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Anatomy of success, hierarchy, and inequality I. Domains with a lot of uncertainty have the highest likelihood of skilled people failing Success Success comes with perseverance and improvements, opposing luck, over which we have zero control. Success While the chances of getting lucky might be as small as p0 ≪ 1 – success results from the subsequent deliberate actions and skill acquisition with the aim to work on the opportune occasion for boosting its chances of recurrence in the future, The positive probability increment δp > 0 in can describe the effectiveness of learning, or the gain of advantageous skill contributing toward a favorable outcome. Enhancing success chances by persistent learning and skill acquisition We suppose that the efficiency of learning process can be described by the probability gain of getting success in the future, after every successful trial, pn = 0 if pn−1+ω < 0. The positive probability increment ω > 0 describes a positive feedback on the motivation to perform further trials after the previous success. Enhancing success chances by persistent learning and skill acquisition Enhancing success chances by persistent learning and skill acquisition With no effect of learning, pi = p, all ui are independent identically distributed random variables and PN is given by the binomial distribution Enhancing success chances by persistent learning and skill acquisition The effect of positive feedback for the bimodal model is revealed by the geometric distribution of distances Di between sequent success events, Enhancing success chances by persistent learning and skill acquisition This marginal distribution satisfies an intuitively plausible Pascal type recurrence relation for the probabilities PN(n), expressing the simple idea of that n successes in N trials can be reached either from n successes in N − 1 trials plus a final failure, or from n − 1 or from n − 1 successes in N − 1 trials and a final success: Enhancing success chances by persistent learning and skill acquisition This marginal distribution satisfies an intuitively plausible Pascal type recurrence relation for the probabilities PN(n), expressing the simple idea of that n successes in N trials can be reached either from n successes in N − 1 trials plus a final failure, or from n − 1 or from n − 1 successes in N − 1 trials and a final success: Multiplying it by xn and summing over all n = 0, . . . ,∞, Enhancing success chances by persistent learning and skill acquisition Enhancing success chances by persistent learning and skill acquisition Enhancing success chances by persistent learning and skill acquisition Enhancing success chances by persistent learning and skill acquisition p = 1 − e−ωt, and r = p0/ω. Enhancing success chances by persistent learning and skill acquisition • The probability density plot for the number of successful trials in the process with persistent learning, in which the initial probability of success is p0 = 0.1 and the probability increment ω = 0.02. • When learning matters, the number of tries is attributed to skill. • The probability gradients shown by arrows ”worsen” the chances for success if the number of trials is small, but ”enhance” these chances for longer trial sequences. Driving down cycle time in trials allows for more experiments, which can produce better results for those with early luck compounded Like a squirrel in a wheel. Freud's repetition compulsion When cause and effect are not well understood, or the environment is permanently changing even those who do everything ”right”, will fail more likely than not, as acquired skill does not necessarily pay off over time. The natural mechanism that can help to improve the long-term performance under uncertainty is diversification of activities. By getting involved in many different projects, e.g. by making a portfolio of many investments, or by bearing and raising many children, one can dramatically increase the chance of that advantageous skill and persistent efforts will redeem over time. Like a squirrel in a wheel. Freud's repetition compulsion Let us consider a large group of individuals engaged in many different activities, each being characterized by some probability of initial success p0 > 0 and by some probability increment ω > 0, after every successful trial. The state of getting precisely n < nmax successful outcomes after t < tmax trials in every activity is then characterized by the probability where nmax and tmax are precisely determined by p0 and ω given. Like a squirrel in a wheel. Freud's repetition compulsion Let us consider a large group of individuals engaged in many different activities, each being characterized by some probability of initial success p0 > 0 and by some probability increment ω > 0, after every successful trial. The state of getting precisely n < nmax successful outcomes after t < tmax trials in every activity is then characterized by the probability where nmax and tmax are precisely determined by p0 and ω given. The equilibrium state of such a group striving for success in a variety of activates can be determined as a state of maximum entropy: Like a squirrel in a wheel. Freud's repetition compulsion The equilibrium state of such a group striving for success in a variety of activates can be determined as a state of maximum entropy: It is worth a mention that the number of different (n, t)–states of having precisely n successful outcomes after t trials grows unboundedly when the probability increment tends to zero ω → 0 and therefore entropy of success does either. Like a squirrel in a wheel. Freud's repetition compulsion The equilibrium state of such a group striving for success in a variety of activates can be determined as a state of maximum entropy: The value of entropy increases as the number of states grows, as nmax, tmax ∼ ω−1 and tends to infinity for reducing probability increments ω → 0. The maximum entropy gradient is observed for the initial probability of success p0 ≈ 1/3. Like a squirrel in a wheel. Freud's repetition compulsion The equilibrium state of such a group striving for success in a variety of activates can be determined as a state of maximum entropy: Therefore, when success might breed success, but initial success was random, the most likely behavior to be observed over the large enough group of individuals engaged in a variety of activities is to make things into a matter of routine, by continuously repeating actions over and over again, without searching for any improvement of the future chances for success. Repetition compulsion Being in a group in the face of uncertainty, we are forced to repeat always the same behavior pattern, without any improvement – as no lesson can been learnt from the old experience. The endless repetition of behavior, or life patterns, which were difficult (or distressing) in previous life was a key concept in Freud’s understanding of mental life – repetition compulsion. Repetition compulsion Being in a group in the face of uncertainty, we are forced to repeat always the same behavior pattern, without any improvement – as no lesson can been learnt from the old experience. The endless repetition of behavior, or life patterns, which were difficult (or distressing) in previous life was a key concept in Freud’s understanding of mental life – repetition compulsion. The essential character-traits which remain always the same and which are compelled to find expression in a repetition of the same experience appeared to Freud as ultimately contradicting with the organism’s search for pleasure: ”hypothesis of a compulsion to repeat - something that seems more primitive, more elementary, more instinctual than the pleasure principle which it overrides”. In the later editions of his work, Freud had extended this point, by stating that ”such the repetitions are of course the activities of instincts intended to lead to satisfaction; but no lesson has been learnt from the old experience of these activities having led only to unpleasure”. The rich get richer. Pareto principle The self-reinforcing behavior of certain probability distributions and stochastic processes are known since the early works of Gibrat and Yule: A stochastic urn process discrete units of wealth, usually called ”balls” ◦ , are added continuously as an increasing function of the number of balls already present in a set of cells, usually called ”urns” | |, arranged in linear order The rich get richer. Pareto principle At each round of the urn process, either a bar or a ball is selected with probability α and 1 − α, respectively. If a ball is selected, it is thrown in such a way that each space in all cells has an equal chance of receiving it. If a bar is selected, it is placed next to an existing bar, so that the new cell of unit size emerges at a rate α. The rich get richer. Pareto principle The size of the k-th cell, tk, be the number of balls in this cell plus one, i.e. the number of spaces existing in the cell: between two balls, or between two bars, or between a ball and a bar. The aggregate size of all cells is increased steadily by one at the end of the round, regardless of whether a bar or a ball is selected at any given round, either because the size of one of the cells is increased by one or because a new cell of size 1 is added. The rich get richer. Pareto principle Let p(x, t) be the expected value of the number of cells with size x when the aggregate size of all cells is t. Then, for x = 1, we have where α is the probability that p(1, t) is increased by one and (1 − α)p(1, t)/t is the probability that p(1, t) is decreased by one as a result of a ball falling in one of the unit-sized cells. The rich get richer. Pareto principle Let p(x, t) be the expected value of the number of cells with size x when the aggregate size of all cells is t. Then, for x = 1, we have where α is the probability that p(1, t) is increased by one and (1 − α)p(1, t)/t is the probability that p(1, t) is decreased by one as a result of a ball falling in one of the unit-sized cells. At the steady state, for all x = 1, 2, ..., it should be The rich get richer. Pareto principle Let p(x, t) be the expected value of the number of cells with size x when the aggregate size of all cells is t. Then, for x = 1, we have The rich get richer. Pareto principle Let p(x, t) be the expected value of the number of cells with size x when the aggregate size of all cells is t. Then, for x = 1, we have The rich get richer. Pareto principle for any time t, and therefore, for the stationary distribution, it will be also true that The rich get richer. Pareto principle for any time t, and therefore, for the stationary distribution, it will be also true that The Yule distribution, The rich get richer. Pareto principle The Yule distribution, The cumulative distribution function for the Yule distribution x The exponent ρ is the inverse probability to add a ball (a unit of wealth) at a round that is nothing else but the average wealth per cell in the urn model. The processes of accumulated advantage lead to the skewed, heavy-tailed (Pareto) distributions of wealth. Inequality rising from risk taking under uncertainty ”Anyone who bet any part of his fortune, however small, on a mathematically fair game of chance acts irrationally,” wrote Daniel Bernoulli in 1738. People’s preferences with regard to choices that have uncertain outcomes are described by the expected utility hypothesis. This hypothesis states that under the quite general conditions the subjective value associated with an uncertain outcome is the statistical expectation of the individual’s valuations over all outcomes. In particular, a decision maker could use the expected value criterion as a rule of choice in the presence of risky outcomes. Inequality rising from risk taking under uncertainty The individual’s risk aversion is accounted by a mathematical function called the utility function Utility refers to the perceived value of a good (or wealth), and the utility function (viewed as a continuous function of actual wealth) describes the attitudes towards risky projects of a ”rational trader”, whose objective is to maximize growth of his wealth in the long term. Such a trader would attach greater weight to losses than he would do to gains of equal magnitude. Inequality rising from risk taking under uncertainty The risk aversion implies that the utility functions of interest are concave. The plausible example of utility functions is given by where 0 < λ < 1 is the risk tolerance parameter – as λ decreases, traders become more risk-averse and vice versa. Inequality rising from risk taking under uncertainty The risk aversion implies that the utility functions of interest are concave. The plausible example of utility functions is given by where 0 < λ < 1 is the risk tolerance parameter – as λ decreases, traders become more risk-averse and vice versa. In the limit of maximum risk avoidance, λ → 0, Inequality rising from risk taking under uncertainty Inequality rising from risk taking under uncertainty Accordingly the maximum entropy principle, the system would evolve toward the state of maximum entropy characterized by the probability distribution which can be achieved in the largest number of ways, being the most likely distribution to be observed. Inequality rising from risk taking under uncertainty Accordingly the maximum entropy principle, the system would evolve toward the state of maximum entropy characterized by the probability distribution which can be achieved in the largest number of ways, being the most likely distribution to be observed. Inequality rising from risk taking under uncertainty We are interested in the probability distribution of wealth over the population pw with maximum entropy under the condition of maximum risk avoidance. Inequality rising from risk taking under uncertainty We are interested in the probability distribution of wealth over the population pw with maximum entropy under the condition of maximum risk avoidance. The Pareto distribution of wealth Wealth inequality can be viewed as a direct statistical consequence of making decisions under uncertainty, under the condition of zero risk tolerance Inequality rising from risk taking under uncertainty We are interested in the probability distribution of wealth over the population pw with maximum entropy under the condition of maximum risk avoidance. Inequality rising from risk taking under uncertainty We are interested in the probability distribution of wealth over the population pw with maximum entropy under the condition of maximum risk avoidance. The more risk is taken by traders investing under uncertainty, the more unequal distribution of assets among them is likely to be observed in the long term. ... the more adventurous traders, the more their fortune, the less the number of lucky ones. Conclusions • We have introduced and studied the probability model of success. The probability increments of getting success in the future can be maximized over time when the consistent efforts are made in a direction of highest positive impact and when the personal role of an actor within a team is increasingly important. • We have also demonstrated that being in a group facing uncertainty, we may be trapped within the repetition compulsion and forced to repeat always the same behavior pattern, without any improvement – as no lesson can been learnt from the old experience. • Wealth inequality among the population rises from taking risky decisions under uncertainty by the vital few: the more adventurous traders, the more their fortune, the less the number of lucky ones.