What is conflict?
This is a situation, in which the interests of the parties collide and a conflict of interests occurs. Each of the participants wants something, not what others want.
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How game theory can be used?
Game theory is a mathematical theory that is tightly connected with the social sciences. Modern socio-economic studies are based on game theory, but game theory can be useful for every person. Game theory enters into the life of each of us unnoticed. The application of game theory can be understood in two ways. It can be like an understanding of the surrounding situation, political, economic or transport situation. It can also be of direct benefit to the person who uses it.
The simplest examples of conflict are games (checkers, chess, card games, etc.). They differ in that they are conducted according to certain rules. The rules of the game are a system of conditions that indicate what opportunities are given to players (a list of possible moves); what result (gain, loss) each given set of moves leads to. But in real-life rules not always defined.
In this essay I want to find how game theory might be used in real life. Not every conflict that occurs in practice proceeds according to the rules. To make a mathematical analysis possible of a conflict, one need to present the conflict in a game form, that is, indicate the strategies (actions) that are possible for the participants, and specify what the result will be if the players choose a certain strategy.
Thus, the game is a conflict with clearly defined conditions.
Game theory gives an opportunity to predict the consequences of our actions and the actions of other people. It often happens that the result of a conflict - even with quite certain strategies of the participants - is impossible to predict exactly, since it depends on the case. Such random circumstances interfering in the course of the game can be, for example, shuffling and dealt cards, hitting or not hitting the target when shooting, etc. Then, instead of “game result”, you need to talk about the average result, i.e. average per game if a sufficiently large number of games is played. Indeed, in one game, it may accidentally “get lucky” to a player who applies a clearly unwise strategy. If there are many games, on average, the one who behaves intelligently wins.
When we talk about the result, or the average result, of the game, we assume that this result is expressed by a certain number. Does it always happen like that? Not always. For example, in chess we do not always express the result in numbers, but simply say: win, lose, draw. But you can agree and translate them into a numerical form, for example, to assign a gain to a value of + 1, to a loss of –1, a draw of 0.
We will assume that in any conflict, the gain (loss) of each player is expressed by a number. Then the main task of game theory can be formulated as follows: how should a reasonable player in a conflict with a reasonable opponent (or opponents) behave (what strategy should he use) in order to ensure, on average, the greatest possible gain for himself/herself? But one off the main paradoxes of game theory is a problem that if people do not coordinate with each other, occur situation when all people lose.
2. Two-person zero-sum game
If there are two parties in the conflict, the game is called two-person, if several - multiple. Two-person games are simpler than multiple ones and have more practical significance. We will consider only two-person games.
Each game will be considered as a conflict between two players: R (“red”) and B (“blue”). For convenience of reasoning, in order to have some particular point of view, we will usually take the side of one of the players (let it be R) and talk about him/her “we”, and about the other - “opponent”. This does not mean that side R will have any real advantage. It will just be more convenient for us.
A game is called a zero-sum game if one side wins what the other loses, i.e., the sum of the R and B winnings is zero. An example of a zero-sum game is the question of who pays for a taxi: the income of one person for another is a loss. In life, there are often conflicts in which this condition is not fulfilled. For example, in a military confrontation or beneficial cooperation like in any non zero-sum game, it is quite possible that both sides lose or win.
So, let us assume that the interests of R and B are strictly opposite and that the sum of their winnings is zero. It will be very convenient for us in the computational sense. After all, if the winnings R are equal in magnitude and opposite in sign to the winnings B, then you can consider the winnings of only one of the players: the winnings of the other will be determined automatically.
Let's choose R as the winning player. Player R is interested in turning his/her winnings (we denote it by R) to the maximum (to make it the greatest). Player B, on the contrary, is interested in turning it into a minimum (to make it the smallest). Each of the players R and B, pursuing its own goal, takes all measures to make themselves better, and the opponent - worse.
As a result of the struggle of interests, if both opponents are equally reasonable, there must be some equilibrium position, in which each player gets what is due to him, no more and no less. This equilibrium average gain, which player R can count on, if both sides carry on with reasonably, i.e., stick to their optimal (best) strategies, called the price of the game. For example, football match, if one team win, than other team lose, but every team need to have optimal strategy and when they create their strategy they must take into consideration actions, strategy of other team. If not to take into account strategy of opponent , the best strategy to score the most goals will simply run in a straight line to the opponent’s goal and score the ball. But it will be impossible because of actions of opponents and this strategy will be losing. This applies to almost any game with a zero sum, in order to get our optimal strategy in any business, we must create it taking into account the fact that the opponent will adhere to his optimal strategy
If the price of the game is zero, then this is a fair game, that is, it is equally profitable or unprofitable to both sides, if the price of the game is positive, then the game is beneficial to R. If negative, you will have to admit that it is beneficial for B
To solve a game is to find a pair of optimal strategies (for R and B) and the price of the game, i.e., the average player's gain R, if both — and R and B — behave reasonably.
And if only R will behave reasonably but not B? Well, so much the worse for B! Gain for R can not decrease. In the worst case, it will remain the same, and in the best - increase.
The goal of game theory is to develop recommendations for the reasonable behavior of players in conflict situations, i e. definition of the 'optimal strategy' of each of them.
3. Normal-form game
3.1 Matrix game
We will consider only finite games, i.e., those in which each participant has a finite number of strategies.
If player R has m strategies and player B has n strategies a game is called a game mXn.
The rules of the game can be written in the form of a table (or matrix) with m rows and n columns. Rows match the “red” strategies that we denote: R_1, R_2, ..., R_m, and the columns - blue strategies: B_1, B_2, ..., B_n.
B_1 B_2 … B_n
R_1 k_11 k_12 … k_1n
R_2 k_21 k_22 … k_21
… … … … …
R_m k_m1 k_m2 … k_mn
Winnings (or average winnings) are placed in the cells of the table with the corresponding pair of strategies. For example, k_12 is a win, if the 'red' choose the strategy R_1, and the 'blue' - B_2; in general, k_ij is a gain in combining the strategy of R_i and B_j.
Such a table is called a payoff matrix or just a game matrix.
If the final game is written in the form of such a matrix, then they say that it reduced to normal form.
But try, for example, to record and normal form ordinary chess! You will immediately find that the number of possible strategies is vastly large - so large that listing them goes beyond the capabilities of not only man, but also modern computers. It's a pity! Because if building a chess game matrix was possible it would have a very curious consequences ... But let's not get ahead of ourselves.
4. Examples of finite games. Minimax principle
Example 1. The game 'Three fingers.' Two players R and B simultaneously and without saying a word show each other one, two or three fingers. If the total number of fingers shown (the first and second player together) is an even number, then R wins: he gets as many points as the total number of fingers, if odd - B wins on the same conditions. It is required to record the game in normal form.
Solution. We renumber the strategies according to the number of fingers shown. Game 3x3.
The matrix of the game will be:
B_1 B_2 B_3 min
R_1 2 -3 4 -3*
R_2 -3 4 -5 -5
R_3 4 -5 6 -5
Max 4* 4* 6*
Suppose we have chosen strategy R_1. What will do enemy? He will choose strategy B_2; our gain will be equal to –3, that is, we will lose 3 points. Too bad! Write the number –3 against the first row in the additional column. Let's try another strategy - R_2. A reasonable opponent will respond to it with B_3. Then we will lose 5 points. Even worse! The third strategy - R_3 - gives exactly the same result: winning (-5).
What to do? Perhaps, the R_1 strategy will still be better than others - having chosen it, we guarantee to ourselves that no matter how the opponent behaves, we won’t win less than (-3) points). The value (-3) is the maximum guaranteed gain that we (the “reds') can secure for ourselves by applying only one strategy. Such a strategy should be R_1. How did we get (-3)? We found the minimum of each line and took the maximum of these minimums. This value is called the maximin or lower price of the game. We will denote it by α.
α=max┬i〖(min┬j〖k_ij)〗 〗
Now let's think about the enemy. He also wants to give less, and get more. But no matter what strategy he chooses, we (the Reds) will behave in such a way as to get more from him. So the adversary (“blue”) should write out in each column not the minimum, but the maximum number.
From these maxima, he must find the minimum, the so-called minimax or the upper price of the game, which we denote β. This value in our case is 4 and is marked with an asterisk. In order not to lose more than 4, the “blue” must adhere to either of their two strategies B_1, B_2
β=min┬j(max┬i〖k_ij 〗 )
So, if each participant is invited to choose a single strategy, then these strategies should be: R_1 for the “red” and B_1 or B_2 for the “blue”.
How did we choose these strategies? Guided by the principle which says - in the game, behave in such a way as to get the most benefit from the worst actions of the enemy for you. This principle is called the minimax principle and is the main one in game theory. Applying this principle, we have so far recommended player R to show one finger, player B to show one or two fingers.
But have we found a solution to the game, such a pair of strategies that are in equilibrium? It’s easy to make sure not. The strategies we found have an annoying property: they are unstable. Indeed, let both players stick to the recommended clean strategies: R_1 and, for example, B_1
As long as both adhere to these strategies, the gain will be 2, i.e., for each game B loses two points. Let's say he found out from somewhere that we adhere to the R_1 strategy. What will he do? Of course, he will immediately switch to B_2 strategy and will receive 3 points each, that is, he will reduce our win to -3. But if we will find out the behavior of B? We will move to R_2 strategy. In response to this, B will switch to B_3 , etc.
We were convinced that a couple of strategies stemming from the minimax principle are unstable: once one player finds out what the other does, the balance is disturbed...
Will it always be like this? Not always.
Example 2.
B_1 B_2 B_3 min
R_1 3 1 2 1
R_2 6 8 5 5
R_3 2 4 4 2
Max 6 8 5
It is required to find the lower price of the game α, the upper price of the game β and minimax strategies and check if they are stable.
Solution. From the analysis of the additional column and row we get: α = 5, β = 5. Maximin is equal to minimax! What follows from this?
Take a couple of minimax strategies: R_2 and B_3. If both adhere to these strategies, then the gain will be equal to 5. Now, let's say, we find out the behavior of the enemy. What will we do? Nothing! We will continue to adhere to the R_2 strategy, because any deviation from it is unprofitable for us. Whether we know or don’t know about the enemy’s behavior, we will still adhere to the R_2 strategy! The same applies to the 'blue' - it makes no sense to change their B_3strategy.
In this example, a pair of strategies R_2 and B_3 is stable, that is, it represents the equilibrium position and gives a solution to the game.
If α = β = v, then the number v is called the price of the game.
A game for which α = β is called a saddle point game.
For a game with a saddle point, finding a solution consists in choosing a maximin and minimax strategy that are optimal.
If the game given by the matrix does not have a saddle point, then mixed strategies are used to find its solution.
The class of games with a saddle point is of great importance in game theory. In particular, it was proved that if according to the rules of the game each player knows the result of all previous moves, both his own and the opponent's (the so-called game with full information), then the game has a saddle point and, therefore, has a solution in pure strategies.
Example 3.
Consider the application of game theory in economic science as an example of a sewing enterprise that produces children's dresses and suits, selling its products through a store. Sales of products in our case depend on the weather. Using data from past observations, the company can sell 600 suits and 1975 dresses in warm weather during April - May, and 1000 suits and 625 dresses in cool weather. Unit costs during the months indicated amounted to $ 27 for suits, $ 8 for dresses, and the selling price is $ 48 and $ 16, respectively.
The task is to maximize the average profit from the sale of manufactured products, taking into account the uncertainty of the weather in the months under consideration. Thus, the marketing service of the enterprise must in these conditions determine the optimal strategy of the enterprise, providing in any weather a certain average income. We will solve this problem by the methods of game theory, the game in this case will be related to the type of games with nature.
Under these conditions, an enterprise has two clean strategies: Strategy A, based on warm weather, and Strategy B, based on cold weather. We will consider nature as a second player also with two strategies: cool weather (strategy C) and warm weather (strategy D). If the company chooses strategy A, then in the case of cool weather (nature strategy C), the income will be
600 (48–27) 625 (16 - 8) - (1975 - 625) 8 = 6800$
In the case of warm weather (nature strategy D), the income is
600 (48 - 27) + 1975 (16 - 8) = 28400$
If the company chooses strategy B, then the sale of products in cool weather (C) will receive an income equal to:
10000 (48-27) + 625 (16-8)=26000$,
and in warm weather(D):
600 (48-27)+625(16-8)-(1000-600)27=6800$
Therefore, the matrix of the original game has the form:
C D
A 6800 28400
B 26000 6800
The first and second rows of this matrix correspond to the strategies A and B of the enterprise, and the first and second strategies C and D of nature.
The payment matrix shows that the first player (company) will never receive an income of less than 6800. But if the weather conditions coincide with the chosen strategy, then the revenue (win) will be 26000 or 28400. It follows that in conditions of uncertain weather, the company will provide the greatest guaranteed income, if it is applied alternately, then A, then strategy B. Such a strategy is called mixed. Optimization of the mixed strategy will allow the first player to always receive winnings regardless of the strategy of the second player.
Let x be the probability of the first player applying strategy A, then the probability of using his B strategy equal (1 – x). In the case of an optimal mixed strategy, the first player (company) will receive the same average income if second player use both the strategy C (cold weather) or the strategy D (warm weather):
6800x+26000(1-x)=6800(1-x)+28400x
From this we get that x=8⁄17; 1-x=9⁄17
Therefore, the first player, using pure strategies A and B in a ratio of 8: 9, will have an optimal mixed strategy, providing him in any case with an average income of
(6800-8⁄17)+(26000-9⁄17)=16965$ - this value will be the price of the game in this case. It is easy to calculate how many suits and dresses an enterprise should produce with an optimal strategy:
(600suits + 1975dresses) 8⁄17+(1000suits-625dresses) 9⁄17=
=812suits+1260dresses
Thus, the optimal strategy of the enterprise is to produce 812 suits and 1260 dresses, which will provide an average income of 16,965$ in any weather. In the analyzed example, we showed finding the optimal enterprise strategies by means of calculating the frequencies of pure strategies, which are weather conditions here. The significance and role of the necessary conditions is reduced to finding a payment matrix whose further solution exists and is possible in the presence of a saddle point.
Conclusion
In conclusion, it should be emphasized that game theory is a very complex field of knowledge. When handling it, it is necessary to observe certain caution and clearly know the boundaries of application. Too simple interpretations conceal a hidden danger. Analysis and consultations based on game theory, because of their complexity, are recommended only for particularly important problem areas. The experience of firms shows that the use of appropriate tools is preferable when making single, fundamentally important planned strategic decisions, including the preparation of large cooperative agreements. Applying game theory to everyday decisions, using it spontaneously is not always the right decision. To use game theory, one need to take into account many different factors, thoroughly calculate the different probabilities, etc., it takes a lot of time and effort. However, the application of game theory makes it easier for us to understand the essence of what is happening, and the versatility of this branch of science allows us to successfully use the methods and properties of this theory in various areas of our activity.
Game theory instills the discipline of the mind. It requires a decision maker to systematically formulate possible alternatives to behavior, evaluate their results, and most importantly, take into account the behavior of other objects. A person familiar with game theory is less likely to think others are more stupid than himself - and therefore avoids many unforgivable mistakes. However, game theory cannot, and is not designed to give decisiveness, perseverance in achieving goals, despite the uncertainty and risk. Knowing the basics of game theory does not give us a clear gain, but protects us from making stupid and unnecessary mistakes
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