DECISION PROBLEMS WITH MULTIPLE DECISION MAKERS

Consider the decision problem

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The objective function (14.1) can be interpreted in terms of a profit depending on two decision variables, x1 and x2, which must stay within feasible sets S1 and S2, respectively. Note that, even though the constraints on x1 and x2 are separable, we cannot decompose the overall problem, since the two decisions interact through the two profit functions π1(x1x2) and π2(x1x2). Nevertheless, using the array of optimization methods we should be able to find optimal decisions, images and images, yielding the optimal total profit

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In doing so, we assume that there is either a single stakeholder in charge of making both decisions, or a pair of cooperative decision makers, in charge of choosing x1 and x2, respectively, but sharing a common desire to maximize the overall sum of profits. But how about the quite realistic case of two noncooperative decision makers, associated with profit functions π(x1x2) and π2(x1x2), respectively?

Decision maker 1 wishes to solve the problem

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whereas decision maker 2 wishes to solve the problem

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Unfortunately, these two problems, stated as such, make no sense. Which value of x2 should we consider in problem (14.2)? Which value of x1 should we consider in problem (14.3)? We must clarify how the two decision makers make their moves.

  1. One possibility is that the two decision makers act sequentially. For instance, decision maker 1 might select x1 ∈ S1 before decision maker 2 selects x2 ∈ S2. In this case, we may say that decision maker 1 is the leader, and decision maker 2 is the follower. In making her choice, decision maker 1 could try to anticipate the reaction of decision maker 2 to each possible value of x1.
  2. Another possibility is that the two decisions are made simultaneously. Unfortunately, the conceptual tools that we have developed so far do not help us in making any sensible prediction about the overall outcome of such a simultaneous decision.

In Section 14.3 we illustrate the first case with a concrete, although stylized, example. Then, in section 14.4 we consider the second case as well, introducing a general theory of noncooperative games. Game theory aims at finding a sensible prediction of an equilibrium solution images, which depends on the precise assumptions that we make about the structure of the game. Whatever equilibrium solution we obtain, it cannot yield an overall profit larger than images, as the following inequality necessarily holds:

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If this inequality were violated, images would not be the optimal solution of problem (14.1). This means that if decentralize decisions, the overall system is likely to lose something.


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