The Twins Part 2

So as she drove them home that Tuesday afternoon, my neighbor began interrogating the boys and demanding an explanation from them. They stuck to their story and seemed like they were not going to budge and then my neighbour used a tactic better known as strategic moves in SDA.
In analyzing the moves she made (she wasn't even aware she was playing a game...all she wanted was to get to the bottom of the story), I realized that some calculation of the payoffs was made. The payoffs of the players is below with 4 being the highest payoff and 1 being the lowest payoff:

The mother
Punish the kids if they confess = 1
Don't punish the kids if they don't confess = 2
Punish the kids if they confess = 3
Don't punish the kids if they confess = 4

The twins
Punish the kids if theyconfess = 1
Punish the kids if they don't confess = 2
Don't punish the kids they confess = 3
Don't punish the kids if they don't confess = 4

The strategies and their payoffs:

Upper Row
Mom won't punish if kids confess: 4(mom), 3 (kids)
Mom won't punish if kids don't confess: 2(mom), 4 (kids)

Lower Row
Mom will punish if kids confess: 3 (mom), 1 (kids)
Mom will punish if kids don't confess: 1 (mom), 2 (kids)

The Nash Equilibrium in this case is 2,4 where the kids don't confess and the mother doesn't punish them. This is the best payout for all players of the game. Though it is the highest payoff for the kids, the mother can then threaten to play down unless the kids played left i.e the mother could threaten to punish them unless they confessed or punished them if they refused to confess. In evaluating the payoffs, the first strategic move their mother makes is already credible because it is her highest payoff, but the other strategic move (to punish the kids if they refused to confess) needed more credibility. If their mother had already made up in her mind that she was going to punish them reagardless of their decision to confess or not, she would have to show her moves as credible. Her payoffs would be way less than punishing them if they confessed than punishing them if they refused to confess.

I guess she had no problem giving them the credibility of her actions because before they pulled through the driveway, they had recanted the whole episode as it actually occurred. I guess mothers have that innate ability to get the most difficult confessions out of their beloved kids....