Optimal mechanism for the recognition of the solution of the jealousy. Types of peer-to-peer: peer-to-peer according to Nash, Shtekelberg, Pareto-optimal peer-to-peer, peer-to-peer strategies
Alternately, in a single graph, the lines will drink those propositions, omitting the graphical representation of the equality in coordinates P, Q(Figure 2.6). Cross point line maj coordinates (P*, Q*), de R* - equal price, Q*- equally important obsyag of virobnitstva that spozhivannya.
Rinkova jealous- tse such a camp of the market, for which, for the same price, I obligated to drink a more expensive proposition.
Less at the point of rіvnovaga E market balancing, no market agents have no incentives to change the situation. Tse means that the market is equal in power durability - in times of guilt of an unimportant turn to the market, agents are motivated to turn to the market for equality. To confirm the stability, call on the logic of L. Walras and A. Marshall.
According to L. Walras, for the sake of high prices, too much proposition is blamed - superficial A-B in fig. 2.6i), such a market is called buy market, oskіlki kupets maє mozhlivіst pіd o'clock stowage favors vimagati price reduction. In such a situation, we are not in front of the seller, who is embarrassed to lower the prices and quickly reduce the price. The world has a reduction in prices, it will drink more, vodrіzok A-B quickly, the docks become a point of jealousy Є.
For low prices, there is an excess of drinking - a shortage seller's market. Buyer zmushuє-
den fast savings and overpaying for a scarce product, for increased prices of growth obsyag propositions, shortages shorten, docks of the market come to the river.
According to A. Marshall (small. 2.66), with small commitments, the price will change the price of the seller, for larger commitments - navpaki. In any situation, the imbalance is stimulated by the adoption of prices, or I will oblige to drink that proposition from the bik of jealousy. Equity (a) according to Walras - the price regulates the imbalance of obsyagiv drink and propositions, (b) according to Marshall - the prices of the purchase and the seller are changed.
Rice. 2.6. Establishment of the market price: c) for L. Walras; b) for A. Marshall
Change the market demand for chi propositions to lead to a change in the level (Fig. 2.7). If, for example, the drink of the drink grows, then the line of the drink is right-handed, even though the price is equal to the increase. As the market proposition changes, the proposition line shifts to the left, which leads to an increase in the price of that fast contract.
Tsya model of the market is static;
"Pavutin-like" model
As an example of a dynamic model of market equivalence, we will induce the simplest “pavutin-like” model. Admissible, obsyag drink to fall in the same price of the flow period t, and general propositions - in the price of the previous period t-1:
Q d i = Q d i (P t) , Q s i = Q s i (P t -1) ,
de t = 0.1 .... T-discrete value of the time period.
Rice. 2.7. Change of market rіvnovagi:
a) as a result of increased drinking; b) after the change
propositions
Rink price P t may not match with an equally important price R *, moreover, there are three options for dynamics P t(Figure 2.8).
The variant of the trajectory of the development in this model is to lie in the spivvіdnoshennia nahilіv liniy drink and propositions.
Rice. 2.8. "Pavutinopodіbna" model of the market rivnovaga:
a) changes are being made; 5) respite
vіd rіvnovagi zbіshuєtsya ("catastrophe" model); c) market
cyclically kolivaetsya around the point of equal, ale equal
Vivchimo mechanizm of the establishment of the market rіvnovagi, if under the influence of the change of chinniki in the drink of propositions the market will come out of this camp. Іsnuyut two main variant of disproporcії mіzh drink and propozitsієyu: surplus and shortage of goods.
too much(superfluous) product - tse taka situation above the market, if the value of the proposition of the product in cіy cіnі exceeds the value of the drink of the new one. In the times between virobniks, competition, struggle for buyers is blamed. The vigrashі overflows with that, hto proponuє vigіdnіshі umovi realіzacії goods. In this rank, the market will not turn into a jealous camp.
Deficit to the goods - at the time I will drink the goods at the cіy cіnі perevishuє proponovane kіlkіst of the product. This situation is to blame for the increase in purchases due to the possibility of supplying a scarce product. Remagaet the one who propagates a high price for the entire product. The price has grown up schodo nyogo respect virobnikіv, yakі pochinayut rozshiryuvati virobnitstvo, zbіlshyuchi cim propozitsіyu goods. As a result, the system rotates at the rank of equal.
Vykhodyachi z usogo above what has been said, we come to the visnovka, sho cіna realіzuє funktsіyu, scho vrіvnovazhuє, stimuliyuyuchi expansion of the range of vіrobnitstvа і propositionії product with deficitі і strimuyuchi propositionіyu, pozbavlyuchi rynok vіd nadlishkіv.
Vrіvnovazhuє role tsіni awake yak through drink, and through proposition.
It seems that, due to the fact that it was equal, as it stood on our market, the bula was destroyed - under the influence of any factors (for example, an increase in income) led to an increase in drink, as a result of this, the curve was reversed D1 in D2(Fig. 4.3 a), and the proposition was left unchanged.
If the price of this product did not change immediately after the change of the crooked drink, then the next step for the increase in drink is the situation, if for the front price P1 kіlkіst merchandise, like leather z pokuptsіv now can pridbati (QD) revisit that obsyag, which can be propagated at a given price product (QS). The value of drink is now revisiting the value of the proposition of a given product, which means a vindication shortage of goods at rosemary Df = QD - Qs on this market.
Deficiency of goods, as we know, to bring to competition mіzh purchases for mozhlivіst pridbannya cgogo goods, scho to bring up to the growth of market price cіn. At zv'yazku z the law propozitsії response prodavtsіv pіdvishchennya cіni bude zbіlshennya obyagu zaproponovanogo goods. On the graph of which will be shown the transfer of the points of the market equalization E1 vzdovzh crooked propositions to її peretina with a new crooked drink D2 de i will reach a new level of this market E2 h equally important number of goods Q2 at an equal price R2.
Rice. 4.3. Usunennya points of equal value.
Let's see the situation, if the equal camp will be destroyed on the side of the river.
It seems that due to the fact that, under the influence of some factors, there was an increase in propositions, as a result, the curve shifted to the right from the position S1 in S2 and the drink is filled with the inevitable (Fig. 4.3 b).
For washing the savings of the market price on a large equal (R1) increase the propositions to too much goods at rosemіrі Sp = Qs - QD. As a result, blame seller competition, what to bring down the market price (s P1 before P2)і srostannya obsjagu goods that are sold. On the graph of which, the moving points of the market level will be displayed E1 vzdovzh crooked drink to її peretina with a new crooked proposition, which will lead to the establishment of a new level E2 with parameters Q2і R2.
Similarly, it is possible to influence the equal price and the equal number of goods to change the drink and change the proposition.
In the primary literature, the rules for mutual intercourse and propositions are formulated.
The increase in the drink is a result of the increase in the equal price and the equal quantity of the goods.
The change in drink is a result of the fall and the equal price and the equal number of goods.
An increase in propositions attracts a change in an equally important price and an increase in an equally important number of goods.
Short propositions attract an increase in the equal price and a change in the equal number of goods.
Varto say - koristuyuchis tsimi rules, you can know an equally important point for any changes drink that proposition.
The turn of the price to the market equal level can basically be changed in the following way:
administrative regulation of prices;
monopoly the virobnik, otherwise, it’s slower, which allows you to reduce the monopoly price, as if it were a piece-dependant, so it’s underestimated.
The optimal strategies of the theory of conflicts are considered to be such strategies, as to bring gravity to stable equals, tobto. such situations that please all the gravity.
The optimality of the variant of the theory of igor is based on the understanding equally important situation:
1) it is not obvious to any one of the graves in an equally important situation, as all other things are lost to them,
2) sense of jealousy - in case of bugoriasis recurrence, gravitation to look at the situation of jealousy, raising the risk of being in a strategic situation.
In skin interactions, they can also be seen equally:
1. jealous in defensive strategies . Viznaєtsya strategies, yakі zabezpechuyut engraving guarantees the result;
2. jealous at dominant strategies .
Dominating strategy such a plan is called, which ensures the participant the maximum win regardless of the other. Therefore, the equal dominating strategies will be overthrown by the dominating strategies of both participants.
If the optimal strategies of gravity dominate over other strategies, then we can be equal in dominant strategies. At grі "dilemma uz'yaznenih" equally important for Nesh set of strategies will be ("vyznavat - vyznavati"). Moreover, it is important to note that for grave A, so for grave B, “to recognize” is a dominating strategy, just like “not to recognize” - dominated;
3. jealous Nesha . Jealous Nesha the type of solution is called two and more gravity, for which the participant is unable to win more money, changing his decision in a unilateral order, if other participants do not change the decision.
Permissible - gra n osіb in normal form, de - collection of pure strategies, and - collection of winnings.
If skin gravel chooses a strategy from the profile of strategies, gravel will win. Moreover, you win to lay a deposit in the light of the strategy’s profile: not only in the case of the strategy developed by the Grave himself, but also in the form of other people’s strategies. The profile of the strategy is equal according to Nash, as the change of one’s strategy is not visible to the same gravity, so for whatever
Grass can mother jealous of Nesha and in pure strategies, and zmishanih.
Nesh doviv, what should I allow mixed strategies tody in skin gr n Gravtsіv would want only one jealous Nesha.
In the situation, equal to Neshu, the strategy of skin engraving is the best for the strategy of smaller engravings;
4. Rivnavaga Stackelberg. Stackelberg model- Game-theoretic model of an oligopolistic market for the obviousness of informational asymmetry. In this model, the behavior of firms is described in a dynamic manner with new and thorough information, in which the behavior of firms is modeled for help static play with new information. The main feature of the gr є presence of the leading firms, as the first to establish the release of goods, other firms are oriented at their own retailers. Main change of mind:
· Galuzy viroblyaє odnorodny goods: vіdmіnnostі produktії ії raznih іznih znimatіlі іnіmіlі, otzhe, kupetets pіd hіborіv іv, yak іrmі kupuvati, orientiruetsya selієntuієsі tsіnu;
· Galusi dіє has a small number of firms;
· Firms install the quantity of manufactured products, and the price for it is determined according to the drink;
· This is the name of the leading firm, and other firms are guided by the choice of the company.
In this rank, the Stackelberg model is victorious for the significance of the optimal solution in dynamic games and influencing the maximum gravity gain, out of minds, which were formed after the choice of one or more gravity. Rivnovagi according to Stackelberg.- a situation, if you can’t win your wins unilaterally, and decisions are taken on the back of one grave and become led to another grave. In grі "the dilemma of the aggravated" the jealousy of Shtakelberg will be reached in the square (1; 1) - "reveal the guilt" of obm with evil deeds;
5. Pareto optimality- such a camp of the system, according to the meaning of the skin private criterion, describes the camp of the system, it can be completed without lowering the camp of other graves.
Pareto's principle to say this: "Be it a change, like not to bring zbitkiv, but like to bring misfortune to some people (for their own assessment of power), є polypshennyam." In this rank, the right to all changes is recognized, so as not to give anyone a dodatkovo shkodi.
Bagato becoming systems that are Pareto optimal is called the “Pareto multiple”, “the multiple of alternatives that are optimal for the Pareto sense”, or “the multiple of optimal alternatives”.
The situation, if Pareto efficiency is achieved, is the situation, if everything is possible in the exchange of vices.
Efficiency for Pareto is one of the central ones to understand for modern economics. On the basis of this understanding, the first and the other fundamental theorems will be good.
One of the additions to Pareto-optimality is Pareto-allocation of resources (labor resources and capital) for international economic integration, tobto. economic union of two and more countries. Tsikavo, that Pareto-rozpodіl until that time of international economic integration was adequately described mathematically (Dalimov R.T., 2008). Analіz showing scho Dolan vartіst sektorіv that revenues of the Labor resursіv ruhayutsya protispryamovano vіdpovіdno to good vіdomogo rіvnyannya teploprovіdnostі analogіchno Gas Åbo rіdini in prostorі scho daє mozhlivіst zastosuvati technique analіzu scho vikoristovuєtsya in fіzitsі, schodo ekonomіchnih zavdan of mіgratsії ekonomіchnih parametrіv.
Pareto Optimum it seems that the good supply of the economy reaches the maximum, and the distribution of the resources becomes optimal, as if the change of which one rose the good will be good if you want one subject of the economic system.
Pareto-optimal market stance- Situation, if it is not possible to expand the camp of a participant in the economic process, at the same time not lowering the goodness of at least one of the others.
Based on the Pareto criterion (criterion for the growth of a suspenseful kindness), the rush at the optimum is less possible for such a distribution of resources, which is the best way to do good by hiring one person, not having a problem with anyone else.
To say that Situation S* dominates Pareto Situation S, i.e.:
for some grave yoga wingrash y S<=S*
· If you want one grave, for some reason you can win the situation S*>S
The task of the "dilemma of ties" is equal to Pareto, if the station of the same gravity is polypshit, without losing the station of the other one, it is not possible, depending on the situation of the square (2; 2).
Look at butt 1.
Let's take a look at the mechanism of establishing a market balance, if under the influence of a change in factors, the market will go out of the way. Іsnuyut two main variant of disproporcії mіzh drink and propozitsієyu: surplus and shortage of goods.
too much(superfluous) goods - tse taka situation on the market, if the value of propositions to the goods for a cієyu cіnoy exceeds the value of the drink by a new one. And here among the virobniks blame the competition, the struggle of buyers. The vigrashі overflows with that, hto proponuє vigіdnіshі umovi realіzacії goods. In this rank, the market will not turn into a jealous camp.
Deficit goods - at times I will drink the goods for a given price, I will transfer the quantity of the goods. This situation is to blame for the increase in purchases due to the possibility of supplying a scarce product. Remagaet the one who propagates a high price for the entire product. The price has grown to a new level of respect for virobnikіv, yakі pochinayut rozshiryuvati virobniktvo, zbіlshyuchi cim proposition of goods. As a result, the system rotates at the rank of equal.
In this way, cіna vykonuє vrіvnovavazhuyuchu funktsіyu, stimuliyuyuchi razshirennya vyrobnitstva and propozії product with defіtsіtі and strimuyuchi propozitsіyu, pozbavlyuyuchi market vіd nadlishkіv.
The vrіvnovazhuє role of price is manifested as through drink, and through proposition.
Admittedly, scho rivnova, as it stood on our market, the bula was destroyed - under the influence of any factors (for example, an increase in income) there was an increase in drink, as a result of this, the curve was reversed D1 in D2(Fig. 4.3 a), and the proposition was left unchanged.
If the price of this product did not change immediately, then for the increase in the cost of the situation, if for the front price P1 kіlkіst goods, yak leather iz pokuptsіv now can pridbati (QD) revisit that obligation, which can be propagated for this price by the scribes of this product (QS). The value of drink is now revisiting the value of the proposition of a given product, which means a vindication shortage of goods at rosemary Df = QD - Qs on this market.
Deficiency of goods, as we know, to bring to competition mіzh purchases for mozhlivіst pridbannya cgogo goods, scho to bring up to the growth of market price cіn. Vіdpovіdno to the law propozitsії response prodavtsіv on pіdvishchennya cіni bude zbіlshenny obyagu zaproponovanogo goods. On the chart, you will see the transfer of the market equalization points E1 vzdovzh crooked propositions to її peretina with a new crooked drink D2 de i will reach a new level of this market E2 h equally important number of goods Q2 at an equal price R2.
Rice. 4.3. Usunennya points of equal value.
Let's look at the situation, if the equally important camp is destroyed from the side of the proposition.
Admittedly, under the influence of some factors, there was an increase in the proposition, as a result of which the curve shifted right-handed from the position S1 in S2 and the drink is filled with the inevitable (Fig. 4.3 b).
For washing the savings of the market price on a large equal (R1) increase the propositions to too much goods at rosemіrі Sp = Qs - QD. As a result, blame seller competition, what to bring down the market price (s P1 before P2)і srostannya obsjagu goods that are sold. On the graph, the moving points of the market level will be displayed E1 vzdovzh crooked drink to її peretina with a new crooked proposition, which will lead to the establishment of a new level E2 with parameters Q2і R2.
Similarly, it is possible to influence the equal price and the equal number of goods to change the drink and change the proposition.
In the primary literature, the rules for mutual intercourse and propositions are formulated.
1. Increasing the number of drinks and increasing the equal price and the equal quantity of goods.
2. The change in the drink for the fall and the equal price and the equal quantity of the goods.
3. Increasing the propositions of pulling behind a change in the equal price and an increase in the equal number of goods.
4. Short-term propositions attract for themselves an increase in an equally important price and a change in an equally important number of goods.
Koristuyuchis tsimi rules, you can know an equally important point for whether any changes drink that proposition.
The turn of the price to the market equal level can basically be changed in the following way:
1) administrative regulation of prices
2) monopoly virobnik, otherwise, it is gentle, which allows you to reduce the monopoly price, as if it were piece-based, so it is underestimated.
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In an antagonistic context, it is natural to consider such a result as optimal, with any kind of gravity, it is not visible in any way. A similar result (x *, y *) is called the situation of equality, and the principle of optimality, grounding on the basis of the situation of equality, is called the principle of equality.
Appointment. For the matrix graph with the expansion matrix, the result is є the situation of jealousy but a saddle point, like
At a single point, the matrix element is at the same time a minimum for its row and a maximum for its column. Have grі z butt 2 element a 33є saddle point. Optimal for this group is the third strategy for both graves. Like the first grave, it turns into the third strategy, you start to win less, lower a 33. Like other graves, they are guided by the third strategy, we start to grow more, lower a 33. In this order, for both graves there is nothing better, lower in sequence to finish the third strategy.
The principle of optimal behavior: if in the matrix graph there is a sad point, then the optimal one is the choice of a strategy, which confirms the sad point. What will happen, if more than one saddle point appears in the Greek?
Theorem. Come on 
two dovіlnі sіdlovі specks at the matrix grі. Todi:
proof. From the point of view of the situation, it is possible:
We represent in the left part of the unevenness (2.8) , and to the right - , to the left part of the unevenness (2.9) - , to the right - . Todi we take:
The sounds sing out the equanimity:
From the theorem it is clear that the function wins the same and the same value in all situations of equality. The same number is called at the price of a gri. And the strategies that show whether there are any sad points are called optimal strategies graves 1 and 2, obviously. Looking back at (2.7), all optimal strategies are mutually reinforcing.
The optimality of the behavior of the gravity does not change, as in the case of impersonal strategies they are left unchanged, and the function wins to multiply by a positive constant (otherwise, a constant number is added to it).
Theorem. For the basis of the saddle point (i *, j *) in the matrix grid, it is necessary and sufficient, so that maximin and minimax are:
(2.10) 
proof. Necessity. If (i*,j*) is a single point, then it is correct (2.6) :
(2.11) 
Vodnocha may:
(2.12) 
From (2.11) and (2.12) we take:
(2.13) 
Rozmirkovuchi similarly, we come to equanimity:
in such a manner,
On the other hand, the reverse unevenness (2.5) is always victorious, then (2.10) is justified.
Prosperity. Let it be fair (2.10). We bring the presence of a saddle point. Maemo:
Vidpovidno to equivalence (2.10), unevenness (2.15) and (2.16) are transformed into equivalence. If anything is possible:
The theorem has been completed. It has been arguably brought to light that the maximum and minimum value of the cost is significant.
Zmishane extension gri
Let's take a look at the matrix group G. If the situation is equal, then minmax is better than maximin. Moreover, leather from gravel can provide other gravel information about your optimal strategy. Yogo supernik cannot be taken away from the same information of any supplementary benefit. Now it is acceptable that G has no situation of equality. Todi:
In my opinion, the minimax and the maximin strategies are not stable. Gravity can mother stimuli until the release of their protective strategies, due to the possibility of winning more, but I will program it with a risk, so that I will win less, lower if the protective strategy is stuck. When risky strategies are blocked, the transfer of information about them to the enemy can be fatal: gravel automatically takes less winnings, lower when protective strategies are blocked.
butt 3. Let the matrix gri maє look:
For such a matrix, tobto. the situation is not equal. Protective strategies of graves є i * = 1, j * = 2. Let grave 2 take strategy j*=2, and grave 1 choose strategy i=2. then the rest take away the win 3, if the two ones have more, the lower is the maximum. However, Grave 2 thinks about Gravity Plan 1, so change your strategy to j=1, and then the first win is 0, so the maximum is less. Analogous mirroring can be carried out in another grave. Zagal can create a non-trivial vysnovka, which zastosuvannya adventurous strategy can bring a bigger result, lower guarantees, that її zastosuvannya pov'yazanі z rizikom. Blame nutrition, why can't you combine an adventurous strategy with an adventurous one in such a way as to increase your average win? In fact, food is about those, how to add gravel between gravels (2.17)?
It appears that sensible decisions are made for a different strategy, that is a vipadkovy choice of pure strategies. Guess what strategy engraving 1 is called mixed, yakscho vybіr i-th row to be carried out by him with deykoyu ymovіrnistyu p i. Such a strategy can be identified with a rozpodіl omіvіrnosti 
on faceless rows. Assume that the first one has m pure strategies, and the other has n pure strategies. Todі їх їх їмішіні strategії – tse imovіrnіsnі vectors:
(2.18) 
Let's take a look at two possible changes in the strategy of the first engraving from the butt 3: 
. These strategies are divided by the difference between the pure strategies. Just as in the first view the rows of the matrix are selected with equal weights, then in the other view they are different. If we are talking about a change in strategy, then maybe on the basis of the vipadkovy choice not the choice of “navmannya”, but the choice of foundations on the work of the vipadkovy mechanism, which will ensure the safety of the needs of us rozpodіlnosti. So for the implementation of the first of the worst strategies, it is good to drop coins. Gravetets rob the first row, or the other one is fallow, as the coin falls. In the middle gravel, however, it is often chosen as the first row, as well as the other, prote viber on a specific iteration, if not in accordance with the original fixed rule, and may have the maximum level of transparency: before the implementation of the drop-down mechanism, the faults should not be cast on the first grave. For the implementation of another mixed strategy, it is good to use the mechanism of foaling. Gravets take these same paprits, commemorating three of them with a cross, and throwing them into a hat. Let's sweat, navmannya, take one of them. Vіdpovіdno up to the classical theory of imitations of wines with a cross of imitation 3/7, and pure papirets - with imitation 4/7. A similar mechanism of foaling is built to realize whether it is rational feasibility.
Let the gravitational pull out of mixed strategies (2.18). Todi won the first engraving on a well-taken iteration gry є vypadkovoy value: v(X,Y). Oscilki gravitsi choose strategies independently one or the other, then, according to the multiplicity theorem, the flexibility of choosing the result (i, j) will win more expensive additionalness. Then the law of rozpodіlu vipadkovoї magnitude v(X,Y) tasks with an advancing table
Let it play now for a long time. Todi average gain for such a group is more than mathematical estimation of the value v(X,Y).
(2.19) 
At the end, but to achieve a great number of iterations, the average wins three times in size (2.19).
Butt 4. Razrahuemo average win (2.19) for gris with butt 3 for an hour of victoriousness against the gravity of offensive strategies: 
. The wins matrix and the wins matrix look like this:
We know the average:
In this way, the average win (2.20) may have an intermediate value between the maximum and the minimum.
Oskіlki be-like a bet of mixed strategies X and Y, you can improve the average value of the game, blaming the search for an optimal strategy. Naturally rose from the follow-up of protective strategies. Protective strategy of the first engraving secures this maximin. Protective strategy of another grave does not allow the first to win more, lower minimax. The most significant result of the theory of problems with common interests is the offensive:
Theorem. Whether it is a matrix game, the situation is equal in different strategies. The proof of the theorem is not easy. Which course is going down.
Lasts: Іsnuvannya situatsії rivnovagi means that maximin dorivnyuє minіmax, and, then, be a matrix graph maє tsіnu. The optimal strategy of the first grave is the maximum strategy. The optimal strategy of the other is minimax. The shards of the task for the sake of optimal strategies have been violated, it seems that there should be a matrix graph you can untie on impersonal mixed strategies.
Gris Cherry 2x2
Butt 5. Virishiti group. It doesn't matter to change, there is no single point. Significantly optimal first-gravity strategy (x, 1-x)- Tse vector stovpets, but for clarity, write it down in front of the row. The optimal strategy of another grave is significant (y,1-y).
Vigrash of the first grave with a vipadical value with an advancing rozpodil:
| v(x,y) | 2 | -1 | -4 | 7 |
| p | xy | x(1-y) | (1x)y | (1-x)(1-y) |
We know the average win for the iteration of the first grave - mathematical estimation of the fall value v(x,y):
Let's remake this viraz:
This mathematical calculation is summed up by the constants (5/7) and the changeable part: 14(x-11/14)(y-8/14). What is the meaning y starts on 8/14, then the first grave can be chosen X in such a rank, to build a change of the positive part, to increase the power of winning. What is the meaning X checks on 11/14, then another grave can be chosen y in such a manner, to change the negative part, changing the first gravel. In this order, the single point is represented by equalities: x * = 11/14, y * = 8/14.
2.5 Igor's Celebration
The way of displaying similar images is shown on the butt.
Butt 6. Virishity group 
. Let's reconsider that there is no single point. Significantly changed the strategy of the first engraving X=(x, 1-x)- Tse vector stovpets, but for clarity, write it down in front of the row.
Let the first one take zastosovuyu strategy X, and the other - his pure j-th strategy. Significantly average win of the first gravel in this situation. Maemo:
Let us represent the graphs of functions (2.21) on the reverse side.
The ordinate of the point, which is located on the opposite of the straight lines, wins the first gravity in the situation, if the fault is stagnant, the strategy is reversed (x, (1-x)) and the other one is a pure strategy. Guarantees the result of the first engraving - the price of the bottom is the home of straight lines (lamana ABC). Nayvischa point tsієї lamanoї (point B) є the maximum guaranteed result of engraving 1. The abscissa of the point indicates the optimal strategy of the first engraving.
Oskіlki shukana dot є peretina linіy i , then її abscissa can be found as a solution:
Also, the optimal strategy of the first engraving is (5/9, 4/9). The ordinate of the point is the price gry. Won dorivnyuє:
(2.22) 
Respectfully, that the line, which follows another strategy of another grave, passes more than point B. It means that if the first grave is stagnating its optimal strategy, and grave 2 is to a friend, then the program of the other will be more equal due to the stasis of strategy 1. another strategy can take the fate of the optimal strategy of another grave. The optimal strategy of engraving 2 is the fault of the mother looking: 
. Pure strategies 1 and 3 suttivimi. Strategy 2 is called nesutteva. From the little one higher, and from the equality (2.22) it can be seen that, with the first gravel, its optimal strategy won the other grave, to lie in the same way, like from its original strategies of the fault. Vіn can zastosuvati so be it zmіshanu strategіy, scho s іstotnyh (zokrema optimal), win and in this way not change. Absolutely analogous to the firmness є fair і for protilazhny vipadku. If another gravel zastosovuє its optimal strategy, then the winnings of the first gravel cannot be deposited, in fact, because of their own strategies, wines zastosovuє that good price. Vikoristovuyuchi tse firming, we know the optimal strategy of another engraving.
