Game Theory: Games with Nature

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

Closer to practice

In a previous article on game theory, situations were considered in which the logic of the actions of two players was assumed, each of which 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, should begin with the construction of a payment matrix, which is, in essence, the most time-consuming step in 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 it consciously acts only one of the participants, in most cases called one player. 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 specific player, state policy, real nature.

Types of tasks and selection criteria

There are two types of tasks in games with nature:

• The task of making decisions in risk conditions when the probabilities with which nature accepts each of the possible states are known;
• Decision-making tasks in conditions of uncertainty when it is not possible to obtain information about the probabilities of the occurrence 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 Samsung with its Galaxy S5. The second player to play "nature" will be Apple and its iPhone 6. The

time for the release of a new smartphone is coming up, a presentation has passed, experts have expressed their opinion, and the player alone must make an important decision when to release the product? Having simplified the situation, we will have three options: before the competitor (A ₁ ), with him (A ₂ ) or after (A ₃ ). Naturally, until the new iPhone comes out, we don’t know if it will be much better than ours (B ₁ ), the same (B ₃ ) or much inferior in quality (B ₃) Having calculated the profit in all cases, we finally get the matrix:

	In 1	In 2	In 3
A 1	5	5	7
A 2	3	4	6
A 3	2	4	8

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

1. Wald's criterion (maximin). The player expects that nature will follow the worst path for him, and you should choose the option with maximum profit for the worst outcome, so this criterion is considered pessimistic. You can imagine it in the form 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 minimum profit 3 after action B ₁
for A ₃ minimum profit 2 after action B ₁

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

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

for A ₁ maximum profit 7
for A ₂ maximum profit 6
for A ₃ maximum profit 8

Out of 7, 6 and 8 maximum profit will bring option A ₃

3. The Hurwitz criterion recommends a strategy defined 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 yields a result that takes into account the possibility of being the worst, and the best behavior of nature. At A = 1, this criterion can be replaced by the maximum criterion, and at A = 0, by the Wald criterion. The value of A depends on the degree of responsibility of the player: one: the higher it is, the closer A is to unity. 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

Of the answers received, the maximum value brings the action of A ₁

4. Savage criterion (minimax). Its essence lies in choosing 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 action, and the smallest is selected 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 B ₃
for A ₂ maximum profit 6 after action B ₃
for A ₃ maximum profit 8 after action B ₃

Thus out of 7 , 6 and 8 minimum profit (6) will give us option A 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 the greatest. The criterion has one drawback: it is not always possible to accurately determine the probability of an event from the side of nature. The formula for it is max (Σ q * i).

First, we put the probability of occurrence of each of the events of nature equal to 0.33, and obtained

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 experts, we got the probability of events for nature 0.5; 0.4; 0.1; respectively. Thus,

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 it is pointless to comment on the result.

The main task is to find the optimal (or at least rational) strategies that best lead the system to the goal under given external conditions. To select strategies in conditions of uncertainty, any criteria can be applied, in conditions of risk, the 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, it is better to use several criteria. In the event that the recommendations are the same, you can confidently choose the best solution. If the recommendations are contradictory, the decision should be made more carefully, taking into account the strengths and weaknesses.