# Value of the game from payoff matrix

I am absolutely new to decision theory . I came across this following payoff matrix in the book.(Math. Stats : John E Freund).

Player A
I    II

Player B 1       7    -4
2       8     10

The value of the game is given as 8 units.However i . have a question . I agree the optimal strategy for Play B is "2" . But i want to argue that the optimal choice for player A is 2 . This is because in decision theory the assumption is made that "each player must choose a strategy without knowing what the opponent is going to do and that once a player has made a choice it can not be changed" ..Going by this logic, the optimal strategy for player A should be II , because i see in that case , the loss would be less overall for the moves of player B . This , logic if true gives the value of the game to be 10 , which is not correct. I want to know , where i am wrong.

• Is this a zero-sum game? Which player is the minimiser and which is the maximiser? – Theoretical Economist Apr 21 '18 at 2:48

The statement that "each player must choose a strategy without knowing what the opponent is going to do and that once a player has made a choice it can not be changed" just means the choice in each time the game is played are made without information of what the opponent played this time. The players are allowed to examine the whole payoff matrix and think about what the opponent might do. In this game Player B finds that move $2$ dominates move $1$. No matter what A does, B is better off playing $2$ than playing $1$, so he should play $2$ all the time. For A, neither strategy dominates the other, but A "knows" B will play $2$, so by playing $1$ A can lose $8$ instead of $10$. If B were so foolish to play $1$ it is true that A could do better by playing $2$, but we don't count on foolishness.

• Thanks a lot.. Your answer is a perfect clarification. – warrior_monk Apr 21 '18 at 2:55

The basic assumptions in a game theory are:

1) Each player is rational and tries to maximize his/her payoff in the game.

2) Each player knows that his/her opponent is rational and tries to maximize his/her payoff in the game.

You can solve this problem by Reducing by Dominance.

Since the entries in Row 2 are greater (or equal to) the corresponding ones in Row 1 (it implies Row 2 dominates Row 1, i.e. Player 2 is better off with strategy 2), we can eliminate Row 1 to get: $$\begin{array}{cc|cc} &&Player \ 1 \\ &&A&B \\ \hline Player \ 2 & 2 & 8 & 10 \end{array}$$ Since the entry in Column A is less than (or equal to) the corresponding one in Column B (it implies Column A dominates Column B, i.e. the Player 1 is better off with strategy A), we can eliminate Column B to get: $$\begin{array}{cc|c} &&Player \ 1 \\ &&A \\ \hline Player \ 2 & 2 & 8 \end{array}$$ So, the Player 2 has an advantage of $8$ units over Player 1.

• What if you begin with player 1(Player A in original ques)? In that case . "II" is the optimal strategy for him , which is Player 1 – warrior_monk Apr 22 '18 at 4:26
• Option 1: If P1 first chooses the strategy A, then P2 will choose the strategy 2 and P2 will win 8 units. Option 2: If P1 first chooses the strategy B, then P2 will choose the strategy 2 and P2 will win 10 units. Which option is rational (better) for P1? Certainly, Option 1 (because P1 loses less: 8 instead of 10). So, P1 knows that P2 is rational and P2 will maximize his/her payoff. For P2 the strategy 2 is better than strategy 1 (i.e. strategy 2 dominates strategy 1). Now, can you work out the two options when P2 starts and P1 ends and which option is rational for P2? – farruhota Apr 22 '18 at 5:22