Game Theory: Nash Equilibrium

It's no big secret that after watching A Beautiful Mind, I became very interested in Game Theory. I sometimes pick up on games as they occur in real life. Things start move a little slower and I start analyzing minute details of the events and interactions before me. Actually, I have picked up on these situations before I even watched the movie, but I didn't know that there was an actual term for it.

Recently, J at work brought up something with which he has an issue and I think I can classify it as a GT example - specifically a Nash Equilibrium example. In J's example, there is a task that needs to be carried out by a group and one person from that group has to always be available to execute the task. The problem is that this task has to be executed around lunchtime so not everyone from the group is around and, specifically, one of the group members always gets up and leaves as soon as it's 12:00. One of the other group members always find a way to be there when it needs to be executed.

The Nash Equilibrium is a game "in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy (i.e., by changing unilaterally). If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium."

Therefore, in our example, it comes down to a question of what each member of the group believes is beneficial. The guy who leaves at 12:00PM on the dot would probably think that as long as he gets to go to lunch, that's all that matters since the other members of the group will execute the task in his absence. The rest of the group may believe that as long as the task is executed, it's all that matters. Since that one guy is always leaving anyways, the rest have grown accustomed carrying out the task.

So long as the individual and the rest of the group hold onto their strategies, does that constitute a Nash Equilibrium with two recognized goals? They would, of course, have to hold onto their belief of what is beneficial. If another member of the group changed his mind as to what is beneficial, then that would disrupt the equilibrium. Or, even better, if the individual changed his belief to coincide with what the rest of the group believes is beneficial, then that would create a new equilibrium with only one beneficial outcome instead of the two I've presented.

No one stated that it is the right thing to do for the individual to consistently leave at lunchtime - he's obviously not a team player. But regardless, he's found an equilibrium state in which he is definitely benefiting since the rest of the group's goals are being met. Maybe he shouldn't be considered a part of the group being analyzed in the equilibrium. But, no, he's still a part of that team. Ah, I digress.