The Bertrand paradox is a problem in the classical interpretation of probability theory. Joseph introduced it in his work Calcul des probabilités (1889) as an example that probabilities cannot be clearly defined if a mechanism or method produces a random variable.
Formulation of the problem
Bertrand's paradox is as follows.
First you need to consider an equilateral triangle inscribed in a circle. The diameter is randomly selected. What is the likelihood that it is longer than the side of the triangle?
Bertrand made three arguments, all of which are apparently true, but yielded different results.
Random Endpoints Method
You need to select two places on the circle and draw an arc connecting them. For the calculation, the Bertrand probability paradox is considered. It is necessary to imagine that the triangle is rotated so that its vertex coincides with one of the end points of the chord. It is worth paying attention that if the other part is on an arc between two places, the circumference is longer than the side of the triangle. The length of the arc is one third of the circle, so the probability that the random chord is longer is 1/3.
Selection method
You must select the radius of the circle and a point on it. After that, you need to build a chord through this place, perpendicular to the diameter. In order to calculate the Bertrand paradox of probability theory under consideration, it is necessary to imagine that the triangle is rotated so that the side is perpendicular to the radius. The chord is longer than the leg, if the selected point is closer to the center of the circle. And in this case, the side of the triangle bisects the radius. Therefore, the probability that the chord is longer than the side of the inscribed figure is 1/2.
Random chords
Midpoint method. You must select a place on the circle and create a chord with a given middle. The axis is longer than the edge of the inscribed triangle, if the selected place is within a concentric circle of radius 1/2. The area of ââthe smaller circle is one fourth of the larger figure. Therefore, the probability of a random chord is longer than the side of the inscribed triangle, and equal to 1/4.
As presented above, the selection methods differ in the weight they give to certain chords, which are diameters. In method 1, each chord can be selected in exactly one way, regardless of whether it is a diameter.
In method 2, each straight line can be selected in two ways. While any other chord will be selected only one of the possibilities.
In method 3, each midpoint selection has a single parameter. Except for the center of the circle, which is the middle of all diameters. These problems can be avoided by âorderingâ all issues in order to exclude parameters without affecting the resulting probabilities.
Selection methods can also be visualized as follows. A chord that is not a diameter is uniquely identified by its midpoint. Each of the three selection methods presented above gives a different distribution of the middle. And options 1 and 2 provide two different heterogeneous separations, while method 3 gives an even distribution.
The classic paradox of solving the Bertrand problem depends on the method by which the chord is chosen âat randomâ. It turns out that if a random selection method is specified in advance, the problem has a clearly defined solution. This is due to the fact that each individual method has its own distribution of chords. The three judgments demonstrated by Bertrand correspond to different selection methods, and in the absence of additional information there is no reason to give preference to one over the other. Accordingly, the claimed problem does not have a unique solution.
An example of how to make the general answer unique is to indicate that the endpoints of the chord are evenly distributed between 0 and c, where c is the circle circumference. This distribution is the same as in Bertrand's first argument, and the resulting unique probability will be 1/3.
This paradox of Bertrand Russell and other uniqueness of the classical interpretation of the possibility justify more rigorous formulations. Including probability frequency and subjective Bayesian theory.
What is the basis of the Bertrand paradox
In his 1973 article, A Well-posed Problem, Edwin Jaynes proposed his own unique solution. He noted that Bertrandâs paradox is based on a premise based on the principle of âmaximum ignorance.â This means that you should not use any information that is not provided in the wording of the problems. Janes pointed out that Bertrand's task does not determine the position or size of the circle. And he argued that therefore any definite and objective decision should be âindifferentâ to size and position.
For illustration
It is worth assuming that all chords are randomly placed on a circle with a diameter of 2 centimeters, now it is necessary to throw straws into it from afar.
Then you need to take another circle with a smaller diameter (for example, 1 centimeter), which fits into a large figure. Then the distribution of chords on this smaller circle should be the same as on the maximum. If the second figure also moves inside the first, the probability, in principle, should not change. It is very easy to see that the following change will occur for method 3: the distribution of chords on a small red circle will be qualitatively different from the division on a large circle.
The same thing happens for method 1. Although this is more difficult to see in the graphical representation.
Method 2 is the only one that turns out to be both a large-scale and translational invariant.
Method number 3 seems simply extensible.
But method 1 is neither one nor the other.
Nevertheless, Janes did not easily use invariants to accept or reject these methods. This would leave the possibility that there is another indescribable method that is consistent with its aspects of reasonable value. Janes applied integral equations describing invariance. To directly determine the probability distribution. In his problem, the integral equations really have a unique solution, and this is exactly what was called the second method of random radius above.
In a 2015 article, Alon Drori argues that the Janes principle can also provide two other Bertrand solutions. The author assures that the mathematical implementation of the above-mentioned invariance properties is not unique, but depends on the basic random selection procedure that the person decided to use. He shows that each of Bertrand's three solutions can be obtained using rotational, scaling, and translational invariance. At the same time concluding that the principle of Jaynes is just as subject to interpretation as the method of indifference itself.
Physical experiments
Method 2 is the only solution that satisfies the invariants of transformations that are present in specific physiological concepts, such as statistical mechanics and gas structure. And also in the experiment proposed by Janes on throwing straws from a small circle.
Nevertheless, other practical experiments can be developed that give answers in accordance with other methods. For example, in order to arrive at a solution to the first random endpoint method, you can attach a counter to the center of the area. And let the results of two independent rotations highlight the final places of the chord. In order to arrive at a solution to the third method, you can cover the circle with molasses, for example, and mark the first point on which the fly lands as the middle chord. Several contemplators created research to extract different conclusions and confirmed the results empirically.
Recent events
In his 2007 article, Bertrand's Paradox and the Principle of Indifference, Nicholas Schackel argues that after more than a century the task is still unresolved. She continues to refute the principle of indifference. In addition, in his 2013 article, âBertrand Russellâs Paradox Revised: Why All Decisions Are Not Applicable in Practice,â Darrell R. Robot shows that all the proposed resolutions are in no way related to his own question. So it turned out that the paradox would be much more difficult to solve than previously thought.
Shakel emphasizes that so far many scientists and people far from science have tried to resolve the Bertrand paradox. He is still overcome with the help of two different approaches.
Those in which the difference between nonequivalent problems was considered, and those in which the task was always considered correct. Shakel cites Louis Marinoff (as a typical representative of the demarcation strategy) and Edwin Janes (as the author of a well-thought-out theory) in his books.
Nevertheless, in a recent work, âSolving a Complex Problem,â Dideric Aerts and Massimiliano Sassoli de Bianchi believe that in order to solve Bertrandâs paradox, premises must be sought in a mixed strategy. According to these authors, first you need to fix the problem by clearly indicating the nature of the entity that is being randomized. And only after this is done, can any task be considered correct. That is exactly what Janes thinks.
So the principle of maximum ignorance can be used to solve it. For this purpose, and since the problem does not determine how a chord should be selected, the principle is applied not at the level of various possible options, but at a much deeper one.
Parts selection
This part of the problem requires the calculation of the meta-average for all possible methods, which the authors call the universal average. To deal with this, they use the sampling method. Inspired by what is being done in defining the law of probability in Wiener processes. Their result is consistent with the numerical consequence of Jaynes, although their well-posed task differs from the original, author's.
In economics and commerce, the Bertrand paradox, named after its creator Joseph Bertrand, describes a situation in which two players (firms) achieve a Nash equilibrium state. When both enterprises set a price equal to marginal cost (MS).
Bertrandâs paradox is based on a premise. It lies in the fact that in models such as Cournot competition, an increase in the number of firms is associated with the convergence of prices with marginal costs. In these alternative models, the Bertrand paradox is in the oligopoly of a small number of firms that make a positive profit, charging prices above cost.
To begin with, it is worth assuming that two firms A and B sell homogeneous goods, each of which has the same cost of production and distribution. It follows that buyers choose a product solely on the basis of price. And this means that demand is infinitely flexible in value. Neither A nor B will set a higher price than the others, because this will lead to the fact that the whole Bertrand paradox will collapse. One of the market participants will yield to its competitor. If they set the same price, companies will share the profit.
On the other hand, if any company at least slightly reduces its price, it will receive the entire market and significantly greater returns. Since A and B are aware of this, each of them will try to undercut the competitor until the product is sold at zero economic profit.
Recent work has shown that there may be an additional equilibrium in the Bertrand paradox with a mixed strategy, with a positive economic profit, provided that the monopoly amount is infinite. For the case of final profit, it was shown that a positive increase in conditions of price competition is impossible in mixed equilibria and even in the more general case of correlated systems.
In fact, the Bertrand paradox in economics is rarely found in practice, because real products almost always differentiate in some other way than the price (for example, overpayment for a label). Firms have restrictions on their production and distribution capabilities. That is why two enterprises rarely have the same costs.
Bertrand's result is paradoxical, because if the number of firms increases from one to two, the price decreases from monopolistic to competitive and remains at the same level as the number of enterprises increasing in the future. This is not very realistic, because in reality markets with a small number of firms with bargaining power usually set prices that exceed marginal costs. Empirical analysis shows that in most industries with two competitors, positive returns are obtained.
In the modern world, scientists are trying to find solutions to the paradox that are more consistent with the Cournot competition model. Where two companies in the market make a positive profit, which lies somewhere between completely competitive and monopoly levels.
Some reasons why the Bertrand paradox is not directly related to the economy:
- Capacity limitations. Sometimes firms do not have sufficient capacity to satisfy all demand. This moment was first raised by Francis Edgeworth and gave rise to the Bertrand-Edgeworth model.
- Integer prices. Prices higher than MC are excluded because one firm can cut another for an arbitrarily small amount. If prices are discrete (for example, must take integer values), then one firm must trim another by at least one ruble. This implies that the value of the small currency is higher than MC. If another company sets the price for it higher, another company can lower it and capture the entire market, Bertrandâs paradox is precisely this. This will not bring her any profit. This company will prefer to share 50/50 sales with another company and receive purely positive revenue.
- Product differentiation. If the products of different firms differ from each other, then consumers may not fully switch to products with a lower price.
- Dynamic competition. Repeated interaction or repeated price competition can lead to a balance of value.
- More products for a higher amount. This follows from repeated interaction. If one company sets its price a little higher, it will still receive approximately the same number of purchases, but a large profit for each product. Therefore, another company will increase its margin, etc. (Only in repeat games, otherwise the dynamics are going in the other direction).
Oligopoly
If two companies can agree on a price, then it is in their long-term interests to keep the agreement: the income from the reduction in value is less than twice the revenue from compliance with the agreement and lasts only until the other company reduces its own prices.
Probability theory (like the rest of mathematics) is actually a recent invention. And the development was not smooth. The first attempts to formalize the calculus of probability were made by the Marquis de Laplace, who proposed defining the concept as the ratio of the number of events leading to the outcome.
This, of course, only makes sense if the number of all possible activities is finite. And, in addition, all events are equally probable.
Thus, at that time, these concepts did not seem to have a solid foundation. Attempts to extend the definition to the case of an infinite number of events have led to even greater difficulties. The Bertrand paradox is one such discovery that made mathematicians wary of the whole concept of probability.