The Grid Problem

Determining if a problem/game has a Nash Equilibrium is not an easy problem.  In fact, it is harder than the classical NP problems and it is considered to be PPAD.  In this posting, I will look and classify the Grid Problem to be the following:

Grid is a shared infrastructure with a limited number of resources.  The limit could be high, but it must be less than what two players (users of the Grid) are able to consume.  We want to look at the Grid Problem and see if there exists a configuration where all the users/players are satisfied.  In short, we want to see if there exists a Nash’s Equilibrium point for the Grid Problem. 

Lets take a step back, and look at the classes of games that exist.  I want to do this to level-set and clarify my position about where various infrastructure types will fit into the different classes of games.

We have the following:

  1. Strategic Games (a.k.a Games in Normal Form)
  2. Extensive Games with perfect information
  3. Extensive Games without perfect information
  4. Coalitional Games

Strategic games are rather interesting in that both players (or N-players) make the decision in advance, and stick to that decision for the duration of the games.  Extensive games, however, players can modify their strategy between moves.

Lets look at a Grid infrastructure:

A. Resource are limited
B. Jobs are submitted to the Queue to be processed
C. Queues are most likely FIFO
D. A Player wants to get results asap without regards to the other players

Based on the above properties, we can deduce the following:

  • Property A makes this game a Zero-sum game.
  • Property B makes this game an imperfect information game
  • Properties C + D makes this game a non cooperative strictly competitive game

In a Strategic game, the player chooses his plan of action once and for all and independently of the other players.  Each plan is made based on a set of preferences, but those preferences *must* be selfish as we are dealing with a Zero-sum game.

If this problem can be reduced to the Matching Pennies problem, we can deduce that the Grid Problem does not have a Nash Equilibrium.


Art Sedighi