Game Theory Pt. 2

In Oligopolies, and in game theory, there are also sequential games. Chess is a good example of a sequential game. In sequential games, there is time-sensitive sequencing, OR simultaneous knowledge of the other player's decision by both players. As such, we use decision trees to mark off the outcomes of sequential games.


DIFFERENT PATHS: The first player to move can use BACKWARDS INDUCTION to predict which moves their opponent will make given their move. The first mover here can predict all of the outcomes, and will probably choose the "large" strategy, because they will receive 30 points in every outcome for the large scenario. Given that the first player will always choose the "large" strategy, the second mover will always choose the large strategy as well, because they prefer having 3 points to having 0.3 points. AS SUCH, we know that there is a NASH EQUILIBRIUM, because both players are playing their best strategy given the strategy of the other play. Additionally, this is Pareto, as we cannot make either player better off.

ULTIMATUM BARGAINING GAME: In an ultimatum, the first player imposes a "take it or leave it offer". For an example, lets say that my mom gives my sister a dollar. My mom tells my sister that she must take that dollar and share some of it with me, or else she will take it away. In other words, my sister will offer me a portion of the money she has received, and I can accept it, or decline it. If I reject the offer, then neither me nor my sister will get a dollar. This is the payoff tree:

SISTER will propose $X for herself, and $(1-X) for me. If I accept this offer, I will get $(1-X), and my sister gets $X. If I reject this offer, we both get nothing.

Nash Equilibrium Occurs where I accept my sister's offer (regardless of the offer). This is because I would rather get a little bit of money than no money. Neither me nor my sister has any incentive to use any strategy other than this.

WHAT SHOULD MY SISTER'S STRATEGY BE? She should offer me the smallest amount as possible, because it is to my advantage to accept ANY offer. SO...

If my sister offers me 1 cent, it is still in my best interest to accept it, because 1 cent is better than nothing. In this scenario, my sister will get to keep 99 cents, and I will get 1 cent!

ULTIMATUM BARGAINING WITH AN ACCEPTANCE THRESHOLD: This is a version of ultimatum bargaining, but here, the second mover (me) can declare a minimum acceptance threshold (Y) in advance. This changes the payoff tree.

My sister can either propose an offer greater or equal to my minimum acceptance threshold (100-X > or = Y), or lower than it (100-X < Y). If she offers me an amount equal to or greater than my minimum acceptance threshold, then I will get $1-X, and she will get $X. If she offers me an amount lower than my minimum acceptance threshold, then I will reject the offer, and we will both get nothing.

Here, Nash Equilibrium occurs where my sister accepts my minimum acceptance threshold. This is because she would rather have a little bit of money than no money. Given my minimum acceptance threshold, it is always in my sister's best interests to offer an amount which complies with it.

SO WHAT IS MY BEST STRATEGY? Well, because it is always in my sister's best interest to accept my threshold, I stand to make the most money by setting my threshold as high as possible (99 cents). If I do this, then I will make 99 cents, and my sister will only make one cent.

KIDNAPPER GAMES ARE ALSO IMPORTANT, AS ARE COMPETITIVE MARKETS, but my internet just died and deleted all of the previous crap I typed up, and I am NOT spending another hour and typing it all up again. FORGET IT!