## Wednesday, October 23, 2019

### Berlekamp Memorial Workshop Talks, Day 2

On the second (and final) day of the MSRI workshop, we had more talks, which I've attempted to summarize.  As always, please feel free to comment to correct what I got wrong and add things I didn't include!

Carlos Santos: "A Universal Ruleset"

Carlos reprised a talk about a subject Elwyn really enjoyed: finding a universal ruleset.  I didn't know this before, but this was a topic that Elwyn even inspired when he asked the question: What is the habitat of *2?  This means, which rulesets contain *2?  Even more general: which rulesets contain positions that equal all elements of the Short Conway Group?  Any such ruleset can be called "Universal".

Carlos mentioned multiple different procedural strategies he used to try to find such a ruleset.  Although, he could have come up with a new one, as he reasoned, "If I invent something, I am cheating."  So instead he attempted to use Amazons, Traffic Lights, Konane, then found victory with Generalized Konane, constructing everything in the SCG from that.  He was very happy that Generalized Konane already existed in CGSuite, so he could use it without "cheating".

Afterwards, Carlos spoke briefly about the supposed separation between recreational mathematics and "serious" mathematics, finishing with, "The opposite of fun is not serious."

Svenja Huntemann: "Bounding the Boiling Point"

(Joint work with Carlos Santos and Richard Nowkowski.)

Svenja talked about temperature and "How much urgency is necessary to play at a specific position.  This is the first talk where I really got an education about Thermography and thermographs.  Svenja's work uses thermic versions of hot games--which are games with the same temperature but with only one option for each player.

This thermic version makes it easier to find bounds on the temperature.  The boiling point, then is the suprenum of temperatures of a class of games.  If we can find bounds on the confusion intervals of games in the class, we can use that to find the boiling point!

Svenja discussed the boiling point of Domineering and Elwyn's conjecture that it is 2.  There is a bunch of work on this, including an example position with temperature 2.  A new result here is that long "snakes" cannot have temperature above 3.

Neil McKay: "Yellow-Brown Hackenbush"

Neil presented work on a topic Elywn had published in GONC3.  Yellow-Brown Hackenbush is different than Blue-Red Hackenbush because players cannot play on components where all the edges are yellow or all the edges are brown.  (The three colors of edges: YeLlow edges are takeable by Left, BRown are takeable by Right, and OlivE are takeable by Everyone.)  I felt very lucky that I could distinguish been the three colors in the slides.  Neil stuck to those three colors to preserve the choices, but admitted that it was hard to see the difference on the actual slides.

Unlike B-R Hackenbush, every Y-B Hackenbush position is dicotic.  Thus, all positions are infinitesimals.

Additionally, if playing a restricted version on stalks (path graphs), there are some cool simplifications you can do.  Neil has done work and solved the outcomes for the game, but not yet for the actual canonical form values.  This is something that Elwyn had left as an open problem.  Neil concluded by listing a bunch of open problems remaining with both Yellow-Brown and Blue-Red Hackenbush.

Richard Nowakowski: "Pic Arête"

Richard spoke about Pic Arête, which is Dots & Boxes without the extra move for completing a box.  Though it's actually played like Strings-and-Coins.  The game was solved on the grid by Meyniel and Roudnoff in 1988.  Richard considered playing on a general graph.  It turns out that a French team (Blanc, Duchêne, and Gravier) solved this in 2006, except for some cases.  Apparently these solutions don't always find the optimal move, but find a move good enough to win anyways.

Richard extended this to say that each edge contains a game instead of a single move to cut it so that the edge doesn't disappear until the game there becomes { | }, or exactly zero.  Then the overall game uses the same scoring mechanism as Strings & Coins.  E.g. edge weight games of +1 and -1 give you a Blue-Red version of Pic Arête.  Richard showed some neat examples with simplications.

"I claim I solved it last night in a dream," Richard said of one of the problems he's looking at.

Mike Fisher: "Beatty Games: Big and Small"

(Joint work with Mike Fraboni, Svenja, Urban, RJN, and Carlos)

Mike talked about Beatty Sequences and Games, and actually found the values for positions using different values of alpha.  He then dove into the atomic weights. As Mike kept saying, a lot of this was just personal exercises he gave to himself to see what was going on.  He found a relationship to the atomic weights and the number of consecutive numbers in the Beatty sequences below the number.

Mike then turned his attention to Octal Games, using the octal codes to describe Beatty games.  These infinite octal codes cause interesting patterns of Grundy values that Mike plotted.  Beyond all this, Mike then took Beatty sequences in another direction, investigating a partizan variant of the Beatty game.  He discovered that Richard, Angela, Urban and other had already explained the Golden-Ratio version, so he looked at values and Reduced Canonical Forms of games based on other "Metallic Ratios", e.g. Silver, and Bronze.  (I didn't know these existed!)

David Wolfe: "Distinguishing Gamblers from Investors at the Blackjack Table"

David finished up the talks with some work back from 2002/2003.  He first started by explaining basic Blackjack strategies, then talked about how card counting can enter into the mix.  Counting perfectly can shift the advantage a player has from -.5% to +.5%  David wanted to try to gauge the skill of a player by watching them play.  He catered to us a bit by asking: "How can we assess the skill of people playing combinatorial games?"  He added that the whole thing could be extended to evaluate anyone working in a competitive "gaming" field.

Since the variance is very high, you need to watch a player for a long time in order to try to evaluate this.  Also, instead of awarding points for the money a player actually earns, you instead credit them for the expected value of each play.  This is then compared to the baseline strategy for the game that doesn't include card counting.  This was a really cool approach!

These talks have been an excellent tribute to Elwyn.  I was exposed to so much more about what he accomplished than I had known existed.  I'm really fortunate that I was able to attend this meeting.  I'm very appreciative to MSRI and all of the organizers, especially Aaron Siegel.

## Tuesday, October 22, 2019

### Berlekamp Memorial Workshop Talks, Day 1

Aaron Siegel: "Elwyn Berlekamp and Combinatorial Game Theory"

Aaron gave an amazing history of Elwyn's 56 years of publishing about games.  As he later mentioned, probably none of us at the workshop would have been in the field if it weren't for him.  This was such a great talk.  Here are some of the things I learned:
• At 10 years old, Elwyn was allowed to play Dots-and-Boxes with his friends in the back of the room during Math class, as he was bored by the lectures.  Thank goodness for understanding teachers!  In 2002, nearly 50 years later, he published a book on Dots-and-Boxes.  In it there are four theorems.  As Aaron summarized, if you only know theorem n and your opponent knows n+1, it's hopeless.
• As an undergrad at MIT, Elwyn wrote one of the first chess-playing programs and published columns about Bridge in the newspaper.  He would later publish about bridge-playing programs.
• He learned about Nim at Bell Labs, where he spent three summers also during college.
• He lost to a Dots-and-Boxes-playing program created by other students, so he learned more about it, then came back and won.
• He worked at Bell Labs again later and wrote a paper connecting Dots and Boxes to Sprague-Grundy theory.  His supervisors challenged him, asking why this was useful.  He wrote a six-page response, which earned him vindication from the VP.
• Winning Ways took a long time to complete.  It began when Elwyn met Richard Guy in North Carolina in 1967.  Elwyn proposed writing the book, and Guy suggested they team up with John Conway.  The three of them met in 1969 at a conference at Oxford.  They spent the next 13 years working on Winning Ways, which was published in 1982.  I really had known nothing of the history of the creation of Winning Ways before this talk!
• Elwyn got into Go in 1988 while working with David Wolfe.  Elwyn created the 9-dan stumping problem, on which he could defeat some of the best Go players as either Black or White.  As David explained, this position was about halfway between a natural endgame and the completely contrived boards that they had worked out a theory for using infinitesimals.
• In 1996, Elwyn started using Coupon Stacks as a way to get a professional opinion on the temperature of a game.
• In 2018, Elwyn published his final paper, this time about Entrepreneurial Chess, which we played at the game.
Elwyn's amazing career spanned multiple fields, but, as Aaron mentioned, "I think games were always what he loved best."

Svenja Huntemann: "Introduction to CGT"

Svenja gave an introduction to CGT to fill an unexpectedly empty slot.  Svenja covered impartial games, outcome classes, and basic partisan notions explained via Domineering: fractions, switches, and infinitesimals.  Her off-the-cuff version of this introduction is better than my planned and practiced version.

Melissa Huggan: "The Cheating Robot"

Melissa talked about work with Richard Nowakowski that continues her great PhD work on simultaneous games.  In the simultaneous positions, what if Right knows what Left is going to do before they both make their move?  (The name is derived from a Rock-Paper-Scissors robot that can cheat and win by very quickly reacting to what the human player does.)

In this cool model, it turns out that Zero is unique!  It can only occur when both players can only move to zero (causing a draw).  There can be other drawing scenarios, but they don't act like Zero when included in sums!  Melissa took a closer look at Topping Dominoes and used this to show some interesting examples and how to play optimally in these scenarios.

Urban Larsson: "The Fragility of Golden Games"

Urban talked about some joint work with Yakov Babichenko on non-combinatorial games where payoffs (0 or 1) are on the leaves of a binary tree.  Players alternate choosing between the two subtrees at each level, so the value of the game is easily decided recursively.  If the payoffs are assigned at random, how often does the overall value change?  If the Pr[leaf payoff is 1] is the golden ratio, then we call this a Golden Game.

Fragility is then based on the Hamming Distance to change the outcome of the game.  It turns out that Golden Games are very fragile, while non-Golden games are more robust.  As Urban pointed out, "Golden Games are special with high probability."  They want to look at some applications to machine learning with an interfering adversary.  E.g. how many pixels need to be changed until a machine can no longer recognize that a photo is of either a dog or a wolf?  How fragile are these algorithms?

I'm definitely looking forward to the Day 2 talks!