GAME 2010 (part 3)

Today is raining. Usual.
In the past days Eva Tardos has presented a lot of interesting stuff.
For instance, she has talked about congestion games. In these games
a meaningful function (e.g., a cost, a response time, a delay) depends
on the load of some structure. We can think about the time necessary
to travel on a road depending on the traffic level, or the delay of running
of a process on a CPU in a load balancing scenario.
These kinds of problems turn out to be equivalent to the problems using
a potential function. Which have the nice property that a Nash equilibrium
exists for the game, if and only if the potential has a minimum. Moreover
Nash equilibria are local minima of the potential. Apart from the equivalences,
since the change in potential can be expressed using the change of
congestion seen by a single user (the change in the traffic of a road seen
by a user, when he changes the road he is travelling on), and since the
potential can be used to approximate the cost of Nash equilibrium,
using a congestion function is a natural choice in order to bound the quality
of the equilibrium.
Moreover, we can consider an infinite amount of infinitesimally small users,
moving the problem in the field of continuous. Besides being able to
analyze the problem with the tools of continuous math, in this framework
are clearly, under a mathematical point of view, identifiable the selfish and
the altruistic contribution to the welfare of the users, when a user moves,
modifying in this way the congestion of the structure. (so when an
infinitesimally part of flow of traffic moves from one road to another one).

In two days I will be at the Βόσπορος again.

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