# HIERARCHICAL decision making problems arise naturally

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9 GARG AND NARAHARI: MECHANISM DESIGN FOR SINGLE LEADER STACKELBERG PROBLEMS 385 Substituting the values from (24), we get the following relation: It is easy to see that is the same for all and it contains only those pure strategy profiles of the followers which are part of a weakly dominant strategy equilibrium in the game. Therefore, the above inequality can be written as Using relation (25), the above can be written as This is precisely the condition (23). Part II: Now, we take up proving the condition (19). For this, we start with condition (26). Note that for the leader, the condition (26) implies that which is precisely the condition (19). (Q.E.D.). G. Bayesian Stackelberg Incentive Compatibility (Basic) Unlike the dominant strategy and Bayesian incentive compatibility, the incentive compatibility of a social choice function for an SLRF problem needs to be defined separately for the leader and the followers. 1) BaSIC for the Leader: Definition 3.5: An SCF is said to be Bayesian Stackelberg incentive compatible (BaSIC) for the leader (or truthfully implementable in Bayesian Stackelberg equilibrium for the leader) if the direct revelation mechanism has a Bayesian Stackelberg equilibrium in which (27) That is, truth revelation is a Bayesian Stackelberg equilibrium strategy for the leader in the game induced by. Following is a necessary and sufficient condition for an SCF to be BaSIC for the leader (28)

10 386 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 (29) (30) (31) Recall that the implicit assumption here is that all the type sets are finite. If the sets are infinite, then the above condition will transform to the following condition: 3) Bayesian Stackelberg Incentive Compatibility of a Social Choice Function: Definition 3.7: We call an SCF to be Bayesian Stackelberg incentive compatible (BaSIC) if it is BaSIC for both the leader and the followers. The following proposition provides a sufficient condition for BaSIC of a social choice function in the presence of hierarchical decision making. Proposition 3.2: If an SCF is dominant strategy incentive compatible then it will be BaSIC. Proof: Let be a dominant strategy incentive compatible SCF. By the definition of dominant strategy incentive compatibility, we can say that the direct revelation mechanism has a dominant strategy equilibrium in which (37) (32) assuming that the infimum is attained for some. 2) Basic for the Followers: Definition 3.6: An SCF is said to be Bayesian Stackelberg incentive compatible (BaSIC) for the followers (or truthfully implementable in Bayesian Stackelberg equilibrium for the followers) if the direct revelation mechanism has a Bayesian Stackelberg equilibrium in which (33) Invoking Proposition 3.1, we can say that the direct revelation mechanism has a Bayesian Stackelberg equilibrium in which Substituting the values of (37) in (38) and (38), we get the following Bayesian Stackelberg equilibrium of the direct revelation mechanism, which suffices to prove that is BaSIC (34) That is, truth revelation is a Bayesian Stackelberg equilibrium strategy for the followers in the game induced by. Following is a necessary and sufficient condition for an SCF to be BaSIC for the followers: (35) Recall that the implicit assumption here is that all the type sets are finite sets. If the sets are infinite, then the above condition will reduce to the following condition: (36) (Q.E.D.). H. Revelation Principle for Followers Analogous to the revelation principles for dominant strategy equilibrium and Bayesian Nash equilibrium, we also have the revelation principle for Bayesian Stackelberg equilibrium. Proposition 3.3: (Revelation Principle for Followers): Suppose that there exists a mechanism that implements the SCF in Bayesian Stackelberg equilibrium. Then, is BaSIC for the followers. Proof: If implements SCF in Bayesian Stackelberg equilibrium, then there exists a pure strategy Bayesian Stackelberg equilibrium of the game induced by such that. Such an equilibrium must satisfy conditions (19) (23). Also, we have seen that a SCF is BaSIC for the followers iff it satisfies the condition (35). Therefore, in order to prove the above revelation principle, we must show that the conditions

13 GARG AND NARAHARI: MECHANISM DESIGN FOR SINGLE LEADER STACKELBERG PROBLEMS 389 Note that Observation 4.1 immediately implies the first part of the function. For the second part, note that Observation 4.2 implies that 1) must lie in the interval ; 2). In order to compute for this scenario, we would first compute the expected gain of the seller, denoted by,ifhe reports his type to be, while his actual type is under the case when. Remember that at the same time all the other sellers are also reporting their types according to the strategy. The following expression is a direct consequence of the right-hand side curves in the Fig. 1 6 : otherwise that Substituting the value of. Thus, we can conclude, the above relation becomes (40) By change of variables in double integration, we get (38) In order to compute an optimal strategy for seller, we must apply the first-order necessary condition, which would imply that We know that the function which results from the above relation is an optimal strategy for the seller. Therefore, it must also satisfy that. This fact would transform the above relation into the following first-order differential equation: (39) (41) In order to compute an optimal reserve pricing strategy for the buyer, we must apply the first order necessary condition, which would imply that It is straightforward to show that the solution of the above o.d.e. would be Buyer s Secure Strategy: We have shown that for any action taken by the buyer, there is only one strategy profile with which the sellers will respond. Thus, in order to find out the secure strategy for the buyer, we must compute the expected payoff of the buyer when she announces a reserve price of, while her actual type is. Notice that due to the monotonicity of the function, the minimum reported type is given by. It is easy to see that if then 6 In this expression, we have made use of the fact that s ( 1 ) is a strictly increasing, hence invertible, and differentiable function. We have also made use of some results about the CDF of order statistics of independent random variables [23]. Substituting the values of, and from the appendix, we will get Replacing with, we get the following expression: Proof of Part 2: We have already shown in Part 1 that for any announced reserve price, a seller will report as his type if his actual type is. It was also shown that which essentially implies that this seller cannot

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