Which Strategy is the right strategy?

Further assume, to start with, that Pa = .09, Pb = .7, Pc = .2.

There are 8 different possible strategies in true. The strategy is the decision whom to shoot when there are still three people left. We intuitively decided that

Must be the right strategy. Now we want to prove or disprove that assumption.

The task is to show that this strategy is Nash Equilibrium.

This means that if a player let us say A, chooses to shoot at the different person in order to improve winning chances than the other players can decide to maximize their wining strategies accordingly.

For example let us say that the best outcome for A is if it shoots at C. Then B makes its decision based on this fact, i.e., B chooses a person to shoot such that it maximizes his winning strategy knowing that A will shoot at C. Once B has chosen his target (let us say A) it is player C turn to choose. The player C chooses such that it maximizes his winning strategy knowing that A is shooting at C and B is shooting at A. Once player C determines his target, let us say A, we go back to A. Player A has a chance to change his strategy if there is a better choice knowing that B decided to shoot at A and C at A. We proceed like this till we find a strategy at which none of the players would change their choice. This strategy is called the Nash equilibrium.

Project Goal: Find Nash equilibrium.

Follow the steps:

Step 1. Compute the Wa,Wb,Wc the winning probabilities for A, B, and C respectively for all 8 strategies.

Step 2. Run the algorithm of finding the Nash Equilibrium explained above.