Convex Cooperative Games
Introduced by Shapley in (Shapley 1971), convex cooperative games capture the intuitive property some games have of "snowballing". Specifically, a game is convex if its characteristic function is supermodular:
It can be shown (see, e.g., Section V.1 of (Driessen 1988)) that the supermodularity of is equivalent to
that is, "the incentives for joining a coalition increase as the coalition grows" (Shapley 1971), leading to the aforementioned snowball effect. For cost games, the inequalities are reversed, so that we say the cost game is convex if the characteristic function is submodular.
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Famous quotes containing the words cooperative and/or games:
“Then we grow up to be Daddy. Domesticated men with undomesticated, frontier dreams. Suddenly life—or is it the children?—is not as cooperative as it ought to be. It’s tough to be in command of anything when a baby is crying or a ten-year-old is in despair. It’s tough to feel a sense of control when you’ve got to stop six times during the half-hour ride to Grandma’s.”
—Hugh O’Neill (20th century)
“As long as lightly all their livelong sessions,
Like a yardful of schoolboys out at recess
Before their plays and games were organized,
They yelling mix tag, hide-and-seek, hopscotch,
And leapfrog in each other’s way all’s well.”
—Robert Frost (1874–1963)