nhaliday + communication-complexity   16

Correlated Equilibria in Game Theory | Azimuth
Given this, it’s not surprising that Nash equilibria can be hard to find. Last September a paper came out making this precise, in a strong way:

• Yakov Babichenko and Aviad Rubinstein, Communication complexity of approximate Nash equilibria.

The authors show there’s no guaranteed method for players to find even an approximate Nash equilibrium unless they tell each other almost everything about their preferences. This makes finding the Nash equilibrium prohibitively difficult to find when there are lots of players… in general. There are particular games where it’s not difficult, and that makes these games important: for example, if you’re trying to run a government well. (A laughable notion these days, but still one can hope.)

Klarreich’s article in Quanta gives a nice readable account of this work and also a more practical alternative to the concept of Nash equilibrium. It’s called a ‘correlated equilibrium’, and it was invented by the mathematician Robert Aumann in 1974. You can see an attempt to define it here:
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july 2017 by nhaliday
6.896: Essential Coding Theory
- probabilistic method and Chernoff bound for Shannon coding
- probabilistic method for asymptotically good Hamming codes (Gilbert coding)
- sparsity used for LDPC codes
mit  course  yoga  tcs  complexity  coding-theory  math.AG  fields  polynomials  pigeonhole-markov  linear-algebra  probabilistic-method  lecture-notes  bits  sparsity  concentration-of-measure  linear-programming  linearity  expanders  hamming  pseudorandomness  crypto  rigorous-crypto  communication-complexity  no-go  madhu-sudan  shannon  unit  p:**  quixotic 
february 2017 by nhaliday
Computational Complexity: Favorite Theorems: The Yao Principle
The Yao Principle applies when we don't consider the algorithmic complexity of the players. For example in communication complexity we have two players who each have a separate half of an input string and they want to compute some function of the input with the minimum amount of communication between them. The Yao principle states that the best probabilistic strategies for the players will achieve exactly the communication bounds as the best deterministic strategy over a worst-case distribution of inputs.

The Yao Principle plays a smaller role where we measure the running time of an algorithm since applying the Principle would require solving an extremely large linear program. But since so many of our bounds are in information-based models like communication and decision-tree complexity, the Yao Principle, though not particularly complicated, plays an important role in lower bounds in a large number of results in our field.
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january 2017 by nhaliday

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