In this video, I'll talk about two-person zero-sum games and pure strategies.
In a previous video titled Decision Trees and Decision Making under Uncertainty, we
assume a single decision maker chooses the optimal strategy without reference to the
effect of decisions of others.
In many business situations, there are usually more than one decision makers.
The decision made by one decision maker would affect how other decision makers react.
We can use game theory to making decisions in cases where two or more decision makers
have conflicting or common interests.
In game theory, the decision makers are usually called players, and the decision making process
is usually called a game.
In this video, we will only consider the games with two players, player 1 or the row player,
and player 2 or the column player.
In this example, Player 1 must choose 1 of 3 strategies, 1, 2, and 3.
Simultaneously, player 2 must choose 1 of 3 strategies, A, B, and C.
If player 1 chooses strategy 1 and player 2 chooses strategy A, then player 1 will win
$2 and player 2 will lose $2.
If player 1 chooses strategy 1 and player 2 chooses strategy B, then player 1 will lose
$1 and player 2 will win $1.
And so on.
This matrix is called the payoff matrix or reward matrix.
It only shows the payoffs of player 1.
The payoff of player 2 in each cell is just the opposite of the payoff of player 1.
So we simply omit it in this matrix.
Some basic assumptions.
In a two-person zero-sum game, every dollar that one player wins comes out of the other
player's pocket, so the two players have totally conflicting interests and the payoffs
always add up to zero.
Both players are rational and each chooses a strategy that enables him or her to do the
best he or she can.
Both players have the knowledge about the payoff matrix.
Now, let's try to find the optimal strategy for player 1 and the optimal strategy for
player 2.
For player 1, if he chooses strategy 1, then player 2 will choose strategy C so player
1 will lose $3 to her.
Note that, -3 is the smallest number in row 1 of the payoff matrix.
Similarly, if player 1 chooses strategy 2, then player 2 will choose strategy B so player
1 will lose $5 to her.
-5 is the smallest number in row 2.
If player 1 chooses strategy 3, then player 2 will choose strategy B so player 1 will
win only $3, instead of $4 or $5 from her.
3 is the smallest number in row 3.
Out of these three possible outcomes, player 1 should choose the strategy with the largest
value, 3, which is the maximum value of the row min.
Therefore, the best strategy for player 1 is strategy 3.
By choosing strategy 3, player 1 can ensure that he will win at least $3.
For player 2, if she chooses strategy A, then player 1 will choose strategy 3 to win $4.
Note that, 4 is the largest number in column 1 of the payoff matrix.
If player 2 chooses strategy B, then player 1 will choose strategy 3 to win $3.
3 is the largest number in column 2.
If player 2 chooses strategy C, then player 1 will choose strategy 2 to win $6.
6 is the largest number in column 3.
Out of these three possible outcomes, player 2 should choose the strategy so that player
1 will win the smallest amount from her.
It is 3, which is the minimum value of the column max.
Therefore, the best strategy for player 2 is strategy B. By choosing strategy B, player
2 can ensure that player 1 will win at most $3.
This game has a saddle point, 3.
A saddle point is a payoff that is simultaneously equal to the max of row min and the min of
column max.
If a two-person zero-sum game has a saddle point, then we call the common value the value
of the game to the row player.
So, this game has a value of 3 to player 1.
A saddle point can also be thought of as an equilibrium point.
Neither player would want to move away from this saddle point.
The two-person zero-sum game is a special case of the two-person constant-sum game.
A two-person constant-sum game is a game in which, for any choice of both player's strategies,
the sum of the two players' payoffs is a constant value c.
A two-person zero-sum game is just a two-person constant-sum game with c=0.
If the sum of the two players' payoffs is not a constant value, then this type of games
are called two-person nonconstant-sum games.
Here is an example of a two-person constant-sum game.
Two radio programs are competing for an audience of 10 million listeners.
They each have three options, broadcasting news, music, or celebrity interviews.
If radio program 1 broadcasts news and radio program 2 also broadcasts news, program 1
will have 6 million listeners and program 2 will have 4 million listeners.
If program 1 broadcasts news and program 2 broadcasts music, program 1 will have 5 million
listeners and program 2 will have 5 million listeners.
And so on.
For every cell, we only show the number of listeners of program 1 because we can calculate
the number of listeners of program 2 by subtracting the number of listeners of program 1 from
the total of 10 million listeners.
The minimum values of the rows are 5, 2, and 1, respectively.
The maxmin is 5.
The maximum values of the columns are 9, 5, and 8, respectively.
The minmax is 5.
So the max of row min = the min of column max = 5.
This is a saddle point, and the value of the game to radio program 1 is 5.
Some games may have more than one saddle point.
For example, this is a zero-sum game.
The minimum values of the rows are -1, 0, and -2, respectively.
The maxmin is 0.
The maximum values of the columns are 0, 0, and 3, respectively.
The minmax are 0 and 0.
So the max of row min = the min of column max = 0.
We have two saddle points, and the value of the game to player 1 is 0.
Some games may have no saddle point.
For example, this is a constant-sum game.
The sum is 10.
The minimum values of the rows are 5, 2, and 1, respectively.
The maxmin is 5.
The maximum values of the columns are 9, 6, and 7, respectively.
The minmax is 6.
Because 5 is not equal to 6, we don't have a saddle point.
That means, at least one player will benefit by moving away from maxmin or minmax.
For example, for this minmax 6, player 2 will benefit by moving from column B to column
A so the payoff of player 1 will be reduced from 6 to 5.
Similarly, for this maxmin 5, player 1 will benefit by moving from row 1 to row 2 so the
payoff of player 1 will be increased from 5 to 9.
For this type of games, we will introduce a method for finding an equilibrium using
mixed strategies in another video.
Okay, that is about two-person zero-sum games and two-person constant-sum games.
Thanks for watching.
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