Casey Warmbrand's Home Page
Casey Warmbrand
Graduate Teaching Assistant
Department of Mathematics
The University of Arizona
Tucson, AZ 85721
Office: Math Bldg, Room 217
Phone: (520) 626-4996
Email: caseyw@math.arizona.edu
I am currently a TA for Math215-005 - Linear Algebra
Click me for Linear Algebra (Math 215) information and resources
Research Interests
Some of my research interests include Game Theory, Poker Models, Network Theory, Voting/Ranking Models, Combinatorics, Tilings, Random Walks, Random Matrix Theory, Asymptotic Analysis, Non-Intersecting Paths, Orthogonal Polynomials, and Graph Theory.
Poker Models - Best-Response Rules to Off-Equilibrium Strategies by Irrational Opponents
My current focus and the topic of my dissertation is on Poker Models and Game Theory. I am looking at Best-Response rules to non-equilibrium strategies played by irrational opponents in various simplified poker models. A common practice in Game Theory and the study of Nash equilibria is to assume rational players in the game. This leads to what I dub "perfect play," i.e. players who will behave in a manor that is in their own best interest. However, I feel that this is a poor assumption. When one goes to a casino, possibly here in Tucson, and sits down at the poker table to play against other locals, it is clear almost immediately that many of the opponents are NOT playing rationally. People are, in general, irrational, and irrational behaviour is observed in almost all real world situations. A good model should consider this.
So, what does it mean to be an irrational poker player? Well, it means you are not playing the Nash equilibrium. You are employing some strategy other than the optimal one. In regards to poker, this could mean you are either playing too few, or too many hands. In poker terminology, this would be considered playing too "tight" or too "loose," respectively. My analysis involves the finding of Best-Response rules to these off-equilibrium strategies that such irrational player might be using. From the Best-Response rules, some simple asymptotic analysis allows us to translate this in to genral statements about poker, which are in fact applicable to real-world poker games, not just the simplified models we are studying. For example, one such result says that against a "tight" caller, a player who is not calling as much as he should or would be if he was playing equilibrium, you, as the bettor, should bet more! This is intuitive, in the sense that if he is folding too much when you bet, you should bet even more to take advantage of his weak or tight play. Another example is against a "loose" bettor, someone who is betting frequently, more than he should be or would be if he was playing the equilibrium, you should call slightly more than you would normally! The thinking here being, that if he is betting too many hands, he has to be betting some hands that are weaker than the rational, equilibrium playing opponent would be, and by calling with slightly more hands than you would normally, you will be able to catch him with weaker hands and win more pots!
Currently, I have examine both the Borel and von Neumann models of poker, and plan to extend my analysis to both multi-stage and multi-player poker models. I also have the intention of examining how this analysis might be adaptable to a model which allows for a varying bet size, this would model something like a no-limit or pot-limit poker game. If you are interested in this research, I am always happy to talk about poker models and game theory. Feel free to email me or drop by my office to discuss this anytime!
Asymptotic Analysis of Random Tiling Problems Related to the Aztec Diamond
Up until very recently, I had been doing research on domino tilings of the Aztec Diamond. Specifically working on deformed boundaries of the Aztec Diamond, in hopes of producing an inscribed cardioid as the asymptotic boundary curve of the new shape that is being tiled.
The picture below on the left is a random tiling of an order 50 Aztec Diamond using 2-by-1 dominos. The one on the right is a order 20 regular hexagon tiled by lozenges, three different types of unit area rhombi shown below in green, blue, and red.
Of particular interest is the inscribed "circle" that is appearing as the number of tiles increases.
Props to Jim Propp (pun intended) and others for the creation of these tilings.
Over the first few weeks of the Fall semester of 2007 I gave a series of talks summarizing the work of Kurt Johansson and his application of technique developed by Lindström, Gessel and Viennot to compute asymptotic probabilities of objects related to the Aztec diamond.
Here are the slides from the first 3 weeks:
Talk #1 - Aztec Diamonds.
Talk #2 - Determinants and Non-Intersecting Paths.
Talk #3 - The Lindström-Gessel-Viennot Method.
Here is a paper I co-authored a while back (pdf):
On Network Theory - Published: PNAS | May 17, 2005 | vol. 102 | no. 20
Here is a paper I wrote recently on Poker Models - Under Revisions
Here is a copy of my Master's Thesis, a large N asymptotic analysis of partitions of N, based on the work of A. Vershik and S. Kerov.
The partitions are obtained using the Robinson-Schensted algorithm on a uniformly distributed permutation of N objects.
The measure on partitions, induced from the uniform measure on permutations by the Robinson-Schensted bijection, is called the Plancherel measure.
In order to visualize the partitions and eventually describe them with a function, it is useful to use Young diagrams and Young Tableaux.
A limiting shape for partitions is found through rescaling the boundary function as we take the limit N to infinity.
Some other papers I've written for various classes:
Self-Testing and Self-Correcting Algorithms
Ideals defining curves with generic zero
Here is a more recent copy of my Curriculum Vitae (mosly up to date).
I am an avid sports fan; I am always willing to debate why I feel baseball is the greatest sport ever! If a major sport has a team with a 4 letter name ending in "ets", you can bet I am rooting for them.
Barry Bonds may have passed Hank Aaron for the all-time MLB record for career HRs, but Barry still doesn't have a well-defined Erdös-Bacon number!