Zubov's method

From formulasearchengine
Jump to navigation Jump to search

Parrondo's paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more explanatory description is:

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.

Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman. However, the paradox disappears when rigorously analyzed.

Illustrative examples

The saw-tooth example

File:Parrandos Paradox Unbiased Games.PNG
Figure 1

Consider an example in which there are two points A and B having the same altitude, as shown in Figure 1. In the first case, we have a flat profile connecting them. Here, if we leave some round marbles in the middle that move back and forth in a random fashion, they will roll around randomly but towards both ends with an equal probability. Now consider the second case where we have a saw-tooth-like region between them. Here also, the marbles will roll towards either ends with equal probability (if there were a tendency to move in one direction, marbles in a ring of this shape would tend to spontaneously extract thermal energy to revolve, violating the second law of thermodynamics). Now if we tilt the whole profile towards the right, as shown in Figure 2, it is quite clear that both these cases will become biased towards B.

Now consider the game in which we alternate the two profiles while judiciously choosing the time between alternating from one profile to the other.

File:Parrandos Paradox Biased Games.PNG
Figure 2

When we leave a few marbles on the first profile at point E, they distribute themselves on the plane showing preferential movements towards point B. However, if we apply the second profile when some of the marbles have crossed the point C, but none have crossed point D, we will end up having most marbles back at point E (where we started from initially) but some also in the valley towards point A given sufficient time for the marbles to roll to the valley. Then we again apply the first profile and repeat the steps (points C, D and E now shifted one step to refer to the final valley closest to A). If no marbles cross point C before the first marble crosses point D, we must apply the second profile shortly before the first marble crosses point D, to start over.

It easily follows that eventually we will have marbles at point A, but none at point B. Hence for a problem defined with having marbles at point A being a win and having marbles at point B a loss, we clearly win by playing two losing games.

The coin-tossing example

A second example of Parrondo's paradox is drawn from the field of gambling. Consider playing two games, Game A and Game B with the following rules. For convenience, define Ct to be our capital at time t, immediately before we play a game.

  1. Winning a game earns us $1 and losing requires us to surrender $1. It follows that Ct+1=Ct+1 if we win at step t and Ct+1=Ct−1 if we lose at step t.
  2. In Game A, we toss a biased coin, Coin 1, with probability of winning P1=(1/2)−ϵ. If ϵ>0, this is clearly a losing game in the long run.
  3. In Game B, we first determine if our capital is a multiple of some integer M. If it is, we toss a biased coin, Coin 2, with probability of winning P2=(1/10)−ϵ. If it is not, we toss another biased coin, Coin 3, with probability of winning P3=(3/4)−ϵ. The role of modulo M provides the periodicity as in the ratchet teeth.

It is clear that by playing Game A, we will almost surely lose in the long run. Harmer and Abbott[1] show via simulation that if M=3 and ϵ=0.005, Game B is an almost surely losing game as well. In fact, Game B is a Markov chain, and an analysis of its state transition matrix (again with M=3) shows that the steady state probability of using coin 2 is 0.3836, and that of using coin 3 is 0.6164.[2] As coin 2 is selected nearly 40% of the time, it has a disproportionate influence on the payoff from Game B, and results in it being a losing game.

However, when these two losing games are played in some alternating sequence - e.g. two games of A followed by two games of B (AABBAABB...), the combination of the two games is, paradoxically, a winning game. Not all alternating sequences of A and B result in winning games. For example, one game of A followed by one game of B (ABABAB...) is a losing game, while one game of A followed by two games of B (ABBABB...) is a winning game. This coin-tossing example has become the canonical illustration of Parrondo's paradox – two games, both losing when played individually, become a winning game when played in a particular alternating sequence. The apparent paradox has been explained using a number of sophisticated approaches, including Markov chains,[3] flashing ratchets,[4] Simulated Annealing[5] and information theory.[6] One way to explain the apparent paradox is as follows:

  • While Game B is a losing game under the probability distribution that results for Ct modulo M when it is played individually (Ct modulo M is the remainder when Ct is divided by M), it can be a winning game under other distributions, as there is at least one state in which its expectation is positive.
  • As the distribution of outcomes of Game B depend on the player's capital, the two games cannot be independent. If they were, playing them in any sequence would lose as well.

The role of M now comes into sharp focus. It serves solely to induce a dependence between Games A and B, so that a player is more likely to enter states in which Game B has a positive expectation, allowing it to overcome the losses from Game A. With this understanding, the paradox resolves itself: The individual games are losing only under a distribution that differs from that which is actually encountered when playing the compound game. In summary, Parrondo's paradox is an example of how dependence can wreak havoc with probabilistic computations made under a naive assumption of independence. A more detailed exposition of this point, along with several related examples, can be found in Philips and Feldman.[7]

A simplified example

For a simpler example of how and why the paradox works, again consider two games Game A and Game B, this time with the following rules:

  1. In Game A, you simply lose $1 every time you play.
  2. In Game B, you count how much money you have left. If it is an even number, you win $3. Otherwise you lose $5.

Say you begin with $100 in your pocket. If you start playing Game A exclusively, you will obviously lose all your money in 100 rounds. Similarly, if you decide to play Game B exclusively, you will also lose all your money in 100 rounds.

However, consider playing the games alternatively, starting with Game B, followed by A, then by B, and so on (BABABA...). It should be easy to see that you will steadily earn a total of $2 for every two games.

Thus, even though each game is a losing proposition if played alone, because the results of Game B are affected by Game A, the sequence in which the games are played can affect how often Game B earns you money, and subsequently the result is different from the case where either game is played by itself.

Application

Parrondo's paradox is used extensively in game theory, and its application in engineering, population dynamics,[8] financial risk, etc., are also being looked into as demonstrated by the reading lists below. Parrondo's games are of little practical use such as for investing in stock markets[9] as the original games require the payoff from at least one of the interacting games to depend on the player's capital. However, the games need not be restricted to their original form and work continues in generalizing the phenomenon. Similarities to volatility pumping and the two-envelope problem[10] have been pointed out. Simple finance textbook models of security returns have been used to prove that individual investments with negative median long-term returns may be easily combined into diversified portfolios with positive median long-term returns.[11] Similarly, a model that is often used to illustrate optimal betting rules has been used to prove that splitting bets between multiple games can turn a negative median long-term return into a positive one.[12]

Name

In the early literature on Parrondo's paradox, it was debated whether the word 'paradox' is an appropriate description given that the Parrondo effect can be understood in mathematical terms. The 'paradoxical' effect can be mathematically explained in terms of a convex linear combination.

However, Derek Abbott, a leading Parrondo's paradox researcher provides the following answer regarding the use of the word 'paradox' in this context: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

Parrondo's paradox does not seem that paradoxical if one notes that it is actually a combination of three simple games: two of which have losing probabilities and one of which has a high probability of winning. To suggest that one can create a winning strategy with three such games is neither counterintuitive nor paradoxical.

See also

Further reading

References

  1. ↑ G. P. Harmer and D. Abbott, "Losing strategies can win by Parrondo's paradox", Nature 402 (1999), 864
  2. ↑ D. Minor, "Parrondo's Paradox - Hope for Losers!", The College Mathematics Journal 34(1) (2003) 15-20
  3. ↑ G. P. Harmer and D. Abbott, "Parrondo's paradox", Statistical Science 14 (1999) 206-213
  4. ↑ G. P. Harmer, D. Abbott, P. G. Taylor, and J. M. R. Parrondo, in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations, D. Abbott, and L. B. Kish, eds., American Institute of Physics, 2000
  5. ↑ G. P. Harmer, D. Abbott, and P. G. Taylor, The Paradox of Parrondo's games, Proc. Royal Society of London A 456 (2000), 1-13
  6. ↑ G. P. Harmer, D. Abbott, P. G. Taylor, C. E. M. Pearce and J. M. R. Parrondo, Information entropy and Parrondo's discrete-time ratchet, in Proc. Stochastic and Chaotic Dynamics in the Lakes, Ambleside, U.K., P. V. E. McClintock, ed., American Institute of Physics, 2000
  7. ↑ Thomas K. Philips and Andrew B. Feldman, Parrondo's Paradox is not Paradoxical, Social Science Research Network (SSRN) Working Papers, August 2004
  8. ↑ V. A. A. Jansen and J. Yoshimura "Populations can persist in an environment consisting of sink habitats only". Proceedings of the National Academy of Sciences USA, 95(1998), 3696-3698 .
  9. ↑ R. Iyengar and R. Kohli, "Why Parrondo's paradox is irrelevant for utility theory, stock buying, and the emergence of life," Complexity, 9(1), pp. 23-27, 2004
  10. ↑ Winning While Losing: New Strategy Solves'Two-Envelope' Paradox at Physorg.com
  11. ↑ M. Stutzer, The Paradox of Diversification, The Journal of Investing, Vol. 19, No.1, 2010.
  12. ↑ M. Stutzer, "A Simple Parrondo Paradox", Mathematical Scientist, V.35, 2010.

Earlier than you decide whether stainless steel cookware is worth buying, lets first discuss what stainless steel cookware is. Chrome steel is manufactured from an alloy, or a mix of metals. Mostly, basic iron with chromium, nickel or another minor metals. The chromium supplies rust safety and gives your cookware durability. The nickel supplies rust safety as effectively, and adds a elegant look. Most properly made chrome steel cookware has copper or aluminum added to the bottom of the pan or pot. That is performed to increases the ability of the pot or pan to conduct heat.
The perfect stainless-steel cookware is the principle category, but nonetheless it is divided into several subcategories primarily based on the standard and the value range. It can be confusing to choose the best chrome steel cookware out of the classes that may meet your necessities. That is where we took a step ahead to explain you all the data that will be useful for you to understand how to choose one of the best stainless steel cookware. The most effective chrome steel cookware set is manufactured from cheap to expensive and high quality constructed pots and pans.
You can find magnetic chrome steel in the layer on the outside of some quality pieces of stainless steel. This is to make it suitable with induction stovetops, which involve the use of a quickly charging electromagnetic area to warmth cookware. Excessive-quality stainless-steel, like All-Clad , makes use of three layers of steel—the austenite layer of metal on the within, ferrite steel on the skin, and a layer of aluminum sandwiched between the 2 for optimum warmth conductivity (metal alone does not conduct heat evenly). Lesser-quality stainless steel is usually just one layer of austenitic chrome steel.
Aesthetically talking, stainless steel is a smart alternative should you prefer to display or hang pots or pans. The clean, crisp look of all stainless-steel kitchenware can remodel a mishmash of cookware into a classy décor assertion. Chrome steel kettles, such as the Cuisinart Tea Kettle will mix particular person kitchenware into a cohesive and nice entity. Take into account purchasing stainless-steel utensils as well. Already acquired a stunning stainless-steel cookware collection? The Cuisinart Chef’s Assortment stainless pot rack is perhaps the of completion for a kitchen, liberating up house and making those pots and pans readily accessible. Get the stainless-steel cookware of your culinary desires at Macy’s!
Arduous-anodized aluminum cookware is likely one of the hottest varieties of materials, despite the fact that many people don't quite perceive the construction. Onerous-anodized aluminum is plain aluminum that has been processed in a collection of chemical baths charged with an electric present. The result's a fabric that has the same superior heat conductivity as aluminum but is non-reactive with acidic foods, such as tomatoes, and twice as exhausting as stainless-steel. Two drawbacks to laborious-anodized cookware are that it is not dishwasher-protected and, because it is not magnetic, it is not going to work with induction vary tops.
The enamel over steel technique creates a bit that has the warmth distribution of carbon steel and a non-reactive, low-stick floor. Such pots are much lighter than most different pots of similar measurement, are cheaper to make than stainless-steel pots, and don't have the rust and reactivity problems with cast iron or carbon steel. citation wanted Enamel over metal is ideal for big stockpots and for other giant pans used largely for water-based cooking. Because of its mild weight and simple cleanup, enamel over metal is also common for cookware used while camping. For more about stainless steel cookware reviews look at our web site. Clad aluminium or copper edit
Distinctive specialty cookware pieces served a la carte to go with any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior end. Designed with an impact bonded, aluminum disk encapsulated base which distributes warmth quickly and evenly to allow precise temperature control. Handles are riveted for durability and performance. The New Specialty Cookware is compatible for all range sorts including induction. Along with the multi use operate, another distinctive characteristic is bottom to prime interior volume markings in both quarts and metric measurement; and every bit comes with a tempered glass lid, oven safe to 350°F.
Whether or not you're a cooking enthusiasts, knowledgeable chef or just cooking for your family you already know the importance of having a totally stocked kitchen. Not solely do you need the appropriate substances, however you additionally need the proper tools to get the job done. In any sort of fundamental cooking training lesson, you will learn that stainless steel is your new greatest friend with regards to kitchen cookware. What you will also study is that high quality cooking tools does not often come at a discounted price. Because of this, it is very important take good care of your cookware! Listed below are some fundamentals for chrome steel care.
To fight the uneven heating downside, most stainless steel pans are laminations of aluminum or copper on the underside to spread the heat around, and stainless steel inside the pan to provide a cooking floor that is impervious to no matter you would possibly put inside. In my expertise, this stainless-steel floor continues to be too sticky to fry on, and should you ever burn it you get a everlasting bother spot. However, sometimes a stainless steel cooking floor turns out to be useful when you can't use aluminum (see under) so I maintain some around. Choose one thing with a fairly thick aluminum layer on the underside.
Effectively, until you’re a metals professional and go inspect the factory where the steel is made to see whether or not their manufacturing process creates a pure austenite with out corrosive materials fashioned, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the best quality. I believe your finest guess is to easily purchase excessive-quality stainless-steel from the beginning, from a brand with a status for good quality. But, I think I have discovered one way which you could determine if the stainless cookware you have already got is doubtlessly reactive.

Retrieved from "https://en.formulasearchengine.com/w/index.php?title=Zubov%27s_method&oldid=12974"