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mechanism design without money using visual .net toattach qr code 2d barcode on web,windows application ASP.NET Web Form Project Bibliography H. Adachi. On a characterizat ion of stable matchings.

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Exercises 10.1 To what extent is Lemma 10.1 sensitive to the richness of the preference domain For example, does the result hold if the preference domain is even smaller, e.

g., containing only symmetric single-peaked preferences 10.2 Suppose that an anonymous rule described in Theorem 10.

2 has parameters (ym)n 1 . Express this rule as a generalized median voter scheme with parameters m=1 ( S ) S N . 10.

3 Suppose that a rule f is strategy-proof and onto, but not necessarily anonymous. Fix the preferences of agents 2 through n, ( 2 ,. .

., n ), and denote the outcomes obtainable by agent 1 as O= f( ,. ,. . .

,. ) = {x [0,1]: . R s.t. f (.

,. . .

,. exercises Show that O = [a, b] for some a, b [0, 1] (without appealing directly to Theorem 10.4). 10.

4 Prove Theorem 10.4. 10.

5 For the case of three agents, generalize Theorem 10.2 to a 3-leaved tree. Speci cally, consider a connected noncyclic graph (i.

e., a tree) with exactly three leaves, 1 , 2 , 3 . Preferences over such a graph are single-peaked if there is a peak pi such that for any x in the graph, and any y in the (unique shortest) path from x to pi , y i x.

The concepts of strategy-proofness, onto, and anonymity generalize in the straightforward way to this setting. Describe all the rules that satisfy these conditions for the case n = 3. (Hint: rst show that when all agents peaks are restricted to the interval [ 1 , 2 ], the rule must behave like one described in Theorem 10.

2.) For the nonanonymous case with n 3, see Schummer and Vohra (2004). 10.

6 Prove that the TTCA returns an outcome in the core of the house allocation game. 10.7 The TTC mechanism is immune to agents misreporting their preferences.

Is it immune to agents misreporting the identity of their houses Speci cally, suppose a subset of agents trade among themselves rst before participating in the TTC mechanism. Can all of them be strictly better off by doing so 10.8 Consider an instance of the stable matching problem.

Let be a matching (not necessarily stable) and the male optimal stable matching. Let B = {m : (m) >m (m)}. Show that if B = then there is a m B and woman w such that (m, w) is a blocking pair for .

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