Seminar: Topics in Cooperative Game Theory

Transcription

1 Seminar: Topics in Cooperative Game Theory PD Dr. André Casajus LSI Leipziger Spieltheoretisches Institut November

2 General remarks Write a well-structured essay: introduction/motivation, main part (possibly structured itself), conclusion. The starred themes can be worked on and be presented by two students, i.e., one joint essay and a joint presentation. The same theme can be presented at both of the sessions, i.e., any theme can be chosen by to students/pairs of students as long as you intend to present the essay at di erent sessions. Hand in the paper as PDF le. In the presentation, it might be necessary to focus on the main results/aspects of your essay. You can present your slides via beamer. Just have a PDF le of your slides at hand (on a USB stick, preferably not infected by viruses and the like). Handouts would be nice.

3 ule Deadline for choice of topics:???; at least two topics plus preferences Presentation session:??? Deadline for essays:???

4 Di erential marginality and the Shapley value* Give a detailed proof of the characterization of the Shapley value by E, N, and DM on the full domain V (N)! Simplify the proof of Casajus (2011, Theorem 1 and Proposition 3) along the following remarks: The rst part of the proof is not needed on V (N). Case 2(ii) can be simpli ed using the idea of Case 2(ii) in the proof of Casajus (2010, Theorem 2), the remarks at the end of this proof, and Casajus (2010, Remark 1).

5 A new characterization of the Shapley value* Consider the following axiom: Weak coalitional independence, WCI. If i, j 2 T or i, j /2 T for some T N and v, w 2 V (N) are such that v (S) = w (S) and for all S N, S 6= T, then ϕ i (N, v ) = ϕ j (N, v ) i ϕ i (N, w ) = ϕ j (N, w ). Discuss WCI and prove the following Theorem: Theorem. The Shapley value is characterized by E, NG, M, and WCI. In order to do so, you may wish to make use of the following lemmas. Of course, you have to prove them too. Lemma. E, NG, and M imply D. Lemma. D and WCI imply S.

6 2-e ciency, proxy neutrality, and additivity Prove or disprove the following claim: Claim. If ϕ is additive for two-player games and meets 2E, then ϕ satis es A.

7 A new characterization of the chi-value* Consider the following axioms: Order preserving splitting, OSP. If P 0 is ner than P and j 2 P (i), then ϕ i (N, v, P) ϕ j (N, v, P) i ϕ i (N, v, P 0 ) ϕ j (N, v, P 0 ). Modularity, MO. For all x 2 R N, ϕ (N, m x, P) = x, where m x 2 V (N) and m x (K ) = i2k x i for all K N. Prove the following Theorem on the χ-value (Casajus 2009): Theorem. The χ-value is the unique CS-value that satis es CE, A, CS, GN, MO and OSP.

8 The weighted chi-value: characterization* In Casajus & Tutic (2008, Theorem 4.2), IOO (which imho is not too nice) can be replaced by DSP below. Proof! Directed splitting, DSP. If j 2 P (i), i 2 N, and P is ner than P 0, then we have ϕ i (N, v, P) ϕ i N, v, P 0 0 i ϕ j (N, v, P) ϕ j N, v, P 0 0 for all coalition functions v, v 0 2 V (N). Unfortunately, DSP cannot be relaxed into DSP. Proof! Weak directed splitting, DSP. If j 2 P (i), i 2 N, and P is ner than P 0, then we have ϕ i (N, v, P) ϕ i N, v, P 0 ϕ j (N, v, P) ϕ j N, v, P 0 0.

9 For technical reasons, on the next slides :-(

10 Casajus, A. (2009). Outside options, component e ciency, and stability, Games and Economic Behavior 65(1): Casajus, A. (2010). Another characterization of the Owen value without the additivity axiom, Theory and Decision 69(4): Casajus, A. (2011). Di erential marginality, van den Brink fairness, and the Shapley value, Theory and Decision 71(2): Casajus, A. & Tutic, A. (2008). Nash bargaining, Shapley threats, and outside options, working paper, Wirtschaftswissenschaftliche Fakultät, Universität Leipzig, Germany.