Tuesday, 22 July 2008
Saturday, 19 July 2008
Poisoning pigeons in the park
According to Wikipedia, Tom Lehrer said: "Some of you may have met mathematicians and wondered how they got that way."
Der Screamer
This afternoon I built myself a Tube Screamer Clone called Der Screamer available as a kit at Musikding. Below you find the very first, well, screams, recorded using a Fender Mexican Standard Stratocaster (upgraded with Lace Sensor Pickups) and a Roland Micro Cube (with the "Black Panel" model). (Sorry for my bad playing skills and the recording quality, I just finished the project and I took whatever lay around to record it). Now I'm thinking about doing an amp kit from Tube Amp Doctor.
Friday, 18 July 2008
Game theory and empirical observations (this might be interesting for you, Parcival)
Encouraged by Parcival's comment on my post about modal logic, I'll pass on some wisdom of Joe Halpern's recent course in Lausanne (see also this post).
The short version is this: In game theory, the concept of Nash equilibrium (now famous by the movie A Beautiful Mind) is considered as the "best" solution (in some sense), but empirical observations showed that players may use completely different solutions to a particular game than the Nash equilibrium supposes.
Consider for example the following game: Two people are asked a (natural) number between 2 and 100. If both answer the same number, both will get as many dollars, as the number they said. If they answer different numbers, whoever said the lower number will get as many dollars as the number he said and additionally 2$ bonus, while the other one also gets the lower number, but minus 2$ penalty. Say for example, the two players both say 100, then they will get 100$. If the first player says 98 and the second says 99, then the first one will get 100$ and the second one will get 96$. Are we still together? (to quote Joe Halpern.) By the way, this is known as Traveler's dilemma.
Now game theory tells us, both players should say 2, as this is the only Nash equilibrium (for details, see below). But even without empirical studies, you clearly see that almost no player will say 2. Even members of the Game Theory Society didn't stick to this strategy, as a survey shows, the most popular strategies were to play 90 or higher. What's the problem about playing 2? It's quite simple: you will regret it, because you could have made a lot more money (even though you might make less than you opponent). So, people will usually choose strategies, that minimize their regret: If you play 2 and your opponent plays 95, you best response would have been 94. So you could have won 96$, but as you played 2, you will win 4$. So the amount of money you regret is 92$. People seem to be very sensitive to regret, it's not only important, what they won, but also what they could have won. So, you're trying to minimize your regret. And some simple calculations later you will see, that the optimal strategy to minimize your regret is to play 97. And this also agrees with the empirical studies.
More details can be found in this paper by Halpern and Pass.
The short version is this: In game theory, the concept of Nash equilibrium (now famous by the movie A Beautiful Mind) is considered as the "best" solution (in some sense), but empirical observations showed that players may use completely different solutions to a particular game than the Nash equilibrium supposes.
Consider for example the following game: Two people are asked a (natural) number between 2 and 100. If both answer the same number, both will get as many dollars, as the number they said. If they answer different numbers, whoever said the lower number will get as many dollars as the number he said and additionally 2$ bonus, while the other one also gets the lower number, but minus 2$ penalty. Say for example, the two players both say 100, then they will get 100$. If the first player says 98 and the second says 99, then the first one will get 100$ and the second one will get 96$. Are we still together? (to quote Joe Halpern.) By the way, this is known as Traveler's dilemma.
Now game theory tells us, both players should say 2, as this is the only Nash equilibrium (for details, see below). But even without empirical studies, you clearly see that almost no player will say 2. Even members of the Game Theory Society didn't stick to this strategy, as a survey shows, the most popular strategies were to play 90 or higher. What's the problem about playing 2? It's quite simple: you will regret it, because you could have made a lot more money (even though you might make less than you opponent). So, people will usually choose strategies, that minimize their regret: If you play 2 and your opponent plays 95, you best response would have been 94. So you could have won 96$, but as you played 2, you will win 4$. So the amount of money you regret is 92$. People seem to be very sensitive to regret, it's not only important, what they won, but also what they could have won. So, you're trying to minimize your regret. And some simple calculations later you will see, that the optimal strategy to minimize your regret is to play 97. And this also agrees with the empirical studies.
More details can be found in this paper by Halpern and Pass.
Machines rule the world
I'm just reading Yde Venema's Lectures on the modal μ-calculus and it's quite interesting but all those automata, like the Büchi automaton, the Muller automaton or the Parity automaton and their variants (think streams, trees, flows, biflows, nondeterministic, alternating, logical, etc.) make me feel dizzy sometimes. I really like the coalgebraic approach, universal coalgebra seems to be a nice thing, but I don't know that many things about it yet, so I will read Rutten's paper Universal coalgebra: a theory of systems when I find some time for it.
By the way (for those who are interested), there's also a thing called universal algebra and this is really nice theory (which I can tell you because I know some things about it). There's an excellent free book on this topic, namely A Course in Universal Algebra by Burris and Sankappanavar.
By the way (for those who are interested), there's also a thing called universal algebra and this is really nice theory (which I can tell you because I know some things about it). There's an excellent free book on this topic, namely A Course in Universal Algebra by Burris and Sankappanavar.
Monday, 14 July 2008
Modal logic & all that
As some of you know I recently began working on my PhD thesis. I have a lot of fun (really!) reading about new (to me) stuff such as modal logic, epistemic logic, common knowledge and μ-calculus. I already did some work in the field of logic, but this was mainly classical mathematical logic and proof theory and had only some basic knowledge in fields mentioned above. As I see now, I should have looked at them earlier, they are lots of fun for several reasons. First of all: epistemic logic has various (natural!) connections to lots of different fields: philosophy, game theory, mathematics and computer science. Second: This is the first time in my mathematical career that I'm able to explain at least some basic ideas to my non-mathematical friends (if they'd only ask...). Third: there are all this funny riddles, like the wise men puzzle (although I prefer this solution), the muddy children puzzle (which is the less violent version of what is called Josephine's puzzle in the first source to the wise men puzzle) or the mind tingling surprise test paradox.
Interested? I'll try to keep you up to date on this topic, but in the meantime I recommend the excellent book Reasoning about Knowledge by Fagin, Halpern, Moses and Vardi. For a start you might also spend some thoughts on Plato's famous tripartite definition of knowledge as justified true belief.
Interested? I'll try to keep you up to date on this topic, but in the meantime I recommend the excellent book Reasoning about Knowledge by Fagin, Halpern, Moses and Vardi. For a start you might also spend some thoughts on Plato's famous tripartite definition of knowledge as justified true belief.
Saturday, 5 July 2008
HDRing around
Today I just felt like taking my trustworthy old Sony Alpha 100 and making some HDR images of my Framus Renegade Pro.
By the way: I'm quite busy at the moment as I'm currently attending the Logic Colloquim 08 in Bern.
By the way: I'm quite busy at the moment as I'm currently attending the Logic Colloquim 08 in Bern.
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