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I have no background and Economics and am trying to teach myself about some basic things in Economics. 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

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!

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  • $\begingroup$ I am sure this is more appropriate in math.stackexchange.com Almost no thing here is economical. $\endgroup$ Commented Mar 18, 2022 at 20:10
  • $\begingroup$ it's a sequential game so NE won't help much to understand it. The relevant equilibrium concept is subgame perfect equilibrium. $\endgroup$
    – user39962
    Commented Mar 18, 2022 at 20:45

1 Answer 1

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Your question is more appropriate for Mathematics SE. But anyway we going to make some differentiation.

  1. A saddle point is a unstable point, clearly distinguished from the Nash Equilibrium, in that the last is a stable estate.
  2. Is the important relevance that the Nash equilibrium like the Saddle Point are states, meanwhile the Optimal Strategy is some algorithm to move between states, characterized to be able to get to the Nash Equilibrium. I hope that this helps you.
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