DIMENSIÓN 3: CAPACIDAD DE RESPUESTA
4.1. DESARROLLO DE LA PROPUESTA
In this work we considered a player-specific graphical resource allocation game, a resource allocation game played over an influence graph. The game models a system in which every node can choose a subset of resources, and the value of a selected resource to a node is decreased if any of its neighbors chooses the same resource. We showed that pure strategy Nash equilibria exist in the game for arbitrary influence graph topologies even though the game does not admit a potential function, and gave a bound on the complexity of finding equilibria. We then considered the problem of reaching an equilibrium when nodes update their allocations one at a time to their best allocation, that is, every node performs a best reply. We showed that for non-complete influence graphs there might be cycles in best replies but the system would reach an equilibrium eventually if updates are done in a random order. For complete influence graphs there are no cycles, and hence an equilibrium is reached after a finite number of steps. We also showed that if neighboring nodes try to allocate disjoint sets of resources then there are no cycles, even if the nodes’ update steps are improvements instead of best replies. Finally, we showed that the convergence properties hold even if improvement steps are performed simultaneously by nodes that form an independent set of the influence graph, and proposed an efficient algorithm to reach a NE over sparse influence graphs that only requires local coordination between neighboring nodes.
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Proof of Lemma 2. According to the structure of the utility functionUi(ai, a−i) = P
{r|ar i=1}U
r
i(1, a−i), a best reply ai(t) can stop to be such in two situations:
(i) The payoffs of one or more allocated resources{r∈ R|ar
i(t) = 1}decrease;
(ii) The payoffs of one or more not allocated resources{r∈ R|ar
i(t) = 0}increase;
According to the definition of cost saving in (1), case (i) can happen only if somei- freeresources allocated byibecomei-busy. This requires that some playerj∈ N(i) starts allocating some j-busy resources. Similarly, case (ii) can happen only if a neighborj∈ N(i) allocating an resourcerevicts it, making resourcer i-free. Proof of Lemma 3. We prove the lemma by showing that a best reply satisfying condition 1) must occur in a best reply cycle in order for the cycle to exist. The utility of at least one player must decrease at least once in a best reply cycle. According to the definition of cost saving in (1), in order for the utility Uj(a(t))
of a player j to decrease, some neighbor i ∈ N(j) needs to start allocating some i-busyresource allocated byj. It follows that, from Table 1, in a best reply cycle there has to be at least one best reply satisfying condition 1) or 3). Let r∈Ei(t)
andr0∈I
i(t) be two resources in the evicted and inserted sets, respectively, during
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of inserted and evicted set that ar
i(t−1) = 1 and ar 0
i (t−1) = 0. Assume now
that this best reply satisfies 3). This impliescir< cir0δi. Note that this inequality
also implies that playerialways prefersr0 overr (r0 always provides a higher cost
saving) and thus ifar
i(t−1) = 1, thenar 0
i (t−1) = 1. Since there cannot be a best
reply satisfying 3), there must be at least one best reply satisfying 1). This proves the lemma.
Proof of Lemma 4. Sincer0 isi-busyit yields to playerithe payoffcir0δi. Consider
an objectr00 for whichcir00> cir0. Since playeriis performing a best reply, ifr00 is
i-free then its payoff iscir00> cir0δi and consequentlyar 00
i (t) = 1. Similarly, ifr00 is
i-busythen its payoff iscir00δi> cir0δi, and consequentlyar 00 i (t) = 1.
Proof of Lemma 5. Part A: First we show that |Bi(t−1)| ≥ |Bi(t)|. Playeri can
only increase the number ofi-busyallocated resources if she evicts at least onei-free resourcer0froma
i(t−1) and inserts ani-busyresourcerat stept. Thus by (1) we
have
cir0 < cirδi (10)
Since we are in a best reply cycle, at some stept0 > tthe strategy ai(t−1) must
become a best reply for playeri, i.e. ai(t0) =ai(t−1). This requires eithercir0> cir
orcir0 > cirδi, and both contradict (10).
Part B: Second we show that for every step in a best reply cycle |Bi(t−1)| =
|Bi(t)|must hold. We denote byC(t) the set of the chosen resources, the resources
allocated by at least one player ina(t),C(t) ={r|ar
j(t) = 1 for somej ∈N } ⊆ R.
Similarly, we denote by C(t)−i the set of resources allocated by the players not
includingi,C(t)−i ={r|arj(t) = 1 for somej ∈N\ {i} }. It is easy to see that the
sets|Fi(t)| and |C(t)| are related and on a complete influence graph it holds that
|C(t)|=|C(t)−i|+|Fi(t)|. On one hand, a best reply for which|Fi(t−1)|=|Fi(t)|
does not affect|C(t)|since|C(t−1)−i|=|C(t)−i|. On the other hand, a best reply
for which|Fi(t−1)|>|Fi(t)| decreases the size of setC
|C(t)| = |C(t)−i|+|Fi(t)|=|C(t−1)−i|+|Fi(t)|
< |C(t−1)−i|+|Fi(t−1)|=|C(t−1)|
Since best replies for which|Fi(t−1)| >|Fi(t)| do not exist in a cycle (Part A),
best replies for which|Fi(t−1)|<|Fi(t)| cannot exist either, as otherwise the size
of setC would increase indefinitely. Hence |Fi(t−1)| =|Fi(t)|, which proves the
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Valentino Pacificistudied computer engineering at Politecnico di Milano in Milan, Italy. In October 2010, he completed a joint M.Sc. degree in computer engineering, between Politecnico di Milano and KTH, Royal Institute of Technology in Stockholm, Sweden. Now he is a researcher at the Laboratory for Communi- cation Networks in KTH, Royal Institute of Technology and he is pursuing his Ph.D. His research interests include the modeling, design and analysis of content management systems using game theoretical tools.
György Dánreceived the M.Sc. degree in computer engineer- ing from the Budapest University of Technology and Economics, Hungary in 1999 and the M.Sc. degree in business administration from the Corvinus University of Budapest, Hungary in 2003. He worked as a consultant in the field of access networks, streaming media and videoconferencing 1999-2001. He received his Ph.D. in Telecommunications in 2006 from KTH, Royal Institute of Technology, Stockholm, Sweden, where he currently works as an assistant professor. He was visiting researcher at the Swedish In- stitute of Computer Science in 2008. His research interests include the design and analysis of distributed and peer-to-peer systems and cyber-physical system security.