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:
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"):
- 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:
- 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
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.
- 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 :
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!