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Опубликовано в Forex diversification is | Октябрь 2, 2012

In this post we will have a look at Parrondos paradox. In a paper* entitled “Information Entropy and Parrondo's Discrete-Time Ratchet”** the. Alternatively, what happens if it takes a pass on all the small winners and scales into positions? Yes, it will scale into losers, but it should. By modelling organisms that alternate between individual and colonial lifestyles, the well-known Parrondo's paradox can emerge in an ecological setting.
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Using the notation from Equation 1 , switching rates can then be expressed as follows:. A variety of mechanisms might trigger this switching behavior in biological systems. For example, since the nomadic organisms are highly mobile, they could frequently re-enter their original colonial habitat after leaving it, and thus be able to detect whether resource levels are high enough for recolonization.

Equation 4 thus becomes:. All other population sizes and capacities can then be understood as ratios with respect to this critical population size. Simulation results revealed population dynamics that could be categorized into the following regimes:. The following sections describe the listed regimes in greater detail, with a focus upon the regimes involved in the paradox. Figures generated via numerical simulation are provided as examples of behavior within each regime.

As described earlier, both nomadic and colonial behaviors can be modelled as losing strategies given the appropriate parameters. Simulations across a range of parameters elucidated the conditions which resulted in extinction for both strategies. Figure 1a shows an example when both strategies are losing, resulting in extinction, while Figure 1b shows an example where only the colonial sub-population survives. Hence, nomadism is always a losing strategy.

However, the conditions under which colonial behavior is a losing strategy are more complicated. Complex dynamics occur when the critical capacity A is just below 1 that can result in either survival or extinction. That is:. The intuition behind this is straightforward. For a formal proof, refer to Theorem A. Under this condition, both nomadism Game A and colonialism Game B are losing strategies when played individually. Paradoxically, it is possible to combine these two strategies through behavioral switching such that population survival is ensured, thereby producing an overall strategy that wins.

Simulation results over a range of parameters have predicted this paradoxical behavior, and also elucidated the conditions under which it occurs. Figure 2a is a typical example where the population becomes extinct, even though it undergoes behavioral switching, while Figure 2b is a typical example where behavioral switching ensures population survival.

Conceptually, this paradoxical survival is possible because the colonial strategy, or Game B, is history-dependent. Behavioral switching to a nomadic strategy decreases the colonial population size, allowing the resources in the colonial environment, represented by K , to recover.

Switching back to a colonial strategy then allows the population to take advantage of the newly generated resources. Because switching occurs periodically, as can be seen in Figure 2b , it should be noted that the organisms need not even detect the amount of resources present in the environment to implement this strategy. A biological clock would be sufficient to trigger switching behavior.

The exact process by which survival is ensured can be understood by analysing the simulation results in detail. In the nomadic phase, the colonial population n 2 is close to zero, the nomadic population n 1 undergoes slow exponential decay, and the carrying capacity K undergoes slow linear growth.

K increases until it reaches L 2 , which triggers the switch to colonial behavior. The population thus enters the colonial phase. If the colonial population n 2 exceeds the critical capacity A at this point, then n 2 will grow until it slightly exceeds the carrying capacity K. Subsequently, n 2 decreases in the tandem with K until K drops to L 1 , triggering the switch back to the nomadic phase.

This implies that, by the end of the nomadic phase, n 1 needs to be greater by a certain amount than A as well. Otherwise, there will be insufficient nomads to form a colony which can overcome the Allee effect. A full derivation is provided in the Appendix Theorem A. The greater the difference between the switching levels, the longer the nomadic phase will last, because it takes more time for K to increase to the requisite value for switching, L 2.

And the longer the nomadic phase lasts, the more n 1 will decay. If, at the end of the nomadic phase, the value that n 1 decays to happens to be less than B , then the population will fail to survive. It follows that there should be some constraint on the difference between the switching levels L 1 and L 2. In other words, n 2 has to grow sufficiently quickly during the colonial phase such that it exceeds both K and L 1 before switching begins. This can be seen occurring in Figure 2b.

The Figures also show that r 1 close to 1, but this is not strictly necessary. Collectively, Equations 10—11 are sufficient conditions for population survival. Mathematical details are provided in the Appendix Theorems A.

Note that Equation 10 contains an implicit lower bound on L 1. The following bound is thus obtained:. On the other hand, under the assumptions made, there is no upper bound for L 1 , and hence no absolute upper bound for L 2 either. This suggests that given a sufficiently well-designed switching rule, K can grow larger over time while ensuring population survival. Such a rule is investigated in the following section. Suppose that, in addition to being able to detect the colonial carrying capacity, nomads and colonists are able to detect or estimate their current population size.

This might happen by proxy, by communication, or by built-in estimation of the time required for growth or decay to a certain population level. The following switching rule then becomes possible:. That is, L 1 is set to the carrying capacity K whenever n 2 rises to K , resulting immediately in a switch to nomadic behavior, and that L 2 is in turn set to K whenever n 1 falls to B , resulting in an immediate switch to colonial behavior.

This switching rule is optimal according to several criteria. As such, it avoids the later portion of the colonial phase where K and n 2 decrease in tandem, and maximizes the ending value n 2. Consequently, it also maximizes the value of n 1 at the start of each nomadic phase.

Furthermore, by switching to colonial behavior right when n 1 decays to B , the rule maximizes the duration of the nomadic phase while ensuring survival. This in turn means that the growth of K is maximized, since the longer the nomadic phase, the longer that K is allowed to grow. It plays Game A, the nomadic strategy, for as long as possible, in order to maximize K and hence the returns from Game B. Suppose that K grows more during each nomadic phase than it falls during each colonial phase.

Then the switching rule is not just optimal, but it also enables long-term growth. Simulation results predict that this can indeed occur. Together with K , the per-phase maximal values of n 1 and n 2 increase as well. In the cases shown, long-term growth is achieved because K indeed grows more during each nomadic phase than it falls during the subsequent colonial phase. As can be seen from Figure 3a , this is, in turn, because the nomadic phase lasts much longer than the colonial phase, such that the amount of environmental destruction due to colonialism is limited.

An interesting phenomenon that can be observed from Figure 3b is how the nomadic population size n 1 , which peaks at the start of each nomadic phase, eventually exceeds the carrying capacity K , and then continues doing so by increasing amounts at each peak. This is, in fact, a natural consequence of the population model. When n 2 grows large, the assumption that switching is much faster than colonial growth starts to break down.

The result is that when a large colonial population begins switching to nomadism, a significant number of colonial offspring are simultaneously being produced. These offspring also end up switching to a nomadic strategy, resulting in more nomadic organisms than there were colonial organisms before.

A particularly pronounced example of this is shown in Figure 4. When n 2 is large, switching takes longer, causing a drop in K , and a large increase in n 1. However, this same phenomenon also introduces a limiting behavior to the pattern of long-term growth. This occurs because sufficiently high levels of n 2 cause a qualitative change in the dynamics of behavioral switching.

Normally, switching to nomadic behavior starts when K falls below L 1 , and ends when K rises above it again. But when n 2 is sufficiently large, the faster production of colonial offspring drags out the duration of switching, as seen in Figure 4. The higher levels of n 2 , combined with the longer switching duration, causes an overall drop in K by the end of the switching period.

Because the increase in K during the subsequent nomadic phase is unable to overcome this drop, K stops increasing in the long-run. Nonetheless, it is clear that significant long-term gains can be achieved via the optimal switching rule. Limiting behavior eventually emerges, but this is to be expected in any realistic biological system.

Our proposed model is convenient for the functional understanding of growth and survival, and can be easily modified for a variety of applications. Additional constraints can be imposed under which survival and long-term growth are still observed. For example, in many biological systems, the dynamics of habitat change might occur on a slower timescale than both colonial and nomadic growth i.

It can clearly be seen that survival is still possible under such conditions. Another practical constraint that can be imposed is limiting the growth of the carrying capacity to some maximal value K m a x , capturing the fact that the resources in any one habitat do not grow infinitely large.

This can be achieved by modifying Equation 6 as follows:. As Figure 7 shows, even long-term growth is possible, under both fast habitat change Figure 7a and slow habitat change Figure 7b. In both cases, the carrying capacity K converges towards a maximum value as it approaches K max.

In particular, any growth model that is devoid of competitive or collaborative effects is readily captured by Equation 2 nomadism , while any logistic growth model which includes both the Allee effect and habitat destruction can be described using Equations 3 and 4 colonialism. Many organisms also exhibit behavioral change or phenotypic switching in response to changing environmental conditions. By incorporating this into our model, we have demonstrated that nomadic-colonial alternation can ensure the survival of a species, even when nomadism or colonialism alone would lead to extinction.

Furthermore, it has been demonstrated that an optimal switching rule can lead to long-term population growth. If one views the carrying capacity K as the capital of the population, then it is clear that Equation 5 is a capital-dependent switching rule. By setting the appropriate amounts of capital at which switching should occur, survival and growth can be achieved.

Survival is achieved by ensuring that Game A, or nomadism, is never played beyond the point where extinction is inevitable, that is, the point where n 1 falls below the critical level B. The history-dependent dynamics of Game B are thus optimally exploited. Several limitations of the present study should be noted. Firstly, the study only focuses on cases where nomadism and colonialism are individually losing strategies, despite the abundance of similar strategies that do not lose in the real world.

This is because assuming individually losing strategies in fact leads to a stronger result — if losing variants of nomadism and colonialism can be combined into a winning strategy, it follows that non-losing variants can be combined in a similar way too see Theorem A.

Secondly, the population model does not encompass all variants of qualitatively similar behavior. For example, many other equations can be used to model the Allee effect Boukal and Berec, Nonetheless, our proposed model is general enough that it can be adapted for use with other equations and be expected to produce similar results.

Even the presence of the Allee effect is not strictly necessary, since the colonial population might die off at low levels because of stochastic fluctuations, rather than because of the effect. Theorem A. Thirdly, though it is trivially the case that pure nomadism and pure colonialism cannot out-compete a behaviorally-switching population, a more complex analysis of the evolutionary stability of behavioral switching is beyond the scope of this paper.

Finally, spatial dynamics are not accounted for in this study. Exploring such dynamics is a goal for future work. Both the relative error tolerance and absolute error tolerance were determined to be 10 After each detection, the parameters were automatically modified as per the switching rule, and the simulation continued with the new parameters. Broad regimes of model behavior were observed by running simulations across a wide range of parameters and initial conditions.

General trends and conditions observed within each regime were formalized analytically, the details of which can be found in the Appendix. In these derivations, reasonable assumptions were made in order to make the model analytically tractable. Initial conditions corresponding to unstable equilibria e. Our comprehensive model captures both capital and history-dependent dynamics within a realistic ecological setting, thereby exhibiting Parrondo's paradox without the need for exogenous environmental influences.

Not only could it provide evolutionary insight into strategies analogous to nomadism, colonialism, and behavioral diversification, it potentially also explains why environmentally destructive species, such as Homo sapiens , can thrive and grow despite limited environmental resources. By providing a theoretical model under which such paradoxes occur, our approach may enable new insights into the evolution of cooperative colonies, as well as the conditions required for sustainable population growth.

Let t 1 and t 2 respectively be the start and end of a period of switching from sub-population i to sub-population j. Then either of the following must hold:. Switching to nomadism begins when K falls below L 1 and stops when it rises above L 1 again.

Similarly, switching to colonialism begins when K rises above L 2 and stops when it falls below L 2 again. In both cases then:. Let x be the colonial population at the end of a colonial phase, and y be the colonial population at the start of the subsequent nomadic phase.

Let z be the nomadic population at the start of the subsequent nomadic phase. Note that z is an increasing function of x , and that z converges to x as x grows larger. The roots of this equation are given by the Lambert W function. Equation A3 follows. This completes the proof. Let x be the nomadic population at the end of a nomadic phase, and y be the nomadic population at the start of the subsequent colonial phase.

Let z be the colonial population at the start of the subsequent colonial phase. We have. Note that in the first case, z is an increasing function of x. Let t C denote the start of the subsequent colonial phase. This also implies a lower bound on L 1 , given by:. During this period, the colonial population n 2 is close to or equal to 0. Combining this with the two equations above, we obtain. For survival to occur, the colonial population at the start of CP 2 needs to be greater than A.

By Theorem A. Substituting into the above, we have. Notice that by Theorem A. Then by Equation A4 in Theorem A. Substituting this into Equation A11 gives us Equation A8 , as desired. It thus needs to be the case that. Solving this gives us the lower bound on L 1.

Let CP 1 denote the colonial phase in question. Given Equation A12 , all switching to the nomadic phase only happens after n 2 exceeds K during CP 1. We have already presumed in Theorem A. Thus, given our assumptions, Equation A8 and Equation A12 collectively ensure survival. Suppose that pure nomadism or pure colonialism or both result in population survival. That is,. Then there are conditions under which behavioral alternation between the two pure strategies will result in a total population size with higher long-term periodic maxima.

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses. Thank you for submitting your article "Nomadic-colonial alternation can enable population growth despite habitat destruction: An ecological Parrondo's paradox" for consideration by eLife.

Your article has been favorably evaluated by Ian Baldwin Senior Editor and three reviewers, one of whom is a member of our Board of Reviewing Editors. The reviewers have opted to remain anonymous. The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

All reviewers agree that this is an interesting paper that introduces Parrondo's paradox to an ecological setting. In this setting, the paradox consists of the fact that even though each of two sub-populations go extinct when left alone, migration between the sub-populations can lead to persistence. The mechanism enabling persistence is that one of the habitats can regenerate itself while the population spends time declining in the other habitat, so that after some time the regenerated habitat can harbor a growing population again for some time, before that habitat deteriorates again, and so on.

I think this phenomenon is worth being brought to the attention of ecologists, and will likely enhance the conceptual toolbox of people concerned with conservation issues. As you will see the reviewers have raised a number of concerns about the paper. Chief among those is the biological realism and relevance of some of the model assumptions and interpretations. These concerns need to be addressed in a major revision. After we have received the revision, we will make a decision as to the suitability of your paper for eLife based on how each of the following points were addressed:.

Specifically, the paper assumes that the carrying capacity in the habitat of the colonizers changes on the same rapid ecological time scale as the population density itself changes. This seems unrealistic. Is it possible to formulate an alternative model, e. If not, what is the rationale behind assuming the same time scales for those two dynamics?

Clearly, this is not realistic, and the question is whether it is possible to obtain the same results with more realistic assumptions, which would e. At the very least, this kind of switching behavior needs to be justified biologically. Are there other, biologically more relevant switching behaviors that would lead to the same results? This is an interesting paper that introduces Parrondo's paradox to an ecological setting. In this setting, the paradox consists of the fact that even though each of two subpopulations go extinct when left alone, migration between the subpopulations can lead to persistence.

The mechanism enabling persistence is that one of the habitats can regenerate itself while the population spends time declining in the other habitat, so that after some time the regenerated habitat can harbour a growing population again for some time, before that habitat deteriorates again, and so on. I think this phenomenon is worth being brought to the attention of ecologists, and will e. The paper appears to be well executed technically, but I do have some concerns regarding the biological significance of the work.

Chief among those is the fact that the carrying capacity is assumed to change on the same time scale as the population density i. On intuitive grounds, I would expect that the dynamics of the carrying capacity to be much slower than the population dynamics.

Also, in the absence of any population in the habitat in which the carrying capacity is changing, the carrying capacity will grow without bounds according to eq. I think one could obtain a biologically more reasonable model by assuming some sort of time scale separation between carrying capacity dynamics and population dynamics e.

The question would then be whether the phenomenon observed by the authors would still be present under such conditions. I am also not really convinced by the proposed switching behaviour subsection "Behavioral switching" under "Population model". The authors purpose that individuals switch from nomadic to colonial behaviour when the carrying capacity for the colonial dynamics is high enough. But how would nomadic individuals be able to assess the carrying capacity for the colonial life style I can understand how the colonial individuals would assess that carrying capacity?

The more elaborate switching rule given by 13 is even less intuitive: why would colonial individuals switch to monadic life style once they reach carrying capacity? After all, this means that individuals trade a per capita growth rate of 0 because they treat carrying capacity with a negative per capita growth rate as is the case for nomadic life style by assumption.

I think it would make more sense to switch once the colonial growth rate is "negative enough", e. Again, the question then becomes whether persistence can still be observed under such more realistic assumptions. Overall, I think the paper is interesting, but needs a much more careful treatment of the biological realism underlying various assumptions, and as a consequence, a much more careful discussion of the biological relevance of the results obtained.

This is a cleverly laid out paper that will introduce a relatively new perspective to ecological and evolutionary biologists. That is, this paper lays out an example of how switching "colonial and nomadic" behavior can lead to persistence despite each singular case not allowing persistence.

While I think the general idea of organisms responding to variation in a manner that yields persistence or coexistence is far from new to ecologists and evolutionary ecologists, the perspective put forth is interesting and novel. My guess is that there are numerous existing models that may, in hindsight and with a little work, fit into this "capital" or "history dependent" version of Parrondo's paradox.

I do think people studying complex systems may not be overwhelmingly surprised by the notion that an overall average growth rate less than zero in two strategies, or two of whatever say competitors , can respond to variation environmentally driven or internally driven in a manner that yields persistence or coexistence but, again, the paper framed within the Parrondo's paradox seems potentially profitable for understanding and looking for this type of persistence outcome.

I have no major comments and I think this paper is well written, well analyzed at least for the ideas laid out , and so very close to publishable as is. Choose your reason below and click on the Report button. This will alert our moderators to take action.

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Probitas partners investing in infrastructure funds investors | Figure 1a shows an example when both strategies are losing, resulting in extinction, while Figure 1b shows an example where only the colonial sub-population survives. Flipping the trades still yields a random outcome. Commun Nonlinear Sci Numer Simul. We develop a dynamic population model by introducing an additional dormitive prey population to the existing predator-prey model which can be active active form and enter dormancy dormant form. Shaun, The above description does not indicate the time frame being utilized, only the use of the MA. Competition between two types of prey under different carrying capacities. Based on the scale-free network, the average fitness of the population of game B, d Bis 0. |

What is the full form of ipo cycle | That is:. The favorable neighboring environment of the nodes with large degree and the unfavorable neighboring environment of the nodes with small degree are constantly strengthened in the course of play. Hi, Well your approach is interesting as it provides a visible continuum to a dilemma. Numerous organisms exhibit analogous behavioral diversity, from slime moulds amoeba vs. This is because, in the random mode, the subject chooses the cooperative strategy with probability of 0. There is no easy solution as there are way too many variables, including the use of stops, targets, trailling stops etc. |

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People drop the word "apparent" in these cases as it is a mouthful, and it is obvious anyway. So no one claims these are paradoxes in the strict sense. In the wide sense, a paradox is simply something that is counterintuitive. Parrondo's games certainly are counterintuitive—at least until you have intensively studied them for a few months. The truth is we still keep finding new surprising things to delight us, as we research these games.

I have had one mathematician complain that the games always were obvious to him and hence we should not use the word "paradox. In either case, it is not worth arguing with people like that. From Wikipedia, the free encyclopedia. This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: There are an excessive number of references to non-notable research articles. Only original notable and significant contributions should be included.

Please help improve this article if you can. March Learn how and when to remove this template message. Paradox in game theory. S2CID Scientific Reports. Bibcode : NatSR PMC PMID Bibcode : PNAS Minor, "Parrondo's Paradox - Hope for Losers! Statistical Science. Harmer, D. Abbott , P. Taylor, and J. Parrondo , in Proc. Unsolved Problems of Noise and Fluctuations , D. Abbott , and L. Kish , eds. Proceedings of the Royal Society of London A. Taylor, C.

Pearce and J. Parrondo, Information entropy and Parrondo's discrete-time ratchet , in Proc. McClintock , ed. Philips and Andrew B. Retrieved 28 August Journal of Theoretical Biology. Bibcode : JThBi. ISSN ISSN X. Proceedings of the National Academy of Sciences. Fluctuation and Noise Letters. Bibcode : FNL The Mathematical Scientist. The University of Adelaide. Archived from the original on 21 June Well-known paradoxes. Theseus' ship List of examples Sorites.

Petersburg Thrift Toil Tullock Value. List Category. Topics in game theory. Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Game description language Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game.

Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand.

Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Purification theorem Revelation principle Zermelo's theorem. There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately.

There are 3 problems with this Parrondo's paradox concerbning forex: 1. Market is changed all the time so it can not be just one unchanged rule for "this game" 2. Any combination of A and B can be separated system 3. Price is not randomly moved on the chart.

It may work with winning systems. For example - I change manual systems all the time You agree to website policy and terms of use. What can be done to make a very highly unprofitable EA Profitable? New comment. Ruben Couto. The best thing is to try out! Pawel Wojnarowski. This topic is very helpful. Thanks lot. Enigma71fx : This statement is extremely far from truth.

By the way, if it was so simple just by reverting a failing strategy , why aren't everyone in the world using such method? Oluwadare Paul Oguntosin. For a lossing EA, Martingale is Evil?! Do you have any. Sergey Golubev. Hi forexgoshen , The situation with reversal is more complicated.

There is spread 2. We will receive completely different system - it is RSI using on breakout way: 4. Parrondo's paradox? Valerii Mazurenko. I don't remember, who and when said, that unprofitable EA could stay profitable using simple reverse, if average loss more then two spread.