# Task 6. Game Theory

Test problem. Companies Samotlor and BaltikaPromNeft are striving to enter new markets and compete with each other in terms of the amount of oil extracted. At this stage, a forecast of production prospects in millions of barrels for a given period is being calculated. The prepared analytical report indicates four possible strategies for the players - they are presented in the payoff matrix:

|  |  | BaltikaPromNeft Strategies |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 1 | 2 | 3 | 4 |
| Samotlor Strategies | 1 | 3 | 3 | 2 | 2 |
|  | 2 | 2 | 5 | 3 | 3 |
|  | 3 | 4 | 3 | 4 | 3 |
|  | 4 | 2 | 4 | 5 | 4 |

Given that some strategies may be unsuitable for both players, it makes sense to conduct dominance ${ }^{1}$ and exclude such strategies from the analysis. Find the optimal strategies ${ }^{2}$ for the volume of production for both players and the value of the game.

Note. The answer is the analytically derived correction coefficients for the production indicators of the players specified in the payoff matrix.
Solution to the problem. To find the best strategy for Samotlor company, we will write down the minimum number in each row: $(2,2,3,2)$. The maximum of these is 3 - this number is the lower value of the game and pertains to the strategy СЗ. Next, for BalticPromNeft, we highlight the maximum values of the payoff: $(4,5,5,4)$ - the minimum will be the number 4. Thus, the upper value of the game is determined, pertaining to strategies Б1 and Б4.

Since the lower and upper values do not coincide, we can conclude that there is no solution in pure strategies. The value of the game lies in the following range $\mathbf{3}<\boldsymbol{v}<\mathbf{4}$.

Note that the game has a large dimension - the matrix is $4 * 4$. Let's try to apply dominance - exclude unprofitable strategies from it. They can be identified by comparing columns or rows with each other - if all elements of some column are greater than or equal to the elements of another column, then it can be excluded from consideration. A similar situation arises when comparing rows - rows with values less than or equal to other rows are removed. As a result, we get the matrix:

$$
\left(\begin{array}{llll}
3 & 3 & 2 & 2 \\
2 & 5 & 3 & 3 \\
4 & 3 & 4 & 3 \\
2 & 4 & 5 & 4
\end{array}\right) \rightarrow\left(\begin{array}{ll}
3 & 2 \\
2 & 3 \\
4 & 3 \\
2 & 4
\end{array}\right) \rightarrow\left(\begin{array}{ll}
4 & 3 \\
2 & 4
\end{array}\right)
$$

Let's solve the problem analytically. The average payoff for Samotlor company, under the conditions (a) using its optimal mixed strategy $x^{*}=\left(x_{1}^{*}, x_{2}^{*}\right)$ and (b) using a pure strategy corresponding to the first column by BalticPromNeft, is equal to the value of the game:

$$
4 x_{1}^{*}+2 x_{2}^{*}=v
$$

Similarly, we can calculate the average payoff when BalticPromNeft uses a pure strategy corresponding to the second column of the payoff matrix. We also consider that the sum of the correction coefficients is $100\%$, i.e., one: $x_{1}^{*}+x_{2}^{*}=1$. Based on this, we will construct a system of linear equations and represent its solution.

$$
\left\{\begin{array} { c } 
{ 4 x _ { 1 } ^ { * } + 2 x _ { 2 } ^ { * } = v } \\
{ 3 x _ { 1 } ^ { * } + 4 x _ { 2 } ^ { * } = v } \\
{ x _ { 1 } ^ { * } + x _ { 2 } ^ { * } = 1 }
\end{array} \quad \left\{\begin{array}{l}
x_{1}^{*}=2 / 3 \\
x_{2}^{*}=1 / 3 \\
v=10 / 3
\end{array}\right.\right.
$$

Determine the mixed strategy for BalticPromNeft, here the average loss of this player, for any pure strategy of Samotlor company, is equal to the value of the game:

![](https://cdn.mathpix.com/cropped/2024_05_06_c5abfa1befa3791cfd2eg-11.jpg?height=211&width=1158&top_left_y=1545&top_left_x=506)

The game is solved, the optimal mixed strategies are determined.