Game Theory: Games with Nature

The protracted continuation of a series of articles on game theory.


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Closer to practice

In the last article on the theory of games, situations were considered in which the logic of the actions of two players was assumed, each of whom wants to get the maximum benefit for himself. The next stage is the so-called games with nature. Formally, the study of games with nature, as well as strategic ones, must begin with the construction of a payment matrix, which is essentially the most time-consuming step of preparing a decision. Errors in the payment matrix cannot be compensated by any computational methods and will lead to an incorrect final result.

A distinctive feature of the game with nature is that only one of the participants consciously acts in it, in most cases the player is called one. Player two (nature) does not care about the result, or he is not capable of meaningful decisions. Or, perhaps, the conditions do not depend on the actions of the player, but are determined by external factors: the reaction of the market, which will not harm one particular player, public policy, the real nature.

Types of tasks and selection criteria

There are two types of tasks in games with nature:

• The problem of decision making under risk conditions, when the probabilities with which nature assumes each of the possible states are known;
• Decision making tasks in conditions of uncertainty, when there is no opportunity to obtain information about the probabilities of the appearance of states of nature;

To be closer to real situations, for example, take a relatively real situation. The first player we will make decisions for will be represented by Samsung with its Galaxy S5. The second player to play "nature" will be Apple, and his iPhone 6.

The time of the release of the new smartphone is right, the presentation was held, the experts expressed their opinion, and the player one has to make an important decision when to release the product? Having simplified the situation, we will have three options: to the competitor (A ₁ ), with it (A ₂ ) or after (A ₃ ). Naturally, until the new iPhone comes out, we will not know if it will be much better than ours (B ₁ ), the same (B ₃ ) or much inferior in quality (B ₃ ). Considering the profit in all cases, we end up with a matrix:

	In 1	In 2	At 3
A 1	five	five	7
A 2	3	four	6
A 3	2	four	eight

Now, for making a decision, we have several criteria.

1. Wald criterion (maximin). The player expects that nature will follow the worst path for him, and you should choose the option with the maximum profit for the worst outcome, so this criterion is considered pessimistic. You can present it in the form of max (min i)

With this criterion:

for A _{1 the} minimum profit (5) will result in the actions of nature B ₁ and B ₂
for A _{2, the} minimum profit 3 after action B ₁
for A ₃ minimum profit 2 after action B ₁

Thus from 5, 3 and 2 the maximum profit (5) will give us option A ₁

2. The maximum criterion (maximax) is optimistic, i.e. we hope for the most favorable outcome for us. It is represented as max (max i).

for A _{1 the} maximum profit is 7
for a _{2 the} maximum profit is 6
for A _{3 the} maximum profit is 8

From 7, 6 and 8 the maximum profit will bring option A ₃

3. The Hurwitz criterion recommends a strategy determined by the formula max (A * max i + (1-A) * min i), where A is the degree of optimism and varies from 0 to 1. The criterion gives a result that takes into account the possibility of the worst, so and the best behavior of nature. For A = 1, this criterion can be replaced by the maximum criterion, and for A = 0, the Wald criterion. The value of A depends on the degree of responsibility of the player alone: ​​the higher it is, the closer A is to one. For this example, we take A = 0.4.

for A _{1, the} profit is 0.4 * 7 + 0.6 * 5 = 5.8
for A _{2, the} profit is 0.4 * 6 + 0.6 * 3 = 4.2
for A _{3, the} profit is 0.4 * 8 + 0.6 * 2 = 4.4

From the responses received, the max pribl brings action a ₁

4. Savage Criterion (minimax). Its essence lies in the choice of a strategy that does not allow too high losses. For this, a risk matrix is ​​used, in which the maximum profit is calculated for each variant of the player’s actions, and the smallest one is chosen among the results. Its formula looks like min (max i)

With this criterion:

for A _{1 the} maximum profit (7) will result in the action of nature In ₃
for A _{2, the} maximum profit is 6 after the action B ₃
for A _{3 the} maximum profit is 8 after the action B ₃

Thus from 7, 6 and 8 the minimum profit (6) will give us option ₂

5. According to the Bayesian criterion, it is proposed to give equal probabilities to all the strategies under consideration, and then to accept the one at which the expected gain will be greatest. The criterion has one drawback: it is not always possible to accurately determine the probability of a particular event from the side of nature. The formula for it is max (Σ q * i).

First, we set the probability of the onset of each of the nature of nature equal to 0.33, and got

for A ₁ 5 * 0.33 + 5 * 0.33 + 7 * 0.33 = 5.61
for A ₂ 3 * 0.33 + 4 * 0.33 + 6 * 0.33 = 4.29
for A ₃ 2 * 0.33 + 4 * 0.33 + 8 * 0.33 = 7.63

Obviously, we will get the maximum profit from option A ₃ . However, turning to the experts, we got probabilities of events for nature 0.5; 0.4; 0.1; respectively. In this way

for A ₁ 5 * 0.5 + 5 * 0.4 + 7 * 0.1 = 5.2
for A ₂ 3 * 0.5 + 4 * 0.4 + 6 * 0.1 = 3.7
for A ₃ 2 * 0.5 + 4 * 0.4 + 8 * 0.1 = 3.4

I think the result is pointless to comment.

The main task is to find the optimal (or at least rational) strategies that best bring the system to the goal under the given external conditions. For the choice of strategies in conditions of uncertainty, you can apply any criteria, in terms of risk, Bayes criterion is more effective. However, the choice between the criteria themselves is usually based on intuition, depending on the nature of the decision maker (in particular, his risk appetite).

If the decision is made in the face of uncertainty, then it is better to use several criteria. In the event that the recommendations are the same, it is safe to choose the best solution. If the recommendations are contradictory, the decision should be taken more carefully, taking into account the strengths and weaknesses.