Formal explanation of Nash equilibrium
A situation in which, if the other party’s behavior remains unchanged, your gain will decrease but not increase if you take any other action.
This is hard to make sense of in the end, isn’t it? Since I have a test coming up, I tried to understand it with my own intuitive explanation.
An Intuitive Explanation of Nash Equilibrium
This is a situation where you can’t change your back.
You arrive at a resort at night, but you forgot to get a hotel room. Let’s consider the situation.
Let’s say the person you are dealing with is the front desk clerk of the hotel. Let’s say that the person at the front desk of the hotel has made a rational decision to let the traveler stay when he or she arrives. (Because whether I enter or not, the hotel will receive money, but not less, right?)
This time, let’s put aside the other party’s strategy and focus on the possible actions I can choose.
Right now, my possible strategies are as follows.
(i) Stay at a luxury hotel at the drop of a hat
(ii) Stay in a park in the harbor.
If I do (i), the hotel will get 12,000 yen, but I will lose money.
In case of (ii), the hotel gains 12,000 yen, but I lose money. In case of (ii), the hotel gains 0 yen (no change), but I am exposed to the sea breeze of the port, and the quality of my sleep becomes worst, and I suffer damage.
This damage is greater than the financial damage in (i).
In this case, the strategy of staying at a luxury hotel in (i) is a Nash equilibrium.
Why is this?
If you choose a strategy other than (i), i.e. (ii), you will suffer more damage.
Now, the person at the front desk of the hotel assumes the behavior of “accepting people when they come”.
This corresponds to “the other person’s behavior does not change”.
If the other person’s behavior does not change, if you take any other action than that action, your gain may decrease, but it will not increase.
In other words, you can’t change your back. In other words, a Nash equilibrium is a pair of actions that we reluctantly choose because we cannot change our back and the actions of the other party.
Note that Nash equilibrium may not exist in the case where a single action can be taken (AKA simple strategy). On the other hand, when a player chooses an action stochastically (AKA mixed strategy), it always exists.
Reference
Introduction to Artificial Intelligence with Illustrations (Second Edition), a textbook for school classes.
Impressions
I found myself in this situation when I went to Fukuejima in the Goto Islands.
When I think about familiar examples in relation to study, I hear that the self-reference effect increases the effectiveness of learning.
Also, explaining to others is highly effective for learning, and there is an element of recall learning, so writing articles on Qiita is highly beneficial for myself as well.