arXiv subjects

Yakov Babichenko

Publications and source records attributed to Yakov Babichenko.

46 records · Page 3Linked to original sources

arXiv2014

Empirical Distribution of Equilibrium Play and Its Testing Application

We show that in any $n$-player $m$-action normal-form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three types of equilibria: Nash, correlated and coarse correlated. For each one of them we obtain upper and lower bounds on the number of samples required for the empirical distribution over the sampled action profiles to form an approximate equilibrium with probability close to one. These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where $n$ and $m$ are large. In addition, our results substantially improve previously known upper bounds on the support size of approximate equilibria in games with many players. In particular, for all the three types of equilibria we show the existence of approximate equilibrium with support size polylogarithmic in $n$ and $m$, whereas the previously best-known upper bounds were polynomial in $n$.

cs.GT↗

arXiv2014

Query Complexity of Approximate Nash Equilibria

We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players $n$. Our main result states that for $n$-player binary-action games and for constant $\varepsilon$, the query complexity of an $\varepsilon$-well-supported Nash equilibrium is exponential in $n$. One of the consequences of this result is an exponential lower bound on the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium.

cs.GT↗

arXiv2014

Axiomatic Approach to Solutions of Games

We consider solutions of normal form games that are invariant under strategic equivalence. We consider additional properties that can be expected (or be desired) from a solution of a game, and we observe the following: - Even the weakest notion of individual rationality restricts the set of solutions to be equilibria. This observation holds for all types of solutions: in pure-strategies, in mixed strategies, and in correlated strategies where the corresponding notions of equilibria are pure-Nash, Nash and coarse-correlated. An action profile is (strict) simultaneous maximizer if it simultaneously globally (strictly) maximizes the payoffs of all players. - If we require that a simultaneous maximizer (if it exists) will be a solution, then the solution contains the set of pure Nash equilibria. - There is no solution for which a strict simultaneous maximizer (if it exists) is the unique solution.

cs.GT↗

arXiv2013

Small-Support Approximate Correlated Equilibria

We prove the existence of approximate correlated equilibrium of support size polylogarithmic in the number of players and the number of actions per player. In particular, using the probabilistic method, we show that there exists a multiset of polylogarithmic size such that the uniform distribution over this multiset forms an approximate correlated equilibrium. Along similar lines, we establish the existence of approximate coarse correlated equilibrium with logarithmic support. We complement these results by considering the computational complexity of determining small-support approximate equilibria. We show that random sampling can be used to efficiently determine an approximate coarse correlated equilibrium with logarithmic support. But, such a tight result does not hold for correlated equilibrium, i.e., sampling might generate an approximate correlated equilibrium of support size Ω(m) where m is the number of actions per player. Finally, we show that finding an exact correlated equilibrium with smallest possible support is NP-hard under Cook reductions, even in the case of two-player zero-sum games.

cs.GT↗

arXiv2013

Approximate Nash Equilibria via Sampling

We prove that in a normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium where each player randomizes uniformly among a set of O(log(m) + log(n)) pure strategies. This result induces an $N^{\log \log N}$ algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the number of players (m=poly(n)), where $N=nm^n$ is the size of the game (the input size). In addition, we establish an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approximate Nash equilibrium using the random sampling algorithm.

cs.GT↗

arXiv2013

Query Complexity of Correlated Equilibrium

We study lower bounds on the query complexity of determining correlated equilibrium. In particular, we consider a query model in which an n-player game is specified via a black box that returns players' utilities at pure action profiles. In this model we establish that in order to compute a correlated equilibrium any deterministic algorithm must query the black box an exponential (in n) number of times.

cs.GT↗

arXiv2013

Small Support Equilibria in Large Games

In this note we provide a new proof for the results of Lipton et al. on the existence of an approximate Nash equilibrium with logarithmic support size. Besides its simplicity, the new proof leads to the following contributions: 1. For n-player games, we improve the bound on the size of the support of an approximate Nash equilibrium. 2. We generalize the result of Daskalakis and Papadimitriou on small probability games from the two-player case to the general n-player case. 3. We provide a logarithmic bound on the size of the support of an approximate Nash equilibrium in the case of graphical games.

cs.GT↗

arXiv2012

Musical chairs

In the {\em Musical Chairs} game $MC(n,m)$ a team of $n$ players plays against an adversarial {\em scheduler}. The scheduler wins if the game proceeds indefinitely, while termination after a finite number of rounds is declared a win of the team. At each round of the game each player {\em occupies} one of the $m$ available {\em chairs}. Termination (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be {\em in conflict}. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

math.CO↗

arXiv2012

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with minimal area, (up to constant), namely $Θ(1/\log n)$. As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$ and $\{W(s): s \ge 0\}$. Let $τ_t:=\min\{s \ge 0: B(s)=t\}$. Then $X_t=W(τ_t)$ is a Cauchy process, and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ is a Kakeya set of zero area. The area of the $ε$-neighborhood of $K$ is as small as possible, i.e., almost surely of order $Θ(1/|\log ε|)$.

math.PR↗

arXiv2011

Oblivious Collaboration

Communication is a crucial ingredient in every kind of collaborative work. But what is the least possible amount of communication required for a given task? We formalize this question by introducing a new framework for distributed computation, called {\em oblivious protocols}. We investigate the power of this model by considering two concrete examples, the {\em musical chairs} task $MC(n,m)$ and the well-known {\em Renaming} problem. The $MC(n,m)$ game is played by $n$ players (processors) with $m$ chairs. Players can {\em occupy} chairs, and the game terminates as soon as each player occupies a unique chair. Thus we say that player $P$ is {\em in conflict} if some other player $Q$ is occupying the same chair, i.e., termination means there are no conflicts. By known results from distributed computing, if $m \le 2n-2$, no strategy of the players can guarantee termination. However, there is a protocol with $m = 2n-1$ chairs that always terminates. Here we consider an oblivious protocol where in every time step the only communication is this: an adversarial {\em scheduler} chooses an arbitrary nonempty set of players, and for each of them provides only one bit of information, specifying whether the player is currently in conflict or not. A player notified not to be in conflict halts and never changes its chair, whereas a player notified to be in conflict changes its chair according to its deterministic program. Remarkably, even with this minimal communication termination can be guaranteed with only $m=2n-1$ chairs. Likewise, we obtain an oblivious protocol for the Renaming problem whose name-space is small as that of the optimal nonoblivious distributed protocol. Other aspects suggest themselves, such as the efficiency (program length) of our protocols. We make substantial progress here as well, though many interesting questions remain open.

cs.DC↗