Computational Mechanism Design Chapter 3

Notice that in each of the preceding examples an

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Unformatted text preview: an item that is substitutable for A?" In addition to solving realistic problem instances without complete information revelation, it is also important that dynamic methods allow an agent to respond to requests for information with an approximate solution to its own valuation function. Notice that in each of the preceding examples an agent can respond without rst computing its exact value for all bundles. Examples: Complete Information is Not Necessary Examples 1 3 are simple problems instances in which the optimal allocation and the Vickrey payments can be computed without complete information from agents. Although there is no consideration of agent incentives at this stage, a well structured iterative auction can compute optimal outcomes without complete information from agents and provide incentives for agents to reveal truthful information. Example 1. Single-item auction with 3 agents, and values v1 = 16; v2 = 10; v3 = 4. The Vickrey outcome is to sell the item to agent 1 for agent 2's value, i.e. for 10. Instead of information fv1 ; v2 ; v3 g it is su cient to know fv1  10; v2 = 10; v3  10g to compute this outcome. Example 2. Consider a combinatorial auction problem in which we ask every agent for the bundle that maximizes their value. If the response from each agent is non-overlapping, as illustrated in Figure 3.1 then we cam immediately compute the outcome of the GVA. The e cient allocation is to give each agent its favorite bundle; every agent gets its valuemaximizing bundle so there can be no better solution. The Vickrey payments in this example are zero, intuitively because there is no competition between agents. We do not need any information about the value of an agent for any other bundles, and we do not need even need an agent's value for its favorite bundle. Example 3. Consider the simple combinatorial allocation problem instance in Table 3.1, with items A, B and agents 1, 2, 3. The values of agent 1 for item B and bundle AB are stated as a  b and b  15, but otherwise left unde ned. Consider the following cases: 77 4 1 3 2 Figure 3.1: A simple combinatorial allocation problem. Each disc represents an item, and the selected bundles represent the bundles with maximum value for agents 1, 2, 3 and 4. In this example this is su cient information from agents to compute the e cient solution and the Vickrey payments. a 5 In this case the GVA assigns bundle AB to agent 3, with V  = 15, V,3 = max 10+ a; b , so that the payment for agent 3 is pvick 3 = 15 , 15 , max 10+ a; b  = max 10 + a; b . It is su cient to know fa  5; b  15; max 10 + a; b g to compute the outcome. a  5 In this case the GVA assigns item B to agent 1 and item A to agent 2, with V  = 10 + a, V,1 = 15, and V,2  = 15. The payment for agent 1 is pvick1 = a,10+a,15 = 5 and the payment for agent 2 is pvick 2 = 10,10+a,15 = 15,a. It is su cient to know fa; b  15g to compute the outcome. Notice that it is not necessary to compute the value of the optimal allocation S to compute Vickrey payments; we only need to...
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