Control in Games and Ratchets

Toledo, Spain, April 28th-May 1st

	Overview
	Participants
	Program
	Abstracts
	Travel info

Waiting for Godot

A hard life

The team

Overview

Control theory is an old and well developed field of research in engineering. One of the aims of this theory is to device algorithms that optimize the operation of a dynamical system, which may have elements of stochasticity. Nevertheless both the issues of collective stochastic motion and the surprising new mechanisms and concepts that were developed recently in stochastic dynamics have not been explored from the point of view of control theory. The meeting is organized around a small number of lectures by experts in stochastic control, games, and ratchets, followed by extensive discussions where participants will try to explore or establish new connections or new research directions linking stochastic dynamics with optimization and game theory.

Participants

Pau Amengual (Universitat de les Illes Balears, Spain)
Ehrhard Behrends (Free University of Berlin, Germany)
Ricardo Brito (Universidad Complutense de Madrid)
Christian Van den Broeck (Limburg Universitair Centrum, Belgium)
Bart Cleuren (Limburg Universitair Centrum, Belgium)
Francisco J. Cao (Universidad Complutense de Madrid, Spain)
José Cuesta (Universidad Carlos III de Madrid, Spain)
Luis Dinís (Universidad Complutense de Madrid, Spain)
Juan MR Parrondo (Universidad Complutense de Madrid, Spain)
Esteban Moro (Universidad Carlos III de Madrid, Spain)
Raúl Toral (Universitat de les Illes Balears, Spain)

Program

Wednesday 28th: Dinner at the Parador (21:00 h).

	Thursday 29th	Friday 30th	Saturday 1st
9:30-10:00	Parrondo	Cuesta	Van den Broeck
10:00-10:45	Dinís	Cuesta	Van den Broeck
10:45-11:30	Cleuren	Amengual
11:30-12:00	Coffee break
12:00-13:00	Behrends	Toral
13:00-16:00	Lunch and discussions
16:00-17:00	Cao	Moro

Abstracts

Pau Amengual: Parrondo's games and the zipping algorithm.
We study the relation between the discrete--time version of the flashing ratchet known as Parrondo's games and a compression technique used very recently with thermal ratchets for evaluating the transfer of information -- negentropy -- between the Brownian particle and the source of fluctuations. We present some results concerning different versions of Parrondo's games showing, for each case,a good qualitative agreement between the gain and the inverse ofthe entropy.

Ehrhard Behrends: Optimal stochastic and adaptive strategies for Parrondo's games.
Parrondo's paradox states that there are losing gambling games which, when being combined stochastically or in a suitable deterministic way, give rise to winning games. We investigate the probabilistic background. We show how the properties of the equilibrium distributions of the Markov chains under considerationgive rise to the paradoxical behaviour, and we provide methods how to find the best stochastic and adaptive strategies.

Christian Van den Broeck: Concluding Remarks.

Bart Cleuren: Primary Parrondo Paradox.
A primary model exhibiting the Parrondo paradox is introduced. It requires two rather than three states, involving only superstable fixed points, resulting in very important technical simplifications. Due to these simplifications it is possible to study in detail more complicated scenarios, such as multiplayer games involving strategy. We calculate the gain as a function of the number N of players, including exact analytic results for small values of N and in the limit N going to infinity. We show that the socalled greedy strategy is not always the best choice. The same calculations are repeated for the original Parrondo games. For these original games, we demonstrate that the greedy strategy is optimal for N=1 and N=2 but not for N=3. In the limit N going to infinity our analysis reveals a very rich behavior including the possibility of phase transitions as a function of the chosen strategy.

Francisco J. Cao: Control in Ratchets.
Controlled ratchets, in addition to their theoretical relevance as rectifiers of thermal fluctuations, are technologically feasible and have potential applications in biology, condensed matter, and nanotechnology. We present a feedback controlled flashing ratchet, and the results for the current induced by a protocol which maximizes the instant velocity of the center of mass at any time. This protocol is optimal for one particle and performs better than any periodic flashing for ensembles of moderate size, whereas is defeated by a random or periodic switching for large ensembles. These results prompt open questions as finding the optimal protocol to maximize the current or the efficiency.

José Cuesta: Individual selection, strong reciprocity and human altruism.
In recent years, strong reciprocity, i.e., the predisposition to cooperate with others and to punish those who violate the norms of cooperation at personal cost, has been proposed as a schema for predicting and understanding altruism in humans. While evidence from behavioral experiments supports this claim, the evolutionary origins of this trait remain unclear, and group selection is generally invoked to explain the existence of strong reciprocators. Here we present a simple agent model, based on the so-called ultimatum game, which shows that, although strong reciprocity could in principle be regarded as disadvantageous from the viewpoint of individuals, it can spontaneously appear and be maintained by individual evolution only. In addition, our model agrees with the available experimental data.

Luis Dinís: Strategies in collective Parrondo games.
Typing ``game theory and strategy'' in Google internet search engine, more than 900000 web pages will show up. This simple experiment shows that the notion of strategy is central in the study of games. Moreover, it is also of interest to optimal control theory, where the problem of finding a strategy that maximizes a given quantity is also widely treated.
In the talk, the topic of strategies in the so-called paradoxical games---two gambling games based on the Brownian flashing ratchet---is addressed. Optimal strategies for one player can be found, giving trivial winning games. However, we find unexpected results when collective games are considered. For instance, maximizing the winning probability in each turn of the game can yield steady losses in average, whereas a blind strategy, such as selecting the game at random, increases the average capital. The same effect can be found if the players are allowed to choose the game by voting and in a controlled deterministic dynamical system, helping us to identify the general underlying mechanisms and understand how they could be translated back into a controlled collective Brownian ratchet, where similar phenomena may appear.

Juan MR Parrondo: The physics of gambling.
Brownian motion and gambling have been related since the first works by Einstein and Bachelier in 1905. The talk is a brief introduction to some of the topics that will be covered at the workshop, from ratchets to gambling and strategy games.

Esteban Moro: The minority Game as a Learning Problem.
In this talk I will present a brief introduction to the Minority Game (MG) focusing on its main ingredients and phenomenology. In the MG a set of agents compete through adaptation for a finite resource. After clearing up the spurious parts of the model I will show that the MG is in fact a learning problem in artificial neural networks in which agents tend to store time sequences of patterns as well as to unlearn other agents' strategies to differentiate. Possible extensions and generalizations of this "unlearning game" will be presented.

Raúl Toral: A Fokker-Plack description for Parrondo's games.
It is well known that the Parrondo's paradox can be thought of as a discretization of a flashing ratchet. In this talk I will show how this relationship can be made quantitative by writing the master equation describing the Parrondo's games as a consistent discretization of the Fokker--Planck equation for an overdamped Brownian particle describing a ratchet. Our expressions, besides giving further insight on the relation between ratchets and Parrondo's games, allow us to precisely relate the games probabilities and the ratchet potential such that periodic potentials correspond to fair games and winning games produce a tilted potential. Using this relation, we obtain expressions for the stationary probability and current (games gain) in terms of an effective potential. We also demonstrate that the expressions obtained are nothing but a discretised version of the equivalent expressions in terms of the solution of the Fokker--Planck equation with multiplicative noise.

Travel info

The workshop will be held at the Parador de Toledo.

From the airport one can take the metro (line 8) to Nuevos Ministerios and then take the train to Toledo. The timetable is the following:

Nuevos Ministerios departure	Toledo arrival
8:35	10:00
10:19	11:44
15:30	16:57
18:00	19:32
19:28	20:49

From the Toledo train station one must take a taxi to the Parador.

Another possibility is the bus Madrid-Toledo (Continental Auto). Buses depart from the Mendez Alvaro bus station (metro Mendez Alvaro, line 6) every half an hour. To get to Méndez Alvaro metro station, take the metro at the airport (line 8) to Nuevos Ministerios and then take line 6 to Méndez Álvaro.