Infinite Action Space
It may be that a player has an infinite number of possible actions to choose from at a particular decision node. The device used to represent this is an arc joining two edges protruding from the decision node in question. If the action space is a continuum between two numbers, the lower and upper delimiting numbers are placed at the bottom and top of the arc respectively, usually with a variable that is used to express the payoffs. The infinite number of decision nodes that could result are represented by a single node placed in the centre of the arc. A similar device is used to represent action spaces that, whilst not infinite, are large enough to prove impractical to represent with an edge for each action.
The tree on the left represents such a game, either with infinite action spaces (any real number between 0 and 5000) or with very large action spaces (perhaps any integer between 0 and 5000). This would be specified elsewhere. Here, it will be supposed that it is the latter and, for concreteness, it will be supposed it represents two firms engaged in Stackelberg competition. The payoffs to the firms are represented on the left, with q1 and q2 as the strategy they adopt and c1 and c2 as some constants (here marginal costs to each firm). The subgame perfect Nash equilibria of this game can be found by taking the first partial derivative (reference?) of each payoff function with respect to the follower's (firm 2) strategy variable (q2) and finding its best response function, . The same process can be done for the leader except that in calculating its profit, it knows that firm 2 will play the above response and so this can be substituted into its maximisation problem. It can then solve for q1 by taking the first derivative, yielding . Feeding this into firm 2's best response function, and (q1*,q2*) is the subgame perfect Nash equilibrium.
Read more about this topic: Extensive-form Game
Famous quotes containing the words infinite, action and/or space:
“The process of writing has something infinite about it. Even though it is interrupted each night, it is one single notation.”
—Elias Canetti (b. 1905)
“Therefore all just persons are satisfied with their own praise. They refuse to explain themselves, and are content that new actions should do them that office. They believe that we communicate without speech, and above speech, and that no right action of ours is quite unaffecting to our friends, at whatever distance; for the influence of action is not to be measured by miles.”
—Ralph Waldo Emerson (1803–1882)
“In the United States there is more space where nobody is is than where anybody is.”
—Gertrude Stein (1874–1946)