pro t maximization in mechanism design in Visual Studio .NET Creation Code39 in Visual Studio .NET pro t maximization in mechanism design

pro t maximization in mechanism design Using Barcode creator for VS .NET Control to generate, create Code 39 Extended image in .NET applications. PDF-417 13.5.1 Competitive Framework As with our w orst-case bounds from the previous section, the rst issue that must be addressed to study frugality is the competitive framework and in particular the benchmark for comparison, which in this case is a cost benchmark. We would like the frugality ratio to capture the overpayment of a mechanism with respect to a natural lower bound. One natural choice for this lower bound is the minimum payment by a nontruthful mechanism, in which case, the frugality ratio would characterize the cost of insisting on truthfulness.

Consider the mechanism N which, given the bids b, selects the cheapest feasible set with respect to these bids, and pays each winning agent his bid (ties are broken in favor of the ef cient allocation). This mechanism is a pay-your-bid auction and is not truthful. However, it does have at least one (full information) pure Nash equilibrium, i.

e., a bid vector b such that, for each agent i, given the bids b i by all other agents, i maximizes his pro t by bidding bi . A Nash equilibrium can be considered a natural outcome of the mechanism N , and the resulting net payments are thus a good cost benchmark.

As we are interested in a lower bound, we de ne the cheapest Nash value N (v) to be the minimum payments by N over all of its Nash equilibria.3 To illustrate this de nition, consider the case of an s-t path auction in which there are k parallel paths, as in our k = 2 path example above. Then, N (v) is precisely the cost of the second-cheapest path the agents on the cheapest path will raise their bids until the sum of their bids equals the cost of the second-cheapest path, at which point they can no longer raise their bids.

None of the other edges have incentive to raise their bids (as they are losing either way), nor to lower their bids, as they would incur a negative pro t. Thus, the metric in this case makes perfect sense it is the cost of the second cheapest solution disjoint from the actual cheapest. With a cost benchmark in hand, we can now formalize a competitive framework for these problems.

De nition 13.52 The frugality ratio of truthful mechanism M for buying a feasible set in set system (E, F) is sup. M(v) , N (v). where M(v) de notes the total payments of M when the actual private values are v, and N (v) is the cost benchmark, the cheapest Nash value with respect to the true values v.. 13.5.1.

1 Boun .NET Code 3/9 ds on the Frugality Ratio The example we saw earlier shows that the VCG mechanism does not, in general, have small frugality ratio. There is, however, one class of set systems for which VCG is.

Here we consi der only Nash eqilibria where nonwinners bid their true value, and ties are broken according to ef ciency. We refer the reader to the relevant references for a justi cation of this restriction..

frugality known to have optimal frugality ratio equal to 1, and is given in the following theorem (see Exercise 13.12). Theorem 13.

53 VCG has frugality ratio one if and only if the feasible sets of the set system are the bases of a matroid. On the other hand, for path auctions, say when there are two parallel paths, each consisting of many agents, VCG can have frugality ratio (n). The following lower bound shows that this bad case is not unique to the VCG mechanism.

Theorem 13.54 Consider the path auction problem on a graph G consisting of two vertex disjoint s-t paths, P and P , where . P . = n, (. P . is the number of edges on the path P ), and P . = n . Then a ny truthful mechanism for buying a path in this graph has frugality ratio at least ( nn ). proof De ne v(P ,i) to be the vector of private values for agents in P , in which edge i on P has cost 1/ n (so its value is vi = 1/ n), and all the rest of the edges in P have cost zero.

Similarly, let v(P ,j ) be the vector of private values for agents in P in which edge j on P has cost 1/ n and all the rest of the edges have cost zero. Let M be an arbitrary deterministic truthful path auction applied to this graph. De ne a bipartite graph G with a node for each edge in G and directed edges de ned as follows: there is an edge from node i (corresponding to edge i in P ) to node j (corresponding to edge j in P ) (respectively an edge from j to i), if when running M on bid vector (v(P ,i) , v(P ,j ) ) path P wins (resp.

P wins). Since there are nn directed edges in this graph, there must be either a node i in P with at least n /2 outgoing edges or a node j in P with at least n/2 outgoing edges. In the former case, observe that, by the monotonicity of any truthful mechanism, P must still win even if all edges in P bid 0, and the payments to each of the relevant edges equal their threshold bid which is at least 1/ n .

Thus the total payments are at least n /2. Since in this case the cheapest Nash equilibrium is 1/ n, we obtain the desired lower bound. The analysis for the second case proceeds mutatis mutandis.

The previous lower bound can be generalized to randomized mechanisms. An immediate corollary of this lower bound is that any truthful mechanism has frugality ratio n on a graph consisting of two vertex disjoint paths of length n. Thus, for this graph, VCG achieves the optimal frugality ratio.

On the other hand, if n = 1, the above lower bound on the frugality ratio of any mechanism is n. However, for the case of two parallel paths, one of length 1 and one of length n, VCG has a frugality ratio of n the worst case is when the long path wins. This raises the question of whether or not there is a better truthful mechanism for this graph.

The answer to this question is yes. The principle is fairly simple: if a large set is chosen as the winner, each of its elements will have to be paid a certain amount (depending on the other agent s bids). Hence to avoid overpayment, a mechanism.

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