Conservation Law of Utility and Equilibria in Non-Zero Sum Games

This short note demonstrates how one can define a transformation of a non-zero sum game into a zero sum, so that the optimal mixed strategy achieving equilibrium always exists. The transformation is equivalent to introduction of a passive player into a game (a player with a singleton set of pure strategies), whose payoff depends on the actions of the active players, and it is justified by the law of conservation of utility in a game. In a transformed game, each participant plays against all other players, including the passive player. The advantage of this approach is that the transformed game is zero-sum and has an equilibrium solution. The optimal strategy and the value of the new game, however, can be different from strategies that are rational in the original game. We demonstrate the principle using the Prisoner's Dilemma example

The Value of Information and Circular Settings

We present a universal concept for the Value of Information (VoI), based on the works of Claude Shannon's and Ruslan Stratonovich that can take into account very general preferences of the agents and results in a single number. As such it is convenient for applications and also has desirable properties for decision theory and demand analysis. The Shannon/Stratonovich VoI concept is compared to alternatives and applied in examples. In particular we apply the concept to a circular spatial structure well known from many economic models and allow for various economic transport costs.Comment: 23 pages, 5 figure

On emotion, learning and uncertainty: a cognitive modelling approach

A problem of emotion and cognition is considered within a unified theory of cognition. There is a strong case for modern cognitive models to take arousal component of emotion into account because of its significant influence on performance (e.g. the inverted-U effect). Several hypotheses have been proposed to explain this effect, but they have not been integrated into cognitive architectures. Based on the analysis of the ACT-R (Anderson & Lebiere, 1998) cognitive architecture the mechanisms that can be used to model this effect are identified. Then a model of the classical Yerkes and Dodson experiment is introduced. The model matches the data by modifying several parameters, particularly noise and goal value in the conflict resolution strategy. Thus, the model supports the idea that the character of decision making changes for different arousal and motivational states. The effect of these changes on learning is analysed using information theory. In particular, randomness in behaviour due to a noise increase leads to a faster entropy reduction. Thus, noise can improve learning in the initial stage of problem exploration or upon changes in the environment. Furthermore, dynamic motivation can optimise the expenditure of effort. Therefore, emotion may play an important role in adaptation of cognitive processes. It is argued that the current conflict resolution mechanism in ACT-R does not explain the dynamics suggested by the model. A new theory and algorithm are proposed that use posterior estimation of expected costs. There are three main contributions of the thesis: 1) Ways of including the effects of emotion and motivation into cognitive models; 2) The analysis of the role of emotion in learning and intelligence; and 3) The introduction of a new machine learning algorithm suitable for applications not only in cognitive modelling, but in other areas of computer science

The duality of utility and information in optimally learning systems

The paper considers learning systems as optimisation systems with dynamical information constraints, and general optimality conditions are derived using the duality between the space of utility functions and probability measures. The increasing dynamics of the constraints is used to parametrise the optimal solutions which form a trajectory in the space of probability measures. Stochastic processes following such trajectories describe systems achieving the maximum possible utility gain with respect to a given information. The theory is discussed on examples for finite and uncountable sets and in relation to existing applications and cognitive models of learning

On emotion, learning and uncertainty: a cognitive modelling approach

A problem of emotion and cognition is considered within a unified theory of cognition. There is a strong case for modern cognitive models to take arousal component of emotion into account because of its significant influence on performance (e.g. the inverted-U effect). Several hypotheses have been proposed to explain this effect, but they have not been integrated into cognitive architectures. Based on the analysis of the ACT-R (Anderson & Lebiere, 1998) cognitive architecture the mechanisms that can be used to model this effect are identified. Then a model of the classical Yerkes and Dodson experiment is introduced. The model matches the data by modifying several parameters, particularly noise and goal value in the conflict resolution strategy. Thus, the model supports the idea that the character of decision making changes for different arousal and motivational states. The effect of these changes on learning is analysed using information theory. In particular, randomness in behaviour due to a noise increase leads to a faster entropy reduction. Thus, noise can improve learning in the initial stage of problem exploration or upon changes in the environment. Furthermore, dynamic motivation can optimise the expenditure of effort. Therefore, emotion may play an important role in adaptation of cognitive processes. It is argued that the current conflict resolution mechanism in ACT-R does not explain the dynamics suggested by the model. A new theory and algorithm are proposed that use posterior estimation of expected costs. There are three main contributions of the thesis: 1) Ways of including the effects of emotion and motivation into cognitive models; 2) The analysis of the role of emotion in learning and intelligence; and 3) The introduction of a new machine learning algorithm suitable for applications not only in cognitive modelling, but in other areas of computer science

Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence

We discuss a relation between the Kantorovich-Wasserstein (KW) metric and the Kullback-Leibler (KL) divergence. The former is defined using the optimal transport problem (OTP) in the Kantorovich formulation. The latter is used to define entropy and mutual information, which appear in variational problems to find optimal channel (OCP) from the rate distortion and the value of information theories. We show that OTP is equivalent to OCP with one additional constraint fixing the output measure, and therefore OCP with constraints on the KL-divergence gives a lower bound on the KW-metric. The dual formulation of OTP allows us to explore the relation between the KL-divergence and the KW-metric using decomposition of the former based on the law of cosines. This way we show the link between two divergences using the variational and geometric principles

Monotonicity of Fitness Landscapes and Mutation Rate Control

A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher's work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. First, we consider the geometric case of fitness being isomorphic to distance from an optimum, and show how problems of optimal mutation rate control can be solved exactly or approximately depending on additional constraints of the problem. Then we consider the general case of fitness communicating only partial information about the distance. We define weak monotonicity of fitness landscapes and prove that this property holds in all landscapes that are continuous and open at the optimum. This theoretical result motivates our hypothesis that optimal mutation rate functions in such landscapes will increase when fitness decreases in some neighbourhood of an optimum, resembling the control functions derived in the geometric case. We test this hypothesis experimentally by analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. Our findings support the hypothesis and find that the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We discuss the relevance of these findings to living organisms

Opposing effects of final population density and stress on Escherichia coli mutation rate

Evolution depends on mutations. For an individual genotype, the rate at which mutations arise is known to increase with various stressors (stress-induced mutagenesis-SIM) and decrease at high final population density (density-associated mutation-rate plasticity-DAMP). We hypothesised that these two forms of mutation-rate plasticity would have opposing effects across a nutrient gradient. Here we test this hypothesis, culturing Escherichia coli in increasingly rich media. We distinguish an increase in mutation rate with added nutrients through SIM (dependent on error-prone polymerases Pol IV and Pol V) and an opposing effect of DAMP (dependent on MutT, which removes oxidised G nucleotides). The combination of DAMP and SIM results in a mutation rate minimum at intermediate nutrient levels (which can support 7 × 10  cells ml ). These findings demonstrate a strikingly close and nuanced relationship of ecological factors-stress and population density-with mutation, the fuel of all evolution