Finding the optimal strategy and the value for strictly determined games
Put an asterisk(*) next to the minimum entry in each row.
Put a box around the maximum entry in each column.
The entry that has both an asterisk and a box represents the value of the game and is called a
saddle point .
The row that is associated with the saddle point represents the best strategy for the row player, and the column that is associated with the saddle point represents the best strategy for the column player.
A game matrix can have more than one saddle point, but all saddle points have the same value.
If no saddle point exists, the game is not strictly determined. Non-strictly determined games are the subject of
[link] .
Find the saddle points and optimal strategies for the following game.
We find the saddle point by placing an asterisk next to the minimum entry in each row, and by putting a box around the maximum entry in each column as shown below.
Since the second row, first column entry, which happens to be 10, has both an asterisk and a box, it is a saddle point. This implies that the value of the game is 10, and the optimal strategy for the row player is to always play row 2, and the optimal strategy for the column player is to always play column 1. If both players play their optimal strategies, the row player will win 10 units each time.
The row player's strategy is written as
$\left[\begin{array}{cc}0& 1\end{array}\right]$ indicating that he will play row 1 with a probability of 0 and row 2 with a probability of 1. Similarly the column player's strategy is
written as
$\left[\begin{array}{c}1\\ 0\end{array}\right]$ implying that he will play column 1 with a probability of 1, and column 2 with a probability of 0.
In this section, we study games that have no saddle points. Which means that these games do not possess a pure strategy. We call these games
non-strictly determined games . If the game is played only once, it will make no difference what move is made. However, if the game is played repeatedly,
a mixed strategy consisting of alternating random moves can be worked out.
We consider the following example.
Suppose Robert and Carol decide to play a game using a dime and a quarter. At a given signal, they simultaneously show one of the two coins. If the coins match, Robert gets both coins, but if they don't match, Carol gets both coins. Determine whether the game is strictly determined.
We write the payoff matrix for Robert as follows:
To determine whether the game is strictly determined, we look for a saddle point. Again, we place an asterisk next to the minimum value in each row, and a box around the maximum value in each column. We get
Since there is no entry that has both an asterisk and a box, the game does not have a saddle point, and thus it is non-strictly determined.
We wish to devise a strategy for Robert. If Robert consistently shows a dime, for example, Carol will see the pattern and will start showing a quarter, and Robert will lose. Conversely, if Carol repeatedly shows a quarter, Robert will start showing a quarter, thus resulting in Carol's loss. So a good strategy is to throw your opponent off by showing a dime some of the times and showing a quarter other times. Before we develop an optimal strategy for each player, we will consider an arbitrary strategy for each and determine the corresponding payoffs.
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