Conclusion and Future Work

In the last chapter, we’re going to briefly conclude our analysis, describe additional pos­

sibly extended models, and discuss some aspects that can be improved in the future.

In a two­player file­sharing game, we detailedly examine all Nash equilibria includ­

ing their stability. When the need for resources is almost not limited, there are two non­

collapsing Nash equilibria, one of which with a greater contribution is stable. When the need is a little limited, the contribution of the Nash equilibrium with a greater contribu­

tion will be lowered and it will become unstable. Besides, there exist two additional side Nash equilibria in this case. When the limitation is drastic, the system will collapse. In a three­player file­sharing game, we still examine all Nash equilibria, yet without stability.

In a multi­player file­sharing game, we only examine symmetric Nash equilibria without stability. The conclusion of the PoA and PoS remains the same when the number of play­

ers increases from two to three. It remains the same for an arbitrary number of players if we only consider the symmetric Nash equilibria.

We give an intuitive explanation of the PoS and PoA here. The PoS and PoA both increase with K since the Nash equilibria (the consequence of selfishness) naturally falls behind the maximum total utility (which increases with the amount of resources). We also discover that the two parameters b and n both represent the flexibility of the model. If we increase b and n, ideally the best Nash equilibrium will be improved and the worst Nash equilibrium will be deteriorated. Hence the PoS decreases with b, n but the PoA increases with b, n. When the need for resources is a little limited, the PoS can remain 1 because

the maximum total utility is not too far away such that the best Nash equilibrium is able to catch up.

After the analysis in multi­player file­sharing games, the reader may make a guess of the following conjectures. First, (d_L, d_L, ..., d_L) is always unstable, and (d_H, d_H, ..., d_H) is always stable in a multi­player file­sharing game. Second, u(d_L, d_L, ..., d_L) is the least among all non­collapsing Nash equilibria, and u(d_H, d_H, ..., d_H) is the greatest among all non­collapsing Nash equilibria. If the second conjecture is true, the PoA and PoS derived in multi­player file­sharing games are always true even if we take all Nash equilibria into consideration.

In this thesis, we assume each player can provide at most the benefit K of resources to all other players. This is a simple assumption. If we further consider a more realistic situation where they have different limitations K_j, the utility function becomes

u_i(d_i) = −di+X

j̸=i

min{Kj, b d_j p(d_i)}, for 1 ≤ i ≤ n.

If each player has his/her own desired resources of the “total” benefit K distributed on all the other players, the utility function becomes

u_i(d_i) = −di+ min{K, b p(di)X

j̸=i

d_j}, for 1 ≤ i ≤ n.

If the benefits of these resources (K_i) are different, the utility function becomes

ui(di) =−di+ min{Ki, b p(di)X

j̸=i

dj}, for 1 ≤ i ≤ n.

If all players have their unique “files” of different benefits (Kj) and each player will try their best to retrieve all files from all the other players, the utility function becomes

u_i(d_i) =−di+X

j̸=i

if{b dj p(d_i)≥ Kj} · Kj, for 1≤ i ≤ n,

where the value of the “if” function is defined to be 1 if the condition is true, and defined

to be 0 if the condition is false. Since a file is valid only if all portions of it are retrieved, we use the “if” function here. They are also good research problems.

Finally, the reader may discover that in the results of [2] and our thesis, the common problems of P2P such as whitewashing attacks and sybil attacks from malicious users are still not taken into consideration. In fact there are many studies [1, 4, 11, 13, 15] focusing on these problems. Maybe we can study these papers in the future and improve our models to concretely solve the problems.

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