Probably the best and simplest example of the mini-max theory is the Pie Game.
In 1993, I was conducting a negotiating seminar for the Iowa Broadcasters Association (IBA). As I went into the seminar I passed a table that displayed several inviting Iowa home-made pies. I bought one of the pies and took it to the seminar room, but before I entered the room, the executive director of the IBA introduced me to his 15-year-old son and asked if the boy could attend the negotiating seminar.
“Of course,” I said, wanting to be accommodating and realizing that the boy’s presence gave me an excellent opportunity to play the Pie Game.
Before I began the seminar, I put the pie I had purchased on a first-row table and asked the boy to come forward, which he did with some trepidation. I asked him if he liked pie.
“I sure do!”, he replied enthusiastically.
“Me, too. I love pie!”, I gushed.
“Well, I’ll tell you what I’m going to do,” I said. “See this scrumptious pie here? I’m going let you choose how much of it you want. We’re going to play the Pie Game.”
Then I said slowly, pausing between the two rules: “There are only two rules in the Pie Game, which are: Rule #1 is that you can cut the pie any way you want. (pause) Rule #2 is that I get to choose the first piece.”
When I told him the first rule, the boy’s face lit up; he was almost drooling. When I told him the second rule, his face dropped and his brow became furrowed. He pondered for almost a minute. Then, with a big smile on his face, he cut the pie with extreme care exactly in half.
I gave a big sigh of relief because the boy had figured out the correct strategy, as game theory and the mini-max theory had predicted. He had minimized the maximum size of my piece of pie (mini-max) and he had maximized the minimum size of the piece of pie that would be left after I chose (maxi-min). Because the values of the mini-max and the maxi-min strategies must be the same (which is what mathematical calculations prove), he had cut the pie in half.
I doubt if the boy knew about game theory or the mini-max strategy or could work out the mathematical proof, but he liked pie and was rational. So as a reward for being rational and intelligent enough to figure out the correct strategy, I gave him the entire pie, which delighted him, the audience, and me because I had made a dramatic point to begin my negotiating seminar.