8,592 research outputs found
OPTIMAL USE OF COMMUNICATION RESOURCES
We study a repeated game with asymmetric information about a dynamic state of nature. In the course of the game, the better informed player can communicate some or all of his information with the other. Our model covers costly and/or bounded communication. We characterize the set of equilibrium payoffs, and contrast these with the communication equilibrium payoffs, which by definition entail no communication costs.Repeated games, communication, entropy
When Can Limited Randomness Be Used in Repeated Games?
The central result of classical game theory states that every finite normal
form game has a Nash equilibrium, provided that players are allowed to use
randomized (mixed) strategies. However, in practice, humans are known to be bad
at generating random-like sequences, and true random bits may be unavailable.
Even if the players have access to enough random bits for a single instance of
the game their randomness might be insufficient if the game is played many
times.
In this work, we ask whether randomness is necessary for equilibria to exist
in finitely repeated games. We show that for a large class of games containing
arbitrary two-player zero-sum games, approximate Nash equilibria of the
-stage repeated version of the game exist if and only if both players have
random bits. In contrast, we show that there exists a class of
games for which no equilibrium exists in pure strategies, yet the -stage
repeated version of the game has an exact Nash equilibrium in which each player
uses only a constant number of random bits.
When the players are assumed to be computationally bounded, if cryptographic
pseudorandom generators (or, equivalently, one-way functions) exist, then the
players can base their strategies on "random-like" sequences derived from only
a small number of truly random bits. We show that, in contrast, in repeated
two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then
random bits remain necessary for equilibria to exist
Growing Strategy Sets in Repeated Games
A (pure) strategy in a repeated game is a mapping from histories, or, more generally, signals, to actions. We view the implementation of such a strategy as a computational procedure and attempt to capture in a formal model the following intuition: as the game proceeds, the amount of information (history) to be taken into account becomes large and the \quo{computational burden} becomes increasingly heavy. The number of strategies in repeated games grows double-exponentially with the number of repetitions. This is due to the fact that the number of histories grows exponentially with the number of repetitions and also that we count strategies that map histories into actions in all possible ways. Any model that captures the intuition mentioned above would impose some restriction on the way the set of strategies available at each stage expands. We point out that existing measures of complexity of a strategy, such as the number of states of an automaton that represents the strategy needs to be refined in order to capture the notion of growing strategy space. Thus we propose a general model of repeated game strategies which are implementable by automata with growing number of states with restrictions on the rate of growth. With such model, we revisit some of the past results concerning the repeated games with finite automata whose number of states are bounded by a constant, e.g., Ben-Porath (1993) in the case of two-person infinitely repeated games. In addition, we study an undiscounted infinitely repeated two-person zero-sum game in which the strategy set of player 1, the maximizer, expands \quo{slowly} while there is no restriction on player 2's strategy space. Our main result is that, if the number of strategies available to player 1 at stage grows subexponentially with , then player 2 has a pure optimal strategy and the value of the game is the maxmin value of the stage game, the lowest payoff that player 1 can guarantee in one-shot game. This result is independent of whether strategies can be implemented by automaton or not. This is a strong result in that an optimal strategy in an infinitely repeated game has, by definition, a property that, for every , it holds player 1's payoff to at most the value plus after some stageRepeated Games, Complexity, Entropy
Trading locality for time: certifiable randomness from low-depth circuits
The generation of certifiable randomness is the most fundamental
information-theoretic task that meaningfully separates quantum devices from
their classical counterparts. We propose a protocol for exponential certified
randomness expansion using a single quantum device. The protocol calls for the
device to implement a simple quantum circuit of constant depth on a 2D lattice
of qubits. The output of the circuit can be verified classically in linear
time, and is guaranteed to contain a polynomial number of certified random bits
assuming that the device used to generate the output operated using a
(classical or quantum) circuit of sub-logarithmic depth. This assumption
contrasts with the locality assumption used for randomness certification based
on Bell inequality violation or computational assumptions. To demonstrate
randomness generation it is sufficient for a device to sample from the ideal
output distribution within constant statistical distance.
Our procedure is inspired by recent work of Bravyi et al. (Science 2018), who
introduced a relational problem that can be solved by a constant-depth quantum
circuit, but provably cannot be solved by any classical circuit of
sub-logarithmic depth. We develop the discovery of Bravyi et al. into a
framework for robust randomness expansion. Our proposal does not rest on any
complexity-theoretic conjectures, but relies on the physical assumption that
the adversarial device being tested implements a circuit of sub-logarithmic
depth. Success on our task can be easily verified in classical linear time.
Finally, our task is more noise-tolerant than most other existing proposals
that can only tolerate multiplicative error, or require additional conjectures
from complexity theory; in contrast, we are able to allow a small constant
additive error in total variation distance between the sampled and ideal
distributions.Comment: 36 pages, 2 figure
Transforming Monitoring Structures with Resilient Encoders. Application to Repeated Games
An important feature of a dynamic game is its monitoring structure namely,
what the players effectively see from the played actions. We consider games
with arbitrary monitoring structures. One of the purposes of this paper is to
know to what extent an encoder, who perfectly observes the played actions and
sends a complementary public signal to the players, can establish perfect
monitoring for all the players. To reach this goal, the main technical problem
to be solved at the encoder is to design a source encoder which compresses the
action profile in the most concise manner possible. A special feature of this
encoder is that the multi-dimensional signal (namely, the action profiles) to
be encoded is assumed to comprise a component whose probability distribution is
not known to the encoder and the decoder has a side information (the private
signals received by the players when the encoder is off). This new framework
appears to be both of game-theoretical and information-theoretical interest. In
particular, it is useful for designing certain types of encoders that are
resilient to single deviations and provide an equilibrium utility region in the
proposed setting; it provides a new type of constraints to compress an
information source (i.e., a random variable). Regarding the first aspect, we
apply the derived result to the repeated prisoner's dilemma.Comment: Springer, Dynamic Games and Applications, 201
Impermanent Types and Permanent Reputations
We study the impact of unobservable stochastic replacements for the long-run player in the classical reputation model with a long-run player and a series of short-run players. We provide explicit lower bounds on the Nash equilibrium payoffs of a long-run player, both ex-ante and following any positive probability history. Under general conditions on the convergence rates of the discount factor to one and of the rate of replacement to zero, both bounds converge to the Stackelberg payoff if the type space is sufficiently rich. These limiting conditions hold in particular if the game is played very frequently.Reputation, repeated games, replacements, disappearing reputations JEL Classification Numbers: D80, C73
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