Straddling Success: The Power and Peril of Saddle Points
Saddle Point
What are Zero-Sum Games?
ZERO-SUM GAMES AND SADDLE POINT:
‘Zero-Sum Game’ is a term used in this article, and in game theory frequently. In a zero sum game one person’s gain is equivalent to another’s loss, so the net change in wealth or benefit is zero. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero.
‘Saddle point’ refers to the point where maximin value (the highest of a set of minimum values) is equivalent to minimax value (the lowest of a set of maximum values). A saddle point of a pay-off matrix is sometimes called the equilibrium point of the pay-off matrix. In a zero-sum matrix game, an outcome is a saddle point if the outcome is a minimum in its row and maximum in its column.
FINDING THE SADDLE POINT:
There is a convenient algorithm to find the saddle point of any zero-sum two-player game. This is illustrated below.
We begin with the simplest versions of a game played between two players, Rose and Colin. Rose can choose amongst three strategies, whereas Colin chooses amongst two. Their payoffs are represented in table form (a “matrix”):
To play the game, both players secretly write down their chosen strategy on a piece of paper. On cue, then they turn over their papers and determine their payoffs.
If, for example, Rose plays B and Colin plays A, then Rose gets 2 and Colin loses 2. Notice that in the above game, the payoffs for each outcome sum to zero(zero-sum game). Since in such games, Colin will simply lose what Rose has gained (and vice versa), then it is sufficient for us to only write down the payoffs for Rose. Suppose that Rose plays C hoping Colin will play B so she gets 10. Knowing this, Colin realizes that it is better to play A so that he earns 5. Foreseeing this, Rose decides it would be better to play A. Round and round it goes. In this case, we see that this type of reasoning does not leave a resolution that is appealing to either player.
Consider the following zero-sum game:
As we mentioned earlier, it is sufficient for us to only write down the payoffs for Rose. Therefore, we can write the above game matrix as
Notice that the game is biased towards Rose (see Rose C). Moreover, Colin C is a particularly bad strategy since, regardless of what Rose plays, Colin B will always be preferable to Colin C. We say that B dominates C. This leads us to formulate the dominance principle. A rational player should never play a dominant strategy. Moving on, we see that the CB entry (Rose C, Colin B) is special. It is the most cautious strategy: by playing C, Rose is assured to win at least 2 regardless of what Colin does; similarly, by playing Colin B, Colin is assured to lose no more than 2. We call this an equilibrium outcome or saddle point. If Colin knows that Rose will play C, he will play B. If Rose knows that Colin will play B, she will play C. Neither player has any apparent incentive to change their strategy.
This leads us to formulate the following definition “In a zero-sum matrix game, an outcome is a saddle point if the outcome is a minimum in its row and maximum in its column.”
The argument that players will prefer not to diverge from the saddle point leads us to offer the following principle of game theory:
Saddle Point Principle: If a matrix game has a saddle point, both players should play it.
METHOD TO FIND THE SADDLE POINT:
Rose’s strategy is to select her maximin (the maximum of the row minima), while Colin’s strategy is to select his minimax (the minimum of his column maxima). By doing so, each player has chosen the most cautious strategy, for which in Rose’s case, she can guarantee a payoff of at least her maximin value, and Colin can guarantee a loss of at most his minimax value. Assuming that minimax(columns) = maximin(rows) (1) then there exists a saddle point.
Notice that if a saddle point exists, it may not necessarily be unique. However, if multiple saddle points exist, then they must be equal in value. However, not all games have saddle-point solutions and indeed, no optimal pure strategy.
Author — Harshitjain Tj
The Indian Game Theory Society — Delhi Technological University
