Algorithmic Game Theory: 7th International Symposium, SAGT by Ron Lavi

By Ron Lavi

This publication constitutes the refereed lawsuits of the seventh overseas Symposium on Algorithmic video game thought, SAGT 2014, held in Haifa, Israel, in October 2014. The 24 complete papers and five brief papers provided have been conscientiously reviewed and chosen from sixty five submissions. They hide numerous very important points of algorithmic online game idea, reminiscent of matching conception, online game dynamics, video games of coordination, networks and social selection, markets and auctions, cost of anarchy, computational elements of video games, mechanism layout and auctions.

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Extra resources for Algorithmic Game Theory: 7th International Symposium, SAGT 2014, Haifa, Israel, September 30 – October 2, 2014. Proceedings

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Thus summing equality (1) for t from 1 to n − 1 and taking expectation yields that n−1 E t=0 And since E E n−1 t=1 t n−1 ν (O ) t=0 n−t ν (Ot ) = ν (O) − E n−t n−1 Lt . t=1 is the outcome of RSD, we just need to upper-bound Lt now. Let us note that equality (1) can be transformed into Y t+1 Lt+1 = − n−t−1 n−t−1 ν Ot+1 ν (Ot ) − n−t n−t−1 Yt n−t t Yi i=1 n−i for t + 1 < n. Since E Y t+1 Ht = 0, we have E n−1 Thus sequence (Z t )t=0 with Z 0 = 0 and Z t = Ht−1 = 0 as well. is a martingale, and Efficiency of Truthful and Symmetric Mechanisms in One-Sided Matching 23 again from Doob’s Stopping Theorem we get that 0 = E Z 0 = E Z n−1 = E n−1 Y t t=1 n−t , which gives n−1 0=E t=1 Yt =E n−t n−1 t=1 Lt −E n−t n−1 t=1 ν Ot−1 ν (Ot ) − , n−t+1 n−t and since the second sum telescopes we obtain that n−1 E t=1 ν O0 Lt = − E ν On−1 n−t n t ≤ ν (O) .

In any solution N to J every man b ∈ B is matched to a woman from the set P (b). Note that each b ∈ B ranks cn+1 immediately after all acceptable partners and before any unacceptable partners. Now if b is matched to someone outside of P (b), the pair (b, cn+1 ) will be blocking, since b will prefer cn+1 to its current partner, and cn+1 cannot be matched to someone she prefers at least as much as b since we defined Rcn+1 = ∅. The full proof can be found in [5]. The following lemma completes the proof of Theorem 1.

This property was crucial for establishing a correspondence between the pairs that are blocking in I and those that are blocking in J . In order to establish this property here we will need to introduce new agents in J corresponding to each position of woman ci ’s preference list P (ci ) from I. In particular if P (ci ) consists of ti positions, then there will be ti new dogs ai,1 , · · · , ai,ti . We will then define the preference of the dogs as well as the fixed perfect matching M in such a way as to ensure that if man bj appears in position k in P (ci ) then the only dog from the set {ai,1 , · · · , ai,ti } that prefers man bj over his partner in M will be ai,k .

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