Understanding Game Theory Terminology in Probability Based Games I have no background and Economics and am trying to teach myself about some basic things in Game Theory. For example, I am trying to understand the following terms:

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*Nash Equilibrium


*Optimal Strategy


*Saddle Point
To illustrate these concepts, suppose we have the following game (I think the game I have created is called a "Stackelberg Game"):

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*There are 2 players: Player 1 and Player 2

*There are 2 Coins : Coin A and Coin B

*Coin A has a 0.5 Probability of landing on Heads and a 0.5 Probability of landing on Tails

*Coin B has a 0.7 Probability of landing on Heads and a 0.3 Probability of landing on Tails

*If Coin A lands on Heads, a score of +1 is obtained - if Coin A lands on Tails, a score of -1 is obtained.

*If coin B lands on Heads, a score of -2 is obtained - if Coin A lands on Tails, a score of +3 is obtained.

In this game:

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*Player 1 selects a coin and then flips this coin and records his score

*Next, Player 2 selects a coin and then flips this coin and records his score

*The player with the highest score wins

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In this game, Player 2 always has an advantage. He see what coin Player 1 picked and select the more favorable coin based on the choice of Player 1.

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*If Player 1 picked Coin B and got "unlucky", Player 2 automatically wins if he picks Coin A

*If Player 1 picked Coin B and got "lucky", Player 2 can only win if he also picks Coin A

*If Player 1 picked Coin A, regardless of Player 1's result - Player 2 should also pick Coin A if he wants to minimize his chances of loosing

My Question: In this game that I created, I am trying to identify the Nash Equilibrium, Optimal Strategy and Saddle Point :

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*I am confused between the concepts of Nash Equilibrium and Optimal Strategies. Based on the analysis I provided above, it seems like the Optimal Strategy in this game is for both players to always select Coin A - Would this be the Nash Equilibrium?


*I do not understand the concept of a Saddle Point in Game Theory. From Calculus and Optimization, I understand that a Saddle Point is a point on a function in which the first derivatives of the function at that point are 0 but the function does not have a maximum or a minimum of any sort at that point - in Machine Learning, Saddle Points are considered to be obstacles when trying to "fine tune" Machine Learning Models. However, I read a bit about Saddle Points in Game Theory, but I don't quite understand how to identify them or why they are important. Does the game I created have a Saddle Point? If so, what does the Saddle Point in this game "mean" (e.g. Does the Saddle Point simultaneously identify the "best case action for Player 1 and the worst case action for Player 2" )? If this game does not have a Saddle Point - can we "modify this game" (e.g. add more coins, e.g. Coin A, Coin B, Coin C, etc.) such that a Saddle Point can exist?
Thanks!
 A: A Nash equilibrium is when if you "know" the other player in a games strategy and are aware of what they can gain/lose in the game, the choice you both make will remain constant as you gain the most from it regardless of how they choose. In the prisoners dilemma, you know the other person would benefit the most from cooperating so that means that you will cooperate, and even if they for some reason decided not to cooperate anymore, you would still cooperate as you are always going to get your optimum outcome.
The dominant strategy however is the actual best outcome without the knowledge of the other persons move. It can be the same as the Nash equilibrium sometimes, but it isn't always. In the prisoners dilemma the dominant strategy would be to confess because it reduces the average length of a sentence, not because of the fact we know the other person will probably do it which is our Nash equilibrium.
Hopefully this helps a bit it does take a while to wrap your head around feel free to ask anything else!
