Quick question about game theory. Suppose you have a two round game with two players: A and B. There are two strategies, cooperate or cheat.
If both players cooperate, they each receive 10. If one cooperates but the other cheats, then the cooperative player gets 5 while the cheater gets 15. If they both cheat, then both get 7.
If player A commits to cooperating in the first round and adopts a tit-for-tat strategy on player B (if B chooses to cooperate in the first round, A will follow and cooperate in the second round and vice versa for cheating)
What is the likely outcome and payoff of the game?
If we represent the decisions of both players in the 2 rounds as a $2x2$ matrix: $$ M = \left[ {\begin{array}{cc} A_{0} & B_{0} \\ A_{1} & B_{1} \\ \end{array} } \right] $$ and say, $A_{i}$ or $B_{i}$ takes the value 1 if the player co-operates, else 0 if the player cheats. We have these conditions for our problem, that $A_{0} = 1$, and $A_{1} = B_{0}$. So the set of all possible such matrices as outcomes are: $$ \left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} } \right], \left[ {\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} } \right], \left[ {\begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array} } \right], \left[ {\begin{array}{cc} 1 & 1 \\ 1 & 1 \\ \end{array} } \right] $$
We do not really know anything about player $B$'s strategy, so we consider every scenario equally possible. Calculating the payoffs, we get $(12,22)$, $(20,20)$, $(15,25)$ and $(20,20)$ for the above 4 matrices respectively.
So the average payoff should be $(A=14.25, B=21.75)$