http://www.hinduonnet.com/2005/03/12/stories/2005031208552000.htm

The above article talks about the effect that walking has on world cricket. For those of you unfamiliar with the game, check out http://en.wikipedia.org/wiki/Cricket

In short, cricket is a game played between two teams where, much like baseball, there exists a batsman, a bowler who throws the ball at the batsman, and the fielders who try to prevent the batsman from scoring, while trying to get him out. Additionally, there is an umpire who has the authority to make decisions in the game.

Sometimes, a batsman may get out in a manner that is not evident. For example, when trying to hit the ball while batting, the batsman may accidentally only apply a slight nudge on the ball which is then caught by the wicket keeper. In such a situation, it is often difficult for the umpire to realize that the player has hit the ball and has therefore been caught out. In this case, most batsmen tend to wait for the umpire to make his decision. However, there are a handful of players who prefer to preserve their dignity and walk off the pitch when they know they are out.

“Of course, there is still a lot of heated debate in the cricket-playing world about the merits of walking. If you apply game theory to the issue, using something called the Prisoner’s Dilemma, walking is for losers. Winners should almost always stick to the crease until the finger goes up 22 yards in front of them.”

The article refers to a different form of the prisoner’s dilemma, in which the team batting second already knows the outcome of the decision made by the player in the team batting first, i.e. the players in the two teams do not make their decisions simultaneously. However, this does not affect the Nash equilibrium in anyway since players in both teams have a dominant strategy of not walking.

Consider the following payoff matrix for two teams A and B and let P(A) and P(B) denote players from A and B respectively. The two strategies available to players when they have just gotten out are walking (W) and not walking (NW).

A brief explanation for the payoffs:

1) (W, NW) implies that P(A) walks and P(B) does not. In such a situation, Team A would be down by a player and this would positively impact B’s chances of winning the game.

2) (W, W) implies that both P(A) and P(B) walk. In this situation, walking by a player in each team minimizes the impact on the outcome of the game. However, this honesty has a positive impact on how the players are viewed by supporters and therefore improve their public image. Thus both players receive a higher payoff than they would have had had they chosen (NW, NW).

From the above matrix, we can see that the dominant strategy for players from both teams is to not walk. Thus, if game theory were to have its way, the Nash Equilibrium for both players is to not walk and though most players do follow this strategy, the optimal outcome for both the game of cricket and the players themselves would be to follow the ‘walking’ example.