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15.8 Notes Using Barcode printer for .NET framework Control to generate, create Code 39 image in Visual Studio .NET applications. upc Sections 15.1 an .NET framework ANSI/AIM Code 39 d 15.

2. The notion of a cooperative game was rst proposed by von Neumann and Morgenstern (1944). The notion of the core was rst introduced by.

notes Gillies (1959). Code39 for .NET Theorem 15.

6 was independently discovered by Bondareva (1963) and Shapley (1967). Theorem 15.11 is due to Scarf (1967).

Deng et al. (1997) observed the connection between the core of many combinatorial optimization games and the integrality gap of the corresponding LP. Goemans and Skutella (2000) showed this connection for the facility location game, and proved that deciding whether the core of a facility location game is nonempty is NP-complete.

The best lower and upper bound 1 1 on the integrality gap of LP (15.3) are 1.52 , due to Mahdian et al.

(2006), and 1.463 , due to Guha and Khuller (1999) See Immorlica et al. (2006) for an example of a problem modeled using NTU games.

Section 15.3. For a discussion of the NPT, VP, and CS properties of cost sharing mechanisms see Moulin (1999) and Moulin and Shenker (2001).

In our de nition of group-strategyproof mechanisms, we did not allow side payments between members of a coalition. For a discussion of mechanism design in a setting where collusion with side payments is allowed, see Goldberg and Hartline (2005). This cross-monotonicity property for cost sharing is similar to the population monotonicity property introduced by Thomson (1983, 1995) in the context of bargaining.

For cooperative games, this notion was rst introduced by Sprumont (1990). The mechanism M and Theorem 15.16 are due to Moulin (1999), where he also proves a converse to this theorem for submodular games.

Examples on the connection between group-strategyproof mechanisms and cost-sharing schemes, and a partial characterization of such mechanisms are due to Immorlica et al. (2005). Sections 15.

4 and 15.5. For a general introduction to the primal-dual schema from the perspective of approximation algorithms, see the excellent book by Vazirani (2001).

The cost-sharing scheme presented in Section 15.4 for submodular games is due to Dutta and Ray (1989). This scheme was formulated as a primal-dual algorithm and generalized to an algorithm that can increase the dual variables at different rates by Jain and Vazirani (2002).

Both Dutta and Ray (1989) Jain and Vazirani (2002) also prove several fairness properties of their cost-sharing schemes. The technique of using ghost duals and its application to the facility location problem (algorithm in Figure 15.4) and single-source rent-or-buy problem are due to P l and Tardos (2003).

The proof of their a result on the facility location problem (Theorem 15.21) is based on an algorithm that is originally due to Mettu and Plaxton (2000). The rst (non-cross-monotonic) primaldual algorithm for the facility location problem is due to Jain and Vazirani (2001).

The probabilistic technique presented in Section 15.5 and its application to several problems including facility location, vertex cover, and set cover are due to Immorlica et al. (2005).

K nemann et al. (2007) gave a 1/2-budget-balanced mechanism, together o with a matching upper bound for the Steiner forest problem. Section 15.

6. The Shapley value and its axiomatic characterization (Theorem 15.26) are due to Shapley (1953).

In the same paper, Shapley shows that for convex games (which correspond to submodular games in the context of cost sharing) the Shapley value is in the core. The application of Shapley values to the multicast problem is due to Feigenbaum et al. (2000) and is explained in detail in 14.

For other applications of the Shapley value, see the book edited by Roth (1988) or the survey. cost sharing by Winter (2002) VS .NET Code 3 of 9 . The generalization of the Shapley value to games with nonbinary demand is due to Aumann and Shapley (1974).

See the survey by McLean (1994) for various generalizations to NTU games. The result on the computation of Shapley values for submodular games is due to Mossel and Saberi (2006). The axiomatic result of Arrow is given in Arrow (1959).

Axiomatic characterizations of PageRank (Page et al., 1999) are given by Palacois-Huerta and Volij (2004) and by Altman and Tennenholtz (2005). We refer the reader to the excellent survey by Moulin (2002) for further information on the axiomatic approach to cost sharing.

Theorem 15.28 is proved in a seminal paper by Nash (1950). See Moulin (1988) for further discussion of this theorem and its generalization to more than two players.

See Moulin (1988, 2003) for a discussion of various collective utility functions and social choice rules. For more information on the Santa Claus problem see Bansal and Sviridenko (2006) and Asadpour and Saberi (2006)..

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