VII. Maximization Of Conditional Expected Payoff With Goal Setting
I have introduced rules that tell how the planner could gather information to yield more valuable information structures. In that formulation, the probability density function for states of the grand world, , is given and assumed to be fixed. That is, the noncontrollable variables in the payoff function (2) are and . In the current section, we treat the probability density function as controllable in form of goals set by the planner as a way to express desired situations. In this case, the expected payoff function (2) can be rewritten as
where only is the only remaining noncontrollable variable. Assume the set of probability density functions include those functions the planner considered attainable through actions. The planning problem thus becomes one of searching for more valuable information structure such that decision function and probability density function maximize the conditional expected payoff as follows:
Choosing a decision function and a probability density function that maximize U in (19) is equivalent to choosing, for each signal an action () and a goal () that maximize Equation (6). It follows that the action and the goal should be chosen so that the conditional expected payoff given  = is maximized. The criterion of searching for more valuable information structures as depicted in Corollary 1 applies in this case. Put differently, the distinction of planning with and without goal setting is whether selecting a set of goals in form of probability density functions is incorporated in the model of information searching as depicted in Corollary 1. That is, the planner can simply take the probability density functions of independent random variables as given (without goal setting), or seek the optimal functions (with goal setting) in combination of actions that maximize the planner’s expected payoffs.
To illustrate how the model works, consider a land development firm looking for a parcel of land for retailing use in five years. To simplify, the firm considers two location factors: land prices (high or low) and transport networks (convenient or inconvenient) in deciding which land to develop. How should the firm proceed to gather information in order to take the appropriate development action? In this example, combinations of values of location factors are states of the firm’s small world. For example, low land price and convenient network is one of the four exhaustive states of the small world. Each state is associated with a probability that in five years that state would be realized. A payoff table can then be constructed specifying under which state each development action would result in how much payoff. Without goal setting, the probability distribution of states is unchanged, whereas with goal setting, the planner should change that distribution, such as higher probability that the location selected is of lower land prices with more convenient transport networks.
Without further information, the problem is a traditional decision problem in that the best action would be the one maximizes expected payoff. What happens if information gathering is possible and how such planning would yield benefit? Following the present model, the firm should first consider all possible information structures, i. e., all possible partitions (or signals) of the four combinations (or states) of the values of the two location factors. Presumably, some partitions have intuitive meanings, insufficient infrastructure is a signal of low land price and inconvenient transport network. There are totally fifteen such partitions. For each partition, the firm is to select the best development action that maximizes payoff, according to Equations (3) through (7). For all the best development actions under different information structures, the firm is then to choose the globally optimal action that maximizes payoff across all the best development actions. The information structure associated with that optimal action should be the one the firm seeks. The above algorithm of exhaustive enumeration does more than what Corollary 1 claims to do, but the underlying logic is the same. The algorithm can best be implemented by a computer rather than by a human alone.