I will not change unless you do

nimda February 28, 2025

0 0 7 minutes read

In Game Infirmity, how can players end if there may be a better option to decide? Perhaps one player still wants to change their decision. But if they do it, maybe one player wants to change again. How can they hope to escape from this cruel circle? To solve this problem, the idea of NASH equivalent, which I have to explain in this article, is important for the game of the game.

This article is a second part of the second series of chapter's vision. If you have never looked at the first chapter yet, I can encourage you to do what you are familiar with key match goals and ideas. If you do, you're ready for the following steps for our trip on game view. Let's go!

Finding a Solution

Finding a game remedy on the game view can sometimes be tactically. Photo by Mel Poole in Unscwask

Now we will try to find a game solution to Game idea. A solution A collection of actions, where each player increase their work and therefore behaves well. That doesn't mean, that each player wins the game, but that doing the best they can do, they are given to do not know what other players will do. Let's consider the following game:

If you are not familiar with this matrix proportion, you may want to look back in chapter 1 and update your memory. Do you remember that the matrix provides you with each player's reward given for specific actions? For example, if player 1 chooses the action y and player 2 You select an action b, player 1 You will receive 1 reward and player 2 You will receive 3 reward.

Ok, what actions do players decide now? The player 1 doesn't know any player to do, but they can try to find out what will be the best verb according to the player option. If we compare the action of the verb and z (shown blue and red boxes. But what happens, if the player 2 determines the action b (second column)? In that case, action Also if player 2 chooses action c (third column), y is still better than Z (Reward of 2 vs. Reward of 1). That means, that player 1 should use the action of the z, because the action y We are always better.

We compare player rewards 1 Then actions y and z.

For the above consideration, player 2 can expect, that player 1 will not use the Z action so they should not take care of small rewards, because this is a little player, and this helps player 2 determine its action.

We found, that with the player 1 y You're always better than Z, so we don't look at Z again.

When we look at the leaved game, we see, that in Player 2, the option B is better than action A. If player 1 chooses when a 1 player), we've seen that the action z is no longer player 1 Nevertheless.

We compare player rewards for 2 verbs with b.

As a result, the player 2 will never use the action A. Now if player 1 expects the player 2 has never used an action a, the game becomes smaller and will be considered.

We have seen, that by player 2 The action B is always better than the action a, so we don't have to look.

We can continue easily and see that by player 1, X Now you are better than Y (2> 1 and 4> 2). Finally, if player 1 chooses the action A, Player 2 will select the action B, better than C (2> 0). Finally, it is an act only x (player 1) and B (player 2) remaining. That's the solution for our game:

Finally, only one still, which is a player 1 using X and Player 2 Using B.

Would be reasonable for player 1 to choose an action x and player 2 to choose the action B. Note that we reached that conclusion except the conclusion with knowledge Some other player. We just thought that some actions would never be taken, because they always worse than another action. Such actions are called reinstalled firmly. For example, the action of is strongly controlled by the verb y, because y we remain better than z.

The best answer

The scrubble is one of those games, when the best response is searched can take years. Photos by Freystein G. Jononsson on Underwes

Firmly actions are not always, but there is the same importance to us and is called The best answer. Say we know what action one player chooses. In that case, determining the action is a lot of simplest: We only take the active actions. If player 1 knew that player 2 chose the option A, the best answer to player 1 will be y, because Iy has a great reward on that column. Do you see how we always wanted the best answers before? For each possible player verb we are looking for for the best response, if one player choose that action. Officially, player is the best response set of their actions of all other players An action for player 1 that saves the use given by the action of some others. And be careful, that a firmly solid verb will never be the best answer.

Let's get back to the game that follows the first chapter: The prisoner problem. What are the best answers here?

How should the player decide player, if player 2 permits or denies? If Player 2 Agree, Player 1 has to confess, because the reward of 3 is better than 6 reward. And what happens, if a player 2 denies? In that case, confession is also, because it will provide 0, better rewarding than the $ 3 reward. That means admitting a player 1 Acceptance is the best response of player 2. Player 1 We don't have to worry about player's actions at all but should always agree. Because of the match synchronization, the same applies to player 2.

Nash Elilibium

Nash equality is like a large key that allows us to solve the game-theoretic problems. The investigators were very happy when they found. Photo by RC.XYZ NFT Gallery on Undengur

If all players play their best response, we've reached a game solution called Nash equality. This is a key idea of the game's idea, due to an important area: in NASH EXHILIBIM, NO OPPORT ARE YOU A Reason to Change Their Action, Unless any other player does. That means all players are happy as they can and would not change, even if they could. Consider the prison condition from the top: Nash equality is reached when both agree. In this case, no player will change his action without another. They can be better if both Change their action and decided to deny, but as they can't communicate, they do not expect there is a change in another player so they don't change them.

You may wonder if there is always one equal fit of Nash for each game. Let me tell you that it is possible and a lot, such as in Bach vs stravinsky game that already knows in Chapter 1:

This game has two equal nashes: (Bach, Bach) and (Stravinsky, Stravinsky). In both cases, you can easily imagine that there is no reason for any player to change their action. If you live in a bach concert and your friend, you would not leave your seat to go to stravinsky replace, even if you like stravinsky over a bach. In the same way, the Bach fans will not walk away from Chravensky replace if that means leaving his friend alone. In two remaining cases, you can think differently or: If you were in Stravensky Concerts alone, you may want to go out when you join your friend in the Bach. That is, you can change your action even if one player doesn't change. This tells you, that this situation has entered – Nash equality.

However, there can be games without any nutshell at all. Imagine that you are a footballkeeper during the shooting. For convenience, we think you can jump left or right. The opposing group footballer can and shot in the left or right corner, and we think, that he or she decides that he takes the same corners and that he decides that opposition corners. We can show this game as follows:

Matrix of a game of shooting the penalty.

You will not get any NASH equivalent. Each situation has a clear winner (reward 1) and a clear loser (reward -1), and that's why one of the players will always want to change. If you skip right and hold the ball, your opponent will like to change the left corner. But then you will want to change your decision, which we will make your opponent choose any other in common and so on.

Summary

We have learned about getting the point of balance, when no one wants to change. That is a nash equivalent. Photos by Eran Menashri on Undengur

This chapter showed how we can find sport solutions through the idea of NASH. Let us break, what we have learned so far:

The game solution in the Game Theory increases all the work or a player's reward.
The action is called reinstalled firmly If there is another constant action. In this case, it can be irresistible that you have played a tight solution.
The active reward is given actions taken by some players are called The best answer.
A Nash equality It is a situation where the whole player is playing his best response.
According to NASH equality, no player wants to change its action unless another game is working. That way, Nash Equilibria is the right place.
Some sports have multiple Nash Equilibria and other sports have there is no.

If you are sad to the fact that nothing is equivalent to Nash in some games, don't give up! In the next chapter, we will introduce the opportunities and this will allow us to gain additional equality. Stay tuned!