also, here is a slightly revised version of the paper, now with some formalization of expected values as well.

Last Thursday, I finished tidiying up the code for the strategy calculator: it can now also compare strategies, and it is pretty easy to add new ones. I will post the code in a while. Also, I started a bit on a mathematical formalization of the problem and the concepts involved. A first effort on a formalization (due to be released on Cleopatra’s “Schierkamp”), can be read here

Earlier, I described the optimal strategy in the game of 30s if one wants to maximize the number of points. I already mentioned then that there may actually be other things one wants to optimize. Also, since the optimal strategy may be a bit hard to remember, I developed a heuristic strategy.

So, I implemented some simple strategies, and calculated the optimal strategies to optimize various aspects. I calculated the following strategies:

• Maximal – Maximize number of points
• Maximize >30 – Maximize chance of getting 30 points or more
• Minimize loss – Minimize loss (defined as number of points below 30)
• Heuristic – Heuristic strategy resembling the maximal strategy, as described here
• Leave >5 – Put aside 5 or 6, or otherwise the highest dice
• Leave >6 – Put aside 6, or otherwise the highest dice

The various parameters of the different strategies are summarized in the following table:

When interpreting these numbers, one has to keep in mind that the strategies only optimize for the given varible; thus for example, the “over 30s” strategy does not favor a score of 36 over a score of 31. So, if the first five stones are sixes, then the “over 30s” strategy will not discriminate between throwing the last dice or putting it aside. This somewhat troubles the results. Still, it can be noted that for any of the three parameters, there is a somewhat different strategy to reach its optimum.

Next time: a more detailed analysis of the differences between the various strategies. For now, see the source code of the analyser.