The sign of a Latin square is the product of signs of rows (considered as permutations) and the signs of columns. A Latin square is even if its sign is 1, and odd if its sign is -1. It is easy to see that when is odd the numbers of even and odd Latin squares are the same.

**The Alon-Tarsi Conjecture:** When is even the number of even Latin square is different from the number of odd Latin square.

This conjecture was proved when and is prime by Drisko and when and is prime by Glynn. The first open case is .

A stronger conjecture that is supported by known data is that when is even there are actually** more** even Latin squares than odd Latin squares. (This table is taken from the Wolfarm mathworld page on the conjecture.)

**A poll**

Unlike previous “test your intuition” questions the answer is not known.

The Alon-Tarsi conjecture arose in the context of coloring graphs from lists. Alon and Tarsi proved a general theorem regarding coloring graphs when every vertex has a list of colors and the conjecture comes from applying the general theorem to Dinits’ conjecture that can be regarded as a statement about list coloring of the complete bipartite graph . In 1994 Galvin proved the Dinitz conjecture by direct combinatorial proof. See this post. Gian-Carlo Rota and Rosa Huang proved that the Alon-Tarsi conjecture implies the Rota basis conjecture (over ) when is even.

Let be a graph on vertices . Associate to every vertex a variable . Consider the graph polynomial Alon and Tarsi considered the coefficient of the monomial . If this coefficient is non-zero then they showed that for every lists of colors, colors for vertex , there is a legal coloring of the vertices from the lists! Alon and Tarsi went on to describe combinatorially the coefficient as the difference between numbers of even and odd Euler orientations.

Let me mention the paper by Jeannette Janssen The Dinitz problem solved for rectangles that contains a proof based on Alon-Tarsi theorem of a rectangle case of Dinitz conjecture. It is very interesting if the polynomial used by Janssen can be used to prove a “rectangular” (thus a bit weaker) version of the Rota’s basis conjecture. A similar slightly weaker form of the conclusion of Rota’s basis conjecture is conjectured by Ron Aharoni and Eli Berger in much much greater generality.

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Recently Gil asked me whether I would like to contribute to his blog and I am happy to do so. I enjoy both finite and infinite combinatorics and it seems that these fields drifted apart in recent decades. The latter is a comment not possibly substantiated by personal experience yet rather by my perception of the number and kinds of papers published. I believe that there is no mathematical reason for this separation but that it came about for reasons of the sociology of mathematics—maybe one should create a chair for this subject somewhere sometimes. Anyhow, I would like to write a few words about an area where finite and infinite combinatorics come together, the partition calculus for ordinal numbers.

The Partition Calculus, contemporarily classified as 03E02 under Mathematical Logic/Set Theory was initiated in 1956 with the article *A partition calculus in set theory,* published by Paul Erdos and Richard Rado in the Bulletin of the American Mathematical Society. asserts that for every set of -sized subsets of a set of size there is an of size such that all subsets of of size are in or an of size such that no subset of of size is in . Many mathematicians are concerned with the case where are finite. As in the finite realm the notions of cardinal number and ordinal number coincide, it is there unnecessary to differentiate between these notions of size. Furthermore, via the Well-Ordering-Theorem, the Axiom of Choice implies that for cardinalities the statement is equivalent to where denotes the smallest ordinal number of cardinality . It also implies that there is no such that . Hence, believing the Axiom of Choice, one may limit ones study to the cases in which are order-types and is a natural number. Although there are interesting open questions in contexts where the Axiom of Choice fails and also in contexts where not all of are ordinal numbers, I will for now exclude these from the discussion. That is, what I would like to talk about is the subject of transfinite Ramsey Numbers, is to say that but for no .

First I would like to discuss the lower storeys of the transfinite. It is known that generally for a countable ordinal and a natural number the Ramsey number is countable (If no subscript appears it is understood to be ). When is a finite multiple of or , the number’s calculation is similar in character to the case where is a natural number though it tends to be slightly more involved. For example where is the least number without a digraph on vertices without an independent -tuple and without an induced transitive subtournament on vertices. The are the Tournament Ramsey Numbers which have been investigated slightly more intensely in [1], [2] and exact values are known for . The last paper on these problems was published in 1997 by Jean Larson and William Mitchell in the very first issue of the Annals of Combinatorics. A degree argument yields which gives a good idea about the growth rate of as counterexamples to numbers being finite Ramsey numbers easily carry over. Furthermore they found a digraph on thirteen vertices without a transitive triple or independent quadruple thus establishing .

Recently I started to discuss these problems with Ferdinand Ihringer and Deepak Rajendraprasad. The latter found a digraph with the two aforementioned properties but on fourteen vertices and shortly thereafter they could establish by a triangle-count that there is no such digraph on vertices. So we know now that . Generally these problems should provide a nice playground for people in the business of solving Ramsey-type problems with the help of computers, cf. [3].

The situation is slightly more complicated in the case of finite multiples of as a degree argument only yields a cubic upper bound for which is a number defined such that for all natural numbers and . This is elaborated on in detail in [4].

Recently Jacob Hilton has considered problems of this kind involving additional topological structure alone (cf. [5]) and together with Andr\'{e}s Caicedo (cf. [6]).

In the next post I am going to elaborate on results and open questions

regarding Ramsey numbers for larger countable ordinals.

[1] Kenneth Brooks Reid, Jr. and Ernest Tilden Parker. Disproof of a conjecture of

Erdos and Moser on tournaments. J. Combinatorial Theory, 9 (1970).

[2] Adolfo Sanchez-Flores. On tournaments and their largest transitive subtournaments.

Graphs Combin., 10, 1994.

[3] Stanislaw P. Radziszowski. Small Ramsey numbers. Electron. J. Combin., 1:Dynamic

Survey 1, 30 pp. (electronic), 1994,

[4] Thilo Volker Weinert, Idiosynchromatic poetry. Combinatorica, 34 (2014).

[5] Andres Eduardo Caicedo and Jacob Hilton, Topological Ramsey numbers and countable ordinals.

[6] Jacob Hilton. The topological pigeonhole principle for ordinals.

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Timothy Chow launched polymath12 devoted to the Rota Basis conjecture on the polymathblog. A classic paper on the subject is the 1989 paper by Rosa Huang and Gian Carlo-Rota.

Let me mention a strong version of Rota’s conjecture (Conjecture 6 in that paper) due to Jeff Kahn that asserts that if you have bases of an -dimensional vector space then you can choose such that for every , the vectors form a basis of and also the vectors form a basis. This conjecture can be regarded as a strengthening of the Dinits Conjecture whose solution by Galvin is described in this post. Rota’s original conjecture is the case where depends only on .

A very nice special case of Rota’s conjecture was proposed by Jordan Ellenberg in the polymath12 thread: Given trees on vertices, is it always possible to order the edges of each tree so that the th edges in the trees form a tree for every ?

The February 2017 issue of the Notices of the AMS has two beautiful papers on topics we discussed here. Henry Cohn wrote an article A conceptual breakthrough in sphere packing about the breakthrough on sphere packings in 8 and 24 dimensions (see this post) and Art Duval, Carly Klivans, and Jeremy Martin wrote an article on the the Partitionability Conjecture. (That we mentioned here.)

I have some plans to write about the partitionability conjecture (and an even more general conjecture of mine) soon. But now I would like to draw your attention to a weakening of these conjectures, still implying the “numerical” consequences of the original conjectures, that was proved in 2000 by Art Duval and Ping Zhang in their paper Iterated homology and decompositions of simplicial complexes . The partitionability conjecture is about decompositions into subcubes (or intervals in the Boolean lattice) and the result is about decomposition into subtrees of the Boolean lattice. (See here for the massage “trees not cubes!” in another context.)

This pictures was taken by Edna Wigderson in a 50th birthday party for Avi Wigderson. Unfortunately the picture shows us while landing from our 3 meter high (3.28 yards) jumps and thus does not fully capture the achievement.

The guy on the left is Bernard Chazelle, a great computer scientist and geometer, a long time friend of me and Avi, and the father of Damien Chazelle the director and writer of the movie La La Land now nominated for the record 14 Oscar awards. I wish Damien to win a record number of Oscars and to continue writing, directing, and producing wonderful movies so as to keep shattering his own records and giving excitement and joy to hundreds of millions, perhaps even billions, of people. (Update: Six Oscars, Damien the youngest ever director to win.)

For those in Israel let me draw your attention to the Jerusalem Baroque orchestra and especially to the 2017 Bach Festival. (I thanks Menachem Magidor for telling me about this wonderful orchestra.)

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**Theorem** (Hopf and Pannwitz, 1934)**:** Let be a set of points in the plane in general position (no three points on a line) and consider line segments whose endpoints are in . Then there are two disjoint line segments.

**Micha Perles’s proof by Lice:**

Useful properties of lice: A louse lives on a head and wishes to lay an egg on a hair.

Think about the points in the plane as little heads, and think about each line segments between two points as a hair.

The proof goes as follows:

**Step one:** You take lice from your own head and put them on the points of $X$.

**Step two:** each louse examines the hairs coming from the head and lay eggs (on the hair near the head)

**Step three** (not strictly needed)**:** You take back the lice and put them back on your head.

To make it work we need a special type of lice: spoiled-left-wing-louse.

A spoiled-left-wing louse lays an egg on a hair if and only if the area near the head, 180 degrees to the right of this hair is free from other hairs.

**Lemma:** Every louse lays at most one egg.

**Proof of lemma: **As you see from the picture, if the louse lays an egg on one hair, this hair disturbs every other hair.

**Proof of theorem continued:** since there are line segments and only at most eggs there is a hair X between heads A and B with no eggs.

We look at this hair and ask:

Why don’t we have an egg near head A: because there is a hair Y in the angle 180° to the right.

Why don’t we have an egg near head B: because there is a hair Z in the angle 180° to the right.

Y and Z must be disjoint. Q. E. D.

**Remarks:** We actually get a ZIG formed by Y, X, and Z

If we use right-wing-spoiled lice we will get a ZAG.

We can allow the points not to be in general position as long as one hair from a head does not contain another hair from the same head.

The topological version of this problem is the infamous Conway’s thrackle conjecture. See also Stephan Wehner page about it.

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In the very short 2003 paper A simple algorithm for edge-coloring bipartite multigraphs, Information Processing Letters 85 (2003), 301-302, Noga Alon used a similar idea for algorithmic purposes. (He also observed the connection to the camel riddle). Here is how the extra camel idea is used for:

**Theorem:** A bipartite cubic graph has a perfect matching.

(A cubic graph is a 3-regular graph.)

**Proof:** Suppose that has vertices. Multiply each edge times ( large) so that the degree of each vertex is of the form . Now ask your neighbor to give you an additional perfect matching and add it to the graph which now is regular of degree . The new graph has an Eulerian cycle. (If not connected, every connected component has an Eulerian cycle.) When we walk on the Eulerian cycle and take either the even edges or the odd edges we get two subraphs of degree . At least one of them does not use all the edges of the neighbor. We move to this subgraph and give the unused edge back to the neighbor. We repeat, and in each step we move to a regular subgraph of degree a smaller power of two, and give back at least one edge to the kind neighbor. If is large enough to start with we will end with a perfect matching that uses only the original edges of our graph.

(Remark: We can take or . If we are a little more careful and in each step try to give many edges back to the kind neighbor we can use or so.)

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Being invited to give a plenary lecture at the 7ECM was a great honor and, as Keren Vogtmann said in her beautiful opening lecture on outer spaces, it was also a daunting task. I am thankful to Günter Ziegler for his introduction. When I ask myself in what way I am connected to the person I was thirty years ago, one answer is that it is my long-term friendship with Günter and other people that makes me the same person. My lecture deals with the analysis of Boolean functions in relation to expansion (isoperimetric) properties of subsets of the discrete n-dimensional cube. The lecture has made a subjective selection of some results, proofs, and problems from this area.

Yesterday, Leonid Polterovich and I were guests of the exhibition “Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture.” I will start by briefly mentioning the great impact of German-Jewish mathematicians on the early history of the Einstein Institute of Mathematics and Physics at the Hebrew University of Jerusalem, my main academic home since the early seventies. In this picture you can see some early faces of our Institute.

Edmund Landau, the founder and first head of the Institute, moved to Jerusalem from Göttingen in 1927 and moved back to Göttingen a year and a half later. Abraham (Adolf) Halevi Fraenkel moved to Jerusalem from Kiel in 1928 and he can be seen as the father of logic, set theory, and computer science in Israel. My own academic great-grandfather is Michael Fekete, who immigrated to Jerusalem from Budapest in 1928.

I would like to say a few words about two remarkable documents written by Landau in 1925, both related to the inauguration ceremony of the Hebrew University of Jerusalem. You can read more about them in the paper *Zionist internationalism **through number theory: Edmund Landau at the Opening of the Hebrew University **in 1925* by Leo Corry and Norbert Schappacher . The first document is Landau’s toast for the opening ceremonies. Let me quote two sentences:

May great benefit emerge from this house dedicated to pure science, which does not know borders between people and people. And may this awareness emerge from Zion and penetrate the hearts of all those who are still far from this view.

The second document, also from 1925, is probably the first mathematical paper written in Hebrew in modern times. It is devoted to twenty-three problems in number theory and here are its concluding sentences.

At this number of twenty-three problems I want to stop, because

twenty-three is a prime number, i.e., a very handsome number for us. I am certain that I should not fear to be asked by you, for what purpose does one deal with the theory of numbers and what applications may it have. For we deal with science for the sake of it, and dealing with it was a solace in the days of internal and external war that as Jews and as Germans we fought and still fight today.

I wish to make two remarks: First, note that Landau moved from the very ambitious hopes and program of science as a bridge that eliminates borders between nations to a more modest and realistic hope that science and mathematics give comfort in difficult times. Juggling between very ambitious programs and sober reality is in the nature of our profession and we are getting paid both for the high hopes and aims, as well as for the modest results. Second, Landau is famous for his very rigorous and formal mathematical style but his 1925 lecture is entertaining and playful. I don’t know if his move to Jerusalem was the reason for this apparent change of style. Parts of Landau’s lecture almost read like stand-up comedy. Here is, word for word, what Landau wrote about the twin prime conjecture:

Satan knows [the answer]. What I mean is that besides God Almighty no one knows the answer, not even my friend Hardy in Oxford.

These days, ninety years after Landau’s lecture, we can say that besides God Almighty no one knows the answer and not even our friend James Maynard from Oxford. We can only hope that the situation will change before long.

Landau’s hopeful comments were made only nine years after the end of the terrible First World War. He himself died in 1938 in Berlin, after having been stripped of his teaching privileges a few years earlier. I don’t know to what extent the beauty of mathematics was a source of comfort in his last years, but we can assume that this was indeed the case. My life, like the lives of many others of my generation, was overshadowed by the Second World War and the Holocaust and influenced by the quest to come to terms with those horrible events.

Here is the videotaped lecture. (and the slides).

More sources: The home page of my Institute, and a page about its history; An article in the AMS Notices; A blog post in Hebrew about the 1925 Hebrew University events; Shaul Katz: Berlin roots – Zionist incarnation: The ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem. *Science in Context* 17(1/2), 199-234 (2004); A blog post about Yaakov Levitzki and the Amitzur-Levitzki theorem; Schappacher, Norbert: Edmund Landau’s Göttingen — From the life and death of a great mathematical center. *Mathematical Intelligencer* 13 (1991), 12-18. (Talk at the Dedication of the *Landau Center for Research in Mathematical Analysis*, Jerusalem, Feb. 28th, 1989). A recent post on Tao’s blog related to mathematics, science, scientific relations and recent events.

**(Below) First and last page of Hardy and Heilborn obituary on Landau.**

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Boolean functions, Influence, threshold, and Noise

Trying to follow an example of a 1925 lecture by Landau (mentioned in the lecture), the writing style is very much that of a lecture. It goes without saying that I will be very happy for corrections and suggestions of all kinds.

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I am sure that every one of the readers of this blog heard about Laci Babai’s quasi-polynomial algorithm for graph isomorphism and also the recent drama about it: A mistake pointed out by Harald Helfgott, a new sub-exponential but not quasi-polynomial version of the algorithm that Laci found in a couple of days, and then, a week later, a new variant of the algorithm again found by Laci which is quasi-polynomial. You can read the announcement on Babai’s homepage, three excellent Quanta magazine articles by Erica Klarreich** **(I,II,III), Blog posts over Harald’s blog (III,II,I) with links to the video and article (in French), and many blog posts all over the Internet (GLL4,GLL3,GLL2,GLL1,…).

Babai’s result is an off-scale scientific achievement, it is wonderful in many respects, and I truly admire and envy Laci for this amazing breakthrough. I also truly admire Harald for his superb job as a Bourbaki expositor.

Tel Aviv University: Sackler distinguished lectures in Pure Mathematics Wednesday, January 18 (Poster. Sorry, too late, I heard it was very inspiring, don’t miss the other talks!)

Tel Aviv University Combinatorics seminar: Sunday, Jan. 22, 10:00-11:00, Location: Melamed (Shenkar building, ground floor, room 6)

Title: **Canonical partitioning and the emergence of the Johnson graphs:** **Combina****torial aspects of the Graph Isomorphism problem **

(The talk does not depend on Wednesday’s talk)

Hebrew University Colloquium San. Jan 22, 16:00-17:00 Title: **Graph isomorphism and coherent configurations: The Split-or-Johnson routine**

Lecture room 2, Manchester building (Mathematics)

*Local versus global symmetry and the Graph Isomorphism problem I–III*

Lecture I: Monday, January 23, 2017 at 15:30

Lecture II: Tuesday, January 24, 2017 at 15:30

Lecture III: Thursday, January 26, 2017 at 15:30

All lectures will take place at Auditorium 232, Amado Mathematics Building, Technion (Website)

Pekeris lecture, Jan 29, 11:00-12:00 **Hidden irregularity versus hidden symmetry **

EBNER AUDITORIUM (webpage)

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The purpose of this post is to (belatedly) formally announce that the project has ended, to give links to the individual posts and to briefly mention some advances and some thoughts about it.

The posts were

- Polymath10: The Erdos Rado Delta System Conjecture, Posted Nov 2, 2015. (138 comments)
- Polymath10, Post 2: Homological Approach, Posted Nov 10, 2015. (125 comments.)
- Polymath 10 Post 3: How are we doing?, Posted Dec 8, 2015. (103 comments.)
- Polymath10-post 4: Back to the drawing board?, Posted Jan 31, 2016. (11 comments.)
- Polymath 10 Emergency Post 5: The Erdos-Szemeredi Sunflower Conjecture is Now Proven. Posted May 17, 2016. (35 comments.)
- Polymath 10 post 6: The Erdos-Rado sunflower conjecture, and the Turan (4,3) problem: homological approaches, Posted on May 27, 2016. (5 comments.)

The problem was not solved and we did not come near a solution. The posts contain some summary of the discussions, a few results, and some proposals by the participants. Phillip Gibbs found a remarkable relation between the general case and the balanced case. Dömötör Palvolgyi shot down quite a few conjectures I made, and Ferdinand Ihringer presented results about some Erdos-Ko-Rado extensions we considered (In term of upper bounds for sunflower-free families). Several participants have made interesting proposals for attacking the problem.

I presented in the second post a detailed homological approach, and developed it further in the later threads with the help of Eran Nevo and a few others. Then, after a major ingredient was shot down, I revised it drastically in the last post.

Participants made several computer experiments, for sunflower-free sets, for random sunflower-free sets, and also regarding the homologica/algebraic ideas.

The posts (and some comments) give some useful links to literature regarding the problem, and post 5 was devoted to a startling development which occurred separately – the solution of the Erdos-Szemeredi sunflower conjecture for sunflowers with three petals following the cup set developments. (The Erdos-Szemeredi sunflower conjecture is weaker than the Erdos-Rado conjecture.)

A (too) strong version of the homological conjecture appeared in my 1983 Ph. D. thesis written in Hebrew. The typesetting used the Hebrew version of Troff.

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Five years ago I wrote a post entitled Is Backgammon in P? It was based on conversations with Peter Bro Miltersen and Uri Zwick (shown together in the above picture) about the computational complexity of computing the values (and equilibrium points) of various stochastic games, and also on some things I learned from my game theory friends over the years about proving that values exist for some related games. A few weeks ago two former students of Peter, Rasmus Ibsen-Jensen and Kristoffer Arnsfelt Hansen visited Israel and I had a chance to chat with them and learn about some recent exciting advances.

Is there a polynomial time algorithm for chess? Well, if we consider the complexity of chess in terms of the board size then it is fair to think that the answer is “no”. But if we wish to consider the complexity in terms of the number of all possible positions then it is easy to go backward over all positions and determine the outcome of the game when we start with each given position.

Now, **what about backgammon? ** Like chess, backgammon is a game of complete information. The difference between backgammon and chess is the element of luck; at each position your possible moves are determined by a roll of two dice. This element of luck increases the computational skill needed for playing backgammon compared to chess. It can easily be seen that optimal strategy for players in backgammon need not involve any randomness.

**Problem 1: **Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero sum game with perfect information? (Like backgammon)

This question (raised by Ann Condon in 1998) represents one of the most fundamental open problem in algorithmic game theory.

Heads-up poker is just a poker game with two players. To make it concrete you may think about heads-up Texas hold’em poker. This is not a game with complete information, but by according to the minmax theorem it still has a value. The optimal strategies are mixed and involve randomness.

**Problem 2: **Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero-sum game with incomplete information? (like heads-up Texas hold’em poker).

It will be very nice to find even a sub-exponential algorithm for a stochastic zero-sum game with incomplete information like poker.

**Problem 2′: **Is there a subexponential-time algorithm to find the optimal strategy (and thus the value) of a stochastic zero-sum game with incomplete information?

For games with complete information like backgammon, a subexponential algorithm was found by Walter Ludwig and in greater generality by Sergei Vorobyov, Henrik Björklund, and Sven Sandberg. It is related to subexponential simplex-type algorithms for linear programming called RANDOM-FACET, found in the early 90s by Matousek, Sharir and Welzl and myself.

Kristoffer Arnsfelt Hansen (see abstract below) presented a polynomial-time algorithm for 2-persons zero sum stochastic games, when the games have a bounded number of states. (Earlier algorithms were exponential.) The paper is: Exact Algorithms for Solving

Stochastic Games by Kristoffer Arnsfelt Hansen, Michal Koucky, Niels Lauritsen,

Peter Bro Miltersen, and Elias P. Tsigaridas. Slides of the talk are linked here.

As for backgammon there are very good computer programs. (We talked about chess-playing computers in this guest post by Amir Ban and since that time Go-playing computers are also available.) The site Cepheus Poker Project and this science paper Heads-up limit hold’em poker is solved are good sources on major achievements by a group of researchers from Alberta regarding two players poker.

**Problem 3: **Is there a polynomial time algorithm to find Nash equilibrium point (or another form of optimal strategy) of a stochastic n-player game with incomplete information? (like Texas holdem poker.) Here *n* is fixed and small.

I think that people are optimistic that even the answer to problem 3 is yes. (There are hardness results for finding equilibrium points in matrix games but the relevance to our case is not clear.) If we want an algorithm which optimally plays poker, it is not clear that finding a Nash equilibrium is the way to go.

**Problem 4:** Find an algorithm for playing Texas hold’em poker when there are more than two players.

When the objective is to maximize revenues against human players I expect that it will be possible to develop computer programs for playing poker better than humans.

**Problem 5:** How to play the game MEDIAN of the previous post?

**Matching pennies** is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Even keeps both pennies, so wins one from Odd (+1 for Even, −1 for Odd). If the pennies do not match (one heads and one tails) Odd keeps both pennies, so receives one from Even (−1 for Even, +1 for Odd) (source wikiPedia)

Variants of this game have been played since ancient times. In Hebrew matching pennies is called ZUG O PERET (even or odd; זוג או פרט). It is played like this: There are two players. Each player in his turn makes an announcement “even” or “odd”. Then each of the two players shows (simultaneously) some number of fingers and the announcing player wins if the sum of fingers has the announced parity.

The big match is a drastic repeated version of matching pennies. The game is played between players Even and Odd. Each player has a penny and in each stage must secretly turn the penny to heads or tails and the payoffs are the same as in matching pennies. If Even plays “head” the game continues to the next stage. However if Even plays “tails” (or tries for the “big match” as it is called) then the payoff in that round is repeated for all future rounds: Namely, if the pennies match Even will get 1 for all future rounds, and if the pennies do not match Even will pay one for all future rounds.

By playing heads with probability 1/2 and tails with probability 1/2, Odd can guarantee an expected payoff of 0. But what about Even? Can he also guarantee an expected payoff of 0? This was an open question for quite some time. The big match was introduced in 1957 by Dean Gillette who asked if the game has a value, namely if Even has a strategy to guarantee a payoff of 0.

**Problem 7:** Does big match has a value?

Here is a blog post on the big match by Presh Talwalkar on his blog “mind your decisions.”* *You also can read about the big match in this post of Kevin Bryan’s economics blog “a fine theorem.”

In 1968, David Blackwell and Thomas S. Ferguson settled Gillete’s question and proved that even can guarantee a zero payoff and thus big match did in fact have a value. This was the first step to showing all zero-sum stochastic games have value under limiting average payoff, which was proven in 1982 by Mertens and Neyman.

Rasmus Ibsen-Jensen presented both positive and negative results on attaining the value for the big match with limited types of strategies and also on complexity issues regarding other stochastic games. Here are the slides for Rasmus’ talk (see full abstract below). Part of the talk is based on the paper The Big Match in Small Space by Kristoffer Arnsfelt Hansen, Rasmus Ibsen-Jensen, and Michal Koucky.

This is a remarkable story with very important results and open questions. Here is the Wikipedia article on stochastic games and this short paper by Eilon Solan. I see now that the post is becoming too long and I will have to talk about it in a different post.

**Problem 8** (informal): Does every stochastic game have ~~a value~~ an equilibrium?

Following a major step by Truman Bewley and Elon Kohlberg (1976), Jean-François Mertens and Abraham Neyman (1981) proved that every two-person zero-sum stochastic game with finitely many states and actions has a uniform value. Nicolas Vieille (2000) has shown that all two-person stochastic games with finite state and action spaces have a limiting-average equilibrium payoff. The big question is to extend Vieille’s result to games with many players.

Kristoffer, Rasmus and Abraham (Merale) Neyman.

Exact algorithms for solving stochastic games

Speaker: Kristoffer Arnsfelt Hansen, Aarhus University

==================================================

In this talk we consider two-player zero sum stochastic games

with finite state and action space from an algorithmic

perspective. Prior to our work, algorithms for solving

stochastic games relied either on generic reductions to decision

procedures for the first order theory of the reals or on value or

strategy iteration. For all these algorithms, the complexity is

at least exponential even when the number of positions is a

constant and even when only a crude approximation is required

We will present an exact algorithm for solving these games based

on a simple recursive bisection pattern. The algorithm runs in

polynomial time when the number of positions is constant and our

algorithms are the first algorithms with this property. While the

algorithm is not based directly on real algebraic geometry, our

algorithm depends heavily on results from the field.

Based on joint work with Michal Koucký, Niels Lauritzen,

Peter Bro Miltersen, and Elias P. Tsigaridas published at STOC’11.

Abstract: The talk will attempt to characterize good strategies for some special cases of stochastic games. For instance, the talk will argue that there might always be a good strategy with a certain property for all games in a special case of stochastic games or that no good strategy exists that has some property for some game. Concretely,

1) for the stochastic game the Big Match, no good strategy (for lim inf) exists that only depends on how long the game has been playing and a finite amount of extra memory (when the extra memory is updated deterministically).

2) for the Big Match there is a good strategy that uses only a single coin flip per round and exponentially less space then previous known good strategies.

3) let x be the greatest reward in a stochastic game. The talk will next give a simple characterization of the states of value equal to x for which there exists either (a) an optimal strategy; (b) for each epsilon>0, a stationary epsilon-optimal strategy; or (c) for each epsilon>0, a finite-memory epsilon-optimal strategy (when the memory is updated deterministically) . The characterization also gives the corresponding strategy.

4) the talk will then consider stochastic games where there exists epsilon-optimal stationary strategies for all epsilon>0. It will argue that the smallest positive probability in a stationary epsilon-optimal strategy must be at least double exponential small for some sub-classes of stochastic games, while for other classes exponential small probabilities suffices.

1) and 2) is based on “The Big Match in Small Space”, 3) is based on “The Value 1 Problem Under Finite-memory Strategies for Concurrent Mean-payoff Games” 4) is based on “Strategy Complexity of Concurrent Stochastic Games with Safety and Reachability Objectives” and “The complexity of ergodic mean-payoff games“. All papers can be found in http://Rasmus.Ibsen-Jensen.com

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