Say that you have two children, or maybe three, and that they fight for what’s theirs. The contested objects are many: cake, Lego sets, the right to various household electronics or to name the family dog. And the children aren’t pleasant about it: they torment each other, and engage in guerrilla tactics distinguishable from those of ruthless insurgents only by their disregard for stealth, which might at least allow you, the parent, a little peace and quiet. Each of them has a story about fairness and what he deserves.

But how much do they really want to win? What would they give up to get dominion over the remote control? It is hard to know anything about these small belligerents other than what they are willing to pay, Paul Raeburn and Kevin Zollman write in “The Game Theorist’s Guide to Parenting: How the Science of Strategic Thinking Can Help You Deal with the Toughest Negotiators You Know—Your Kids” (Scientific American/Farrar, Straus & Giroux). Raeburn, a journalist, and Zollman, a game theorist and philosopher at Carnegie Mellon, maintain that the best move for the mother of these children is to hold an auction—after determining whether market conditions call for an open-cry-first-price auction, a sealed-bid-second-price auction, an English-clock auction, or a Dutch auction—and begin taking bids. (“Mom is an economist, naturally.”) One kid offers four dollars, another ten; the winner bids twelve dollars. The losers walk away satisfied, in the smug conviction that their winning sibling has overpaid. There is really only one problem left for the mother: “What do you do with the money you make in the auction?”

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