Solving a Three-Player Nash Equilibrium

[msmith12]

So I know this isn't exactly a math/physics question, but game theory is close enough so that hopefully someone can help me out.

I am supposed to find ALL Nash Equilibrium for a given three player game such that player 1 has the choices U,D... player 2 has the choices L,R... and player three can choose A,B.

the payoffs are represented by two matrices. The first step is really easy in that player two's dominate strategy is R... this leaves a matrix with the values

09,07|06,15
14,12|05,10

where the first number is the utility to player 1, and the second is the utility to player 3.

This leaves two pure strategy nash equilibrium of 06,15 and 14,12. I need to find the mixed strategy nash equilibrium...

So, setting the probability that player one chooses U is x, and D is 1-x, and the probability that player three chooses A is y, and B is 1-y.

From here I get the two equations
$$y(9x+6(1-x))=(1-y)(14x-5(1-x))... y(3x+6)+y(9x+5)=9x+5... y(12x+11)=9x+5... y=\frac{9x+5}{12x+11}$$
and
$$x(-5y+12)=(1-x)(5y+10)... 22x=5y+10... x=\frac{5y+10}{22}$$
from here, I thought that I could just plug the second equation into the first, and get a value for y, which would give a value for x, and then I would have my probabilities, but this isn't working for me... Am I doing the set-up incorrectly? Am I missing something? Thanks for any help

~Confused

(sorry about the TEX formatting, i couldn't get carriage return to work)

 

[anathema]

Dear Confused,

Thank you for your question. Finding the Nash Equilibrium for a three player game can be a bit tricky, but I can help you through the process.

First, let's define what a Nash Equilibrium is. In game theory, a Nash Equilibrium is a set of strategies for each player in which no player can improve their payoff by unilaterally changing their strategy while the other players keep their strategies constant. In other words, it is the point where all players are playing their best response to each other.

In your given game, we have three players with two choices each. This means there are 2^3 = 8 possible outcomes. Let's list them out:

1) ULA
2) ULB
3) URA
4) URB
5) DLA
6) DLB
7) DRA
8) DRB

Next, we need to determine the payoffs for each player in each of these outcomes. Let's use the notation (P1, P2, P3) to represent the payoffs for player 1, player 2, and player 3 respectively. Using the given matrices, we get the following payoffs for each outcome:

1) (9, 7, 6)
2) (9, 7, 15)
3) (14, 12, 6)
4) (14, 12, 15)
5) (6, 10, 6)
6) (6, 10, 15)
7) (5, 10, 6)
8) (5, 10, 15)

Now, let's analyze each player's best response to each possible outcome. For player 1, their best response is to choose U if player 3 chooses A, and to choose D if player 3 chooses B. Similarly, for player 2, their best response is to choose R if player 1 chooses U, and to choose L if player 1 chooses D. And for player 3, their best response is to choose A if player 2 chooses R, and to choose B if player 2 chooses L.

Using this information, we can see that there are two pure strategy Nash Equilibriums: (U, R, A) and (D, L, B). These are the two outcomes where all players are playing their best response to each other.

Now,

 

[wais]

First of all, great job on finding the pure strategy Nash equilibria! Now, for finding the mixed strategy Nash equilibrium, your set-up seems correct. However, your method of solving for y might be causing some confusion. Instead of plugging in the second equation into the first, try solving for x in the second equation and then substituting that value into the first equation to solve for y. This should give you the probabilities for player 1 and player 3 to choose U and A, respectively.

Another method you could use is to set up a system of equations with x and y as the variables and solve for them simultaneously. For example, using the second equation, you can solve for x in terms of y and then substitute that into the first equation to solve for y. This might be a bit more straightforward and less confusing.

One more thing to keep in mind is that there may be multiple solutions for x and y that satisfy the equations. In this case, you can check each solution by plugging them into the original equations and seeing if they hold true. If they do, then you have found another mixed strategy Nash equilibrium.

I hope this helps and good luck with finding the mixed strategy Nash equilibrium!